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Galilean transformation

2017-07-20T22:55:40Z

NickPercival: 1 revision imported

In [[physics]], a '''Galilean transformation''' is used to transform between the coordinates of two [[reference frames]] which differ only by constant relative motion within the constructs of [[Newtonian physics]]. These transformations together with spatial rotations and translations in space and time form the '''inhomogeneous Galilean group''' (assumed throughout below). Without the translations in space and time the group is the '''homogeneous Galilean group'''. The Galilean group is the [[group of motions]] of [[Galilean relativity]] action on the four dimensions of space and time, forming the '''Galilean geometry'''. This is the [[active and passive transformation|passive transformation]] point of view. The equations below, although apparently obvious, are valid only at speeds much less than the [[speed of light]]. In [[special relativity]] the Galilean transformations are replaced by [[Poincaré transformation]]s; conversely, the [[group contraction]] in the [[classical limit]] {{nowrap|''c'' → ∞}} of Poincaré transformations yields Galilean transformations.

[[Galileo Galilei|Galileo]] formulated these concepts in his description of ''uniform motion''.<ref>{{harvnb|Galilei|1638I|loc=191–196 (in Italian)}} {{harvnb|Galilei|1638E|loc=(in English)}} {{harvnb|Copernicus|Kepler|Galilei|Newton|2002|pp=515–520}}</ref>
The topic was motivated by his description of the motion of a [[ball]] rolling down a [[Inclined plane|ramp]], by which he measured the numerical value for the [[acceleration]] of [[gravity]] near the surface of the [[Earth]].

==Translation==
[[Image:Standard conf.png|right|thumb|300px|Standard configuration of coordinate systems for Galilean transformations.]]
Though the transformations are named for Galileo, it is [[absolute time and space]] as conceived by [[Isaac Newton]] that provides their domain of definition. In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities as [[vector space|vectors]].

This assumption is abandoned in the [[Poincaré transformation]]s. These [[special relativity|relativistic]] transformations are applicable to all velocities, while the Galilean transformation can be regarded as a low-velocity approximation to the Poincaré transformation.

The notation below describes the relationship under the Galilean transformation between the coordinates {{nowrap|1=(''x'', ''y'', ''z'', ''t'')}} and {{nowrap|1=(''x''′, ''y''′, ''z''′, ''t''′)}} of a single arbitrary event, as measured in two coordinate systems S and S', in uniform relative motion ([[velocity]] ''v'') in their common ''x'' and ''x''′ directions, with their spatial origins coinciding at time {{nowrap|1=''t'' = ''t''′ = 0}}:<ref>{{harvnb|Mould|2002|loc=[https://books.google.com/books?id=lfGE-wyJYIUC&pg=PA42 Chapter 2 §2.6, p. 42]}}</ref><ref>{{harvnb|Lerner|2996|loc=[https://books.google.com/books?id=B8K_ym9rS6UC&pg=PA1047 Chapter 38 §38.2, p. 1046,1047]}}</ref><ref>{{harvnb|Serway|2006|loc=[https://books.google.com/books?id=1DZz341Pp50C&pg=PA261 Chapter 9 §9.1, p. 261]}}</ref><ref>{{harvnb|Hoffmann|1983|loc=[https://books.google.com/books?id=JokgnS1JtmMC&pg=PA83 Chapter 5, p. 83]}}</ref>

:<math>x' = x - v t </math>
:<math>y' = y </math>
:<math>z' = z </math>
:<math>t' = t .</math>

Note that the last equation expresses the assumption of a universal time independent of the relative motion of different observers.

In the language of [[linear algebra]], this transformation is considered a [[shear mapping]], and is described with a matrix acting on a vector. With motion parallel to the ''x''-axis, the transformation acts on only two components:
:<math>\begin{pmatrix} x' \\t' \end{pmatrix} = \begin{pmatrix} 1 & -v \\0 & 1 \end{pmatrix}\begin{pmatrix} x \\t \end{pmatrix} </math>
Though matrix representations are not strictly necessary for Galilean transformation, they provide the means for direct comparison to transformation methods in special relativity.

==Galilean transformations==
The Galilean symmetries can be uniquely written as the [[Function composition|composition]] of a ''rotation'', a ''translation'' and a ''uniform motion'' of spacetime.<ref name="mmcm">{{harvnb|Arnold|1989|p=6}}</ref> Let {{math|'''x'''}} represent a point in three-dimensional space, and {{math|''t''}} a point in one-dimensional time. A general point in spacetime is given by an ordered pair {{math|('''x''', ''t'')}}.

A uniform motion, with velocity {{math|'''v'''}}, is given by
:<math>(\bold{x},t) \mapsto (\bold{x}+t\bold{v},t),</math>
where {{math|'''v''' ∈ ℝ3}}. A translation is given by
:<math>(\bold{x},t) \mapsto (\bold{x}+\bold{a},t+s),</math>
where {{math|'''a''' ∈ ℝ3}} and {{math|''s'' ∈ ℝ}}. A rotation is given by
:<math>(\bold{x},t) \mapsto (G\bold{x},t),</math>
where {{math|1=''G'' : ℝ3 → ℝ3}} is an [[orthogonal transformation]].<ref name="mmcm"/>

As a [[Lie group]], the Galilean transformations span 10 dimensions,<ref name="mmcm"/> i.e., comprise 10 generators.

==Galilean group==
Two Galilean transformations {{math| ''G''(''R'', '''v''', '''a''', ''s'')}} [[composition of functions|compose]] to form a third Galilean transformation, {{math| ''G''(''R' '', '''v' ''', '''a' ''', ''s' '') ''G''(''R'', '''v''', '''a''', ''s'') {{=}} ''G''(''R' R'', ''R' '' '''v'''+'''v' ''', ''R' '' '''a'''+'''a' '''+'''v' ''' ''s'', ''s' ''+''s'')}}.
The set of all Galilean transformations {{math|Gal(3)}} on [[space]] forms a [[group (mathematics)|group]] with composition as the group operation.

The group is sometimes represented as a matrix group with spacetime events {{math|( '''x''', ''t'', 1)}} as vectors where {{math|''t''}} is real and {{math|'''x''' ∈ ℝ3}} is a position in space. The action is given by<ref>[http://www.emis.de/journals/APPS/v11/A11-na.pdf]{{harvnb|Nadjafikhah|Forough|2009}}</ref>
:<math>\begin{pmatrix}R & v & a \\ 0 & 1 & s \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x\\ t\\ 1\end{pmatrix} = \begin{pmatrix} R x+vt +a\\ t+s\\ 1\end{pmatrix},</math>
where {{math|''s''}} is real and {{math|''v'', ''x'', ''a'' ∈ ℝ3}} and {{math|''R''}} is a [[rotation matrix]].

The composition of transformations is then accomplished through [[matrix multiplication]]. {{math|Gal(3)}} has named subgroups. The identity component is denoted {{math|SGal(3)}}.

Let {{math|''m''}} represent the transformation matrix with parameters {{math|''v'', ''R'', ''s'', ''a''}}:
:<math>G_1 = \{ m : s = 0, a = 0 \} , </math> uniformly special transformations.
:<math>G_2= \{ m : v = 0, R = I_3 \} \cong (\mathbb{R}^4 , +) ,</math> shifts of origin.
:<math>G_3 = \{ m : s = 0, a = 0, v = 0 \} \cong \mathrm{SO}(3) ,</math> rotations of reference frame (see [[SO(3)]]).
:<math>G_4= \{ m : s = 0, a = 0, R = I_3 \} \cong (\mathbb{R}^3, +) ,</math> uniform frame motions.

The parameters {{math|''s'', ''v'', ''R'', ''a''}} span ten dimensions. Since the transformations depend continuously on {{math|''s'', ''v'', ''R'', ''a''}}, {{math|Gal(3)}} is a [[continuous group]], also called a topological group.

The structure of {{math|Gal(3)}} can be understood by reconstruction from subgroups. The [[semidirect product]] combination (<math>A \rtimes B </math>) of groups is required.
#<math>G_2 \triangleleft \mathrm{SGal}(3)</math> (G2 is a [[normal subgroup]])
#<math>\mathrm{SGal}(3) \cong G_2 \rtimes G_1</math>
#<math>G_4 \trianglelefteq G_1</math>
#<math>G_1 \cong G_4 \rtimes G_3</math>
#<math>\mathrm{SGal}(3) \cong \mathbb{R}^4 \rtimes (\mathbb{R}^3 \rtimes \mathrm{SO}(3)) .</math>

==Origin in group contraction==
Here, we only look at the [[Lie algebra]] of the [[Representation theory of the Galilean group|Galilean group]]; it is then easy to extend the results to the [[Lie group]].

The relevant Lie algebra is [[linear span|spanned]] by {{math|''H'', ''Pi'', ''Ci''}} and {{math|''Lij''}} (an [[antisymmetric tensor]]), subject to [[commutator|commutation relations]], where
:<math>[H,P_i]=0 </math>
:<math>[P_i,P_j]=0 </math>
:<math>[L_{ij},H]=0 </math>
:<math>[C_i,C_j]=0 </math>
:<math>[L_{ij},L_{kl}]=i [\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta_{jl}L_{ik}] </math>
:<math>[L_{ij},P_k]=i[\delta_{ik}P_j-\delta_{jk}P_i] </math>
:<math>[L_{ij},C_k]=i[\delta_{ik}C_j-\delta_{jk}C_i] </math>
:<math>[C_i,H]=i P_i \,\!</math>
:<math>[C_i,P_j]=0 ~.</math>
{{mvar|H}} is the generator of time translations ([[Hamiltonian (quantum mechanics)|Hamiltonian]]), ''Pi'' is the generator of translations ([[momentum operator]]), ''Ci'' is the generator of Galileian boosts, and ''Lij'' stands for a generator of rotations ([[angular momentum operator]]).

This Lie Algebra is seen to be a special [[classical limit]] of the algebra of the [[Poincaré group#Technical explanation|Poincaré group]], in the limit {{math|''c'' → ∞}}. Technically, the Galilean group is a celebrated [[group contraction]] of the Poincaré group (which, in turn, is a [[group contraction]] of the de Sitter group ''SO''(1,4)).<ref>{{harvnb|Gilmore|2006}}</ref>

Renaming the generators of the latter as {{math| ''ϵimn Ji'' ↦ ''Lmn'' ; ''Pi'' ↦ ''Pi'' ; ''P''0 ↦ ''H''/''c'' ; ''Ki'' ↦ ''cCi''}}, where {{math|''c''}} is the speed of light, or any function thereof diverging as {{math|''c'' → ∞}}, the commutation relations (structure constants) of the latter limit to that of the former.

Note the group invariants {{math|''LmnLmn''}}, {{math|''PiPi''}}.

In matrix form, for ''d''=3, one may consider the ''regular representation'' (embedded in ''GL''(5;ℝ), from which it could be derived by a single group contraction, bypassing the Poincaré group),
:
<math>
iH= \left( {\begin{array}{ccccc}
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0\\
\end{array} } \right) , \qquad
</math>
<math>
i\vec{a}\cdot\vec{P}=
\left( {\begin{array}{ccccc}
0&0&0&0 & a_1\\
0&0&0&0 & a_2\\
0&0&0&0 & a_3\\
0 & 0 & 0 & 0& 0\\
0 & 0 & 0 & 0 & 0\\
\end{array} } \right), \qquad
</math>
<math>
i\vec{v}\cdot\vec{C}=
\left( {\begin{array}{ccccc}
0 & 0 & 0 & v_1 & 0\\
0 & 0 & 0 & v_2 & 0\\
0 & 0 & 0 & v_3 & 0\\
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\\
\end{array} } \right), \qquad
</math>
<math>
i \theta_i \epsilon^{ijk} L_{jk} =
\left( {\begin{array}{ccccc}
0& \theta_3 & -\theta_2 & 0 & 0\\
-\theta_3 & 0 & \theta_1& 0 & 0\\
\theta_2 & -\theta_1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\\
\end{array} } \right ) ~. </math>

The infinitesimal group element is then
::<math>
G(R,\vec{v},\vec{a},s)=1\!\!1_5 + \left( {\begin{array}{ccccc}
0& \theta_3 & -\theta_2 & v_1& a_1\\ -\theta_3 & 0 & \theta_1& v_1 & a_2\\
\theta_2 & -\theta_1 & 0 & v_1 & a_3\\ 0 & 0 & 0 & 0 & s\\
0 & 0 & 0 & 0 & 0\\ \end{array} } \right ) + ... ~. </math>

== Central extension of the Galilean group ==
One could, instead,<ref>{{harvnb|Bargmann|1954}}</ref> augment the Galilean group by a [[Lie algebra extension#Central|central extension]] of the Lie algebra spanned by {{math|''H''′, ''P''′''i'', ''C''′''i'', ''L''′''ij'', ''M''}}, such that {{math|''M''}} [[Commutative operation|commute]]s with everything (i.e. lies in the [[center (algebra)|center]]), and
:<math>[H',P'_i]=0 \,\!</math>
:<math>[P'_i,P'_j]=0 \,\!</math>
:<math>[L'_{ij},H']=0 \,\!</math>
:<math>[C'_i,C'_j]=0 \,\!</math>
:<math>[L'_{ij},L'_{kl}]=i [\delta_{ik}L'_{jl}-\delta_{il}L'_{jk}-\delta_{jk}L'_{il}+\delta_{jl}L'_{ik}] \,\!</math>
:<math>[L'_{ij},P'_k]=i[\delta_{ik}P'_j-\delta_{jk}P'_i] \,\!</math>
:<math>[L'_{ij},C'_k]=i[\delta_{ik}C'_j-\delta_{jk}C'_i] \,\!</math>
:<math>[C'_i,H']=i P'_i \,\!</math>
:<math>[C'_i,P'_j]=i M\delta_{ij} ~.</math>

This algebra is often referred to as the '''Bargmann algebra'''.

==See also==
*[[Representation theory of the Galilean group]]
*[[Lorentz group]]
*[[Poincaré group]]
*[[Lagrangian and Eulerian coordinates]]

==Notes==
{{Reflist}}

==References==
*{{cite book|ref=harv|last1=Arnold|first1=V. I.|authorlink=Vladimir Arnold|title=Mathematical Methods of Classical Mechanics|publisher=Springer-Verlag|date=1989|edition=2|isbn=0-387-96890-3|page=6|url=http://www.springer.com/mathematics/analysis/book/978-0-387-96890-2}}
*{{cite journal|ref=harv|last=Bargmann|first=V.|authorlink=Valentine Bargmann|year=1954|title=On Unitary Ray Representations of Continuous Groups|journal=Annals of Mathematics|series=2|volume=59|issue=1|pages=1–46|doi=10.2307/1969831}}
*{{cite book|year=2002|first1=Nicolaus|last1=Copernicus|authorlink1=Nicolaus Copernicus|first2=Johannes|last2=Kepler|authorlink2=Johannes Kepler|first3=Galileo|last3=Galalei|authorlink3=Galileo Galilei|first4=Isaac|last4= Newton|authorlink4=Isaac newton|first5=Albert|last5=Einstein|authorlink5=Albert Einstein|editor-last=Hawking|editor-first=Stephen|editorlink=Stephen Hawking|pages=515–520|title=On the Shoulders of Giants: The Great Works of Physics and Astronomy|isbn=0-7624-1348-4|publisher=[[Running Press]]|location=Philadelphia, London}}
*{{cite book|ref=harv|last=Galilei|first=Galileo|authorlink=Galileo Galilei|year=1638I|title=Discorsi e Dimostrazioni Matematiche, intorno á due nuoue scienze|pages=191–196|publisher=[[Elsevier]]|location=Leiden|language=Italian}}
*{{cite book|ref=harv|last=Galileo|first=Galilei|year=1638E|title=[[Discourses and Mathematical Demonstrations Relating to Two New Sciences]]|trans-title=Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze|others=Translated to English 1914 by [[Henry Crew]] and Alfonso de Salvio}}
*{{cite book|ref=harv|last=Gilmore|first=Robert|year=2006|title=Lie Groups, Lie Algebras, and Some of Their Applications|publisher=[[Dover Publications]]|series=Dover Books on Mathematics|isbn=0486445291}}
*{{citation|title=Relativity and Its Roots|first1=Banesh|last1=Hoffmann|publisher=Scientific American Books|year=1983|isbn=0-486-40676-8|url=https://books.google.com/?id=JokgnS1JtmMC&pg=PA83}}, [https://books.google.com/books?id=JokgnS1JtmMC&pg=PA83 Chapter 5, p. 83]
*{{citation|title=Physics for Scientists and Engineers|volume= 2|first1=Lawrence S.|last1=Lerner|publisher=Jones and Bertlett Publishers, Inc|year=1996|isbn=0-7637-0460-1|url=https://books.google.com/?id=B8K_ym9rS6UC&pg=PA1047}}, [https://books.google.com/books?id=B8K_ym9rS6UC&pg=PA1047 Chapter 38 §38.2, p. 1046,1047]
*{{citation|title=Basic relativity|first1=Richard A.|last1=Mould|publisher=Springer-Verlag|year=2002|isbn=0-387-95210-1|url=https://books.google.com/?id=lfGE-wyJYIUC&pg=PA42}}, [https://books.google.com/books?id=lfGE-wyJYIUC&pg=PA42 Chapter 2 §2.6, p. 42]
*{{cite web|ref=97-105|first=Mehdi|last1=Nadjafikhah|first2=Ahmad-Reza|last2=Forough|year=2009|title=Galilean Geometry of Motions|journal=
Applied Sciences| volume=11|pages= 91-105|url=http://www.emis.de/journals/APPS/v11/A11-na.pdf}}
*{{citation|title=Principles of Physics: A Calculus-based Text|edition=4th|first1=Raymond A.|last1=Serway|first2=John W.|last2=Jewett|publisher=Brooks/Cole - Thomson Learning|year=2006|isbn=0-534-49143-X|url=https://books.google.com/?id=1DZz341Pp50C&pg=PA261}}, [https://books.google.com/books?id=1DZz341Pp50C&pg=PA261 Chapter 9 §9.1, p. 261]

{{Galileo Galilei}}
{{Relativity}}

[[Category:Theoretical physics]]
[[Category:Time in physics]]

Template:Galileo Galilei

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NickPercival: 1 revision imported

{{Navbox
| name = Galileo Galilei
| title = [[Galileo Galilei]]
| state = {{{state|autocollapse}}}
| listclass = hlist
| image = [[File:Galileo-picture.jpg|right|100px]]

| group1 = Scientific career
| list1 =
* [[Observational astronomy]]
* [[Galileo affair]]
* [[Galileo's escapement]]
* [[Galilean invariance]]
* [[Galilean moons]]
* [[Galilean transformation]]
* [[Galileo's Leaning Tower of Pisa experiment|Leaning Tower of Pisa experiment]]
* [[Phases of Venus]]
* [[Celatone]]
* [[Thermoscope]]

| group2 = Works
| list2 =
* ''[[The Assayer]]''
* ''[[De Motu Antiquiora]]''
* ''[[Dialogue Concerning the Two Chief World Systems]]''
* "[[Discourse on the Tides]]"
* "[[Letter to the Grand Duchess Christina]]"
* ''[[Sidereus Nuncius]]''
* ''[[Two New Sciences]]''

| group3 = Family
| list3 =
* [[Vincenzo Galilei]] (father)
* [[Michelagnolo Galilei]] (brother)
* [[Vincenzo Gamba]] (son)
* [[Maria Celeste]] (daughter)
* [[Marina Gamba]] (mistress)

| group4 = Related
| list4 =
* "[[And yet it moves]]"
* [[Villa Il Gioiello]]
* [[Galileo's paradox]]
* [[Sector (instrument)|Sector]]
* [[Museo Galileo]]
** [[Galileo's telescopes]]
* [[Tribune of Galileo]]

| group5 = In popular culture
| list5 =
* [[Life of Galileo|''Life of Galileo'' (1943 play)]]
* [[Lamp At Midnight|''Lamp At Midnight'' (1947 play)]]
* [[Galileo (1968 film)|''Galileo'' (1968 film)]]
* [[Galileo (1975 film)|''Galileo'' (1975 film)]]
* [[Starry Messenger (picture book)|''Starry Messenger'' (1996 book)]]
* [[Galileo's Daughter|''Galileo's Daughter: A Historical Memoir of Science, Faith, and Love'' (1999 book)]]
* [[Galileo Galilei (opera)|''Galileo Galilei'' (2002 opera)]]
* [[Galileo's Dream|''Galileo's Dream'' (2009 novel)]]

}}<noinclude>
{{doc|content=
{{collapsible option}}
[[Category:Italian writer navigational boxes]]
[[Category:Scientist navigational boxes]]
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Galilean invariance

2017-07-20T22:53:11Z

NickPercival: 1 revision imported

{{Refimprove|date=January 2008}}
'''Galilean invariance''' or '''Galilean relativity''' states that the laws of motion are the same in all [[inertial frame]]s. [[Galileo Galilei]] first described this principle in 1632 in his ''[[Dialogue Concerning the Two Chief World Systems]]'' using [[Galileo's ship|the example of a ship]] travelling at constant velocity, without rocking, on a smooth sea; any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary.

== Formulation ==
Specifically, the term ''Galilean invariance'' today usually refers to this principle as applied to [[Newtonian mechanics]], that is, Newton's laws hold in all frames related to one another by a Galilean transformation. In other words, all frames related to one another by such a transformation are inertial (meaning, Newton's equation of motion is valid in these frames). In this context it is sometimes called ''Newtonian relativity''.

Among the axioms from Newton's theory are:
#There exists an ''[[absolute space]]'', in which Newton's laws are true. An inertial frame is a reference frame in relative uniform motion to absolute space.
#All inertial frames share a ''universal time''.

Galilean relativity can be shown as follows. Consider two inertial frames ''S'' and ''S' ''. A physical event in ''S'' will have position coordinates ''r'' = (''x'', ''y'', ''z'') and time ''t'' in ''S'', and ''r' '' = (''x' '', ''y' '', ''z' '') and time ''t' '' in ''S' ''. By the second axiom above, one can synchronize the clock in the two frames and assume ''t'' = ''t' ''. Suppose ''S' '' is in relative uniform motion to ''S'' with velocity ''v''. Consider a point object whose position is given by functions ''r' ''(''t'') '' in ''S' '' and r''(''t'') in ''S''. We see that

:<math>r'(t) = r(t) - v t.\,</math>

The velocity of the particle is given by the time derivative of the position:

:<math>u'(t) = \frac{d}{d t} r'(t) = \frac{d}{d t} r(t) - v = u(t) - v.</math>

Another differentiation gives the acceleration in the two frames:

:<math>a'(t) = \frac{d}{d t} u'(t) = \frac{d}{d t} u(t) - 0 = a(t).</math>

It is this simple but crucial result that implies Galilean relativity. Assuming that mass is invariant in all inertial frames, the above equation shows Newton's laws of mechanics, if valid in one frame, must hold for all frames.<ref>{{cite book|last1=McComb|first1=W. D.|title=Dynamics and relativity|date=1999|publisher=Oxford University Press|location=Oxford [etc.]|isbn=0-19-850112-9|pages=22-24}}</ref> But it is assumed to hold in absolute space, therefore Galilean relativity holds.

=== Newton's theory versus special relativity ===

A comparison can be made between Newtonian relativity and [[special relativity]].

Some of the assumptions and properties of Newton's theory are:
#The existence of infinitely many inertial frames. Each frame is of infinite size (the entire universe may be covered by many linearly equivalent frames). Any two frames may be in relative uniform motion. (The relativistic nature of mechanics derived above shows that the absolute space assumption is not necessary.)
#The inertial frames may move in ''all'' possible relative forms of uniform motion.
#There is a universal, or absolute, notion of time.
#Two inertial frames are related by a [[Galilean transformation]].
#In all inertial frames, Newton's laws, and gravity, hold.

In comparison, the corresponding statements from special relativity are as follows:
#The existence, as well, of infinitely many non-inertial frames, each of which referenced to (and physically determined by) a unique set of spacetime coordinates. Each frame may be of infinite size, but its definition is always determined locally by contextual physical conditions. Any two frames may be in relative non-uniform motion (as long as it is assumed that this condition of relative motion implies a relativistic dynamical effect -and later, mechanical effect in general relativity- between both frames).
#Rather than freely allowing all conditions of relative uniform motion between frames of reference, the relative velocity between two inertial frames becomes bounded above by the speed of light.
#Instead of universal time, each inertial frame possesses its own notion of time.
#The Galilean transformations are replaced by [[Lorentz transformation]]s.
#In all inertial frames, ''all'' laws of physics are the same.

Notice both theories assume the existence of inertial frames. In practice, the size of the frames in which they remain valid differ greatly, depending on gravitational tidal forces.

In the appropriate context, a ''local Newtonian inertial frame'', where Newton's theory remains a good model, extends to, roughly, 107 light years.

In special relativity, one considers ''Einstein's cabins'', cabins that fall freely in a gravitational field. According to Einstein's thought experiment, a man in such a cabin experiences (to a good approximation) no gravity and therefore the cabin is an approximate inertial frame. However, one has to assume that the size of the cabin is sufficiently small so that the gravitational field is approximately parallel in its interior. This can greatly reduce the sizes of such approximate frames, in comparison to Newtonian frames. For example, an artificial satellite orbiting around earth can be viewed as a cabin. However, reasonably sensitive instruments would detect "microgravity" in such a situation because the "lines of force" of the Earth's gravitational field converge.

In general, the convergence of gravitational fields in the universe dictates the scale at which one might consider such (local) inertial frames. For example, a spaceship falling into a black hole or neutron star would (at a certain distance) be subjected to tidal forces so strong that it would be crushed in width and torn apart in length.<ref name="taylowwheeler">Taylor and Wheeler's [http://www.eftaylor.com/pub/chapter2.pdf ''Exploring Black Holes - Introduction to General Relativity'', Chapter 2], 2000, p. 2:6.</ref> In comparison, however, such forces might only be uncomfortable for the astronauts inside (compressing their joints, making it difficult to extend their limbs in any direction perpendicular to the gravity field of the star). Reducing the scale further, the forces at that distance might have almost no effects at all on a mouse. This illustrates the idea that all freely falling frames are locally inertial (acceleration and gravity-free) if the scale is chosen correctly.<ref name="taylowwheeler" />

=== Electromagnetism ===

[[Maxwell's equations]] governing [[electromagnetism]] possess a different [[Symmetry in physics|symmetry]], [[Lorentz invariance]], under which lengths and times ''are'' affected by a change in velocity, which is then described mathematically by a [[Lorentz transformation]].

[[Albert Einstein]]'s central insight in formulating [[special relativity]] was that, for full consistency with electromagnetism, mechanics must also be revised such that Lorentz invariance replaces Galilean invariance. At the low relative velocities characteristic of everyday life, Lorentz invariance and Galilean invariance are nearly the same, but for relative velocities close to [[Speed of light|that of light]] they are very different.

==Work, kinetic energy, and momentum==
Because the distance covered while applying a force to an object depends on the inertial frame of reference, so does the [[Mechanical work|work]] done. Due to [[Newton's laws of motion#Newton's third law|Newton's law of reciprocal actions]] there is a reaction force; it does work depending on the inertial frame of reference in an opposite way. The total work done is independent of the inertial frame of reference.

Correspondingly the [[kinetic energy]] of an object, and even the change in this energy due to a change in velocity, depends on the inertial frame of reference. The total kinetic energy of an [[isolated system]] also depends on the inertial frame of reference: it is the sum of the total kinetic energy in a [[center of momentum frame]] and the kinetic energy the total mass would have if it were concentrated in the [[center of mass]]. Due to the [[conservation of momentum]] the latter does not change with time, so changes with time of the total kinetic energy do not depend on the inertial frame of reference.

By contrast, while the [[momentum]] of an object also depends on the inertial frame of reference, its change due to a change in velocity does not.

==See also==
*[[Absolute time and space]]
*[[Superluminal motion]]

==Notes and references==
<references/>

{{Galileo Galilei}}
{{Relativity|state=expanded}}

{{DEFAULTSORT:Galilean Invariance}}
[[Category:Classical mechanics]]
[[Category:Galileo Galilei|Invariance]]

[[he:מערכת ייחוס#עקרון היחסות של גלילאו]]

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Non-inertial reference frame

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{{Classical mechanics}}
A '''non-inertial reference frame''' is a [[frame of reference]] that is undergoing [[acceleration]] with respect to an [[Inertial frame of reference|inertial frame]].<ref name=Tocaci>{{cite book |title=Relativistic Mechanics, Time, and Inertia |author=Emil Tocaci, Clive William Kilmister |page=251 |url=https://books.google.com/books?id=7dVRL51JRI0C&pg=PA251 |isbn=90-277-1769-9 |year=1984 |publisher=Springer }}</ref> An [[accelerometer]] at rest in a non-inertial frame will in general detect a non-zero acceleration. In a curved [[spacetime]] all frames are non-inertial{{Clarify | date=November 2012}}. The laws of motion in non-inertial frames do not take the simple form they do in inertial frames, and the laws vary from frame to frame depending on the acceleration.<ref>{{cite book |title=Essential Relativity |author=Wolfgang Rindler |page=25 |url=https://books.google.com/books?id=0J_dwCmQThgC&pg=PT43 |isbn=3-540-07970-X |year=1977 |publisher=Birkhäuser}}</ref><ref>{{cite book |title=Basics of Space Flight |author= Ludwik Marian Celnikier |page=286 |url=https://books.google.com/books?id=u2kf5uuaC6oC&pg=PA286 |isbn=2-86332-132-3 |year=1993 |publisher=Atlantica Séguier Frontières}}</ref> To explain the motion of bodies entirely within the viewpoint of non-inertial reference frames, [[fictitious forces]] (also called inertial forces, pseudo-forces<ref name=Iro>{{cite book |author=Harald Iro |title=A Modern Approach to Classical Mechanics |page=180 |url=https://books.google.com/books?id=-L5ckgdxA5YC&pg=PA179 |isbn=981-238-213-5 |year=2002 |publisher=World Scientific }}</ref> and d'Alembert forces) must be introduced to account for the observed motion, such as the [[Coriolis force]] or the [[centrifugal force (fictitious)|centrifugal force]], as derived from the acceleration of the non-inertial frame.<ref name=Shadowitz>{{cite book |author=Albert Shadowitz |url=https://books.google.com/books?id=1axfJqUT6R0C&pg=PA4 |title=Special relativity |isbn=0-486-65743-4 |page=4 |publisher=Courier Dover Publications |edition=Reprint of 1968 |year=1988}}</ref>
As stated by Goodman and Warner, "One might say that '''F''' {{=}} ''m'''''a''' holds in any coordinate system provided the term 'force' is redefined to include the so-called 'reversed effective forces' or 'inertia forces'."<ref name=Goodman>{{cite book |title=Dynamics |author=Lawrence E. Goodman & William H. Warner |url=https://books.google.com/books?id=2z0ue1xk7gUC |isbn=0-486-42006-X |publisher=Courier Dover Publications |year=2001 |edition=Reprint of 1963|page=358}}</ref>

==Avoiding fictitious forces in calculations==
{{see also|Inertial frame of reference|Fictitious force}}
In flat spacetime, the use of non-inertial frames can be avoided if desired. Measurements with respect to non-inertial reference frames can always be transformed to an inertial frame, incorporating directly the acceleration of the non-inertial frame as that acceleration as seen from the inertial frame.<ref name=Alonzo>{{cite book |author= M. Alonso & E.J. Finn |title=Fundamental university physics
|publisher=, Addison-Wesley |year=1992 |url=https://books.google.com/books?id=c5UAAAAACAAJ&dq=isbn=0201565188&lr=&as_brr=0 |isbn= 0-201-56518-8}}</ref> This approach avoids use of fictitious forces (it is based on an inertial frame, where fictitious forces are absent, by definition) but it may be less convenient from an intuitive, observational, and even a calculational viewpoint.<ref name=Price>“The inertial frame equations have to account for ''VΩ'' and this very large centripetal force explicitly, and yet our interest is almost always the small relative motion of the atmosphere and ocean, ''V' '', since it is the relative
motion that transports heat and mass over the Earth. … To say it a little differently—it is the relative velocity that we measure when [we] observe from Earth’s surface, and it is the relative velocity that we seek for most any practical purposes.” [http://ocw.mit.edu/ans7870/resources/price/index.htm MIT essays] by James F. Price, Woods Hole Oceanographic Institution (2006). See in particular §4.3, p. 34 in the [http://ocw.mit.edu/ans7870/resources/price/essay2.pdf Coriolis lecture]</ref> As pointed out by Ryder for the case of rotating frames as used in meteorology:<ref name=Ryder>{{cite book |title=Classical Mechanics |author=Peter Ryder |url=https://books.google.com/books?id=j1Y5FfdQHsQC&pg=PA80 |isbn=978-3-8322-6003-3 |publisher=Aachen Shaker |year=2007 |pages=78–79 }}</ref>
{{quote|A simple way of dealing with this problem is, of course, to transform all coordinates to an inertial system. This is, however, sometimes inconvenient. Suppose, for example, we wish to calculate the movement of air masses in the earth's atmosphere due to pressure gradients. We need the results relative to the rotating frame, the earth, so it is better to stay within this coordinate system if possible. This can be achieved by introducing ''fictitious'' (or "non-existent") forces which enable us to apply Newton's Laws of Motion in the same way as in an inertial frame.|Peter Ryder|''Classical Mechanics'', pp. 78-79}}

==Detection of a non-inertial frame: need for fictitious forces==
That a given frame is non-inertial can be detected by its need for fictitious forces to explain observed motions.<ref name=Serway>{{cite book |title=Physics for scientists & engineers |author=Raymond A. Serway |year=1990 |publisher=Saunders College Publishing |edition=3rd |isbn=0-03-031358-9 |page=135 |url=https://books.google.com/books?lr=&as_brr=0&q=%22fictitious+forces+do+not+exist+when+the+motion+is+observed+in+an+inertial+frame.+The+fictitious+forces+are+used+only+in+an+accelerating%22&btnG=Search+Books}}</ref><ref name="ArnoldQuote">{{cite book |title=Mathematical Methods of Classical Mechanics |page=129 |author=V. I. Arnol'd |isbn=978-0-387-96890-2 |year=1989 |url=https://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=additional+terms+called+inertial+forces.+This+allows+us+to+detect+experimentally&as_oq=&as_eq=&as_brr=0&lr=&as_vt=&as_auth=&as_pub=&as_sub=&as_drrb=c&as_miny=&as_maxy=&as_isbn=|publisher=Springer}}</ref><ref name=Rothman>{{cite book |title=Discovering the Natural Laws: The Experimental Basis of Physics |author= Milton A. Rothman |page=23 |url=https://books.google.com/books?id=Wdp-DFK3b5YC&pg=PA23&vq=inertial&dq=reference+%22laws+of+physics%22&lr=&as_brr=0&source=gbs_search_s&cad=5&sig=ACfU3U33YE3keeD7lDVtQvt-ltW87Lsq2Q
|isbn=0-486-26178-6 |publisher=Courier Dover Publications |year=1989 }}</ref><ref name=Borowitz>{{cite book |title=A Contemporary View of Elementary Physics |page=138 |publisher=McGraw-Hill |year=1968 |url=https://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=The+effect+of+his+being+in+the+noninertial+frame+is+to+require+the+observer+to&as_oq=&as_eq=&as_brr=0&lr=&as_vt=&as_auth=&as_pub=&as_sub=&as_drrb=c&as_miny=&as_maxy=&as_isbn= |asin= B000GQB02A |author=Sidney Borowitz & Lawrence A. Bornstein }}</ref><ref name=Meirovitch>{{cite book |author=Leonard Meirovitch |page=4 |isbn=0-486-43239-4 |publisher=Courier Dover Publications |year=2004 |edition=Reprint of 1970 |url=https://books.google.com/books?id=GfCil84YTm4C&pg=PA4&dq=%22in+accelerated+systems,+we+must%22&lr=&as_brr=0&sig=ACfU3U0UrA5jcOx4pB9QIlyA7BQiXwAV5Q |title =Methods of analytical Dynamics}}</ref> For example, the rotation of the [[Earth]] can be observed using a [[Foucault pendulum]].<ref name=diFrancia>{{cite book |title=The Investigation of the Physical World |author=Giuliano Toraldo di Francia |page=115 |url=https://books.google.com/books?id=cFQ7AAAAIAAJ&pg=PA46&dq=laws+physics+%22+form%22&lr=&as_brr=0&sig=ACfU3U0zI1ZXjyB3G6Z3AI3zM_Z2YfYN6g#PPA115,M1 |isbn=0-521-29925-X |publisher=CUP Archive |year=1981 }}</ref> The rotation of the Earth seemingly causes the pendulum to change its plane of oscillation because the surroundings of the pendulum move with the Earth. As seen from an Earth-bound (non-inertial) frame of reference, the explanation of this apparent change in orientation requires the introduction of the fictitious [[Coriolis effect|Coriolis force]].

Another famous example is that of the tension in the string between [[rotating spheres|two spheres rotating about each other]].<ref>
{{cite book |title=Analytical Mechanics |page=324 |url=https://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA324 |isbn=0-521-57572-9 |publisher=[[Cambridge University Press]] |year=1998 |author=Louis N. Hand, Janet D. Finch}}</ref><ref>{{cite book |title=The Cambridge companion to Newton |url =https://books.google.com/books?id=3wIzvqzfUXkC&pg=PA43 |author=I. Bernard Cohen, George Edwin Smith |page=43 |isbn=0-521-65696-6 |year=2002 |publisher=Cambridge University Press}}</ref> In that case, prediction of the measured tension in the string based upon the motion of the spheres as observed from a rotating reference frame requires the rotating observers to introduce a fictitious centrifugal force.

In this connection, it may be noted that a change in coordinate system, for example, from Cartesian to polar, if implemented without any change in relative motion, does not cause the appearance of fictitious forces, despite the fact that the form of the laws of motion varies from one type of curvilinear coordinate system to another.

==Fictitious forces in curvilinear coordinates==
{{see also|Mechanics of planar particle motion}}
A different use of the term "fictitious force" often is used in [[curvilinear coordinates]], particularly [[polar coordinates]]. To avoid confusion, this distracting ambiguity in terminologies is pointed out here. These so-called "forces" are non-zero in all frames of reference, inertial or non-inertial, and do ''not'' transform as vectors under rotations and translations of the coordinates (as all Newtonian forces do, fictitious or otherwise).

This incompatible use of the term "fictitious force" is unrelated to non-inertial frames. These so-called "forces" are defined by determining the acceleration of a particle within the curvilinear coordinate system, and then separating the simple double-time derivatives of coordinates from the remaining terms. These remaining terms then are called "fictitious forces". More careful usage calls these terms "[[generalized forces|generalized fictitious forces]]" to indicate their connection to the [[generalized coordinates]] of [[Lagrangian mechanics]]. The application of Lagrangian methods to polar coordinates can be found [[Mechanics of planar particle motion#Lagrangian approach|here]].

==Relativistic point of view==
{{unreferenced section|small=y|date=April 2017}}
===Frames and flat spacetime===
{{See|Proper reference frame (flat spacetime)}}
If a region of spacetime is declared to be [[Euclidean space|Euclidean]], and effectively free from obvious gravitational fields, then if an accelerated coordinate system is overlaid onto the same region, it can be said that a ''uniform fictitious field'' exists in the accelerated frame (we reserve the word gravitational for the case in which a mass is involved). An object accelerated to be stationary in the accelerated frame will "feel" the presence of the field, and they will also be able to see environmental matter with inertial states of motion (stars, galaxies, etc.) to be apparently falling "downwards" in the field along curved [[trajectory|trajectories]] as if the field is real.

In frame-based descriptions, this supposed field can be made to appear or disappear by switching between "accelerated" and "inertial" coordinate systems.

===More advanced descriptions===
As the situation is modeled in finer detail, using the [[general principle of relativity]], the concept of a ''frame-dependent'' gravitational field becomes less realistic. In these [[Mach's principle|Machian]] models, the accelerated body can agree that the apparent gravitational field is associated with the motion of the background matter, but can also claim that the motion of the material as if there is a gravitational field, causes the gravitational field - the accelerating background matter "[[light-dragging effects|drags light]]". Similarly, a background observer can argue that the forced acceleration of the mass causes an apparent gravitational field in the region between it and the environmental material (the accelerated mass also "drags light").
This "mutual" effect, and the ability of an accelerated mass to warp lightbeam geometry and lightbeam-based coordinate systems, is referred to as [[frame-dragging]].

Frame-dragging removes the usual distinction between accelerated frames (which show gravitational effects) and inertial frames (where the geometry is supposedly free from gravitational fields). When a forcibly-accelerated body physically "drags" a coordinate system, the problem becomes an exercise in warped spacetime for all observers.

==See also==
*[[Fictitious force]]
*[[Centrifugal force]]
*[[Coriolis effect]]
*[[Inertial frame of reference]]
*[[Free motion equation]]

==References and notes==
{{reflist|30em}}

{{DEFAULTSORT:Non-Inertial Reference Frame}}
[[Category:Frames of reference]]
[[Category:Classical mechanics]]

Module:Other uses

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local mHatnote = require('Module:Hatnote')
local mHatlist = require('Module:Hatnote list')
local mArguments --initialize lazily
local mTableTools --initialize lazily
local libraryUtil = require('libraryUtil')
local checkType = libraryUtil.checkType
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mArguments = require('Module:Arguments')
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mTableTools = require('Module:TableTools')
local x = frame.args[1]
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mArguments.getArgs(frame, {parentOnly = true})
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title = mw.title.getCurrentTitle().prefixedText,
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return p._otheruses(args, options)
end

-- Main generator
function p._otheruses(args, options)
--Type-checks and defaults
checkType('_otheruses', 1, args, 'table', true)
args = args or {}
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error('No default title data provided in "_otheruses" options table', 2)
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end
if emptyArgs then
args = {
options.defaultPage or
mHatnote.disambiguate(options.title, options.disambiguator)
}
end
--Generate and return hatnote
local text = mHatlist.forSeeTableToString({{
use = options.otherText and "other " .. options.otherText or nil,
pages = args
}})
return mHatnote._hatnote(text)
end

return p

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<includeonly>{{Sidebar with collapsible lists
| name = Classical mechanics
| titlestyle = padding-left:0.9em;padding-right:0.9em;
| title = [[Classical mechanics]]
| image = <math>\vec{F} = m\vec{a}</math>
| captionstyle = font-size:90%;padding:0.6em 0;font-style:italic;
| caption = [[Second law of motion]]
| headingstyle = background:#ddf; display:block;margin-bottom:1.0em;
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| listclass = plainlist
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| heading1 = {{startflatlist}}
* [[History of classical mechanics|History]]
* [[Timeline of classical mechanics|Timeline]]
{{endflatlist}}

| list2name = branches
| list2title = Branches
| list2 = {{startflatlist}}

* [[Applied mechanics|Applied]]
* [[Celestial mechanics|Celestial]]
* [[Continuum mechanics|Continuum]]
* [[Analytical dynamics|Dynamics]]
* [[Kinematics]]
* [[Kinetics (physics)|Kinetics]]
* [[Statics]]
* [[Statistical mechanics|Statistical]]
{{endflatlist}}

| list3name = fundamental concepts
| list3title = Fundamentals
| list3 = {{startflatlist}}

* [[Acceleration]]
* [[Angular momentum]]
* [[Couple (mechanics)|Couple]]
* [[D'Alembert's principle]]
* [[Energy]]
** [[Kinetic energy#Newtonian kinetic energy|kinetic]]
** [[Potential energy|potential]]
* [[Force]]
* [[Frame of reference]]
* [[Impulse (physics)|Impulse]]
* {{nowrap|[[Inertia]]{{\}}[[Moment of inertia]]}}
* [[Mass]]
* [[Power (physics)|Mechanical power]]
* [[Work (physics)|Mechanical work]]
* [[Moment (physics)|Moment]]
* [[Momentum]]
* [[Space]]
* [[Speed]]
* [[Time]]
* [[Torque]]
* [[Velocity]]
* [[Virtual work]]
{{endflatlist}}

| list4name = formulations
| list4title = Formulations
| list4 =
* {{longitem|'''[[Newton's laws of motion]]'''}}
* {{longitem|'''[[Analytical mechanics]]''' {{ubl|[[Lagrangian mechanics]] | [[Hamiltonian mechanics]] | [[Routhian mechanics]] | [[Hamilton–Jacobi equation]] | [[Appell's equation of motion]] | [[Udwadia–Kalaba equation]] | [[Koopman–von Neumann classical mechanics|Koopman–von Neumann mechanics]]}}}}

| list5name = core
| list5title = Core topics
| list5 = {{startflatlist}}

* [[Damping]] ([[Damping ratio|ratio]])
* [[Displacement (vector)|Displacement]]
* [[Equations of motion]]
* [[Euler's laws of motion|{{allow wrap|Euler's laws of motion}}]]
* [[Fictitious force]]
* [[Friction]]
* [[Harmonic oscillator]]
{{endflatlist}}
* {{nowrap|[[Inertial frame of reference|Inertial]]{{\}}[[Non-inertial reference frame]]}}
* [[Mechanics of planar particle motion]]
{{startflatlist}}
* [[Motion (physics)|Motion]] ([[Linear motion|linear]])
* [[Newton's law of universal gravitation|{{allow wrap|Newton's law of universal gravitation}}]]
* [[Newton's laws of motion]]
* [[Relative velocity]]
* [[Rigid body]]
** [[Rigid body dynamics|dynamics]]
** [[Euler's equations (rigid body dynamics)|Euler's equations]]
* [[Simple harmonic motion]]
* [[Vibration]]

| list6name = rotational
| list6title = [[Rotation around a fixed axis|Rotation]]
| list6 = {{startflatlist}}
* [[Circular motion]]
* [[Rotating reference frame]]
* [[Centripetal force]]
* [[Centrifugal force]]
** [[Reactive centrifugal force|reactive]]
* [[Coriolis force]]
* [[Pendulum (mathematics)|Pendulum]]
* [[Speed#Tangential speed|Tangential speed]]
* [[Rotational speed]]
{{endflatlist}}
* [[Angular acceleration]]{{\}}[[Angular displacement|displacement]]{{\}}[[Angular frequency|frequency]]{{\}}[[Angular velocity|velocity]]

| list7name = scientists
| list7title = Scientists
| list7 = {{startflatlist}}

* [[Galileo Galilei|Galileo]]
* [[Isaac Newton|Newton]]
* [[Johannes Kepler|Kepler]]
* [[Jeremiah Horrocks|Horrocks]]
* [[Edmond Halley|Halley]]
* [[Leonhard Euler|Euler]]
* [[Jean le Rond d'Alembert|d'Alembert]]
* [[Alexis Clairaut|Clairaut]]
* [[Joseph-Louis Lagrange|Lagrange]]
* [[Pierre-Simon Laplace|Laplace]]
* [[William Rowan Hamilton|Hamilton]]
* [[Siméon Denis Poisson|Poisson]]
* [[Daniel Bernoulli]]
* [[Johann Bernoulli]]
* [[Augustin-Louis Cauchy|Cauchy]]
{{endflatlist}}

| navbarstyle = padding-top:0.15em;

}}</includeonly><noinclude>
{{Documentation
| content = {{Classical mechanics |all}}
{{Collapsible lists option |listnames={{hlist|branches|fundamental concepts|formulations|core|rotational|scientists}}}}
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== See also ==
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Inertial frame of reference

2017-07-20T22:47:10Z

NickPercival: 1 revision imported

{{Other uses|Framing (disambiguation){{!}}Framing}}
{{Use dmy dates|date=September 2010}}
{{Classical mechanics|cTopic=Core topics}}

An '''inertial frame of reference''', in classical physics, is a [[frame of reference]] in which [[Physical body|bodies]], whose [[net force]] acting upon them is zero, are not accelerated, that is they are at rest or they move at a constant [[velocity]] in a straight line.<ref>{{cite| url=http://physics.unm.edu/Courses/Fields/Phys262/lecture17.pdf| accessdate=June 5, 2017| author=Douglas Fields|date=2015| title=Galilean Relativity |work=Physics 262-01 Fall 2015, [[University of New Mexico]]}}</ref> In [[Analytical mechanics|analytical]] terms, it is a frame of reference that describes time and space [[Homogeneity (physics)|homogeneously]], [[isotropic]]ally, and in a time-independent manner.<ref name=LandauMechanics>{{cite book|last=Landau|first=L. D.|last2=Lifshitz|first2=E. M.|title=Mechanics|date=1960|publisher=Pergamon Press|pages=4–6}}</ref> Conceptually, in [[classical physics]] and [[special relativity]], the physics of a system in an inertial frame have no causes external to the system.<ref name="Ferraro">{{cite|title=Einstein's Space-Time: An Introduction to Special and General Relativity|first1=Rafael|last1=Ferraro|publisher=Springer Science & Business Media|date=2007|isbn=9780387699462|pp=209–210}}</ref> An inertial frame of reference may also be called an '''inertial reference frame''', '''inertial frame''', '''Galilean reference frame''', or '''inertial space'''.{{cn|date=June 2017}}

All inertial frames are in a state of constant, [[wiktionary:rectilinear|rectilinear]] motion with respect to one another; an [[accelerometer]] moving with any of them would detect zero acceleration. Measurements in one inertial frame can be converted to measurements in another by a simple transformation (the [[Galilean transformation]] in Newtonian physics and the [[Lorentz transformation]] in special relativity). In [[general relativity]], in any region small enough for the curvature of spacetime and [[tidal forces]]<ref>{{cite book |title=Einstein's Physics: Atoms, Quanta, and Relativity - Derived, Explained, and Appraised |edition=illustrated |first1=Ta-Pei |last1=Cheng |publisher=OUP Oxford |year=2013 |isbn=978-0-19-966991-2 |page=219 |url=https://books.google.com/books?id=thXT19cY9jsC}} [https://books.google.com/books?id=thXT19cY9jsC&pg=PA219 Extract of page 219]</ref> to be negligible, one can find a set of inertial frames that approximately describe that region.<ref name=Einstein0>{{Cite book|title=Relativity: The Special and General Theory |author=Albert Einstein |page= 71 |url= https://books.google.com/?id=YLsSxQqEww0C&pg=PA71 |isbn=0-486-41714-X |publisher=Courier Dover Publications |date=2001 |edition=3rd |origyear= Reprint of edition of 1920 translated by RQ Lawson}}</ref><ref name=Giulini>{{Cite book|title=Special Relativity |author= Domenico Giulini |page =19 |url=https://books.google.com/?id=4U1bizA_0gsC&pg=PA19 |isbn=0-19-856746-4 |date=2005 |publisher=Cambridge University Press}}</ref>

In a [[non-inertial reference frame]] in classical physics and special relativity, the physics of a system vary depending on the acceleration of that frame with respect to an inertial frame, and the usual physical forces must be supplemented by [[fictitious force]]s.<ref name=Rothman>{{Cite book|title=Discovering the Natural Laws: The Experimental Basis of Physics |author= Milton A. Rothman |page=23 |url=https://books.google.com/?id=Wdp-DFK3b5YC&pg=PA23&vq=inertial&dq=reference+%22laws+of+physics%22
|isbn=0-486-26178-6 |publisher=Courier Dover Publications |date=1989}}</ref><ref name=Borowitz>{{Cite book|title=A Contemporary View of Elementary Physics |page=138 |publisher=McGraw-Hill |date=1968 |url=https://books.google.com/books?num=10&btnG=Google+Search|asin= B000GQB02A |author1=Sidney Borowitz |author2=Lawrence A. Bornstein }}</ref> In contrast, systems in non-inertial frames in general relativity don't have external causes, because of the principle of [[geodesic (general relativity)|geodesic motion]].<ref>{{cite|title=Mach's Principle II|first1=James G.|last1=Gilson|date=September 1, 2004|arxiv=physics/0409010}}</ref> In classical physics, for example, a ball dropped towards the ground does not go exactly straight down because the [[Earth]] is rotating, which means the frame of reference of an observer on Earth is not inertial. The physics must account for the [[Coriolis effect]]—in this case thought of as a force—to predict the horizontal motion. Another example of such a fictitious force associated with rotating reference frames is the [[centrifugal effect]], or centrifugal force.

==Introduction==
The motion of a body can only be described relative to something else—other bodies, observers, or a set of space-time coordinates. These are called [[frame of reference|frames of reference]]. If the coordinates are chosen badly, the laws of motion may be more complex than necessary. For example, suppose a free body that has no external forces on it is at rest at some instant. In many coordinate systems, it would begin to move at the next instant, even though there are no forces on it. However, a frame of reference can always be chosen in which it remains stationary. Similarly, if space is not described uniformly or time independently, a coordinate system could describe the simple flight of a free body in space as a complicated zig-zag in its coordinate system. Indeed, an intuitive summary of inertial frames can be given as: In an inertial reference frame, the laws of mechanics take their simplest form.<ref name=LandauMechanics/>

In an inertial frame, [[Newton's first law]], the ''law of inertia'', is satisfied: Any free motion has a constant magnitude and direction.<ref name=LandauMechanics/> [[Newton's second law]] for a [[Point particle|particle]] takes the form:

:<math>\mathbf{F} = m \mathbf{a} \ ,</math>

with '''F''' the net force (a [[Euclidean vector|vector]]), ''m'' the mass of a particle and '''a''' the [[acceleration]] of the particle (also a vector) which would be measured by an observer at rest in the frame. The force '''F''' is the [[vector sum]] of all "real" forces on the particle, such as electromagnetic, gravitational, nuclear and so forth. In contrast, Newton's second law in a [[rotating frame of reference]], rotating at angular rate ''Ω'' about an axis, takes the form:

:<math>\mathbf{F}' = m \mathbf{a} \ ,</math>

which looks the same as in an inertial frame, but now the force '''F'''′ is the resultant of not only '''F''', but also additional terms (the paragraph following this equation presents the main points without detailed mathematics):

:<math>\mathbf{F}' = \mathbf{F} - 2m \mathbf{\Omega} \times \mathbf{v}_{B} - m \mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{x}_B ) - m \frac{d \mathbf{\Omega}}{dt} \times \mathbf{x}_B \ , </math>

where the angular rotation of the frame is expressed by the vector '''Ω''' pointing in the direction of the axis of rotation, and with magnitude equal to the angular rate of rotation ''Ω'', symbol × denotes the [[vector cross product]], vector '''x'''''B'' locates the body and vector '''v'''''B'' is the [[velocity]] of the body according to a rotating observer (different from the velocity seen by the inertial observer).

The extra terms in the force '''F'''′ are the "fictitious" forces for this frame, whose causes are external to the system in the frame. The first extra term is the [[Coriolis force]], the second the [[centrifugal force (rotating reference frame)|centrifugal force]], and the third the [[Euler force]]. These terms all have these properties: they vanish when ''Ω'' = 0; that is, they are zero for an inertial frame (which, of course, does not rotate); they take on a different magnitude and direction in every rotating frame, depending upon its particular value of '''Ω'''; they are ubiquitous in the rotating frame (affect every particle, regardless of circumstance); and they have no apparent source in identifiable physical sources, in particular, [[matter]]. Also, fictitious forces do not drop off with distance (unlike, for example, [[nuclear force]]s or [[electrical force]]s). For example, the centrifugal force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis.

All observers agree on the real forces, '''F'''; only non-inertial observers need fictitious forces. The laws of physics in the inertial frame are simpler because unnecessary forces are not present.

In Newton's time the [[fixed stars]] were invoked as a reference frame, supposedly at rest relative to [[absolute space]]. In reference frames that were either at rest with respect to the fixed stars or in uniform translation relative to these stars, [[Newton's laws of motion]] were supposed to hold. In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of [[fictitious forces]], for example, the [[Coriolis force]] and the [[centrifugal force]]. Two interesting experiments were devised by Newton to demonstrate how these forces could be discovered, thereby revealing to an observer that they were not in an inertial frame: the example of the tension in the cord linking [[rotating spheres|two spheres rotating]] about their center of gravity, and the example of the curvature of the surface of water in a [[bucket argument|rotating bucket]]. In both cases, application of [[Newton's second law]] would not work for the rotating observer without invoking centrifugal and Coriolis forces to account for their observations (tension in the case of the spheres; parabolic water surface in the case of the rotating bucket).

As we now know, the fixed stars are not fixed. Those that reside in the [[Milky Way]] turn with the galaxy, exhibiting [[proper motion]]s. Those that are outside our galaxy (such as nebulae once mistaken to be stars) participate in their own motion as well, partly due to [[expansion of the universe]], and partly due to [[peculiar velocity|peculiar velocities]].<ref name=Balbi>{{Cite book|title=The Music of the Big Bang |author=Amedeo Balbi |isbn=3-540-78726-7 |publisher=Springer |date=2008 |page= 59 |url=https://books.google.com/?id=vEJM7s909CYC&pg=PA58&dq=CMB+%22rotation+of+the+universe%22 }}</ref> The [[Andromeda galaxy]] is on [[Andromeda–Milky Way collision|collision course with the Milky Way]] at a speed of 117 km/s.<ref>{{Cite journal|title=Constraints on the proper motion of the Andromeda galaxy based on the survival of its satellite M33 |pages=894–898 |author1=Abraham Loeb |author2=Mark J. Reid |author3=Andreas Brunthaler |author4=Heino Falcke |journal=The Astrophysical Journal |volume=633 |date=2005 |url=http://www.mpifr-bonn.mpg.de/staff/abrunthaler/pub/loeb.pdf |doi=10.1086/491644 |bibcode=2005ApJ...633..894L|arxiv = astro-ph/0506609|issue=2 }}</ref> The concept of inertial frames of reference is no longer tied to either the fixed stars or to absolute space. Rather, the identification of an inertial frame is based upon the simplicity of the laws of physics in the frame. In particular, the absence of fictitious forces is their identifying property.<ref name=Stachel>{{Cite book|pages= 235–236 |url=https://books.google.com/?id=OAsQ_hFjhrAC&pg=PA235&dq=%22laws+of+nature+took+a+simpler+form%22 |title=Einstein from "B" to "Z" |author=John J. Stachel |isbn=0-8176-4143-2 |publisher=Springer |date=2002}}</ref>

In practice, although not a requirement, using a frame of reference based upon the fixed stars as though it were an inertial frame of reference introduces very little discrepancy. For example, the centrifugal acceleration of the Earth because of its rotation about the Sun is about thirty million times greater than that of the Sun about the galactic center.<ref name=Graneau>{{Cite book|title=In the Grip of the Distant Universe |author1=Peter Graneau |author2=Neal Graneau |page= 147 |url=https://books.google.com/?id=xpIJZxDkWAUC&pg=PA144&dq=universe+%22fixed+stars%22+date:2004-2010 |isbn=981-256-754-2 |publisher=World Scientific |date=2006}}</ref>

To illustrate further, consider the question: "Does our Universe rotate?" To answer, we might attempt to explain the shape of the [[Milky Way]] galaxy using the laws of physics,<ref name=Genz>{{Cite book|title=Nothingness |author=Henning Genz |page= 275 |url= https://books.google.com/?id=Cn_Q9wbDOM0C&pg=PA274&dq=%22rotation+of+the+universe%22 |isbn=0-7382-0610-5 |date=2001 |publisher=Da Capo Press}}</ref> although other observations might be more definitive, that is, provide larger [[Observational error|discrepancies]] or less [[measurement uncertainty]], like the anisotropy of the [[microwave background radiation]] or [[Big Bang nucleosynthesis]].<ref name=Thompson>{{Cite book|title=Advances in Astronomy |url= https://books.google.com/?id=3TrsMTmbr-sC&pg=PA32&dq=CMB+%22rotation+of+the+universe%22 |author=J Garcio-Bellido|editor=J. M. T. Thompson |publisher=Imperial College Press |date=2005 |page= 32, §9 |chapter=The Paradigm of Inflation |isbn=1-86094-577-5}}</ref><ref name=Szydlowski>{{Cite journal|title=Dark energy and global rotation of the Universe |author1=Wlodzimierz Godlowski |author2=Marek Szydlowski |arxiv=astro-ph/0303248 |date=2003 |doi=10.1023/A:1027301723533 |journal=General Relativity and Gravitation |volume=35 |pages=2171|issue=12|bibcode = 2003GReGr..35.2171G }}</ref> The flatness of the Milky Way depends on its rate of rotation in an inertial frame of reference. If we attribute its apparent rate of rotation entirely to rotation in an inertial frame, a different "flatness" is predicted than if we suppose part of this rotation actually is due to rotation of the universe and should not be included in the rotation of the galaxy itself. Based upon the laws of physics, a model is set up in which one parameter is the rate of rotation of the Universe. If the laws of physics agree more accurately with observations in a model with rotation than without it, we are inclined to select the best-fit value for rotation, subject to all other pertinent experimental observations. If no value of the rotation parameter is successful and theory is not within observational error, a modification of physical law is considered, for example, [[dark matter]] is invoked to explain the [[galactic rotation curve]]. So far, observations show any rotation of the universe is very slow, no faster than once every 60·1012 years (10−13 rad/yr),<ref name=Birch>[http://www.nature.com/nature/journal/v298/n5873/abs/298451a0.html P Birch] ''Is the Universe rotating?'' Nature 298, 451 - 454 (29 July 1982)</ref> and debate persists over whether there is ''any'' rotation. However, if rotation were found, interpretation of observations in a frame tied to the universe would have to be corrected for the fictitious forces inherent in such rotation in classical physics and special relativity, or interpreted as the curvature of spacetime and the motion of matter along the geodesics in general relativity.

When [[quantum mechanics|quantum]] effects are important, there are additional conceptual complications that arise in [[quantum reference frame]]s.

==Background==
A brief comparison of inertial frames in special relativity and in Newtonian mechanics, and the role of absolute space is next.

===A set of frames where the laws of physics are simple===
According to the first postulate of [[special relativity]], all physical laws take their simplest form in an inertial frame, and there exist multiple inertial frames interrelated by uniform [[Translation (physics)|translation]]: {{anchor|principle}}<ref name=Einstein>{{Cite book|title=The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity |author=Einstein, A., Lorentz, H. A., Minkowski, H., & Weyl, H. |page=111 |url=https://books.google.com/?id=yECokhzsJYIC&pg=PA111&dq=postulate+%22Principle+of+Relativity%22
|isbn=0-486-60081-5 |publisher=Courier Dover Publications |date=1952 }}</ref>{{quote|Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.|Albert Einstein: ''The foundation of the general theory of relativity'', Section A, §1}}
This simplicity manifests in that inertial frames have self-contained physics without the need for external causes, while physics in non-inertial frames have external causes.<ref name="Ferraro" /> The principle of simplicity can be used within Newtonian physics as well as in special relativity; see Nagel<ref name=Nagel>{{Cite book|title=The Structure of Science |author=Ernest Nagel |page=212 |url=https://books.google.com/?id=u6EycHgRfkQC&pg=PA212&dq=inertial+%22Foucault%27s+pendulum%22 |isbn=0-915144-71-9 |publisher=Hackett Publishing |date=1979 }}</ref> and also Blagojević.<ref name="Blagojević">{{Cite book|title=Gravitation and Gauge Symmetries |author=Milutin Blagojević |page=4 |url=https://books.google.com/?id=N8JDSi_eNbwC&pg=PA5&dq=inertial+frame+%22absolute+space%22 |isbn=0-7503-0767-6 |publisher=CRC Press |date=2002}}</ref>
{{quote|The laws of Newtonian mechanics do not always hold in their simplest form...If, for instance, an observer is placed on a disc rotating relative to the earth, he/she will sense a 'force' pushing him/her toward the periphery of the disc, which is not caused by any interaction with other bodies. Here, the acceleration is not the consequence of the usual force, but of the so-called inertial force. Newton's laws hold in their simplest form only in a family of reference frames, called inertial frames. This fact represents the essence of the Galilean principle of relativity: &ensp;&ensp;&ensp;The laws of mechanics have the same form in all inertial frames.|Milutin Blagojević: ''Gravitation and Gauge Symmetries'', p. 4}}

In practical terms, the equivalence of inertial reference frames means that scientists within a box moving uniformly cannot determine their absolute velocity by any experiment (otherwise the differences would set up an absolute standard reference frame).<ref name=Einstein2>{{Cite book|title=Relativity: The Special and General Theory |author=Albert Einstein |page=17 |date=1920 |publisher=H. Holt and Company |url=https://books.google.com/?id=3H46AAAAMAAJ&printsec=titlepage&dq=%22The+Principle+of+Relativity%22 }}</ref><ref name=Feynman>{{Cite book|title=Six not-so-easy pieces: Einstein's relativity, symmetry, and space-time |author=Richard Phillips Feynman |page=73 |isbn=0-201-32842-9 |date=1998 |publisher=Basic Books |url=https://books.google.com/?id=ipY8onVQWhcC&pg=PA49&dq=%22The+Principle+of+Relativity%22}}</ref> According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the [[Poincaré group]] of symmetry transformations, of which the [[Lorentz transformation]]s are a subgroup.<ref name=Wachter>{{Cite book|title=Compendium of Theoretical Physics |author1=Armin Wachter |author2=Henning Hoeber |page=98 |url=https://books.google.com/?id=j3IQpdkinxMC&pg=PA98&dq=%2210-parameter+proper+orthochronous+Poincare+group%22 |isbn=0-387-25799-3 |publisher=Birkhäuser |date=2006 }}</ref> In Newtonian mechanics, which can be viewed as a limiting case of special relativity in which the speed of light is infinite, inertial frames of reference are related by the [[Galilean group]] of symmetries.

===Absolute space===
{{Main|Absolute space and time}}
Newton posited an absolute space considered well approximated by a frame of reference stationary relative to the [[fixed stars]]. An inertial frame was then one in uniform translation relative to absolute space. However, some scientists (called "relativists" by Mach<ref name=Mach/>), even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced.

Indeed, the expression ''inertial frame of reference'' ({{lang-de|Inertialsystem}}) was coined by [[Ludwig Lange (physicist)|Ludwig Lange]] in 1885, to replace Newton's definitions of "absolute space and time" by a more [[Operational definition#Relevance to science|operational definition]].<ref>{{Cite journal
|author=Lange, Ludwig
|date=1885
|title=Über die wissenschaftliche Fassung des Galileischen Beharrungsgesetzes
|journal=Philosophische Studien
|volume=2}}</ref><ref name=Barbour>{{Cite book|author=Julian B. Barbour |title=The Discovery of Dynamics |edition=Reprint of 1989 ''Absolute or Relative Motion?'' |pages=645–646 |url=https://books.google.com/?id=WQidkYkleXcC&pg=PA645&dq=Ludwig+Lange+%22operational+definition%22
|isbn=0-19-513202-5 |publisher=Oxford University Press |date=2001 }}</ref> As translated by Iro, [https://books.google.com/books?id=9a9KAAAAMAAJ&q=Inertialsystem+inauthor:%22von+Laue%22&dq=Inertialsystem+inauthor:%22von+Laue%22&lr=&as_brr=0&pgis=1 Lange proposed] the following definition:<ref name=Iro>L. Lange (1885) as quoted by Max von Laue in his book (1921) ''Die Relativitätstheorie'', p. 34, and translated by {{Cite book|page=169 |title=A Modern Approach to Classical Mechanics |author=Harald Iro |url=https://books.google.com/?id=-L5ckgdxA5YC&pg=PA179&dq=inertial+noninertial |isbn=981-238-213-5 |date=2002 |publisher=World Scientific}}</ref>

{{quote|A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.}}

A discussion of Lange's proposal can be found in Mach.<ref name=Mach>{{Cite book|title=The Science of Mechanics |page=38 |author=Ernst Mach |url=https://books.google.com/?id=cyE1AAAAIAAJ&pg=PA33&dq=rotating+sphere+Mach+cord+OR+string+OR+rod |publisher=The Open Court Publishing Co. |date=1915}}</ref>

The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by Blagojević:<ref name="Blagojević2">{{Cite book|title=Gravitation and Gauge Symmetries |author=Milutin Blagojević |page=5 |url=https://books.google.com/?id=N8JDSi_eNbwC&pg=PA5&dq=inertial+frame+%22absolute+space%22 |isbn=0-7503-0767-6 |publisher=CRC Press |date=2002}}</ref> {{quote|<ul><li>The existence of absolute space contradicts the internal logic of classical mechanics since, according to Galilean principle of relativity, none of the inertial frames can be singled out. <li>Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial frames.<li>Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.</ul> | Milutin Blagojević: ''Gravitation and Gauge Symmetries'', p. 5}}
The utility of operational definitions was carried much further in the special theory of relativity.<ref name=Woodhouse0>{{Cite book|title=Special relativity |author=NMJ Woodhouse |page=58 |url=https://books.google.com/?id=tM9hic_wo3sC&pg=PA126&dq=Woodhouse+%22operational+definition%22 |isbn=1-85233-426-6 |publisher=Springer |location=London |date=2003}}</ref> Some historical background including Lange's definition is provided by DiSalle, who says in summary:<ref name=DiSalle>{{Cite book
|author =Robert DiSalle
|chapter =Space and Time: Inertial Frames
|title =The Stanford Encyclopedia of Philosophy
|editor=Edward N. Zalta
|url=http://plato.stanford.edu/archives/sum2002/entries/spacetime-iframes/#Oth
|date=Summer 2002}}</ref>
{{quote|The original question, "relative to what frame of reference do the laws of motion hold?" is revealed to be wrongly posed. For the laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.|[http://plato.stanford.edu/archives/sum2002/entries/spacetime-iframes/#Oth Robert DiSalle ''Space and Time: Inertial Frames'']}}

==Newton's inertial frame of reference==
[[Image:Inertial frames.PNG|250px|thumbnail|Figure 1: Two frames of reference moving with relative velocity <math>\stackrel{\vec v}{}</math>. Frame ''S' '' has an arbitrary but fixed rotation with respect to frame ''S''. They are both ''inertial frames'' provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.]]
Within the realm of Newtonian mechanics, an [[inertia]]l frame of reference, or inertial reference frame, is one in which [[Newton's laws of motion#Newton.27s first law|Newton's first law of motion]] is valid.<ref name=Moeller>{{Cite book|author=C Møller |title=The Theory of Relativity |publisher=Oxford University Press |location=Oxford UK |isbn=0-19-560539-X |date=1976 |page=1 |url=http://worldcat.org/oclc/220221617&referer=brief_results |edition=Second}}</ref> However, the [[#principle|principle of special relativity]] generalizes the notion of inertial frame to include all physical laws, not simply Newton's first law.

Newton viewed the first law as valid in any reference frame that is in uniform motion relative to the fixed stars;<ref>The question of "moving uniformly relative to what?" was answered by Newton as "relative to [[absolute space]]". As a practical matter, "absolute space" was considered to be the [[fixed stars]]. For a discussion of the role of fixed stars, see {{Cite book|title=Nothingness: The Science of Empty Space |author=Henning Genz |page= 150 |isbn=0-7382-0610-5 |publisher=Da Capo Press |date=2001 |url=https://books.google.com/?id=Cn_Q9wbDOM0C&pg=PA150&dq=frame+Newton+%22fixed+stars%22
}}</ref> that is, neither rotating nor accelerating relative to the stars.<ref name=Resnick>{{Cite book|title=Physics |page=Volume 1, Chapter 3 |isbn=0-471-32057-9 |url=https://books.google.com/?id=CucFAAAACAAJ&dq=intitle:physics+inauthor:resnick
|publisher=Wiley |date=2001 |edition=5th |author1=Robert Resnick |author2=David Halliday |author3=Kenneth S. Krane |nopp=true }}</ref> Today the notion of "[[absolute space]]" is abandoned, and an inertial frame in the field of [[classical mechanics]] is defined as:<ref name=Takwale>{{Cite book|url=https://books.google.com/?id=r5P29cN6s6QC&pg=PA70&dq=fixed+stars+%22inertial+frame%22 |title=Introduction to classical mechanics |page=70 |author=RG Takwale |publisher=Tata McGraw-Hill|date=1980 |isbn=0-07-096617-6 |location=New Delhi}}</ref><ref name=Woodhouse>{{Cite book|url=https://books.google.com/?id=ggPXQAeeRLgC&printsec=frontcover&dq=isbn=1852334266#PPA6,M1 |title=Special relativity |page=6 |author=NMJ Woodhouse |publisher=Springer |date=2003 |isbn=1-85233-426-6 |location=London/Berlin}}</ref>
{{quote|An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.}}
Hence, with respect to an inertial frame, an object or body [[acceleration|accelerates]] only when a physical [[force]] is applied, and (following [[Newton's laws of motion|Newton's first law of motion]]), in the absence of a net force, a body at [[rest (physics)|rest]] will remain at rest and a body in motion will continue to move uniformly—that is, in a straight line and at constant [[speed]]. Newtonian inertial frames transform among each other according to the [[Galilean transformation|Galilean group of symmetries]].

If this rule is interpreted as saying that [[straight-line motion]] is an indication of zero net force, the rule does not identify inertial reference frames because straight-line motion can be observed in a variety of frames. If the rule is interpreted as defining an inertial frame, then we have to be able to determine when zero net force is applied. The problem was summarized by Einstein:<ref name=Einstein5>{{Cite book|title=The Meaning of Relativity |author=A Einstein |page=58 |date=1950 |url=https://books.google.com/books?num=10&btnG=Google+Search|publisher=Princeton University Press}}</ref>
{{quote|The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration.|Albert Einstein: ''The Meaning of Relativity'', p. 58}}

There are several approaches to this issue. One approach is to argue that all real forces drop off with distance from their sources in a known manner, so we have only to be sure that a body is far enough away from all sources to ensure that no force is present.<ref name=Rosser>{{Cite book|title=Introductory Special Relativity |author=William Geraint Vaughan Rosser |page=3 |url=https://books.google.com/?id=zpjBEBbIjAIC&pg=PA94&dq=reference+%22laws+of+physics%22
|isbn=0-85066-838-7 |date=1991 |publisher=CRC Press }}</ref> A possible issue with this approach is the historically long-lived view that the distant universe might affect matters ([[Mach's principle]]). Another approach is to identify all real sources for real forces and account for them. A possible issue with this approach is that we might miss something, or account inappropriately for their influence, perhaps, again, due to Mach's principle and an incomplete understanding of the universe. A third approach is to look at the way the forces transform when we shift reference frames. Fictitious forces, those that arise due to the acceleration of a frame, disappear in inertial frames, and have complicated rules of transformation in general cases. On the basis of universality of physical law and the request for frames where the laws are most simply expressed, inertial frames are distinguished by the absence of such fictitious forces.

Newton enunciated a principle of relativity himself in one of his corollaries to the laws of motion:<ref name=Feynman2>{{Cite book|title=Six not-so-easy pieces: Einstein's relativity, symmetry, and space-time |author=Richard Phillips Feynman |page=50 |isbn=0-201-32842-9 |date=1998 |publisher=Basic Books |url=https://books.google.com/?id=ipY8onVQWhcC&pg=PA49&dq=%22The+Principle+of+Relativity%22}}</ref><ref name=Principia>See the ''Principia'' on line at [https://archive.org/stream/newtonspmathema00newtrich#page/n7/mode/2up Andrew Motte Translation]</ref> {{quote|The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line.|Isaac Newton: ''Principia'', Corollary V, p. 88 in Andrew Motte translation}}

This principle differs from the [[#principle|special principle]] in two ways: first, it is restricted to mechanics, and second, it makes no mention of simplicity. It shares with the special principle the invariance of the form of the description among mutually translating reference frames.<ref name=note1>However, in the Newtonian system the Galilean transformation connects these frames and in the special theory of relativity the [[Lorentz transformation]] connects them. The two transformations agree for speeds of translation much less than the [[speed of light]].</ref> The role of fictitious forces in classifying reference frames is pursued further below.

==Separating non-inertial from inertial reference frames==

===Theory===
{{Main|Fictitious force}}
{{See also|Non-inertial frame|Rotating spheres|Bucket argument}}
[[Image:Rotating spheres.svg|thumb|180px|Figure 2: Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.]]
[[Image:Rotating-sphere forces.svg|thumb|Figure 3: Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.]]
Inertial and non-inertial reference frames can be distinguished by the absence or presence of [[fictitious force]]s, as explained shortly.<ref name="Rothman"/><ref name="Borowitz"/> {{quote|The effect of this being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations….|Sidney Borowitz and Lawrence A Bornstein in ''A Contemporary View of Elementary Physics'', p. 138}}
The presence of fictitious forces indicates the physical laws are not the simplest laws available so, in terms of the [[#principle|special principle of relativity]], a frame where fictitious forces are present is not an inertial frame:<ref name=Arnold2>{{Cite book|title=Mathematical Methods of Classical Mechanics |page=129 |author=V. I. Arnol'd |isbn=978-0-387-96890-2 |date=1989 |url=https://books.google.com/books?num=10&btnG=Google+Search|publisher=Springer}}</ref>
{{quote|The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.|V. I. Arnol'd: ''Mathematical Methods of Classical Mechanics'' Second Edition, p. 129}}
Bodies in [[non-inertial reference frame]]s are subject to so-called ''fictitious'' forces (pseudo-forces); that is, [[force]]s that result from the acceleration of the [[Frame of reference|reference frame]] itself and not from any physical force acting on the body. Examples of fictitious forces are the [[centrifugal force (fictitious)|centrifugal force]] and the [[Coriolis force]] in [[rotating reference frame]]s.

How then, are "fictitious" forces to be separated from "real" forces? It is hard to apply the Newtonian definition of an inertial frame without this separation. For example, consider a stationary object in an inertial frame. Being at rest, no net force is applied. But in a frame rotating about a fixed axis, the object appears to move in a circle, and is subject to centripetal force (which is made up of the Coriolis force and the centrifugal force). How can we decide that the rotating frame is a non-inertial frame? There are two approaches to this resolution: one approach is to look for the origin of the fictitious forces (the Coriolis force and the centrifugal force). We will find there are no sources for these forces, no associated [[force carrier]]s, no originating bodies.<ref name=note2>For example, there is no body providing a gravitational or electrical attraction.</ref> A second approach is to look at a variety of frames of reference. For any inertial frame, the Coriolis force and the centrifugal force disappear, so application of the principle of special relativity would identify these frames where the forces disappear as sharing the same and the simplest physical laws, and hence rule that the rotating frame is not an inertial frame.

Newton examined this problem himself using rotating spheres, as shown in Figure 2 and Figure 3. He pointed out that if the spheres are not rotating, the tension in the tying string is measured as zero in every frame of reference.<ref name=tension>That is, the universality of the laws of physics requires the same tension to be seen by everybody. For example, it cannot happen that the string breaks under extreme tension in one frame of reference and remains intact in another frame of reference, just because we choose to look at the string from a different frame.</ref> If the spheres only appear to rotate (that is, we are watching stationary spheres from a rotating frame), the zero tension in the string is accounted for by observing that the centripetal force is supplied by the centrifugal and Coriolis forces in combination, so no tension is needed. If the spheres really are rotating, the tension observed is exactly the centripetal force required by the circular motion. Thus, measurement of the tension in the string identifies the inertial frame: it is the one where the tension in the string provides exactly the centripetal force demanded by the motion as it is observed in that frame, and not a different value. That is, the inertial frame is the one where the fictitious forces vanish.

So much for fictitious forces due to rotation. However, for linear acceleration, Newton expressed the idea of undetectability of straight-line accelerations held in common:<ref name=Principia/>
{{quote|If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will continue to move among themselves, after the same manner as if they had been urged by no such forces. |Isaac Newton: ''Principia'' Corollary VI, p. 89, in Andrew Motte translation }}

This principle generalizes the notion of an inertial frame. For example, an observer confined in a free-falling lift will assert that he himself is a valid inertial frame, even if he is accelerating under gravity, so long as he has no knowledge about anything outside the lift. So, strictly speaking, inertial frame is a relative concept. With this in mind, we can define inertial frames collectively as a set of frames which are stationary or moving at constant velocity with respect to each other, so that a single inertial frame is defined as an element of this set.

For these ideas to apply, everything observed in the frame has to be subject to a base-line, common acceleration shared by the frame itself. That situation would apply, for example, to the elevator example, where all objects are subject to the same gravitational acceleration, and the elevator itself accelerates at the same rate.

In 1899 the astronomer [[Karl Schwarzschild]] pointed out an observation about double stars. The motion of two stars orbiting each other is planar, the two orbits of the stars of the system lie in a plane. In the case of sufficiently near double star systems, it can be seen from Earth whether the perihelion of the orbits of the two stars remains pointing in the same direction with respect to the solar system. Schwarzschild pointed out that that was invariably seen: the direction of the [[angular momentum]] of all observed double star systems remains fixed with respect to the direction of the angular momentum of the Solar system. The logical inference is that just like gyroscopes, the angular momentum of all celestial bodies is angular momentum with respect to a universal inertial space.<ref>[http://www.mpiwg-berlin.mpg.de/Preprints/P271.PDF In the Shadow of the Relativity Revolution] Section 3: The Work of Karl Schwarzschild (2.2 MB PDF-file)</ref>

===Applications===
[[Inertial navigation system]]s used a cluster of [[gyroscope]]s and accelerometers to determine accelerations relative to inertial space. After a gyroscope is spun up in a particular orientation in inertial space, the law of conservation of angular momentum requires that it retain that orientation as long as no external forces are applied to it.<ref>{{cite book|last=Chatfield|first=Averil B.|title=Fundamentals of High Accuracy Inertial Navigation, Volume 174|date=1997|publisher=AIAA|isbn=9781600864278}}</ref>{{rp|59}} Three orthogonal gyroscopes establish an inertial reference frame, and the accelerators measure acceleration relative to that frame. The accelerations, along with a clock, can then be used to calculate the change in position. Thus, inertial navigation is a form of [[dead reckoning]] that requires no external input, and therefore cannot be jammed by any external or internal signal source.<ref>{{cite book|last=Kennie|first=edited by T.J.M.|title=Engineering Surveying Technology|date=1993|publisher=Taylor & Francis|location=Hoboken|isbn=9780203860748|page=95|edition=Pbk.|author2=Petrie, G.}}</ref>

A [[gyrocompass]], employed for navigation of seagoing vessels, finds the geometric north. It does so, not by sensing the Earth's magnetic field, but by using inertial space as its reference. The outer casing of the gyrocompass device is held in such a way that it remains aligned with the local plumb line. When the gyroscope wheel inside the gyrocompass device is spun up, the way the gyroscope wheel is suspended causes the gyroscope wheel to gradually align its spinning axis with the Earth's axis. Alignment with the Earth's axis is the only direction for which the gyroscope's spinning axis can be stationary with respect to the Earth and not be required to change direction with respect to inertial space. After being spun up, a gyrocompass can reach the direction of alignment with the Earth's axis in as little as a quarter of an hour.<ref name=l>{{cite journal|title=The gyroscope pilots ships & planes |journal=Life|date=Mar 15, 1943 |pages=80–83|url=https://books.google.com/books?id=YlEEAAAAMBAJ&pg=PA82}}</ref>

==Newtonian mechanics==
{{Main|Newton's laws of motion}}
[[Classical mechanics]], which includes relativity, assumes the equivalence of all inertial reference frames. [[Newton's laws|Newtonian mechanics]] makes the additional assumptions of [[absolute space]] and [[absolute time]]. Given these two assumptions, the coordinates of the same event (a point in space and time) described in two inertial reference frames are related by a Galilean transformation.

:<math>
\mathbf{r}^{\prime} = \mathbf{r} - \mathbf{r}_{0} - \mathbf{v} t
</math>

:<math>
t^{\prime} = t - t_{0}
</math>

where '''r'''0 and ''t''0 represent shifts in the origin of space and time, and '''v''' is the relative velocity of the two inertial reference frames. Under Galilean transformations, the time ''t''2 − ''t''1 between two events is the same for all inertial reference frames and the [[distance]] between two simultaneous events (or, equivalently, the length of any object, |'''r'''2 − '''r'''1|) is also the same.

==Special relativity==
{{Main|Special relativity|Introduction to special relativity}}
[[Albert Einstein|Einstein's]] [[special relativity|theory of special relativity]], like Newtonian mechanics, assumes the equivalence of all inertial reference frames, but makes an additional assumption, foreign to Newtonian mechanics, namely, that in [[free space]] light always is propagated with the [[speed of light]] ''c''0, a defined [http://physics.nist.gov/cgi-bin/cuu/Value?c value] independent of its direction of propagation and its frequency, and also independent of the state of motion of the emitting body. This second assumption has been verified experimentally and leads to counter-intuitive deductions including:
* [[time dilation]] (moving clocks tick more slowly)
* [[length contraction]] (moving objects are shortened in the direction of motion)
* [[relativity of simultaneity]] (simultaneous events in one reference frame are not simultaneous in almost all frames moving relative to the first).

These deductions are [[logical consequence]]s of the stated assumptions, and are general properties of space-time, typically without regard to a consideration of properties pertaining to the structure of individual objects like atoms or stars, nor to the mechanisms of clocks.

These effects are expressed mathematically by the [[Lorentz transformation]]

:<math>x^{\prime} = \gamma \left(x - v t \right) </math>
:<math>y^{\prime} = y</math>
:<math>z^{\prime} = z</math>
:<math>t^{\prime} = \gamma \left(t - \frac{v x}{c_0^{2}}\right)</math>

where shifts in origin have been ignored, the relative velocity is assumed to be in the <math>x</math>-direction and the [[Lorentz factor]] γ is defined by:

:<math>
\gamma \ \stackrel{\mathrm{def}}{=}\
\frac{1}{\sqrt{1 - (v/c_0)^2}} \ \ge 1.
</math>

The Lorentz transformation is equivalent to the [[Galilean transformation]] in the limit ''c''0 → ∞ (a hypothetical case) or ''v'' → 0 (low speeds).

Under [[Lorentz transformation]]s, the time and distance between events may differ among inertial reference frames; however, the [[Lorentz scalar]] distance ''s'' between two events is the same in all inertial reference frames

:<math>
s^{2} =
\left( x_{2} - x_{1} \right)^{2} + \left( y_{2} - y_{1} \right)^{2} +
\left( z_{2} - z_{1} \right)^{2} - c_0^{2} \left(t_{2} - t_{1}\right)^{2}
</math>

From this perspective, the [[speed of light]] is only accidentally a property of [[light]], and is rather a property of [[spacetime]], a [[conversion of units|conversion factor]] between conventional time units (such as [[second]]s) and length units (such as meters).

Incidentally, because of the limitations on speeds faster than the speed of light, notice that in a rotating frame of reference (which is a non-inertial frame, of course) stationarity is not possible at arbitrary distances because at large radius the object would move faster than the speed of light.<ref name=Landau>{{Cite book|title=The Classical Theory of Fields |author1=LD Landau |author2=LM Lifshitz |edition=4th Revised English |pages=273–274 |date=1975 |isbn=978-0-7506-2768-9 |publisher=Pergamon Press }}</ref>

==General relativity==
{{Main|General relativity|Introduction to general relativity}}
{{See also|Equivalence principle|Eötvös experiment}}
General relativity is based upon the principle of equivalence:<ref name=Morin>{{Cite book|title=Introduction to Classical Mechanics |author=David Morin |page=649 |url=https://books.google.com/?id=Ni6CD7K2X4MC&pg=PA469&dq=acceleration+azimuthal+inauthor:Morin |isbn=0-521-87622-2 |publisher=Cambridge University Press |date=2008}}</ref><ref name=Giancoli>{{Cite book|title=Physics for Scientists and Engineers with Modern Physics |author=Douglas C. Giancoli |url=https://books.google.com/?id=xz-UEdtRmzkC&pg=PA155&dq=%22principle+of+equivalence%22
|page=155 |date=2007 |publisher=Pearson Prentice Hall |isbn=0-13-149508-9 }}</ref>{{quote|There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating.|Douglas C. Giancoli, ''Physics for Scientists and Engineers with Modern Physics'', p. 155.}}
This idea was introduced in Einstein's 1907 article "Principle of Relativity and Gravitation" and later developed in 1911.<ref name=General_theory>A. Einstein, "On the influence of gravitation on the propagation of light", ''Annalen der Physik'', vol. 35, (1911) : 898-908</ref> Support for this principle is found in the [[Eötvös experiment]], which determines whether the ratio of inertial to gravitational mass is the same for all bodies, regardless of size or composition. To date no difference has been found to a few parts in 1011.<ref name=NRC>{{Cite book|title=Physics Through the Nineteen Nineties: Overview |page=15 |url=https://books.google.com/?id=Hk1wj61PlocC&pg=PA15&dq=equivalence+gravitation
|isbn=0-309-03579-1 |date=1986 |author=National Research Council (US) |publisher=National Academies Press }}</ref> For some discussion of the subtleties of the Eötvös experiment, such as the local mass distribution around the experimental site (including a quip about the mass of Eötvös himself), see Franklin.<ref name=Franklin>{{Cite book|title=No Easy Answers: Science and the Pursuit of Knowledge |author=Allan Franklin |page=66 |url=https://books.google.com/?id=_RN-v31rXuIC&pg=PA66&dq=%22Eotvos+experiment%22
|isbn=0-8229-5968-2 |date=2007 |publisher=University of Pittsburgh Press }}</ref>

Einstein’s [[general relativity|general theory]] modifies the distinction between nominally "inertial" and "noninertial" effects by replacing special relativity's "flat" [[Minkowski Space]] with a metric that produces non-zero curvature. In general relativity, the principle of inertia is replaced with the principle of [[geodesic (general relativity)|geodesic motion]], whereby objects move in a way dictated by the curvature of spacetime. As a consequence of this curvature, it is not a given in general relativity that inertial objects moving at a particular rate with respect to each other will continue to do so. This phenomenon of [[geodesic deviation]] means that inertial frames of reference do not exist globally as they do in Newtonian mechanics and special relativity.

However, the general theory reduces to the special theory over sufficiently small regions of spacetime, where curvature effects become less important and the earlier inertial frame arguments can come back into play.<ref>{{cite book |title=Information Theory and Quantum Physics: Physical Foundations for Understanding the Conscious Process |first1=Herbert S. |last1=Green |publisher=Springer |date=2000 |isbn=354066517X |page=154 |url=https://books.google.com/books?id=CUJiQjSVCu8C}} [https://books.google.com/books?id=CUJiQjSVCu8C&pg=PA154 Extract of page 154]</ref><ref>{{cite book |title=Theory of Special Relativity |first1=Nikhilendu |last1=Bandyopadhyay |publisher=Academic Publishers |date=2000 |isbn=8186358528 |page=116 |url=https://books.google.com/books?id=qMOyfi_i0j8C}} [https://books.google.com/books?id=qMOyfi_i0j8C&pg=PA116 Extract of page 116]</ref> Consequently, modern special relativity is now sometimes described as only a "local theory".<ref>{{cite book |title=Cosmological Inflation and Large-Scale Structure |first1=Andrew R. |last1=Liddle |first2=David H. |last2=Lyth |publisher=Cambridge University Press |date=2000 |isbn=0-521-57598-2 |page=329 |url=https://books.google.com/books?id=XmWauPZSovMC}} [https://books.google.com/books?id=XmWauPZSovMC&pg=PA329 Extract of page 329]</ref>

==See also==

{{Col-begin}}
{{Col-1-of-2}}
* [[Absolute rotation]]
* [[Diffeomorphism]]
* [[Galilean invariance]]
* [[General covariance]]
{{Col-2-of-2}}
* [[Local reference frame]]
* [[Lorentz invariance]]
* [[Newton's laws of motion#Newton.27s first law|Newton's first law]]
* [[Quantum reference frame]]
{{col-end}}

==References==
{{Reflist|2}}

==Further reading==
* [[Edwin F. Taylor]] and [[John Archibald Wheeler]], ''Spacetime Physics'', 2nd ed. (Freeman, NY, 1992)
* [[Albert Einstein]], ''Relativity, the special and the general theories'', 15th ed. (1954)
* {{cite journal | last1 = Poincaré | first1 = Henri | authorlink = Henri Poincaré | year = 1900 | title = La théorie de Lorentz et le Principe de Réaction | url = | journal = Archives Neerlandaises | volume = V | issue = | pages = 253–78 }}
* [[Albert Einstein]], ''On the Electrodynamics of Moving Bodies'', included in ''The Principle of Relativity'', page 38. Dover 1923

;Rotation of the Universe
* {{Cite book|title=Mach's Principle: From Newton's Bucket to Quantum Gravity |page= 445 |author1=Julian B. Barbour |author2=Herbert Pfister |isbn=0-8176-3823-7 |date=1998 |url=https://books.google.com/?id=fKgQ9YpAcwMC&pg=PA445&dq=Birch++%22rotation+of+the+universe%22+-religion+-astrology+date:1990-2000 |publisher=Birkhäuser}}
* {{Cite book|title=Time Machines |author=PJ Nahin |page= 369; Footnote 12 |url=https://books.google.com/?id=39KQY1FnSfkC&pg=PA369 |date=1999 |isbn=0-387-98571-9 |publisher=Springer }}
* [http://www.nipne.ro/rjp/2008_53_1-2/0405_0416.pdf B Ciobanu, I Radinchi] ''Modeling the electric and magnetic fields in a rotating universe'' Rom. Journ. Phys., Vol. 53, Nos. 1–2, P. 405–415, Bucharest, 2008
* [http://arxiv.org/abs/gr-qc/0206080v1 Yuri N. Obukhov, Thoralf Chrobok, Mike Scherfner] ''Shear-free rotating inflation'' Phys. Rev. D 66, 043518 (2002) [5 pages]
* [http://arxiv.org/abs/astro-ph/0008106v1 Yuri N. Obukhov] ''On physical foundations and observational effects of cosmic rotation'' (2000)
* [http://arxiv.org/abs/astro-ph/9703082v1 Li-Xin Li] ''Effect of the Global Rotation of the Universe on the Formation of Galaxies'' General Relativity and Gravitation, '''30''' (1998) {{doi|10.1023/A:1018867011142}}
* [http://www.nature.com/nature/journal/v298/n5873/abs/298451a0.html P Birch] ''Is the Universe rotating?'' Nature 298, 451 - 454 (29 July 1982)
* [http://www.springerlink.com/content/t13ul36l27222351/fulltext.pdf?page=1 Kurt Gödel] ''An example of a new type of cosmological solutions of Einstein’s field equations of gravitation'' Rev. Mod. Phys., Vol. 21, p. 447, 1949.

==External links==
* [http://plato.stanford.edu/entries/spacetime-iframes/ Stanford Encyclopedia of Philosophy entry]
* {{YouTube|49JwbrXcPjc|Animation clip}} showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.

{{DEFAULTSORT:Inertial Frame Of Reference}}
[[Category:Classical mechanics]]
[[Category:Frames of reference]]
[[Category:Theory of relativity]]
[[Category:Orbits]]

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{{space|hair}}{{#if:{{{3|}}}
|{{{1}}} {{{2}}}&frasl;{{{3}}}
|{{#if:{{{2|}}}
|{{{1}}}&frasl;{{{2}}}
|{{#if:{{{1|}}}
|1&frasl;{{{1}}}
|&frasl;
}}
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Template:Equation box 1

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{{{indent|:}}}{|cellpadding="{{{cellpadding|5}}}" style="border:{{{border|2}}}px solid {{{border colour|#50C878}}};background: {{{background colour|#ECFCF4}}}; text-align: center;"
|{{{title|}}}
{{{equation|<math>z=re^{i\phi}=x+iy \,\!</math> }}}{{#if:{{{ref|}}}|   ({{EquationRef|{{{ref}}}}})}}
|}<noinclude>

{{documentation|content = Type the equation in the blue box. Set box parameters: border thickness and colour, cell padding, and background colour. The full syntax is: 

<nowiki>{{Equation box 1|indent|title|equation|ref|cellpadding|border|border colour|background colour}}</nowiki>

<big>'''Parameters'''</big>

#'''''Indent:''''' either leave blank or type colon (:) to indent the box from the left side of the page.
#'''''Title:''''' leave blank or type title/name of equation
#'''''Equation:''''' type an equation [[WP:«math»|in whatever form]], usually [[Help:Displaying a formula|<tt><math></tt>]] ''equation in LaTeX ...'' <tt></math></tt>. The default equation is the general form of a complex number.
#'''''Ref:''''' (optional) calls {{tl|EquationRef}} with provided value as the parameter.
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#'''''Border:''''' either leave blank (default is 2px) or [[Hex triplet]] to change thickness of border line.
#'''''Border colour:''''' either leave blank (default is light grey-blue, #ccccff) or type [[Hex triplet]] for the border line of the box.
#'''''Background colour:''''' either leave blank (default is white) or type colour code for the background area of the box.

See [[list of colours]] for use with the border and background colours. To use the parameters set the parameter equal to the value by using the equals sign.

<big>'''Examples'''</big>

;Black/white theme

{{Equation box 1
|title='''[[Complex number]]'''
|indent=:
|equation=<math>z=re^{i\phi}=x+iy \,\!</math>
|cellpadding = 6
|border = 1
|border colour = black
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<source lang="moin">{{Equation box 1
|title='''[[Complex number]]'''
|indent=:
|equation=<math>z=re^{i\phi}=x+iy \,\!</math>
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;Green theme

{{Equation box 1
|indent=:
|equation=<math>z=re^{i\phi}=x+iy \,\!</math>
|cellpadding
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<source lang="tex">{{Equation box 1
|indent=:
|equation=<math>z=re^{i\phi}=x+iy \,\!</math>
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;Blue theme

{{Equation box 1
|indent =:
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|equation = <math>z=re^{i\phi}=x+iy \,\!</math>
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<source lang="tex">{{Equation box 1
|indent =:
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|equation = <math>z=re^{i\phi}=x+iy \,\!</math>
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|border colour = #0073CF
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; With reference

{{Equation box 1
|equation = {{math|1=2 × 2 = 4}}
|ref=1}}

<code><nowiki>{{Equation box 1
|equation = {{math|1=2 × 2 = 4}}
|ref=1}}</nowiki></code>

}}
[[Category:Mathematical formatting templates]]
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Template:EquationRef

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<includeonly>{{#if: {{{2|}}}|{{{2}}}|{{{1}}}}}</includeonly><noinclude>{{documentation|content=
The pair of templates {{tl|EquationRef}} and {{tl|EquationNote}} is used for labeling equations in text.

For example, {{markup2|m=<nowiki>:{{EquationRef|Eq. 1}} <math>x^2+x+1=0</math></nowiki>|r=
:{{EquationRef|Eq. 1}} <math>x^2+x+1=0</math>}}
Here, the identifier "'''Eq. 1'''" is a label which can be later referred to with a call to {{tl|EquationNote}}. That is, {{tlb|EquationNote|Eq. 1}} produces a link {{EquationNote|Eq. 1}} to this equation.

Similarly, {{markup2|m=<nowiki>:{{EquationRef|2|Eq. 2:}} <math>x^2-2x+1=0</math></nowiki>|r=
:{{EquationRef|2|Eq. 2:}} <math>x^2-2x+1=0</math>}}
which can be referred to by {{tlb|EquationNote|2}} or even {{tlb|EquationNote|2|Equation 2}} (i.e., {{EquationNote|2|Equation 2}}).

This template can be used together with {{tl|NumBlk}} to produce nicely formatted numbered equations. For instance:

{{markup2|m=<nowiki>{{NumBlk|::|<math>x^2+x+1=0</math>|{{EquationRef|3}}}}</nowiki>|r=
{{NumBlk|::|<math>x^2+x+1=0</math>|{{EquationRef|3}}}}
}}
:which may then be referred to by {{tlb|EquationNote|3}} or even {{tlb|EquationNote|3|Eq. 3}} (i.e., {{EquationNote|3|Eq. 3}}).

{{markup2|m=<nowiki>{{NumBlk|::|<math>x^2+x+1=0</math>|{{EquationRef|4|Eq.4}}}}</nowiki>|r=
{{NumBlk|::|<math>x^2+x+1=0</math>|{{EquationRef|4|Eq.4}}}}
}}
:which may be referred to by {{tlb|EquationNote|4}} (i.e., {{EquationNote|4}}).

{{markup2|m=<nowiki>{{NumBlk|::|<math>x^2+x+1=0</math>|{{EquationRef|Eq.5}}}}</nowiki>|r=
{{NumBlk|::|<math>x^2+x+1=0</math>|{{EquationRef|Eq.5}}}}
}}
:which may be referred to by {{tlb|EquationNote|Eq.5}} (i.e., {{EquationNote|Eq.5}}).

[[Category:Mathematical formatting templates]]

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Template:EquationNote

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'''{{#if:{{{2|}}}|[[#math_{{{1}}}|{{{2}}}]]|[[#math_{{{1}}}|{{{1}}}]]}}'''<noinclude>{{documentation|content=
See [[Template:EquationRef]] for documentation. Note that the link text can be specified via an optional second argument.

[[Category:Mathematical formatting templates]]

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Template:Abs

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<includeonly>|{{#if: {{{1|}}} | {{{1}}} | · }}|</includeonly><noinclude>
{{documentation|content =
This template may be used to enclose text between two pipes, typically denoting the [[absolute value]].

== Usage ==
<pre>
When {{math|{{abs|''x''}} < 7}}, the function is positive.
</pre>
When {{math|{{abs|''x''}} < 7}}, the function is positive.

<pre>
The notation {{math|{{abs}}}} means '''absolute value'''.
</pre>
The notation {{math|{{abs}}}} means '''absolute value'''.

== Alternatives ==
This template is equivalent to enclosing the text between two {{tlx|pipe}} templates (and a hair space &#x200A; around the argument; a hair space being a 1.5 mu space, i.e. a 1/12 em space)
<pre>
When {{math|{{pipe}}&#x200A;''x''&#x200A;{{pipe}} < 7}}, the function is positive.
</pre>
When {{math|{{pipe}} ''x'' {{pipe}} < 7}}, the function is positive.

== See also ==
* {{tl|pipe}}
* {{tl|norm}}
* {{tl|bra-ket}}
* {{tl|langle}} and {{tl|rangle}}

[[Category:Mathematical formatting templates]]
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Lorentz transformation

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{{spacetime|cTopic=Mathematics}}

In [[physics]], the '''Lorentz transformation''' (or '''transformations''') are [[coordinate transformation]]s between two [[coordinate frame]]s that move at constant velocity relative to each other. The transformations are named after the Dutch [[physicist]] [[Hendrik Lorentz]].

Frames of reference can be divided into two groups: [[Inertial frame of reference|inertial]] (relative motion with constant velocity) and [[Non-inertial reference frame|non-inertial]] (accelerating in curved paths, rotational motion with constant [[angular velocity]], etc.). The term "Lorentz transformations" only refers to transformations between ''inertial'' frames, usually in the context of special relativity.

In each reference frame, an observer can use a local coordinate system (most exclusively [[Cartesian coordinates]] in this context) to measure lengths, and a clock to measure time intervals. An observer is a real or imaginary entity that can take measurements, say humans, or any other living organism—or even robots and computers. An [[Event (relativity)|event]] is something that happens at a point in space at an instant of time, or more formally a point in [[spacetime]]. The transformations connect the space and time coordinates of an [[Event (relativity)|event]] as measured by an observer in each frame.<ref group=nb>One can imagine that in each inertial frame there are observers positioned throughout space, each endowed with a synchronized clock and at rest in the particular inertial frame. These observers then report to a central office, where a report is collected. When one speaks of a ''particular'' observer, one refers to someone having, at least in principle, a copy of this report. See, e.g., {{harvtxt|Sard|1970}}.</ref>

They supersede the [[Galilean transformation]] of [[Newtonian physics]], which assumes an absolute space and time (see [[Galilean relativity]]). The Galilean transformation is a good approximation only at relative speeds much smaller than the speed of light. Lorentz transformations have a number of unintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different [[velocity|velocities]] may measure different [[Length contraction|distances]], [[time dilation|elapsed times]], and even different [[Relativity of simultaneity|orderings of events]], but always such that the [[speed of light]] is the same in all inertial reference frames. The invariance of light speed is one of the [[postulates of special relativity]].

Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of [[light]] was observed to be independent of the [[frame of reference|reference frame]], and to understand the symmetries of the laws of [[electromagnetism]]. The Lorentz transformation is in accordance with [[special relativity]], but was derived before special relativity.

The Lorentz transformation is a [[linear transformation]]. It may include a rotation of space; a rotation-free Lorentz transformation is called a '''Lorentz boost'''. In [[Minkowski space]], the mathematical model of spacetime in special relativity, the Lorentz transformations preserve the [[spacetime interval]] between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a [[hyperbolic rotation]] of Minkowski space. The more general set of transformations that also includes translations is known as the [[Poincaré group]].

==History==

{{main|History of Lorentz transformations}}

Many physicists—including [[Woldemar Voigt]], [[George FitzGerald]], [[Joseph Larmor]], and [[Hendrik Lorentz]]<ref>{{Citation
|author=Lorentz, Hendrik Antoon
|year=1904
|title=[[s:Electromagnetic phenomena|Electromagnetic phenomena in a system moving with any velocity smaller than that of light]]
|journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences
|volume=6
|pages=809–831}}</ref> himself—had been discussing the physics implied by these equations since 1887.<ref>{{harvnb|John|O'Connor|1996}}</ref> Early in 1889, [[Oliver Heaviside]] had shown from [[Maxwell's equations]] that the [[electric field]] surrounding a spherical distribution of charge should cease to have [[spherical symmetry]] once the charge is in motion relative to the ether. FitzGerald then conjectured that Heaviside’s distortion result might be applied to a theory of intermolecular forces. Some months later, FitzGerald published the conjecture that bodies in motion are being contracted, in order to explain the baffling outcome of the 1887 ether-wind experiment of [[Michelson–Morley experiment|Michelson and Morley]]. In 1892, Lorentz independently presented the same idea in a more detailed manner, which was subsequently called [[Length contraction|FitzGerald–Lorentz contraction hypothesis]].<ref>{{harvnb|Brown|2003}}</ref> Their explanation was widely known before 1905.<ref>{{harvnb|Rothman|2006|pages = 112f.}}</ref>

Lorentz (1892–1904) and Larmor (1897–1900), who believed the [[luminiferous ether]] hypothesis, also looked for the transformation under which [[Maxwell's equations]] are invariant when transformed from the ether to a moving frame. They extended the FitzGerald–Lorentz contraction hypothesis and found out that the time coordinate has to be modified as well ("[[relativity of simultaneity|local time]]"). [[Henri Poincaré]] gave a physical interpretation to local time (to first order in ''v''/''c'', the relative velocity of the two reference frames normalized to the speed of light) as the consequence of clock synchronization, under the assumption that the speed of light is constant in moving frames.<ref>{{harvnb|Darrigol|2005|pages=1–22}}</ref> Larmor is credited to have been the first to understand the crucial [[time dilation]] property inherent in his equations.<ref>
{{harvnb|Macrossan|1986|pages=232–34}}</ref>

In 1905, Poincaré was the first to recognize that the transformation has the properties of a [[group (mathematics)|mathematical group]],
and named it after Lorentz.<ref>The reference is within the following paper:{{harvnb|Poincaré|1905|pages = 1504–1508}}</ref>
Later in the same year [[Albert Einstein]] published what is now called [[special relativity]], by deriving the Lorentz transformation under the assumptions of the [[principle of relativity]] and the constancy of the speed of light in any [[inertial reference frame]], and by abandoning the mechanical ether as unnecessary.<ref>{{harvnb|Einstein|1905|pages=891–921}}</ref>

== Derivation of the group of Lorentz transformations ==
{{Main|Derivations of the Lorentz transformations|Lorentz group}}

An ''[[Event (relativity)|event]]'' is something that happens at a certain point in spacetime, or more generally, the point in spacetime itself. In any inertial frame an event is specified by a time coordinate ''ct'' and a set of [[Cartesian coordinate]]s {{math|''x'', ''y'', ''z''}} to specify position in space in that frame. Subscripts label individual events.

From Einstein's [[Postulates of special relativity|second postulate of relativity]] follows

{{NumBlk|:|<math>c^2(t_2 - t_1)^2 - (x_2 - x_1)^2 - (y_2 - y_1)^2 - (z_2 - z_1)^2 = 0 \quad \text{(lightlike separated events 1, 2)}</math>|{{EquationRef|D1}}}}

in all inertial frames for events connected by ''light signals''. The quantity on the left is called the ''spacetime interval'' between events {{math|''a''1 {{=}} (''t''1, ''x''1, ''y''1, ''z''1)}} and {{math|''a''2 {{=}} (''t''2, ''x''2, ''y''2, ''z''2)}}. The interval between ''any two'' events, not necessarily separated by light signals, is in fact invariant, i.e., independent of the state of relative motion of observers in different inertial frames, as is [[Derivations of the Lorentz transformations#Invariance of interval|shown using homogeneity and isotropy of space]]. The transformation sought after thus must possess the property that

{{NumBlk|:|<math>c^2(t_2 - t_1)^2 - (x_2 - x_1)^2 - (y_2 - y_1)^2 - (z_2 - z_1)^2 = c^2(t_2' - t_1')^2 - (x_2' - x_1')^2 - (y_2' - y_1')^2 - (z_2' - z_1')^2 \quad \text{(all events 1, 2)}.</math>|{{EquationRef|D2}}}}

where {{math|(''ct'', ''x'', ''y'', ''z'')}} are the spacetime coordinates used to define events in one frame, and {{math|(''ct''′, ''x''′, ''y''′, ''z''′)}} are the coordinates in another frame. First one observes that {{EquationNote|(D2)}} is satisfied if an arbitrary {{math|4}}-tuple {{math|''b''}} of numbers are added to events {{math|''a''1}} and {{math|''a''2}}. Such transformations are called ''spacetime translations'' and are not dealt with further here. Then one observes that a ''linear'' solution preserving the origin of the simpler problem

{{NumBlk|:|<math>c^2t^2 - x^2 - y^2 - z^2 = c^2t'^2 - x'^2 - y'^2 - z'^2 \quad \text{ or } \quad c^2t_1t_2 - x_1x_2 - y_1y_2 - z_1z_2 = c^2t'_1t'_2 - x'_1x'_2 - y'_1y'_2 - z'_1z'_2</math>|{{EquationRef|D3}}}}

solves the general problem too. (A solution satisfying the left formula automatically satisfies the right formula, see [[polarization identity]].) Finding the solution to the simpler problem is just a matter of look-up in the theory of [[classical group]]s that preserve [[bilinear form]]s of various signature.<ref group=nb>It should be noted that the separate requirements of the three equations lead to three different groups. The second equation is satisfied for spacetime translations in addition to Lorentz transformations leading to the [[Poincare group]] or the ''inhomogeneous Lorentz group''. The first equation (or the second restricted to lightlike separation) leads to a yet larger group, the [[conformal group]] of spacetime.</ref> Equation {{EquationNote|(D3)}} can be written more compactly as

{{NumBlk|:|<math>(a, a) = (a', a') \quad \text{or} \quad a \cdot a = a' \cdot a',</math>|{{EquationRef|D4}}}}

where {{math|(·, ·)}} refers to the bilinear form of [[Signature (quadratic form)|signature]] {{math|(1, 3)}} on {{math|ℝ4}} exposed by the right hand side formula in {{EquationNote|(D3)}}. The alternative notation defined on the right is referred to as the ''relativistic dot product''. Spacetime mathematically viewed as {{math|ℝ4}} endowed with this bilinear form is known as [[Minkowski space]] {{math|''M''}}. The Lorentz transformation is thus an element of the group Lorentz group {{math|O(1, 3)}}, the [[Lorentz group]] or, for those that prefer the other [[metric signature]], {{math|O(3, 1)}} (also called the Lorentz group).<ref group=nb>The groups {{math|O(3, 1)}} and {{math|O(1, 3)}} are isomorphic. It is widely believed that the choice between the two metric signatures has no physical relevance, even though some objects related to {{math|O(3, 1)}} and {{math|O(1, 3)}} respectively, e.g., the [[Clifford algebra]]s corresponding to the different signatures of the bilinear form associated to the two groups, are non-isomorphic.</ref> One has

{{NumBlk|:|<math>(a, a) = (\Lambda a,\Lambda a) = (a', a'), \quad \Lambda \in \mathrm O(1, 3), \quad a, a' \in M,</math>|{{EquationRef|D5}}}}

which is precisely preservation of the bilinear form {{EquationNote|(D3)}} which implies (by linearity of {{math|Λ}} and bilinearity of the form) that {{EquationNote|(D2)}} is satisfied. The elements of the Lorentz group are [[Rotation group SO(3)|rotations]] and ''boosts'' and mixes thereof. If the spacetime translations are included, then one obtains the ''inhomogeneous Lorentz group'' or the [[Poincare group]].

==Generalities==
The relations between the primed and unprimed spacetime coordinates are the '''Lorentz transformations''', each coordinate in one frame is a [[linear function]] of all the coordinates in the other frame, and the [[inverse function]]s are the inverse transformation. Depending on how the frames move relative to each other, and how they are oriented in space relative to each other, other parameters that describe direction, speed, and orientation enter the transformation equations.

{{anchor|boost}}Transformations describing relative motion with constant (uniform) velocity and without rotation of the space coordinate axes are called ''boosts'', and the relative velocity between the frames is the parameter of the transformation. The other basic type of Lorentz transformations is rotations in the spatial coordinates only, these are also inertial frames since there is no relative motion, the frames are simply tilted (and not continuously rotating), and in this case quantities defining the rotation are the parameters of the transformation (e.g., [[axis–angle representation]], or [[Euler angle]]s, etc.). A combination of a rotation and boost is a ''homogeneous transformation'', which transforms the origin back to the origin.

The full Lorentz group {{math|O(3, 1)}} also contains special transformations that are neither rotations nor boosts, but rather [[Reflection (mathematics)|reflections]] in a plane through the origin. Two of these can be singled out; [[P-symmetry|spatial inversion]] in which the spatial coordinates of all events are reversed in sign and [[T-symmetry|temporal inversion]] in which the time coordinate for each event gets its sign reversed.

Boosts should not be conflated with mere displacements in spacetime; in this case, the coordinate systems are simply shifted and there is no relative motion. However, these also count as symmetries forced by special relativity since they leave the spacetime interval invariant. A combination of a rotation with a boost, followed by a shift in spacetime, is an ''inhomogeneous Lorentz transformation'', an element of the Poincaré group, which is also called the inhomogeneous Lorentz group.

==Physical formulation of Lorentz boosts==
{{further|Derivations of the Lorentz transformations}}

===Coordinate transformation===
{{anchor|Coordinate transformation}} 
[[File:Lorentz boost x direction standard configuration.svg|thumb|right|300px|The spacetime coordinates of an event, as measured by each observer in their inertial reference frame (in standard configuration) are shown in the speech bubbles. '''Top:''' frame {{math|''F''′}} moves at velocity ''v'' along the {{math|''x''}}-axis of frame {{math|''F''}}. '''Bottom:''' frame {{math|''F''}} moves at velocity −{{math|''v''}} along the {{math|''x''′}}-axis of frame {{math|''F''′}}.<ref>{{harvnb|Young|Freedman|2008}}</ref>]]

A "stationary" observer in frame {{math|''F''}} defines events with coordinates {{math|''t'', ''x'', ''y'', ''z''}}. Another frame {{math|''F''′}} moves with velocity {{math|''v''}} relative to {{math|''F''}}, and an observer in this "moving" frame {{math|''F''′}} defines events using the coordinates {{math|''t''′, ''x''′, ''y''′, ''z''′}}.

The coordinate axes in each frame are parallel (the {{math|''x''}} and {{math|''x''′}} axes are parallel, the {{math|''y''}} and {{math|''y''′}} axes are parallel, and the {{math|''z''}} and {{math|''z''′}} axes are parallel), remain mutually perpendicular, and relative motion is along the coincident {{math|''xx′''}} axes. At {{math|''t'' {{=}} ''t''′ {{=}} 0}}, the origins of both coordinate systems are the same, {{math|(''x, y, z'') {{=}} (''x''′, ''y''′, ''z''′) {{=}} (0, 0, 0)}}. In other words, the times and positions are coincident at this event. If all these hold, then the coordinate systems are said to be in '''standard configuration''', or '''synchronized'''.

If an observer in {{math|''F''}} records an event {{math|''t, x, y, z''}}, then an observer in {{math|''F''′}} records the ''same'' event with coordinates<ref>{{harvnb|Forshaw|Smith|2009}}</ref>

{{Equation box 1
|title='''Lorentz boost''' ({{math|''x''}} ''direction'')
|indent =:
|equation =
:<math>\begin{align}
t' &= \gamma \left( t - \frac{v x}{c^2} \right) \\
x' &= \gamma \left( x - v t \right)\\
y' &= y \\
z' &= z
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

where {{math|''v''}} is the relative velocity between frames in the {{math|''x''}}-direction, {{math|''c''}} is the [[speed of light]], and

:<math> \gamma = \frac{1}{ \sqrt{1 - \frac{v^2}{c^2}}}</math>

(lowercase [[gamma]]) is the [[Lorentz factor]].

Here, {{math|''v''}} is the ''[[parameter]]'' of the transformation, for a given boost it is a constant number, but can take a continuous range of values. In the setup used here, positive relative velocity {{math|''v'' > 0}} is motion along the positive directions of the {{math|''xx''′}} axes, zero relative velocity {{math|''v'' {{=}} 0}} is no relative motion, while negative relative velocity {{math|''v'' < 0}} is relative motion along the negative directions of the {{math|''xx''′}} axes. The magnitude of relative velocity {{math|''v''}} cannot equal or exceed {{math|''c''}}, so only subluminal speeds {{math|−''c'' < ''v'' < ''c''}} are allowed. The corresponding range of {{math|''γ''}} is {{math|1 ≤ ''γ'' < ∞}}.

The transformations are not defined if {{math|''v''}} is outside these limits. At the speed of light ({{math|''v'' {{=}} ''c''}}) {{math|''γ''}} is infinite, and [[faster than light]] ({{math|''v'' > ''c''}}) {{math|''γ''}} is a [[complex number]], each of which make the transformations unphysical. The space and time coordinates are measurable quantities and numerically must be real numbers.

As an [[active transformation]], an observer in F′ notices the coordinates of the event to be "boosted" in the negative directions of the {{math|''xx''′}} axes, because of the {{math|−''v''}} in the transformations. This has the equivalent effect of the ''coordinate system'' F′ boosted in the positive directions of the {{math|''xx''′}} axes, while the event does not change and is simply represented in another coordinate system, a [[passive transformation]].

The inverse relations ({{math|''t'', ''x'', ''y'', ''z''}} in terms of {{math|''t''′, ''x''′, ''y''′, ''z''′}}) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. Here {{math|''F''′}} is the "stationary" frame while {{math|''F''}} is the "moving" frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations from {{math|''F''′}} to {{math|''F''}} must take exactly the same form as the transformations from {{math|''F''}} to {{math|''F''′}}. The only difference is {{math|''F''}} moves with velocity {{math|−''v''}} relative to {{math|''F''′}} (i.e., the relative velocity has the same magnitude but is oppositely directed). Thus if an observer in {{math|''F''′}} notes an event {{math|''t''′, ''x''′, ''y''′, ''z''′}}, then an observer in {{math|''F''}} notes the ''same'' event with coordinates

{{Equation box 1
|title='''Inverse Lorentz boost''' ({{math|''x''}} ''direction'')
|indent =:
|equation =
:<math>\begin{align}
t &= \gamma \left( t' + \frac{v x'}{c^2} \right) \\
x &= \gamma \left( x' + v t' \right)\\
y &= y' \\
z &= z',
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

and the value of {{math|''γ''}} remains unchanged. This "trick" of simply reversing the direction of relative velocity while preserving its magnitude, and exchanging primed and unprimed variables, always applies to finding the inverse transformation of every boost in any direction.

Sometimes it is more convenient to use {{math|''β'' {{=}} ''v''/''c''}} (lowercase [[beta]]) instead of {{math|''v''}}, so that

:<math>\begin{align}
ct' &= \gamma \left( ct - \beta x \right) \,, \\
x' &= \gamma \left( x - \beta ct \right) \,, \\
\end{align}</math>

which shows much more clearly the symmetry in the transformation. From the allowed ranges of {{math|''v''}} and the definition of {{math|''β''}}, it follows {{math|−1 < ''β'' < 1}}. The use of {{math|''β''}} and {{math|''γ''}} is standard throughout the literature.

The Lorentz transformations can also be derived in a way that resembles circular rotations in 3d space using the [[hyperbolic function]]s. For the boost in the {{math|''x''}} direction, the results are

{{Equation box 1
|title='''Lorentz boost''' ({{math|''x''}} ''direction with rapidity'' {{math|''ζ''}})
|indent =:
|equation =
:<math>\begin{align}
ct' &= ct \cosh\zeta - x \sinh\zeta \\
x' &= x \cosh\zeta - ct \sinh\zeta \\
y' &= y \\
z' &= z
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

where {{math|''ζ''}} (lowercase [[zeta]]) is a parameter called ''[[rapidity]]'' (many other symbols are used, including {{math|''θ, ϕ, φ, η, ψ, ξ''}}). Given the strong resemblance to rotations of spatial coordinates in 3d space in the Cartesian xy, yz, and zx planes, a Lorentz boost can be thought of as a [[hyperbolic rotation]] of spacetime coordinates in the xt, yt, and zt Cartesian-time planes of 4d [[Minkowski space]]. The parameter {{math|''ζ''}} is the [[hyperbolic angle]] of rotation, analogous to the ordinary angle for circular rotations. This transformation can be illustrated with a [[Minkowski diagram]].

The hyperbolic functions arise from the ''difference'' between the squares of the time and spatial coordinates in the spacetime interval, rather than a sum. The geometric significance of the hyperbolic functions can be visualized by taking {{math|''x'' {{=}} 0}} or {{math|''ct'' {{=}} 0}} in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varying {{math|''ζ''}}, which parametrizes the curves according to the identity

:<math> \cosh^2\zeta - \sinh^2\zeta = 1 \,. </math>

Conversely the {{math|''ct''}} and {{math|''x''}} axes can be constructed for varying coordinates but constant {{math|''ζ''}}. The definition

:<math> \tanh\zeta = \frac{\sinh\zeta}{\cosh\zeta} \,, </math>

provides the link between a constant value of rapidity, and the [[slope]] of the {{math|''ct''}} axis in spacetime. A consequence these two hyperbolic formulae is an identity that matches the Lorentz factor

:<math> \cosh\zeta = \frac{1}{\sqrt{1 - \tanh^2\zeta}} \,. </math>

Comparing the Lorentz transformations in terms of the relative velocity and rapidity, or using the above formulae, the connections between {{math|''β''}}, {{math|''γ''}}, and {{math|''ζ''}} are

:<math>\begin{align}
\beta &= \tanh\zeta \,, \\
\gamma &= \cosh\zeta \,, \\
\beta \gamma &= \sinh\zeta \,.
\end{align}</math>

Taking the inverse hyperbolic tangent gives the rapidity

:<math> \zeta = \tanh^{-1}\beta \,.</math>

Since {{math|−1 < ''β'' < 1}}, it follows {{math|−∞ < ''ζ'' < ∞}}. From the relation between {{math|''ζ''}} and {{math|''β''}}, positive rapidity {{math|''ζ'' > 0}} is motion along the positive directions of the {{math|''xx''′}} axes, zero rapidity {{math|''ζ'' {{=}} 0}} is no relative motion, while negative rapidity {{math|''ζ'' < 0}} is relative motion along the negative directions of the {{math|''xx''′}} axes.

The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidity {{math|''ζ'' → −''ζ''}} since this is equivalent to negating the relative velocity. Therefore,

{{Equation box 1
|title='''Inverse Lorentz boost''' ({{math|''x''}} ''direction with rapidity'' {{math|''ζ''}})
|indent =:
|equation =
:<math>\begin{align}
ct & = ct' \cosh\zeta + x' \sinh\zeta \\
x &= x' \cosh\zeta + ct' \sinh\zeta \\
y &= y' \\
z &= z'
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The inverse transformations can be similarly visualized by considering the cases when {{math|''x''′ {{=}} 0}} and {{math|''ct''′ {{=}} 0}}.

So far the Lorentz transformations have been applied to ''one event''. If there are two events, there is a spatial separation and time interval between them. It follows from the [[linear transformation|linearity]] of the Lorentz transformations that two values of space and time coordinates can be chosen, the Lorentz transformations can be applied to each, then subtracted to get the Lorentz transformations of the differences;

:<math>\begin{align}
\Delta t' &= \gamma \left( \Delta t - \frac{v \Delta x}{c^2} \right) \,, \\
\Delta x' &= \gamma \left( \Delta x - v \Delta t \right) \,,
\end{align}</math>

with inverse relations

:<math>\begin{align}
\Delta t &= \gamma \left( \Delta t' + \frac{v \Delta x'}{c^2} \right) \,, \\
\Delta x &= \gamma \left( \Delta x' + v \Delta t' \right) \,.
\end{align}</math>

where {{math|Δ}} (uppercase [[delta (letter)|delta]]) indicates a difference of quantities; e.g., {{math|Δ''x'' {{=}} ''x''2 − ''x''1}} for two values of {{math|''x''}} coordinates, and so on.

These transformations on ''differences'' rather than spatial points or instants of time are useful for a number of reasons:

*in calculations and experiments, it is lengths between two points or time intervals that are measured or of interest (e.g., the length of a moving vehicle, or time duration it takes to travel from one place to another),
*the transformations of velocity can be readily derived by making the difference infinitesimally small and dividing the equations, and the process repeated for the transformation of acceleration,
*if the coordinate systems are never coincident (i.e., not in standard configuration), and if both observers can agree on an event {{math|''t''0, ''x''0, ''y''0, ''z''0}} in {{math|''F''}} and {{math|''t''0′, ''x''0′, ''y''0′, ''z''0′}} in {{math|''F''′}}, then they can use that event as the origin, and the spacetime coordinate differences are the differences between their coordinates and this origin, e.g., {{math|Δ''x'' {{=}} ''x'' − ''x''0}}, {{math|Δ''x''′ {{=}} ''x''′ − ''x''0′}}, etc.

===Physical implications===

A critical requirement of the Lorentz transformations is the invariance of the speed of light, a fact used in their derivation, and contained in the transformations themselves. If in {{math|''F''}} the equation for a pulse of light along the {{math|''x''}} direction is {{math|''x'' {{=}} ''ct''}}, then in {{math|''F''′}} the Lorentz transformations give {{math|''x''′ {{=}} ''ct''′}}, and vice versa, for any {{math|−''c'' < ''v'' < ''c''}}.

For relative speeds much less than the speed of light, the Lorentz transformations reduce to the [[Galilean transformation]]

:<math>\begin{align}
t' &\approx t \\
x' &\approx x - vt
\end{align}</math>

in accordance with the [[correspondence principle]]. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".<ref>{{harvnb|Einstein|1916}}</ref>

Three unintuitive, but correct, predictions of the transformations are:
;[[Time dilation]]: Suppose there is a clock at rest in {{math|''F''}}. If a time interval (say a "tick") is measured at the same point so that {{math|Δ''x'' {{=}} 0}}, then the transformations give this tick in {{math|''F''′}} by {{math|Δ''t''′ {{=}} ''γ''Δ''t''}}. Conversely, suppose there is a clock at rest in {{math|''F''′}}. If a tick is measured at the same point so that {{math|Δ''x''′ {{=}} 0}}, then the transformations give this tick in F by {{math|Δ''t'' {{=}} ''γ''Δ''t''′}}. Either way, the boosted observer measures longer time intervals than the observer in the other frame.
;[[Relativity of simultaneity]]: Suppose two events occur simultaneously ({{math|Δ''t'' {{=}} 0}}) along the x axis, but separated by a nonzero displacement {{math|Δ''x''}}. Then in {{math|''F''′}}, we find that <math>\Delta t' = \gamma \frac{-v\Delta x}{c^{2}} </math>, so the events are no longer simultaneous according to a moving observer.
;[[Length contraction]]: Suppose there is a rod at rest in {{math|''F''}} aligned along the x axis, with length {{math|Δ''x''}}. In {{math|''F''′}}, the rod moves with velocity {{math|-''v''}}, so its length must be measured by taking two simultaneous ({{math|Δ''t''′ {{=}} 0}}) measurements at opposite ends. Under these conditions, the inverse Lorentz transform shows that {{math|Δ''x'' {{=}} ''γ''Δ''x''′}}. In {{math|''F''}} the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in {{math|''F''}}. We conclude that the boosted observer measures a shorter length, by a factor of {{math|''γ''}}, than the observer in the rest frame of the rod. Length contraction affects any geometric quantity related to lengths, so from the perspective of a moving observer, areas and volumes will also appear to shrink along the direction of motion.

===Vector transformations===

{{further|Euclidean vector|vector projection}}

[[File:Lorentz boost any direction standard configuration.svg|300px|thumb|An observer in frame {{math|''F''}} observes {{math|''F''′}} to move with velocity {{math|'''v'''}}, while {{math|''F''′}} observes {{math|''F''}} to move with velocity {{math|−'''v'''}}. The coordinate axes of each frame are still parallel and orthogonal. The position vector as measured in each frame is split into components parallel and perpendicular to the relative velocity vector {{math|'''v'''}}. '''Left:''' Standard configuration. '''Right:''' Inverse configuration.]]

The use of vectors allows positions and velocities to be expressed in arbitrary directions compactly. A single boost in any direction depends on the full relative [[velocity vector]] {{math|'''v'''}} with a magnitude {{math|{{abs|'''v'''}} {{=}} ''v''}} that cannot equal or exceed {{math|''c''}}, so that {{math|0 ≤ ''v'' < ''c''}}.

Only time and the coordinates parallel to the direction of relative motion change, while those coordinates perpendicular do not. With this in mind, split the spatial [[position vector]] {{math|'''r'''}} as measured in {{math|''F''}}, and {{math|'''r'''′}} as measured in {{math|''F′''}}, each into components perpendicular (⊥) and parallel ( ‖ ) to {{math|'''v'''}},

:<math>\mathbf{r}=\mathbf{r}_\perp+\mathbf{r}_\|\,,\quad \mathbf{r}' = \mathbf{r}_\perp' + \mathbf{r}_\|' \,, </math>

then the transformations are

:<math>\begin{align}
t' &= \gamma \left(t - \frac{\mathbf{r}_\parallel \cdot \mathbf{v}}{c^{2}} \right) \\
\mathbf{r}_\|' &= \gamma (\mathbf{r}_\| - \mathbf{v} t) \\
\mathbf{r}_\perp' &= \mathbf{r}_\perp
\end{align}</math>

where · is the [[dot product]]. The Lorentz factor {{math|''γ''}} retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definition {{math|'''β''' {{=}} '''v'''/''c''}} with magnitude {{math|0 ≤ ''β'' < 1}} is also used by some authors.

Introducing a [[unit vector]] {{math|'''n''' {{=}} '''v'''/''v'' {{=}} '''β'''/''β''}} in the direction of relative motion, the relative velocity is {{math|'''v''' {{=}} ''v'''''n'''}} with magnitude {{math|''v''}} and direction {{math|'''n'''}}, and [[vector projection]] and rejection give respectively

:<math>\mathbf{r}_\parallel = (\mathbf{r}\cdot\mathbf{n})\mathbf{n}\,,\quad \mathbf{r}_\perp = \mathbf{r} - (\mathbf{r}\cdot\mathbf{n})\mathbf{n}</math>

Accumulating the results gives the full transformations,

{{Equation box 1
|title='''Lorentz boost''' (''in direction'' {{math|'''n'''}} ''with magnitude'' {{math|''v''}})
|indent =:
|equation =
<math>\begin{align}
t' &= \gamma \left(t - \frac{v\mathbf{n}\cdot \mathbf{r}}{c^2} \right) \,, \\
\mathbf{r}' &= \mathbf{r} + (\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n} - \gamma t v\mathbf{n} \,.
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The projection and rejection also applies to {{math|'''r'''′}}. For the inverse transformations, exchange {{math|'''r'''}} and {{math|'''r'''′}} to switch observed coordinates, and negate the relative velocity {{math|'''v''' → −'''v'''}} (or simply the unit vector {{math|'''n''' → −'''n'''}} since the magnitude {{math|''v''}} is always positive) to obtain

{{Equation box 1
|title='''Inverse Lorentz boost''' (''in direction'' {{math|'''n'''}} ''with magnitude'' {{math|''v''}})
|indent =:
|equation =
<math>\begin{align}
t &= \gamma \left(t' + \frac{\mathbf{r}' \cdot v\mathbf{n}}{c^{2}} \right) \,, \\
\mathbf{r} &= \mathbf{r}' + (\gamma-1)(\mathbf{r}'\cdot\mathbf{n})\mathbf{n} + \gamma t' v\mathbf{n} \,,
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The unit vector has the advantage of simplifying equations for a single boost, allows either {{math|'''v'''}} or {{math|'''β'''}} to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacing {{math|''β''}} and {{math|''βγ''}}. It is not convenient for multiple boosts.

The vectorial relation between relative velocity and rapidity is<ref>{{harvnb|Barut|1964|page=18–19}}</ref>

:<math> \boldsymbol{\beta} = \beta \mathbf{n} = \mathbf{n} \tanh\zeta \,,</math>

and the "rapidity vector" can be defined as

:<math> \boldsymbol{\zeta} = \zeta\mathbf{n} = \mathbf{n}\tanh^{-1}\beta \,, </math>

each of which serves as a useful abbreviation in some contexts. The magnitude of {{math|'''ζ'''}} is the absolute value of the rapidity scalar confined to {{math|0 ≤ ''ζ'' < ∞}}, which agrees with the range {{math|0 ≤ ''β'' < 1}}.

===Transformation of velocities===

{{further|differential of a function|velocity addition formula}}

[[File:Lorentz transformation of velocity including velocity addition.svg|300px|thumb|The transformation of velocities provides the definition [[velocity addition formula|relativistic velocity addition]] {{math|&oplus;}}, the ordering of vectors is chosen to reflect the ordering of the addition of velocities; first {{math|'''v'''}} (the velocity of F′ relative to F) then {{math|'''u'''′}} (the velocity of X relative to F′) to obtain {{math|'''u''' {{=}} '''v''' &oplus; '''u'''′}} (the velocity of X relative to F).]]

Defining the coordinate velocities and Lorentz factor by

:<math>\mathbf{u} = \frac{d\mathbf{r}}{dt} \,,\quad \mathbf{u}' = \frac{d\mathbf{r}'}{dt'} \,,\quad \gamma_\mathbf{v} = \frac{1}{\sqrt{1-\dfrac{\mathbf{v}\cdot\mathbf{v}}{c^2}}}</math>

taking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to

:<math>\mathbf{u}' = \frac{1}{ 1 - \frac{\mathbf{v}\cdot\mathbf{u}}{c^2} }\left[\frac{\mathbf{u}}{\gamma_\mathbf{v}} - \mathbf{v} + \frac{1}{c^2}\frac{\gamma_\mathbf{v}}{\gamma_\mathbf{v} + 1}\left(\mathbf{u}\cdot\mathbf{v}\right)\mathbf{v}\right] </math>

The velocities {{math|'''u'''}} and {{math|'''u'''′}} are the velocity of some massive object. They can also be for a third inertial frame (say F′′), in which case they must be ''constant''. Denote either entity by X. Then X moves with velocity {{math|'''u'''}} relative to F, or equivalently with velocity {{math|'''u'''′}} relative to F′, in turn F′ moves with velocity {{math|'''v'''}} relative to F. The inverse transformations can be obtained in a similar way, or as with position coordinates exchange {{math|'''u'''}} and {{math|'''u'''′}}, and change {{math|'''v'''}} to {{math|−'''v'''}}.

The transformation of velocity is useful in [[stellar aberration]], the [[Fizeau experiment]], and the [[relativistic Doppler effect]].

The [[Acceleration (special relativity)#Three-acceleration|Lorentz transformations of acceleration]] can be similarly obtained by taking differentials in the velocity vectors, and dividing these by the time differential.

===Transformation of other quantities===

In general, given four quantities {{math|''A''}} and {{math|'''Z''' {{=}} (''Z''x, ''Z''y, ''Z''z)}} and their Lorentz-boosted counterparts {{math|''A''′}} and {{math|'''Z'''′ {{=}} (''Z''′x, ''Z''′y, ''Z''′z)}}, a relation of the form

:<math>A^2 - \mathbf{Z}\cdot\mathbf{Z} = {A'}^2 - \mathbf{Z}'\cdot\mathbf{Z}'</math>

implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates;

:<math>\begin{align}
A' &= \gamma \left(A - \frac{v\mathbf{n}\cdot \mathbf{Z}}{c} \right) \,, \\
\mathbf{Z}' &= \mathbf{Z} + (\gamma-1)(\mathbf{Z}\cdot\mathbf{n})\mathbf{n} - \frac{\gamma A v\mathbf{n}}{c} \,.
\end{align}</math>

The decomposition of {{math|'''Z'''}} (and {{math|'''Z'''′}}) into components perpendicular and parallel to {{math|'''v'''}} is exactly the same as for the position vector, as is the process of obtaining the inverse transformations (exchange {{math|(''A'', '''Z''')}} and {{math|(''A''′, '''Z'''′)}} to switch observed quantities, and reverse the direction of relative motion by the substitution {{math|'''n''' ↦ −'''n'''}}).

The quantities {{math|(''A'', '''Z''')}} collectively make up a ''[[four vector]]'', where {{math|''A''}} is the "timelike component", and {{math|'''Z'''}} the "spacelike component". Examples of {{math|''A''}} and {{math|'''Z'''}} are the following:

:{| class="wikitable"
|-
! Four vector
! {{math|''A''}}
! {{math|'''Z'''}}
|-
| Position [[four vector]]
| [[Time]] (multiplied by {{math|''c''}}), {{math|''ct''}}
| [[Position vector]], {{math|'''r'''}}
|-
| [[Four momentum]]
| [[Energy]] (divided by {{math|''c''}}), {{math|''E''/''c''}}
| [[Momentum]], {{math|'''p'''}}
|-
| Wave [[four vector]]
| [[angular frequency]] (divided by {{math|''c''}}), {{math|''ω''/''c''}}
| [[wave vector]], {{math|'''k'''}}
|-
| [[Four spin]]
| (No name), {{math|''s''t}}
| [[spin (physics)|Spin]], {{math|'''s'''}}
|-
| [[Four current]]
| [[Charge density]] (multiplied by {{math|''c''}}), {{math|''ρc''}}
| [[Current density]], {{math|'''j'''}}
|-
| [[Electromagnetic four potential]]
| [[Electric potential]] (divided by {{math|''c''}}), {{math|''φ''/''c''}}
| [[Magnetic potential]], {{math|'''A'''}}
|}

For a given object (e.g., particle, fluid, field, material), if {{math|''A''}} or {{math|'''Z'''}} correspond to properties specific to the object like its [[charge density]], [[mass density]], [[spin (physics)|spin]], etc., its properties can be fixed in the rest frame of that object. Then the Lorentz transformations give the corresponding properties in a frame moving relative to the object with constant velocity. This breaks some notions taken for granted in non-relativistic physics. For example, the energy {{math|''E''}} of an object is a scalar in non-relativistic mechanics, but not in relativistic mechanics because energy changes under Lorentz transformations; its value is different for various inertial frames. In the rest frame of an object, it has a [[rest energy]] and zero momentum. In a boosted frame its energy is different and it appears to have a momentum. Similarly, in non-relativistic quantum mechanics the spin of a particle is a constant vector, but in [[relativistic quantum mechanics]] spin {{math|'''s'''}} depends on relative motion. In the rest frame of the particle, the spin pseudovector can be fixed to be its ordinary non-relativistic spin with a zero timelike quantity {{math|''st''}}, however a boosted observer will perceive a nonzero timelike component and an altered spin.<ref>{{harvnb|Chaichian|Hagedorn|1997|page=239}}</ref>

Not all quantities are invariant in the form as shown above, for example orbital [[angular momentum]] {{math|'''L'''}} does not have a timelike quantity, and neither does the [[electric field]] {{math|'''E'''}} nor the [[magnetic field]] {{math|'''B'''}}. The definition of angular momentum is {{math|'''L''' {{=}} '''r''' × '''p'''}}, and in a boosted frame the altered angular momentum is {{math|'''L'''′ {{=}} '''r'''′ × '''p'''′}}. Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns out {{math|'''L'''}} transforms with another vector quantity {{math|'''N''' {{=}} (''E''/''c''2)'''r''' − ''t'''''p'''}} related to boosts, see [[relativistic angular momentum]] for details. For the case of the {{math|'''E'''}} and {{math|'''B'''}} fields, the transformations cannot be obtained as directly using vector algebra. The [[Lorentz force]] is the definition of these fields, and in {{math|''F''}} it is {{math|'''F''' {{=}} ''q''('''E''' + '''v''' × '''B''')}} while in {{math|''F''′}} it is {{math|'''F'''′ {{=}} ''q''('''E'''′ + '''v'''′ × '''B'''′)}}. A method of deriving the EM field transformations in an efficient way which also illustrates the unit of the electromagnetic field uses tensor algebra, [[Lorentz transformation#Transformation of the electromagnetic field|given below]].

==Mathematical formulation==

{{main|Lorentz group}}
{{further|Matrix (mathematics)|matrix product|linear algebra|rotation formalisms in three dimensions}}

Throughout, italic non-bold capital letters are 4×4 matrices, while non-italic bold letters are 3×3 matrices.

===Homogeneous Lorentz group===

Writing the coordinates in column vectors and the [[Minkowski metric]] {{math|''η''}} as a square matrix

:<math> X' = \begin{bmatrix} c\,t' \\ x' \\ y' \\ z' \end{bmatrix} \,, \quad \eta = \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} \,, \quad X = \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix} </math>

the spacetime interval takes the form (T denotes [[transpose]])

:<math> X \cdot X = X^\mathrm{T} \eta X = {X'}^\mathrm{T} \eta {X'} </math>

and is [[Invariant (physics)|invariant]] under a Lorentz transformation

:<math>X' = \Lambda X </math>

where Λ is a square matrix which can depend on parameters.

The [[set (mathematics)|set]] of all Lorentz transformations Λ in this article is denoted <math>\mathcal{L}</math>. This set together with matrix multiplication forms a [[group (mathematics)|group]], in this context known as the ''[[Lorentz group]]''. Also, the above expression {{math|''X·X''}} is a [[quadratic form]] of signature (3,1) on spacetime, and the group of transformations which leaves this quadratic form invariant is the [[indefinite orthogonal group]] O(3,1), a [[Lie group]]. In other words, the Lorentz group is O(3,1). As presented in this article, any Lie groups mentioned are [[matrix Lie group]]s. In this context the operation of composition amounts to [[matrix multiplication]].

From the invariance of the spacetime interval it follows

:<math>\eta = \Lambda^\mathrm{T} \eta \Lambda </math>

and this matrix equation contains the general conditions on the Lorentz transformation to ensure invariance of the spacetime interval. Taking the [[determinant]] of the equation using the product rule<ref group=nb>For two square matrices {{math|''A''}} and {{math|''B''}}, {{math|det(''AB'') {{=}} det(''A'')det(''B'')}}</ref> gives immediately

:<math>[\det (\Lambda)]^2 = 1 \quad \Rightarrow \quad \det(\Lambda) = \pm 1 </math>

Writing the Minkowski metric as a block matrix, and the Lorentz transformation in the most general form,

:<math>\eta = \begin{bmatrix}-1 & 0 \\ 0 & \mathbf{I}\end{bmatrix} \,, \quad \Lambda=\begin{bmatrix}\Gamma & -\mathbf{a}^\mathrm{T}\\-\mathbf{b} & \mathbf{M}\end{bmatrix} \,, </math>

carrying out the block matrix multiplications obtains general conditions on {{math|Γ, '''a''', '''b''', '''M'''}} to ensure relativistic invariance. Not much information can be directly extracted from all the conditions, however one of the results

:<math>\Gamma^2 = 1 + \mathbf{b}^\mathrm{T}\mathbf{b}</math>

is useful; {{math|'''b'''T'''b''' ≥ 0}} always so it follows that

:<math> \Gamma^2 \geq 1 \quad \Rightarrow \quad \Gamma \leq - 1 \,,\quad \Gamma \geq 1 </math>

The negative inequality may be unexpected, because {{math|Γ}} multiplies the time coordinate and this has an effect on [[Time translation symmetry|time symmetry]]. If the positive equality holds, then {{math|Γ}} is the Lorentz factor.

The determinant and inequality provide four ways to classify Lorentz transformations (herein LTs for brevity). Any particular LT has only one determinant sign ''and'' only one inequality. There are four sets which include every possible pair given by the [[Intersection (set theory)|intersection]]s ("n"-shaped symbol meaning "and") of these classifying sets.

{| class="wikitable"
|-
! Intersection, ∩
! '''Antichronous''' (or non-orthochronous) LTs

:<math> \mathcal{L}^\downarrow = \{ \Lambda \, : \, \Gamma \leq -1 \} </math>
! '''Orthochronous''' LTs

:<math> \mathcal{L}^\uparrow = \{ \Lambda \, : \, \Gamma \geq 1 \} </math>
|-
! '''Proper''' LTs

:<math> \mathcal{L}_{+} = \{ \Lambda \, : \, \det(\Lambda) = +1 \} </math>
| '''Proper antichronous''' LTs
:<math>\mathcal{L}_+^\downarrow = \mathcal{L}_+ \cap \mathcal{L}^\downarrow </math>
|'''Proper orthochronous''' LTs
:<math>\mathcal{L}_+^\uparrow = \mathcal{L}_+ \cap \mathcal{L}^\uparrow </math>
|-
! '''Improper''' LTs

:<math> \mathcal{L}_{-} = \{ \Lambda \, : \, \det(\Lambda) = -1 \} </math>
|'''Improper antichronous''' LTs
:<math>\mathcal{L}_{-}^\downarrow = \mathcal{L}_{-} \cap \mathcal{L}^\downarrow </math>
|'''Improper orthochronous''' LTs
:<math>\mathcal{L}_{-}^\uparrow = \mathcal{L}_{-} \cap \mathcal{L}^\uparrow </math>
|-
|}

where "+" and "−" indicate the determinant sign, while "↑" for ≥ and "↓" for ≤ denote the inequalities.

The full Lorentz group splits into the [[Union (set theory)|union]] ("u"-shaped symbol meaning "or") of four [[disjoint set]]s

:<math> \mathcal{L} = \mathcal{L}_{+}^\uparrow \cup \mathcal{L}_{-}^\uparrow \cup \mathcal{L}_{+}^\downarrow \cup \mathcal{L}_{-}^\downarrow </math>

A [[subgroup]] of a group must be [[Closure (mathematics)|closed]] under the same operation of the group (here matrix multiplication). In other words, for two Lorentz transformations {{math|Λ}} and {{math|''L''}} from a particular set, the composite Lorentz transformations {{math|Λ''L''}} and {{math|''L''Λ}} must be in the same set as {{math|Λ}} and {{math|''L''}}. This will not always be the case; it can be shown that the composition of ''any'' two Lorentz transformations always has the positive determinant and positive inequality, a proper orthochronous transformation. The sets <math>\mathcal{L}_+^\uparrow </math>, <math>\mathcal{L}_+</math>, <math>\mathcal{L}^\uparrow</math>, and <math>\mathcal{L}_0 = \mathcal{L}_+^\uparrow \cup \mathcal{L}_{-}^\downarrow</math> all form subgroups. The other sets involving the improper and/or antichronous properties (i.e. <math>\mathcal{L}_+^\downarrow </math>, <math>\mathcal{L}_{-}^\downarrow </math>, <math>\mathcal{L}_{-}^\uparrow </math>) do not form subgroups, because the composite transformation always has a positive determinant or inequality, whereas the original separate transformations will have negative determinants and/or inequalities.

===Proper transformations===

The Lorentz boost is

:<math>X' = B(\mathbf{v})X</math>

where the boost matrix is

:<math> B(\mathbf{v}) = \begin{bmatrix} \gamma&-\gamma\beta n_x&-\gamma\beta n_y&-\gamma\beta n_z\\ -\gamma\beta n_x&1+(\gamma-1)n_x^2&(\gamma-1)n_x n_y&(\gamma-1)n_x n_z\\ -\gamma\beta n_y&(\gamma-1)n_y n_x&1+(\gamma-1)n_y^2&(\gamma-1)n_y n_z\\ -\gamma\beta n_z&(\gamma-1)n_z n_x&(\gamma-1)n_z n_y&1+(\gamma-1)n_z^2\\ \end{bmatrix} \,. </math>

The boosts along the Cartesian directions can be readily obtained, for example the unit vector in the x direction has components {{math|''nx'' {{=}} 1}} and {{math|''ny'' {{=}} ''nz'' {{=}} 0}}.

The matrices make one or more successive transformations easier to handle, rather than rotely iterating the transformations to obtain the result of more than one transformation. If a frame {{math|''F''′}} is boosted with velocity {{math|'''u'''}} relative to frame {{math|''F''}}, and another frame {{math|''F''′′}} is boosted with velocity {{math|'''v'''}} relative to {{math|''F''′}}, the separate boosts are
:<math>X'' = B(\mathbf{v})X' \,, \quad X' = B(\mathbf{u})X </math>
and the composition of the two boosts connects the coordinates in {{math|''F''′′}} and {{math|''F''}},
:<math>X'' = B(\mathbf{v})B(\mathbf{u})X \,. </math>
Successive transformations act on the left. If {{math|'''u'''}} and {{math|'''v'''}} are [[collinear]] (parallel or antiparallel along the same line of relative motion), the boost matrices [[Commutative property|commute]]: {{math|''B''('''v''')''B''('''u''') {{=}} ''B''('''u''')''B''('''v''')}} and this composite transformation happens to be another boost.

If {{math|'''u'''}} and {{math|'''v'''}} are not collinear but in different directions, the situation is considerably more complicated. Lorentz boosts along different directions do not commute: {{math|''B''('''v''')''B''('''u''')}} and {{math|''B''('''u''')''B''('''v''')}} are not equal. Also, each of these compositions is ''not'' a single boost, but still a Lorentz transformation as each boost still preserves invariance of the spacetime interval. It turns out the composition of any two Lorentz boosts is equivalent to a boost followed or preceded by a rotation on the spatial coordinates, in the form of {{math|''R''('''ρ''')''B''('''w''')}} or {{math|''B''({{overline|'''w'''}})''R''({{overline|'''ρ'''}})}}. The {{math|'''w'''}} and {{math|{{overline|'''w'''}}}} are [[velocity addition formula|composite velocities]], while {{math|'''ρ'''}} and {{math|{{overline|'''ρ'''}}}} are rotation parameters (e.g. [[axis-angle representation|axis-angle]] variables, [[Euler angles]], etc.). The rotation in [[block matrix]] form is simply

:<math>\quad R(\boldsymbol{\rho}) = \begin{bmatrix} 1 & 0 \\ 0 & \mathbf{R}(\boldsymbol{\rho}) \end{bmatrix} \,, </math>

where {{math|'''R'''('''ρ''')}} is a 3d [[rotation matrix]], which rotates any 3d vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive transformation). It is ''not'' simple to connect {{math|'''w'''}} and {{math|'''ρ'''}} (or {{math|{{overline|'''w'''}}}} and {{math|{{overline|'''ρ'''}}}}) to the original boost parameters {{math|'''u'''}} and {{math|'''v'''}}. In a composition of boosts, the {{math|''R''}} matrix is named the [[Wigner rotation]], and gives rise to the [[Thomas precession]]. These articles give the explicit formulae for the composite transformation matrices, including expressions for {{math|'''w''', '''ρ''', {{overline|'''w'''}}, {{overline|'''ρ'''}}}}.

In this article the [[axis-angle representation]] is used for {{math|'''ρ'''}}. The rotation is about an axis in the direction of a [[unit vector]] {{math|'''e'''}}, through angle {{math|''θ''}} (positive anticlockwise, negative clockwise, according to the [[right-hand rule]]). The "axis-angle vector"
:<math>\boldsymbol{\theta} = \theta \mathbf{e}</math>
will serve as a useful abbreviation.

Spatial rotations alone are also Lorentz transformations they leave the spacetime interval invariant. Like boosts, successive rotations about different axes do not commute. Unlike boosts, the composition of any two rotations is equivalent to a single rotation. Some other similarities and differences between the boost and rotation matrices include:

*[[matrix inverse|inverse]]s: {{math|''B''('''v''')−1 {{=}} ''B''(−'''v''')}} (relative motion in the opposite direction), and {{math|''R''('''θ''')−1 {{=}} ''R''(−'''θ''')}} (rotation in the opposite sense about the same axis)
*[[identity transformation]] for no relative motion/rotation: {{math|''B''('''0''') {{=}} ''R''('''0''') {{=}} ''I''}}
*unit [[determinant]]: {{math|det(''B'') {{=}} det(''R'') {{=}} +1}}. This property makes them proper transformations.
*[[symmetric matrix|matrix symmetry]]: {{math|''B''}} is symmetric (equals [[transpose]]), while {{math|''R''}} is nonsymmetric but [[orthogonal matrix|orthogonal]] (transpose equals [[matrix inverse|inverse]], {{math|''R''T {{=}} ''R''−1}}).

The most general proper Lorentz transformation {{math|Λ('''v''', '''θ''')}} includes a boost and rotation together, and is a nonsymmetric matrix. As special cases, {{math|Λ('''0''', '''θ''') {{=}} ''R''('''θ''')}} and {{math|Λ('''v''', '''0''') {{=}} ''B''('''v''')}}. An explicit form of the general Lorentz transformation is cumbersome to write down and will not be given here. Nevertheless, closed form expressions for the transformation matrices will be given below using group theoretical arguments. It will be easier to use the rapidity parametrization for boosts, in which case one writes {{math|Λ('''ζ''', '''θ''')}} and {{math|''B''('''ζ''')}}.

====The Lie group SO+(3,1)====

The set of transformations

:<math> \{ B(\boldsymbol{\zeta}), R(\boldsymbol{\theta}), \Lambda(\boldsymbol{\zeta}, \boldsymbol{\theta}) \} </math>

with matrix multiplication as the operation of composition forms a group, called the "restricted Lorentz group", and is the [[special indefinite orthogonal group]] SO+(3,1). (The plus sign indicates positive unit determinant).

For simplicity, look at the infinitesimal Lorentz boost in the x direction (examining a boost in any other direction, or rotation about any axis, follows an identical procedure). The infinitesimal boost is a small boost away from the identity, obtained by the [[Taylor expansion]] of the boost matrix to first order about {{math|''ζ'' {{=}} 0}},

:<math> B_x = I + \zeta \left. \frac{\partial B_x}{\partial \zeta } \right|_{\zeta=0} + \cdots </math>

where the higher order terms not shown are negligible because {{math|''ζ''}} is small, and {{math|''Bx''}} is simply the boost matrix in the ''x'' direction. The [[matrix calculus|derivative of the matrix]] is the matrix of derivatives (of the entries, with respect to the same variable), and it is understood the derivatives are found first then evaluated at {{math|''ζ'' {{=}} 0}},

:<math> \left. \frac{\partial B_x }{\partial \zeta } \right|_{\zeta=0} = - K_x \,. </math>

For now, {{math|''Kx''}} is defined by this result (its significance will be explained shortly). In the limit of an infinite number of infinitely small steps, the finite boost transformation in the form of a [[matrix exponential]] is obtained

:<math> B_x =\lim_{N\rightarrow\infty}\left(I-\frac{\zeta }{N}K_x\right)^{N} = e^{-\zeta K_x} </math>

where the [[Exponential function#Formal definition|limit definition of the exponential]] has been used (see also [[characterizations of the exponential function]]). More generally<ref group="nb">Explicitly,
:<math> \boldsymbol{\zeta} \cdot\mathbf{K} = \zeta_x K_x + \zeta_y K_y + \zeta_z K_z </math>
:<math> \boldsymbol{\theta} \cdot\mathbf{J} = \theta_x J_x + \theta_y J_y + \theta_z J_z </math>
</ref>

:<math>B(\boldsymbol{\zeta}) = e^{-\boldsymbol{\zeta}\cdot\mathbf{K}} \, , \quad R(\boldsymbol{\theta}) = e^{\boldsymbol{\theta}\cdot\mathbf{J}} \,. </math>

The axis-angle vector {{math|'''θ'''}} and rapidity vector {{math|'''ζ'''}} are altogether six continuous variables which make up the group parameters (in this particular representation), and the generators of the group are {{math|'''K''' {{=}} (''Kx, Ky, Kz'')}} and {{math|'''J''' {{=}} (''Jx, Jy, Jz'')}}, each vectors of matrices with the explicit forms<ref group=nb>In [[quantum mechanics]], [[relativistic quantum mechanics]], and [[quantum field theory]], a different convention is used for these matrices; the right hand sides are all multiplied by a factor of the imaginary unit {{math|''i'' {{=}} {{sqrt|−1}}}}.</ref>

:<math>K_x = \begin{bmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{bmatrix}\,,\quad K_y = \begin{bmatrix}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}\,,\quad K_z = \begin{bmatrix}
0 & 0 & 0 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
1 & 0 & 0 & 0
\end{bmatrix}
</math>
:<math>J_x = \begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0 \\
\end{bmatrix}\,,\quad J_y =
\begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0
\end{bmatrix}\,,\quad J_z = \begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
</math>

These are all defined in an analogous way to {{math|''Kx''}} above, although the minus signs in the boost generators are conventional. Physically, the generators of the Lorentz group correspond to important symmetries in spacetime: {{math|'''J'''}} are the ''rotation generators'' which correspond to [[angular momentum]], and {{math|'''K'''}} are the ''boost generators'' which correspond to the motion of the system in spacetime. The derivative of any smooth curve {{math|''C''(''t'')}} with {{math|''C''(0) {{=}} ''I''}} in the group depending on some group parameter {{math|''t''}} with respect to that group parameter, evaluated at {{math|''t'' {{=}} 0}}, serves as a definition of a corresponding group generator {{math|''G''}}, and this reflects an infinitesimal transformation away from the identity. The smooth curve can always be taken as an exponential as the exponential will always map {{math|''G''}} smoothly back into the group via {{math|''t'' → exp(''tG'')}} for all {{math|''t''}}; this curve will yield {{math|''G''}} again when differentiated at {{math|''t'' {{=}} 0}}.

Expanding the exponentials in their Taylor series obtains

:<math> B({\boldsymbol {\zeta }})=I-\sinh \zeta (\mathbf {n} \cdot \mathbf {K} )+(\cosh \zeta -1)(\mathbf {n} \cdot \mathbf {K} )^2</math>
:<math>R(\boldsymbol {\theta })=I+\sin \theta (\mathbf {e} \cdot \mathbf {J} )+(1-\cos \theta )(\mathbf {e} \cdot \mathbf {J} )^2\,.</math>

which compactly reproduce the boost and rotation matrices as given in the previous section.

It has been stated that the general proper Lorentz transformation is a product of a boost and rotation. At the ''infinitesimal'' level the product

:<math> \begin{align} \Lambda
& = (I - \boldsymbol {\zeta } \cdot \mathbf {K} + \cdots )(I + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots ) \\
& = (I + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots )(I - \boldsymbol {\zeta } \cdot \mathbf {K} + \cdots ) \\
& = I - \boldsymbol {\zeta } \cdot \mathbf {K} + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots
\end{align} </math>

is commutative because only linear terms are required (products like {{math|('''θ'''·'''J''')('''ζ'''·'''K''')}} and {{math|('''ζ'''·'''K''')('''θ'''·'''J''')}} count as higher order terms and are negligible). Taking the limit as before leads to the finite transformation in the form of an exponential

:<math>\Lambda (\boldsymbol{\zeta}, \boldsymbol{\theta}) = e^{-\boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J} }.</math>

The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular,

:<math>e^{-\boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J} } \ne e^{-\boldsymbol{\zeta} \cdot\mathbf{K}} e^{\boldsymbol{\theta} \cdot\mathbf{J}},</math>

because the generators do not commute. For a description of how to find the factors of a general Lorentz transformation in terms of a boost and a rotation ''in principle'' (this usually does not yield an intelligible expression in terms of generators {{math|'''J'''}} and {{math|'''K'''}}), see [[Wigner rotation]]. If, on the other hand, ''the decomposition is given'' in terms of the generators, and one wants to find the product in terms of the generators, then the [[Baker–Campbell–Hausdorff formula]] applies.

====The Lie algebra so(3,1)====

Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, the [[set (mathematics)|set]] of all Lorentz generators

:<math>V = \{ \boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J} \} </math>

together with the operations of ordinary [[matrix addition]] and [[matrix multiplication#Scalar multiplication|multiplication of a matrix by a number]], forms a [[vector space]] over the real numbers.<ref group=nb>Until now the term "vector" has exclusively referred to "[[Euclidean vector]]", examples are position {{math|'''r'''}}, velocity {{math|'''v'''}}, etc. The term "vector" applies much more broadly than Euclidean vectors, row or column vectors, etc., see [[linear algebra]] and [[vector space]] for details. The generators of a Lie group also form a vector space over a [[field (mathematics)|field]] of numbers (e.g. [[real number]]s, [[complex number]]s), since a [[linear combination]] of the generators is also a generator. They just live in a different space to the position vectors in ordinary 3d space.</ref> The generators {{math|''Jx, Jy, Jz, Kx, Ky, Kz''}} form a [[basis (linear algebra)|basis]] set of ''V'', and the components of the axis-angle and rapidity vectors, {{math|''θx, θy, θz, ζx, ζy, ζz''}}, are the [[coordinate vector|coordinate]]s of a Lorentz generator with respect to this basis.<ref group=nb>In ordinary 3d [[position space]], the position vector {{math|'''r''' {{=}} ''x'''''e'''''x'' + ''y'''''e'''''y'' + ''z'''''e'''''z''}} is expressed as a linear combination of the Cartesian unit vectors {{math|'''e'''''x'', '''e'''''y'', '''e'''''z''}} which form a basis, and the Cartesian coordinates {{math|''x, y, z''}} are coordinates with respect to this basis.</ref>

Three of the [[commutation relation]]s of the Lorentz generators are

:<math>[ J_x, J_y ] = J_z \,,\quad [ K_x, K_y ] = -J_z \,,\quad [ J_x, K_y ] = K_z \,, </math>

where the bracket {{math|[''A'', ''B''] {{=}} ''AB'' − ''BA''}} is known as the ''[[commutator]]'', and the other relations can be found by taking [[cyclic permutation]]s of x, y, z components (i.e. change x to y, y to z, and z to x, repeat).

These commutation relations, and the vector space of generators, fulfill the definition of the [[Lie algebra]] <math>\mathfrak{so}(3, 1)</math>. In summary, a Lie algebra is defined as a [[vector space]] ''V'' over a [[field (mathematics)|field]] of numbers, and with a [[binary operation]] [ , ] (called a [[Lie bracket]] in this context) on the elements of the vector space, satisfying the axioms of [[Bilinear map|bilinearity]], [[alternatization]], and the [[Jacobi identity]]. Here the operation [ , ] is the commutator which satisfies all of these axioms, the vector space is the set of Lorentz generators ''V'' as given previously, and the field is the set of real numbers.

Linking terminology used in mathematics and physics: A group generator is any element of the Lie algebra. A group parameter is a component of a coordinate vector representing an arbitrary element of the Lie algebra with respect to some basis. A basis, then, is a set of generators being a basis of the Lie algebra in the usual vector space sense.

The [[exponential map (Lie theory)]] from the Lie algebra to the Lie group,

:<math>\mathrm{exp} \, : \, \mathfrak{so}(3,1) \rightarrow \mathrm{SO}(3,1),</math>

provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group. It the case of the Lorentz group, the exponential map is just the [[matrix exponential]]. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is [[surjective function|surjective]] (onto). Hence any group element can be expressed as an exponential of an element of the Lie algebra.

===Improper transformations===

Lorentz transformations also include [[parity inversion]]

:<math> P = \begin{bmatrix} 1 & 0 \\ 0 & - \mathbf{I} \end{bmatrix} </math>

which negates all the spatial coordinates only, and [[T-symmetry|time reversal]]

:<math> T = \begin{bmatrix} - 1 & 0 \\ 0 & \mathbf{I} \end{bmatrix}</math>

which negates the time coordinate only, because these transformations leave the spacetime interval invariant. Here {{math|'''I'''}} is the 3d [[identity matrix]]. These are both symmetric, they are their own inverses (see [[involution (mathematics)]]), and each have determinant −1. This latter property makes them improper transformations.

If {{math|Λ}} is a proper orthochronous Lorentz transformation, then {{math|''T''Λ}} is improper antichronous, {{math|''P''Λ}} is improper orthochronous, and {{math|''TP''Λ {{=}} ''PT''Λ}} is proper antichronous.

=== Inhomogeneous Lorentz group ===

Two other spacetime symmetries have not been accounted for. For the spacetime interval to be invariant, it can be shown<ref>{{harvnb|Weinberg|1972}}</ref> that it is necessary and sufficient for the coordinate transformation to be of the form

:<math>X' = \Lambda X + C </math>

where ''C'' is a constant column containing translations in time and space. If ''C'' ≠ 0, this is an '''inhomogeneous Lorentz transformation''' or '''[[Poincaré transformation]]'''.<ref>{{harvnb|Weinberg|2005|pages=55–58}}</ref><ref>{{harvnb|Ohlsson|2011|page=3–9}}</ref> If ''C'' = 0, this is a '''homogeneous Lorentz transformation'''. Poincaré transformations are not dealt further in this article.

==Tensor formulation==

{{main|Representation theory of the Lorentz group}}
{{For|the notation used|Ricci calculus}}

=== Contravariant vectors ===
Writing the general matrix transformation of coordinates as the matrix equation

:<math>\begin{bmatrix}
{x'}^0 \\
{x'}^1 \\
{x'}^2 \\
{x'}^3
\end{bmatrix} =
\begin{bmatrix}
{\Lambda^0}_0 & {\Lambda^0}_1 & {\Lambda^0}_2 & {\Lambda^0}_3 \\
{\Lambda^1}_0 & {\Lambda^1}_1 & {\Lambda^1}_2 & {\Lambda^1}_3 \\
{\Lambda^2}_0 & {\Lambda^2}_1 & {\Lambda^2}_2 & {\Lambda^2}_3 \\
{\Lambda^3}_0 & {\Lambda^3}_1 & {\Lambda^3}_2 & {\Lambda^3}_3 \\
\end{bmatrix}
\begin{bmatrix}
x^0 \\
x^1 \\
x^2 \\
x^3
\end{bmatrix}
</math>

allows the transformation of other physical quantities that cannot be expressed as four-vectors; e.g., [[tensor]]s or [[spinor]]s of any order in 4d spacetime, to be defined. In the corresponding [[tensor index notation]], the above matrix expression is

:<math>{x^\prime}^\nu = {\Lambda^\nu}_\mu x^\mu,</math>

where lower and upper indices label [[covariance and contravariance of vectors|covariant and contravariant components]] respectively,<ref>{{cite book |title=Mathematics for Physicists |first1=Philippe |last1=Dennery |first2=André |last2=Krzywicki |publisher=Courier Corporation |year=2012 |isbn=978-0-486-15712-2 |page=138 |url=https://books.google.com/books?id=ogHCAgAAQBAJ}} [https://books.google.com/books?id=ogHCAgAAQBAJ&pg=PA138 Extract of page 138]</ref> and the [[summation convention]] is applied. It is a standard convention to use [[Greek alphabet|Greek]] indices that take the value 0 for time components, and 1, 2, 3 for space components, while [[Latin alphabet|Latin]] indices simply take the values 1, 2, 3, for spatial components. Note that the first index (reading left to right) corresponds in the matrix notation to a ''row index''. The second index corresponds to the column index.

The transformation matrix is universal for all [[four-vector]]s, not just 4-dimensional spacetime coordinates. If {{math|''A''}} is any four-vector, then in [[tensor index notation]]

:<math> {A^\prime}^\nu = {\Lambda^\nu}_\mu A^\mu \,.</math>

Alternatively, one writes

:<math> A^{\nu'} = {\Lambda^{\nu'}}_\mu A^\mu \,.</math>

in which the primed indices denote the indices of A in the primed frame. This notation cuts risk of exhausting the Greek alphabet roughly in half.

For a general {{math|''n''}}-component object one may write

:<math>{X'}^\alpha = {\Pi(\Lambda)^\alpha}_\beta X^\beta \,, </math>

where {{math|Π}} is the appropriate [[Representation theory of the Lorentz group|representation of the Lorentz group]], an {{math|''n''×''n''}} matrix for every {{math|Λ}}. In this case, the indices should ''not'' be thought of as spacetime indices (sometimes called Lorentz indices), and they run from {{math|1}} to {{math|''n''}}. E.g., if {{mvar|X}} is a [[bispinor]], then the indices are called ''Dirac indices''.

=== Covariant vectors ===
There are also vector quantities with covariant indices. They are generally obtained from their corresponding objects with contravariant indices by the operation of ''lowering an index''; e.g.,

:<math>x_\nu = \eta_{\mu\nu}x^\mu,</math>

where {{math|''η''}} is the [[metric tensor]]. (The linked article also provides more information about what the operation of raising and lowering indices really is mathematically.) The inverse of this transformation is given by

:<math>x^\mu = \eta^{\nu\mu}x_\nu,</math>

where, when viewed as matrices, {{math|''η''''μν''}} is the inverse of {{math|''η''''μν''}}. As it happens, {{math|''η''''μν'' {{=}} {{math|''η''''μν''}}}}. This is referred to as ''raising an index''. To transform a covariant vector {{math|''A''''μ''}}, first raise its index, then transform it according to the same rule as for contravariant {{math|4}}-vectors, then finally lower the index;

:<math>{A'}_\nu = \eta_{\rho\nu} {\Lambda^\rho}_\sigma \eta^{\mu\sigma}A_\mu.</math>

But

:<math>\eta_{\rho\nu} {\Lambda^\rho}_\sigma \eta^{\mu\sigma} = {\left(\Lambda^{-1}\right)^\mu}_\nu,</math>

I. e., it is the {{math|(''μ'', ''ν'')}}-component of the ''inverse'' Lorentz transformation. One defines (as a matter of notation),

:<math>{\Lambda_\nu}^\mu \equiv {\left(\Lambda^{-1}\right)^\mu}_\nu,</math>

and may in this notation write

:<math>{A'}_\nu = {\Lambda_\nu}^\mu A_\mu.</math>

Now for a subtlety. The implied summation on the right hand side of

:<math>{A'}_\nu = {\Lambda_\nu}^\mu A_\mu = {\left(\Lambda^{-1}\right)^\mu}_\nu A_\mu</math>

is running over ''a row index'' of the matrix representing {{math|Λ−1}}. Thus, in terms of matrices, this transformation should be thought of as the ''inverse transpose'' of {{math|Λ}} acting on the column vector {{math|''A''''μ''}}. That is, in pure matrix notation,

:<math>A' = \left(\Lambda^{-1}\right)^\mathrm{T} A.</math>

This means exactly that covariant vectors (thought of as column matrices) transform according to the [[dual representation]] of the standard representation of the Lorentz group. This notion generalizes to general representations, simply replace {{math|Λ}} with {{math|Π(Λ)}}.

=== Tensors ===
If {{mvar|A}} and {{mvar|B}} are linear operators on vector spaces {{mvar|U}} and {{mvar|V}}, then a linear operator {{math|''A'' ⊗ ''B''}} may be defined on the [[tensor product]] of {{mvar|U}} and {{mvar|V}}, denoted {{math|''U'' ⊗ ''V''}} according to<ref>{{harvnb|Hall|2003|loc=Chapter 4}}</ref>

{{Equation box 1
|indent =:
|equation =
:<math>(A \otimes B)(u \otimes v) = Au \otimes Bv, \qquad u \in U, v \in V, u \otimes v \in U \otimes V.</math>               {{EquationRef|(T1)}}
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7
}}

From this it is immediately clear that if {{mvar|u}} and {{mvar|v}} are a four-vectors in {{mvar|V}}, then {{math|''u'' ⊗ ''v'' ∈ ''T''2''V'' ≡ ''V'' ⊗ ''V''}} transforms as

{{Equation box 1
|indent =:
|equation =
:<math>
u \otimes v \rightarrow \Lambda u \otimes \Lambda v =
{\Lambda^\mu}_\nu u^\nu \otimes {\Lambda^\rho}_\sigma v^\sigma =
{\Lambda^\mu}_\nu {\Lambda^\rho}_\sigma u^\nu \otimes v^\sigma \equiv
{\Lambda^\mu}_\nu {\Lambda^\rho}_\sigma w^{\nu\sigma}.
</math>               {{EquationRef|(T2)}}
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7
}}

The second step uses the bilinearity of the tensor product and the last step defines a 2-tensor on component form, or rather, it just renames the tensor {{math|''u'' ⊗ ''v''}}.

These observations generalize in an obvious way to more factors, and using the fact that a general tensor on a vector space {{math|''V''}} can be written as a sum of a coefficient (component!) times tensor products of basis vectors and basis covectors, one arrives at the transformation law for any [[tensor]] quantity {{math|''T''}}. It is given by<ref>{{harvnb|Carroll|2004|page=22}}</ref>

{{Equation box 1
|indent =:
|equation =
:<math>
T^{\alpha'\beta' \cdots \zeta'}_{\theta'\iota' \cdots \kappa'} =
{\Lambda^{\alpha'}}_\mu {\Lambda^{\beta'}}_\nu \cdots {\Lambda^{\zeta'}}_\rho
{\Lambda_{\theta'}}^\sigma {\Lambda_{\iota'}}^\upsilon \cdots {\Lambda_{\kappa'}}^\zeta
T^{\mu\nu \cdots \rho}_{\sigma\upsilon \cdots \zeta},
</math>               {{EquationRef|(T3)}}
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7
}}

where {{math|''Λχ′ψ''}} is defined above. This form can generally be reduced to the form for general {{math|''n''}}-component objects given above with a single matrix ({{math|Π(Λ)}}) operating on column vectors. This latter form is sometimes preferred; e.g., for the electromagnetic field tensor.

==== Transformation of the electromagnetic field ====
[[File:Lorentz boost electric charge.svg|300px|thumb|Lorentz boost of an electric charge, the charge is at rest in one frame or the other.]]
{{main|Electromagnetic tensor}}
{{Further|classical electromagnetism and special relativity}}

Lorentz transformations can also be used to illustrate that the [[magnetic field]] {{math|'''B'''}} and [[electric field]] {{math|'''E'''}} are simply different aspects of the same force — the [[electromagnetic force]], as a consequence of relative motion between [[electric charge]]s and observers.<ref>{{harvnb|Grant|Phillips|2008}}</ref> The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment.<ref>{{harvnb|Griffiths|2007}}</ref>
*An observer measures a charge at rest in frame F. The observer will detect a static electric field. As the charge is stationary in this frame, there is no electric current, so the observer does not observe any magnetic field.
*The other observer in frame F′ moves at velocity {{math|'''v'''}} relative to F and the charge. ''This'' observer sees a different electric field because the charge moves at velocity {{math|−'''v'''}} in their rest frame. The motion of the charge corresponds to an [[electric current]], and thus the observer in frame F′ also sees a magnetic field.

The electric and magnetic fields transform differently from space and time, but exactly the same way as relativistic angular momentum and the boost vector.

The electromagnetic field strength tensor is given by

:<math>
F^{\mu\nu} = \begin{bmatrix}
0 & -\frac{1}{c}E_x & -\frac{1}{c}E_y & -\frac{1}{c}E_z \\
\frac{1}{c}E_x & 0 & -B_z & B_y \\
\frac{1}{c}E_y & B_z & 0 & -B_x \\
\frac{1}{c}E_z & -B_y & B_x & 0
\end{bmatrix} \text{(SI units, signature }(+,-,-,-)\text{)}.
</math>

in [[SI units]]. In relativity, the [[Gaussian units|Gaussian system of units]] is often preferred over SI units, even in texts whose main choice of units is SI units, because in it the electric field {{math|'''E'''}} and the magnetic induction {{math|'''B'''}} have the same units making the appearance of the [[Electromagnetic tensor|electromagnetic field tensor]] more natural.<ref>{{harvnb|Jackson|1999}}</ref> Consider a Lorentz boost in the {{math|''x''}}-direction. It is given by<ref>{{harvnb|Misner|Thorne|Wheeler|1973}}</ref>

:<math>
{\Lambda^\mu}_\nu = \begin{bmatrix}
\gamma & -\gamma\beta & 0 & 0\\
-\gamma\beta & \gamma & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
\end{bmatrix}, \qquad

F^{\mu\nu} = \begin{bmatrix}
0 & E_x & E_y & E_z \\
-E_x & 0 & B_z & -B_y \\
-E_y & -B_z & 0 & B_x \\
-E_z & B_y & -B_x & 0
\end{bmatrix} \text{(Gaussian units, signature }(-,+,+,+)\text{)},
</math>

where the field tensor is displayed side by side for easiest possible reference in the manipulations below.

The general transformation law {{EquationNote|(T3)}} becomes

:<math>F^{\mu'\nu'} = {\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu}.</math>

For the magnetic field one obtains
:<math>\begin{align}
B_{x'} &= F^{2'3'}
= {\Lambda^2}_\mu {\Lambda^3}_\nu F^{\mu\nu}
= {\Lambda^2}_2 {\Lambda^3}_3 F^{23}
= 1 \times 1 \times B_x \\
&= B_x, \\
B_{y'} &= F^{3'1'}
= {\Lambda^3}_\mu {\Lambda^1}_\nu F^{\mu \nu}
= {\Lambda^3}_3 {\Lambda^1}_\nu F^{3\nu}
= {\Lambda^3}_3 {\Lambda^1}_0 F^{30} + {\Lambda^3}_3 {\Lambda^1}_1 F^{31} \\
&= 1 \times (-\beta\gamma) (-E_z) + 1 \times \gamma B_y
= \gamma B_y + \beta\gamma E_z \\
&= \gamma\left(\mathbf{B} - \boldsymbol{\beta} \times \mathbf{E}\right)_y \\
B_{z'} &= F^{1'2'}
= {\Lambda^1}_\mu {\Lambda^2}_\nu F^{\mu\nu}
= {\Lambda^1}_\mu {\Lambda^2}_2 F^{\mu 2}
= {\Lambda^1}_0 {\Lambda^2}_2 F^{02} + {\Lambda^1}_1 {\Lambda^2}_2 F^{12} \\
&= (-\gamma\beta) \times 1\times E_y + \gamma \times 1 \times B_z
= \gamma B_z - \beta\gamma E_y \\
&= \gamma\left(\mathbf{B} - \boldsymbol{\beta} \times \mathbf{E}\right)_z
\end{align}</math>

For the electric field results

:<math>\begin{align}
E_{x'} &= F^{0'1'}
= {\Lambda^0}_\mu {\Lambda^1}_\nu F^{\mu\nu}
= {\Lambda^0}_1 {\Lambda^1}_0 F^{10} + {\Lambda^0}_0 {\Lambda^1}_1 F^{01} \\
&= (-\gamma\beta)(-\gamma\beta)(-E_x) + \gamma\gamma E_x
= -\gamma^2\beta^2(E_x) + \gamma^2 E_x
= E_x(1 - \beta^2)\gamma^2 \\
&= E_x, \\
E_{y'} &= F^{0'2'}
= {\Lambda^0}_\mu {\Lambda^2}_\nu F^{\mu\nu}
= {\Lambda^0}_\mu {\Lambda^2}_2 F^{\mu 2}
= {\Lambda^0}_0 {\Lambda^2}_2 F^{02} + {\Lambda^0}_1 {\Lambda^2}_2 F^{12} \\
&= \gamma \times 1 \times E_y + (-\beta\gamma) \times 1 \times B_z
= \gamma E_y - \beta\gamma B_z \\
&= \gamma\left(\mathbf{E} + \boldsymbol{\beta} \times \mathbf{B}\right)_y \\
E_{z'} &= F^{0'3'}
= {\Lambda^0}_\mu {\Lambda^3}_\nu F^{\mu\nu}
= {\Lambda^0}_\mu {\Lambda^3}_3 F^{\mu 3}
= {\Lambda^0}_0 {\Lambda^3}_3 F^{03} + {\Lambda^0}_1 {\Lambda^3}_3 F^{13} \\
&= \gamma \times 1 \times E_z - \beta\gamma \times 1 \times (-B_y)
= \gamma E_z + \beta\gamma B_y \\
&= \gamma\left(\mathbf{E} + \boldsymbol{\beta} \times \mathbf{B}\right)_z.
\end{align}</math>

Here, {{math|'''''β''''' {{=}} (''β'', 0, 0)}} is used. These results can be summarized by

:<math>\begin{align}
\mathbf{E}_{\parallel'} &= \mathbf{E}_\parallel \\
\mathbf{B}_{\parallel'} &= \mathbf{B}_\parallel \\
\mathbf{E}_{\bot'} &= \gamma \left( \mathbf{E}_\bot + \boldsymbol{\beta} \times \mathbf{B}_\bot \right) = \gamma \left( \mathbf{E} + \boldsymbol{\beta} \times \mathbf{B} \right)_\bot,\\
\mathbf{B}_{\bot'} &= \gamma \left( \mathbf{B}_\bot - \boldsymbol{\beta} \times \mathbf{E}_\bot \right) = \gamma \left( \mathbf{B} - \boldsymbol{\beta} \times \mathbf{E} \right)_\bot,
\end{align}</math>

and are independent of the metric signature. For SI units, substitute {{math|''E'' → {{frac|''E''|''c''}}}}. {{harvtxt|Misner|Thorne|Wheeler|1973}} refer to this last form as the {{math|3 + 1}} view as opposed to the ''geometric view'' represented by the tensor expression
:<math>F^{\mu'\nu'} = {\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu},</math>
and make a strong point of the ease with which results that are difficult to achieve using the {{math|3 + 1}} view can be obtained and understood. Only objects that have well defined Lorentz transformation properties (in fact under ''any'' smooth coordinate transformation) are geometric objects. In the geometric view, the electromagnetic field is a six-dimensional geometric object in ''spacetime'' as opposed to two interdependent, but separate, 3-vector fields in ''space'' and ''time''. The fields {{math|'''E'''}} (alone) and {{math|'''B'''}} (alone) do not have well defined Lorentz transformation properties. The mathematical underpinnings are equations {{EquationNote|(T1)}} and {{EquationNote|(T2)}} that immediately yield {{EquationNote|(T3)}}. One should note that the primed and unprimed tensors refer to the ''same event in spacetime''. Thus the complete equation with spacetime dependence is
:<math>
F^{\mu' \nu'}\left(x'\right) =
{\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu}\left(\Lambda^{-1} x'\right) =
{\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu}(x).
</math>

Length contraction has an effect on [[charge density]] {{math|''ρ''}} and [[current density]] {{math|'''J'''}}, and time dilation has an effect on the rate of flow of charge (current), so charge and current distributions must transform in a related way under a boost. It turns out they transform exactly like the space-time and energy-momentum four-vectors,
:<math>\begin{align}
\mathbf{j}' &= \mathbf{j} - \gamma\rho v\mathbf{n} + \left( \gamma - 1 \right)(\mathbf{j} \cdot \mathbf{n})\mathbf{n} \\
\rho' &= \gamma \left(\rho - \mathbf{j} \cdot \frac{v\mathbf{n}}{c^2}\right),
\end{align}</math>

or, in the simpler geometric view,
:<math>j^{\mu^\prime} = {\Lambda^{\mu'}}_\mu j^\mu.</math>

One says that charge density transforms as the time component of a four-vector. It is a rotational scalar. The current density is a 3-vector.

The [[Maxwell equations]] are invariant under Lorentz transformations.

=== Spinors ===
Equation {{EquationNote|(T1)}} hold unmodified for any representation of the Lorentz group, including the [[bispinor]] representation. In {{EquationNote|(T2)}} one simply replaces all occurrences of {{math|Λ}} by the bispinor representation {{math|Π(Λ)}},

{{Equation box 1
|indent =:
|equation =
:<math>u \otimes v \rightarrow \Pi(\Lambda) u \otimes \Pi(\Lambda) v =
{\Pi(\Lambda)^\alpha}_\beta u^\beta \otimes {\Pi(\Lambda)^\rho}_\sigma v^\sigma =
{\Pi(\Lambda)^\alpha}_\beta {\Pi(\Lambda)^\rho}_\sigma u^\beta \otimes v^\sigma \equiv
{\Pi(\Lambda)^\alpha}_\beta {\Pi(\Lambda)^\rho}_\sigma w^{\alpha\beta}.
</math>               {{EquationRef|(T4)}}
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7
}}

The above equation could, for instance, be the transformation of a state in [[Fock space]] describing two free electrons.

==== Transformation of general fields ====
A general ''noninteracting'' multi-particle state (Fock space state) in [[quantum field theory]] transforms according to the rule<ref>{{harvnb|Weinberg|2002|loc=Chapter 3}}</ref>

{{NumBlk|:
|<math>\begin{align}
&U(\Lambda, a) \Psi_{p_1\sigma_1 n_1; p_2\sigma_2 n_2; \cdots} \\
= {} &e^{-ia_\mu \left[(\Lambda p_1)^\mu + (\Lambda p_2)^\mu + \cdots\right]}
\sqrt{\frac{(\Lambda p_1)^0(\Lambda p_2)^0\cdots}{p_1^0 p_2^0 \cdots}}
\left( \sum_{\sigma_1'\sigma_2' \cdots} D_{\sigma_1'\sigma_1}^{(j_1)}\left[W(\Lambda, p_1)\right] D_{\sigma_2'\sigma_2}^{(j_2)}\left[W(\Lambda, p_2)\right] \cdots \right)
\Psi_{\Lambda p_1 \sigma_1' n_1; \Lambda p_2 \sigma_2' n_2; \cdots},
\end{align}</math>
| {{EquationRef|1}}
}}
where {{math|''W''(Λ, ''p'')}} is the [[Wigner rotation]] and {{math|''D''(''j'')}} is the {{nowrap|{{math|(2''j'' + 1)}}-dimensional}} representation of {{math|SO(3)}}.

==See also==

{{columns-list|3|

*[[Ricci calculus]]
*[[Electromagnetic field]]
*[[Galilean transformation]]
*[[Hyperbolic rotation]]
*[[Invariance mechanics]]
*[[Lorentz group]]
*[[Representation theory of the Lorentz group]]
*[[Principle of relativity]]
*[[Velocity-addition formula]]
*[[Algebra of physical space]]
*[[Relativistic aberration]]
*[[Prandtl–Glauert transformation]]
*[[Split-complex number]]
*[[Gyrovector space]]

}}

==Footnotes==

{{reflist|group=nb}}

==Notes==

{{reflist|30em}}

==References==

===Websites===

*{{citation |first1 = John J. |last1 = O'Connor |first2 = Edmund F. |last2 = Robertson|title = A History of Special Relativity|url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Special_relativity.html|year=1996}}
*{{citation |first = Harvey R. |last = Brown
|title = Michelson, FitzGerald and Lorentz: the Origins of Relativity Revisited
|url = http://philsci-archive.pitt.edu/id/eprint/987|year=2003}}

===Papers===

{{refbegin|3}}

*{{cite journal | first = J. T. | last = Cushing | title = Vector Lorentz transformations | journal = [[American Journal of Physics]] | year = 1967 |volume = 35 | doi = 10.1119/1.1974267 | pages = 858–862 |url=https://www.deepdyve.com/browse/journals/the-american-journal-of-physics/1967/v35/i9?page=4|bibcode = 1967AmJPh..35..858C }}
*{{cite journal | first = A. J. | last = Macfarlane| title = On the Restricted Lorentz Group and Groups Homomorphically Related to It | journal = [[Journal of Mathematical Physics]] | year = 1962 | volume = 3| issue=6 | pages = 1116–1129| doi=10.1063/1.1703854 | bibcode=1962JMP.....3.1116M}}
*{{citation |first = Tony |last = Rothman
|title = Lost in Einstein's Shadow
|url = http://www.americanscientist.org/libraries/documents/200622102452_866.pdf
|journal = American Scientist |volume = 94 |issue = 2 |pages = 112f. |year = 2006}}

*{{Citation
|author=Darrigol, Olivier
|title=The Genesis of the theory of relativity
|year=2005
|journal=Séminaire Poincaré
|volume=1
|pages=1–22
|url=http://www.bourbaphy.fr/darrigol2.pdf
|doi=10.1007/3-7643-7436-5_1}}

*{{citation |first = Michael N. |last = Macrossan
|title = A Note on Relativity Before Einstein
|url = http://espace.library.uq.edu.au/view.php?pid=UQ:9560
|journal=Brit. Journal Philos. Science |volume = 37 |year=1986 |pages= 232–34 |doi=10.1093/bjps/37.2.232|citeseerx=10.1.1.679.5898}}

*{{citation |first = Henri |last = Poincaré |author-link = Henri Poincaré
|title = [[s:Translation:On the Dynamics of the Electron (June)|On the Dynamics of the Electron]]
|journal = Comptes rendus hebdomadaires des séances de l'Académie des sciences
|volume = 140 |pages = 1504–1508 |year = 1905}}

*{{Citation
|author=Einstein, Albert
|year=1905
|title=Zur Elektrodynamik bewegter Körper
|journal=Annalen der Physik
|volume=322
|issue=10
|pages=891–921
|url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf
|doi=10.1002/andp.19053221004|bibcode = 1905AnP...322..891E }}. See also: [http://www.fourmilab.ch/etexts/einstein/specrel/ English translation].

*{{Cite web|last=Einstein |first=A. |year=1916 |author-link=Albert Einstein|title=Relativity: The Special and General Theory |format=PDF |pages= |url=https://archive.org/stream/cu31924011804774#page/n35/mode/2up |accessdate=2012-01-23}}
*{{cite journal|title=Thomas rotation and the parameterization of the Lorentz transformation group|first=A. A. |last=Ungar|journal=Foundations of Physics Letters|year=1988|volume=1|issue=1|pages=55–89|doi=10.1007/BF00661317|publisher=Kluwer Academic Publishers-Plenum Publishers|issn=0894-9875|subscription=yes|url=http://link.springer.com/article/10.1007/BF00661317|bibcode=1988FoPhL...1...57U}} eqn (55).
*{{cite journal | first = A. A. | last = Ungar | url = http://www.springerlink.com/content/g157304vh4434413/ | title = The relativistic velocity composition paradox and the Thomas rotation| journal = [[Foundations of Physics]] | volume = 19 | pages = 1385–1396 | year = 1989 |bibcode = 1989FoPh...19.1385U |doi = 10.1007/BF00732759 }}
*{{cite journal | first = A. A. | last = Ungar | title = The relativistic composite-velocity reciprocity principle | citeseerx = 10.1.1.35.1131 | journal = [[Foundations of Physics]] | year = 2000 | publisher = Springer | volume = 30 | issue = 2 | pages = 331–342 }}
*{{cite journal | first = C. I. | last = Mocanu | title = Some difficulties within the framework of relativistic electrodynamics| journal = Archiv für Elektrotechnik | year = 1986 | publisher = Springer | volume = 69 | pages = 97–110 | doi=10.1007/bf01574845}}
*{{cite journal | first = C. I. | last = Mocanu | title = On the relativistic velocity composition paradox and the Thomas rotation| journal = [[Foundations of Physics]] | year = 1992 | publisher = Plenum | volume = 5 | pages = 443–456 | doi=10.1007/bf00690425|bibcode = 1992FoPhL...5..443M }}
*{{cite book|ref=harv|last=Weinberg|first=S.|year=2002|title=The Quantum Theory of Fields, vol I|isbn=0-521-55001-7|authorlink=Steven Weinberg|publisher=[[Cambridge University Press]]}}

{{refend}}

===Books===

{{refbegin|3}}

*{{cite book|title=University Physics – With Modern Physics|edition=12th|first1=H. D.|last1=Young|first2=R. A.|last2=Freedman|year=2008|isbn=0-321-50130-6}}
*{{cite book|title=3000 Solved Problems in Physics|series=Schaum Series|first=A.|last=Halpern|publisher=Mc Graw Hill|year=1988|isbn=978-0-07-025734-4|page=688}}
*{{cite book|title=Dynamics and Relativity|first1=J. R.|last1=Forshaw|first2=A. G.|last2=Smith|series=Manchester Physics Series|publisher=John Wiley & Sons Ltd|year=2009|isbn=978-0-470-01460-8|pages=124–126}}
*{{cite book|title=Spacetime Physics|first1=J. A.|last1=Wheeler|first2=E. F|last2=Taylor|author-link1=John Archibald Wheeler|author-link2=Edwin F. Taylor|year=1971|publisher=Freeman|ISBN=0-7167-0336-X}}
*{{cite book|title=Gravitation|first1=J. A.|last1=Wheeler|first2=K. S.|last2=Thorne|first3=C. W. |last3=Misner|author-link1=John Archibald Wheeler|author-link2=Kip Thorne|author-link3=Charles W. Misner|year=1973|publisher=Freeman|ISBN=0-7167-0344-0}}
*{{cite book|title=Spacetime and Geometry: An Introduction to General Relativity |edition=illustrated |first1=S. M. |last1=Carroll |author-link=Sean M. Carroll|publisher=Addison Wesley |year=2004 |isbn=0-8053-8732-3 |page=22 |url=https://books.google.com/books?id=1SKFQgAACAAJ}}
*{{cite book|title=Electromagnetism |edition=2nd |first1=I. S.|last1=Grant|first2=W. R.|last2=Phillips|series=Manchester Physics|publisher=John Wiley & Sons|year=2008|isbn=0-471-92712-0|chapter=14}}
*{{cite book|title=Introduction to Electrodynamics|edition=3rd|first1= D. J.|last1=Griffiths|author-link=David Griffiths (physicist)|publisher=Pearson Education, Dorling Kindersley,|year=2007|isbn=81-7758-293-3}}
*{{cite book|ref=harv|year=2003|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations An Elementary Introduction|publisher=[[Springer Publishing]]|isbn=0-387-40122-9}}
*{{citation |first = S. |last = Weinberg |title = Cosmology |author-link = Steven Weinberg |publisher = Wiley |year = 2008|isbn = 978-0-19-852682-7}}
*{{citation |first = S. |last = Weinberg |author-link = Steven Weinberg |title = The quantum theory of fields (3 vol.)|publisher = Cambridge University Press | year = 2005 |isbn = 978-0-521-67053-1|volume=1}}
*{{citation |first = T. |last = Ohlsson |author-link = Tommy Ohlsson |title = Relativistic Quantum Physics|publisher = Cambridge University Press | year = 2011 |isbn = 978-0-521-76726-2}}
*{{cite book|ref=harv|last=Goldstein|first=H.|author-link=Herbert Goldstein|title=Classical Mechanics|edition=2nd|publisher=[[Addison-Wesley Publishing Company|Addison-Wesley]]|location=Reading MA|isbn=0-201-02918-9|year=1980|orig-year=1950}}
*{{cite book|ref=harv|first=J. D.|last=Jackson|author-link=John David Jackson (physicist)|title=Classical Electrodynamics|pages=542–545|edition=2nd|year=1975|orig-year=1962|isbn=0-471-43132-X|publisher=[[John Wiley & Sons]]|chapter=Chapter 11}}
*{{cite book|ref=harv|last1=Landau|first1=L. D.|author-link1=Lev Landau|last2=Lifshitz|first2=E. M.|author-link2=Evgeny Lifshitz|title=The Classical Theory of Fields|series=[[Course of Theoretical Physics]]|volume=2|edition=4th|publisher=[[Butterworth–Heinemann]]|isbn=0 7506 2768 9|year=2002|orig-year=1939|pages=9–12}}
*{{cite book|ref=harv|last1=Feynman|first1=R. P.|author-link1=Richard Feynman|last2=Leighton|first2= R. B.|author-link2=Robert B. Leighton|last3=Sands|first3=M. |author-link3=Matthew Sands|title=The Feynman Lectures on Physics|volume=1|publisher=Addison Wesley|isbn=0-201-02117-X|year=1977|orig-year=1963|chapter=15}}
*{{cite book|ref=harv|last1=Feynman|first1=R. P.|author-link1=Richard Feynman|last2=Leighton|first2= R. B.|author-link2=Robert B. Leighton|last3=Sands|first3=M. |author-link3=Matthew Sands|title=The Feynman Lectures on Physics|volume=2|publisher=Addison Wesley|isbn=0-201-02117-X|year=1977|orig-year=1964|chapter=13}}
*{{cite book|ref=harv|last1=Misner |first1=Charles W. |authorlink1=Charles W. Misner|last2=Thorne |first2=Kip S. |authorlink2=Kip Thorne|last3=Wheeler |first3=John Archibald |authorlink3=John Archibald Wheeler|year=1973|title=Gravitation|publisher=[[W. H. Freeman]]|location=San Francisco|isbn=978-0-7167-0344-0}}
*{{cite book|ref=harv|first=W.|last=Rindler|author-link=Wolfgang Rindler|year=2006|orig-year=2001|title=Relativity Special, General and Cosmological|chapter = Chapter 9|edition=2nd|publisher=[[Oxford University Press]]|location=Dallas|isbn=978-0-19-856732-5}}
*{{cite book|ref=harv|first=L. H.|last=Ryder|author-link=Lewis Ryder|title=Quantum Field Theory|year=1996|orig-year=1985|isbn=978-0521478144|edition=2nd|publisher=[[Cambridge University Press]]|location=Cambridge}}
*{{cite book|ref=harv|last=Sard|first=R. D.|title=Relativistic Mechanics - Special Relativity and Classical Particle Dynamics|year=1970|publisher=W. A. Benjamin|location=New York|isbn=978-0805384918}}
*{{cite book|ref=harv|author=R. U. Sexl, H. K. Urbantke|title=Relativity, Groups Particles. Special Relativity and Relativistic Symmetry in Field and Particle Physics|year=2001|orig-year=1992|publisher=Springer|isbn=978-3211834435|url=https://books.google.co.uk/books?id=iyj0CAAAQBAJ&printsec=frontcover&dq=sexl+relativity&hl=en&sa=X&ved=0CB8Q6wEwAGoVChMIo7itmMnnxgIVCz4UCh3rVQR1#v=onepage&q=sexl%20relativity&f=false}}
*{{cite book|ref=harv|last=Gourgoulhon|first=Eric|title=Special Relativity in General Frames: From Particles to Astrophysics|year=2013|publisher=Springer|isbn=978-3-642-37276-6|page=213|url=https://books.google.co.uk/books?id=N4HBBAAAQBAJ&pg=PA215&lpg=PA215&dq=mcfarlane+1962+lorentz+transformation&source=bl&ots=KGCT4Mojmz&sig=F6Ed6sgSf1dczXzR13XtA0U1RT0&hl=en&sa=X&ved=0CCQQ6AEwAGoVChMI7Y3Qu92GyQIVBWsUCh25ewBl#v=onepage&q=mcfarlane%201962%20lorentz%20transformation&f=false}}
*{{cite book|ref=harv|last1=Chaichian|first1=Masud|last2=Hagedorn|first2=Rolf|title=Symmetry in quantum mechanics:From angular momentum to supersymmetry|year=1997|publisher=IoP|isbn=0-7503-0408-1|page=239|url=https://books.google.co.uk/books?id=pEhjQgAACAAJ&dq=Symmetry+in+quantum+mechanics&hl=en&sa=X&redir_esc=y}}
*{{cite book|ref=harv|last1=Landau|first1=L.D.|authorlink1=Lev Landau|last2=Lifshitz|first2=E.M.|authorlink2=Evgeny Lifshitz|title=The Classical Theory of Fields|series=Course of Theoretical Physics|volume=2|edition=4th|publisher=[[Butterworth–Heinemann]]|isbn=0 7506 2768 9|year=2002|orig-year=1939}}
{{refend}}

==Further reading==
*{{Citation |first = Albert |last = Einstein |author-link = Albert Einstein |title = Relativity: The Special and the General Theory |place = New York|url = http://www.marxists.org/reference/archive/einstein/works/1910s/relative/ | publisher = Three Rivers Press|year = 1961|publication-date = 1995|isbn = 0-517-88441-0}}
*{{Citation
|first1=A.
|last1=Ernst
|first2=J.-P.
|last2=Hsu
|title=First proposal of the universal speed of light by Voigt 1887
|journal=Chinese Journal of Physics
|volume=39
|issue=3
|url=http://psroc.phys.ntu.edu.tw/cjp/v39/211.pdf
|pages=211–230
|year=2001
|bibcode=2001ChJPh..39..211E
|deadurl=yes
|archiveurl=https://web.archive.org/web/20110716083015/http://psroc.phys.ntu.edu.tw/cjp/v39/211.pdf
|archivedate=2011-07-16
|df=
}}
*{{Citation |first1 = Stephen T. |last1 = Thornton |first2 = Jerry B. |last2 = Marion |title = Classical dynamics of particles and systems |edition = 5th |place = Belmont, [CA.] |publisher = Brooks/Cole |year = 2004 |pages = 546–579 |isbn = 0-534-40896-6}}
*{{Citation |first = Woldemar |last = Voigt |author-link = Woldemar Voigt |title = Über das Doppler'sche princip |journal = Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen |volume = 2 |pages = 41–51 |year = 1887}}

==External links==
{{Wikisource portal|Relativity}}
{{wikibooks|special relativity}}
*[http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf Derivation of the Lorentz transformations]. This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.
*[http://casa.colorado.edu/~ajsh/sr/paradox.html The Paradox of Special Relativity]. This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.
*[http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html Relativity] – a chapter from an online textbook
*[http://www.adamauton.com/warp/ Warp Special Relativity Simulator]. A computer program demonstrating the Lorentz transformations on everyday objects.
*{{YouTube|C2VMO7pcWhg|Animation clip}} visualizing the Lorentz transformation.
*[http://math.ucr.edu/~jdp/Relativity/Lorentz_Frames.html Lorentz Frames Animated] ''from John de Pillis.'' Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, ''etc''.

[[Category:Special relativity]]
[[Category:Theoretical physics]]
[[Category:Mathematical physics]]
[[Category:Spacetime]]
[[Category:Coordinate systems]]
[[Category:Hendrik Lorentz]]

Module:Coordinates

2017-07-19T22:24:32Z

NickPercival: 1 revision imported

--[[
This module is intended to replace the functionality of {{Coord}} and related
templates. It provides several methods, including

{{#invoke:Coordinates | coord }} : General function formatting and displaying
coordinate values.

{{#invoke:Coordinates | dec2dms }} : Simple function for converting decimal
degree values to DMS format.

{{#invoke:Coordinates | dms2dec }} : Simple function for converting DMS format
to decimal degree format.

{{#invoke:Coordinates | link }} : Export the link used to reach the tools

]]

require('Module:No globals')

local math_mod = require("Module:Math")
local coordinates = {};

local current_page = mw.title.getCurrentTitle()
local page_name = mw.uri.encode( current_page.prefixedText, 'WIKI' );
local coord_link = '//tools.wmflabs.org/geohack/geohack.php?pagename=' .. page_name .. '&params='

--[[ Helper function, replacement for {{coord/display/title}} ]]
local function displaytitle(s, notes)
local l = "[[Geographic coordinate system|Coordinates]]: " .. s
local co = '' .. l .. notes .. '';
return '' .. co .. '';
end

--[[ Helper function, Replacement for {{coord/display/inline}} ]]
local function displayinline(s, notes)
return s .. notes
end

--[[ Helper function, used in detecting DMS formatting ]]
local function dmsTest(first, second)
if type(first) ~= 'string' or type(second) ~= 'string' then
return nil
end
local s = (first .. second):upper()
return s:find('^[NS][EW]$') or s:find('^[EW][NS]$')
end

--[[ Wrapper function to grab args, see Module:Arguments for this function's documentation. ]]
local function makeInvokeFunc(funcName)
return function (frame)
local args = require('Module:Arguments').getArgs(frame, {
wrappers = 'Template:Coord'
})
return coordinates[funcName](args, frame)
end
end

--[[ Helper function, handle optional args. ]]
local function optionalArg(arg, supplement)
return arg and arg .. supplement or ''
end

--[[
Formats any error messages generated for display
]]
local function errorPrinter(errors)
local result = ""
for i,v in ipairs(errors) do
local errorHTML = 'Coordinates: ' .. v[2] .. ''
result = result .. errorHTML .. " "
end
return result
end

--[[
Determine the required CSS class to display coordinates

Usually geo-nondefault is hidden by CSS, unless a user has overridden this for himself
default is the mode as specificied by the user when calling the {{coord}} template
mode is the display mode (dec or dms) that we will need to determine the css class for
]]
local function displayDefault(default, mode)
if default == "" then
default = "dec"
end

if default == mode then
return "geo-default"
else
return "geo-nondefault"
end
end

--[[
specPrinter

Output formatter. Takes the structure generated by either parseDec
or parseDMS and formats it for inclusion on Wikipedia.
]]
local function specPrinter(args, coordinateSpec)
local uriComponents = coordinateSpec["param"]
if uriComponents == "" then
-- RETURN error, should never be empty or nil
return "ERROR param was empty"
end
if args["name"] then
uriComponents = uriComponents .. "&title=" .. mw.uri.encode(coordinateSpec["name"])
end

local geodmshtml = ''
.. '' .. coordinateSpec["dms-lat"] .. ' '
.. '' ..coordinateSpec["dms-long"] .. ''
.. ''

local lat = tonumber( coordinateSpec["dec-lat"] ) or 0
local geodeclat
if lat < 0 then
-- FIXME this breaks the pre-existing precision
geodeclat = tostring(coordinateSpec["dec-lat"]):sub(2) .. "°S"
else
geodeclat = (coordinateSpec["dec-lat"] or 0) .. "°N"
end

local long = tonumber( coordinateSpec["dec-long"] ) or 0
local geodeclong
if long < 0 then
-- FIXME does not handle unicode minus
geodeclong = tostring(coordinateSpec["dec-long"]):sub(2) .. "°W"
else
geodeclong = (coordinateSpec["dec-long"] or 0) .. "°E"
end

local geodechtml = ''
.. geodeclat .. ' '
.. geodeclong
.. ''

local geonumhtml = ''
.. coordinateSpec["dec-lat"] .. '; '
.. coordinateSpec["dec-long"]
.. ''

local inner = '' .. geodmshtml .. ''
.. ' / '
.. '';

if not args["name"] then
inner = inner .. geodechtml
.. ' / ' .. geonumhtml .. ''
else
inner = inner .. '' .. geodechtml
.. ' / ' .. geonumhtml .. ''
.. ' ('
.. args["name"] .. ')'
end

return '' ..
'[' .. coord_link .. uriComponents .. ' ' .. inner .. ']' .. ''
end

--[[ Helper function, convert decimal to degrees ]]
local function convert_dec2dms_d(coordinate)
local d = math_mod._round( coordinate, 0 ) .. "°"
return d .. ""
end

--[[ Helper function, convert decimal to degrees and minutes ]]
local function convert_dec2dms_dm(coordinate)
coordinate = math_mod._round( coordinate * 60, 0 );
local m = coordinate % 60;
coordinate = math.floor( (coordinate - m) / 60 );
local d = coordinate % 360 .."°"

return d .. string.format( "%02d′", m )
end

--[[ Helper function, convert decimal to degrees, minutes, and seconds ]]
local function convert_dec2dms_dms(coordinate)
coordinate = math_mod._round( coordinate * 60 * 60, 0 );
local s = coordinate % 60
coordinate = math.floor( (coordinate - s) / 60 );
local m = coordinate % 60
coordinate = math.floor( (coordinate - m) / 60 );
local d = coordinate % 360 .."°"

return d .. string.format( "%02d′", m ) .. string.format( "%02d″", s )
end

--[[
Helper function, convert decimal latitude or longitude to
degrees, minutes, and seconds format based on the specified precision.
]]
local function convert_dec2dms(coordinate, firstPostfix, secondPostfix, precision)
local coord = tonumber(coordinate)
local postfix
if coord >= 0 then
postfix = firstPostfix
else
postfix = secondPostfix
end

precision = precision:lower();
if precision == "dms" then
return convert_dec2dms_dms( math.abs( coord ) ) .. postfix;
elseif precision == "dm" then
return convert_dec2dms_dm( math.abs( coord ) ) .. postfix;
elseif precision == "d" then
return convert_dec2dms_d( math.abs( coord ) ) .. postfix;
end
end

--[[
Convert DMS format into a N or E decimal coordinate
]]
local function convert_dms2dec(direction, degrees_str, minutes_str, seconds_str)
local degrees = tonumber(degrees_str)
local minutes = tonumber(minutes_str) or 0
local seconds = tonumber(seconds_str) or 0

local factor = 1
if direction == "S" or direction == "W" then
factor = -1
end

local precision = 0
if seconds_str then
precision = 5 + math.max( math_mod._precision(seconds_str), 0 );
elseif minutes_str and minutes_str ~= '' then
precision = 3 + math.max( math_mod._precision(minutes_str), 0 );
else
precision = math.max( math_mod._precision(degrees_str), 0 );
end

local decimal = factor * (degrees+(minutes+seconds/60)/60)
return string.format( "%." .. precision .. "f", decimal ) -- not tonumber since this whole thing is string based.
end

--[[
Checks input values to for out of range errors.
]]
local function validate( lat_d, lat_m, lat_s, long_d, long_m, long_s, source, strong )
local errors = {};
lat_d = tonumber( lat_d ) or 0;
lat_m = tonumber( lat_m ) or 0;
lat_s = tonumber( lat_s ) or 0;
long_d = tonumber( long_d ) or 0;
long_m = tonumber( long_m ) or 0;
long_s = tonumber( long_s ) or 0;

if strong then
if lat_d < 0 then
table.insert(errors, {source, "latitude degrees < 0 with hemisphere flag"})
end
if long_d < 0 then
table.insert(errors, {source, "longitude degrees < 0 with hemisphere flag"})
end
--[[
#coordinates is inconsistent about whether this is an error. If globe: is
specified, it won't error on this condition, but otherwise it will.

For not simply disable this check.

if long_d > 180 then
table.insert(errors, {source, "longitude degrees > 180 with hemisphere flag"})
end
]]
end

if lat_d > 90 then
table.insert(errors, {source, "latitude degrees > 90"})
end
if lat_d < -90 then
table.insert(errors, {source, "latitude degrees < -90"})
end
if lat_m >= 60 then
table.insert(errors, {source, "latitude minutes >= 60"})
end
if lat_m < 0 then
table.insert(errors, {source, "latitude minutes < 0"})
end
if lat_s >= 60 then
table.insert(errors, {source, "latitude seconds >= 60"})
end
if lat_s < 0 then
table.insert(errors, {source, "latitude seconds < 0"})
end
if long_d >= 360 then
table.insert(errors, {source, "longitude degrees >= 360"})
end
if long_d <= -360 then
table.insert(errors, {source, "longitude degrees <= -360"})
end
if long_m >= 60 then
table.insert(errors, {source, "longitude minutes >= 60"})
end
if long_m < 0 then
table.insert(errors, {source, "longitude minutes < 0"})
end
if long_s >= 60 then
table.insert(errors, {source, "longitude seconds >= 60"})
end
if long_s < 0 then
table.insert(errors, {source, "longitude seconds < 0"})
end

return errors;
end

--[[
parseDec

Transforms decimal format latitude and longitude into the
structure to be used in displaying coordinates
]]
local function parseDec( lat, long, format )
local coordinateSpec = {}
local errors = {}

if not long then
return nil, {{"parseDec", "Missing longitude"}}
elseif not tonumber(long) then
return nil, {{"parseDec", "Longitude could not be parsed as a number: " .. long}}
end

errors = validate( lat, nil, nil, long, nil, nil, 'parseDec', false );
coordinateSpec["dec-lat"] = lat;
coordinateSpec["dec-long"] = long;

local mode = coordinates.determineMode( lat, long );
coordinateSpec["dms-lat"] = convert_dec2dms( lat, "N", "S", mode) -- {{coord/dec2dms|{{{1}}}|N|S|{{coord/prec dec|{{{1}}}|{{{2}}}}}}}
coordinateSpec["dms-long"] = convert_dec2dms( long, "E", "W", mode) -- {{coord/dec2dms|{{{2}}}|E|W|{{coord/prec dec|{{{1}}}|{{{2}}}}}}}

if format then
coordinateSpec.default = format
else
coordinateSpec.default = "dec"
end

return coordinateSpec, errors
end

--[[
parseDMS

Transforms degrees, minutes, seconds format latitude and longitude
into the a structure to be used in displaying coordinates
]]
local function parseDMS( lat_d, lat_m, lat_s, lat_f, long_d, long_m, long_s, long_f, format )
local coordinateSpec, errors, backward = {}, {}

lat_f = lat_f:upper();
long_f = long_f:upper();

-- Check if specified backward
if lat_f == 'E' or lat_f == 'W' then
lat_d, long_d, lat_m, long_m, lat_s, long_s, lat_f, long_f, backward = long_d, lat_d, long_m, lat_m, long_s, lat_s, long_f, lat_f, true;
end

errors = validate( lat_d, lat_m, lat_s, long_d, long_m, long_s, 'parseDMS', true );
if not long_d then
return nil, {{"parseDMS", "Missing longitude" }}
elseif not tonumber(long_d) then
return nil, {{"parseDMS", "Longitude could not be parsed as a number:" .. long_d }}
end

if not lat_m and not lat_s and not long_m and not long_s and #errors == 0 then
if math_mod._precision( lat_d ) > 0 or math_mod._precision( long_d ) > 0 then
if lat_f:upper() == 'S' then
lat_d = '-' .. lat_d;
end
if long_f:upper() == 'W' then
long_d = '-' .. long_d;
end

return parseDec( lat_d, long_d, format );
end
end

coordinateSpec["dms-lat"] = lat_d.."°"..optionalArg(lat_m,"′") .. optionalArg(lat_s,"″") .. lat_f
coordinateSpec["dms-long"] = long_d.."°"..optionalArg(long_m,"′") .. optionalArg(long_s,"″") .. long_f
coordinateSpec["dec-lat"] = convert_dms2dec(lat_f, lat_d, lat_m, lat_s) -- {{coord/dms2dec|{{{4}}}|{{{1}}}|0{{{2}}}|0{{{3}}}}}
coordinateSpec["dec-long"] = convert_dms2dec(long_f, long_d, long_m, long_s) -- {{coord/dms2dec|{{{8}}}|{{{5}}}|0{{{6}}}|0{{{7}}}}}

if format then
coordinateSpec.default = format
else
coordinateSpec.default = "dms"
end

return coordinateSpec, errors, backward
end

--[[
Check the input arguments for coord to determine the kind of data being provided
and then make the necessary processing.
]]
local function formatTest(args)
local result, errors
local backward, primary = false, false

local function getParam(args, lim)
local ret = {}
for i = 1, lim do
ret[i] = args[i] or ''
end
return table.concat(ret, '_')
end

if not args[1] then
-- no lat logic
return errorPrinter( {{"formatTest", "Missing latitude"}} )
elseif not tonumber(args[1]) then
-- bad lat logic
return errorPrinter( {{"formatTest", "Unable to parse latitude as a number:" .. args[1]}} )
elseif not args[4] and not args[5] and not args[6] then
-- dec logic
result, errors = parseDec(args[1], args[2], args.format)
if not result then
return errorPrinter(errors);
end
result.param = table.concat({args[1], 'N', args[2] or '', 'E', args[3] or ''}, '_')
elseif dmsTest(args[4], args[8]) then
-- dms logic
result, errors, backward = parseDMS(args[1], args[2], args[3], args[4],
args[5], args[6], args[7], args[8], args.format)
if args[10] then
table.insert(errors, {'formatTest', 'Extra unexpected parameters'})
end
if not result then
return errorPrinter(errors)
end
result.param = getParam(args, 9)
elseif dmsTest(args[3], args[6]) then
-- dm logic
result, errors, backward = parseDMS(args[1], args[2], nil, args[3],
args[4], args[5], nil, args[6], args['format'])
if args[8] then
table.insert(errors, {'formatTest', 'Extra unexpected parameters'})
end
if not result then
return errorPrinter(errors)
end
result.param = getParam(args, 7)
elseif dmsTest(args[2], args[4]) then
-- d logic
result, errors, backward = parseDMS(args[1], nil, nil, args[2],
args[3], nil, nil, args[4], args.format)
if args[6] then
table.insert(errors, {'formatTest', 'Extra unexpected parameters'})
end
if not result then
return errorPrinter(errors)
end
result.param = getParam(args, 5)
else
-- Error
return errorPrinter({{"formatTest", "Unknown argument format"}})
end
result.name = args.name

local extra_param = {'dim', 'globe', 'scale', 'region', 'source', 'type'}
for _, v in ipairs(extra_param) do
if args[v] then
table.insert(errors, {'formatTest', 'Parameter: "' .. v .. '=" should be "' .. v .. ':"' })
end
end

local ret = specPrinter(args, result)
if #errors > 0 then
ret = ret .. ' ' .. errorPrinter(errors) .. '[[Category:Pages with malformed coordinate tags]]'
end
return ret, backward
end

--[[
Generate Wikidata tracking categories.
]]
local function makeWikidataCategories()
local ret
if mw.wikibase and current_page.namespace == 0 then
local entity = mw.wikibase.getEntityObject()
if entity and entity.claims and entity.claims.P625 and entity.claims.P625[1] then
local snaktype = entity.claims.P625[1].mainsnak.snaktype
if snaktype == 'value' then
-- coordinates exist both here and on Wikidata, and can be compared.
ret = 'Coordinates on Wikidata'
elseif snaktype == 'somevalue' then
ret = 'Coordinates on Wikidata set to unknown value'
elseif snaktype == 'novalue' then
ret = 'Coordinates on Wikidata set to no value'
end
else
-- We have to either import the coordinates to Wikidata or remove them here.
ret = 'Coordinates not on Wikidata'
end
end
if ret then
return string.format('[[Category:%s]]', ret)
else
return ''
end
end

--[[
link

Simple function to export the coordinates link for other uses.

Usage:
{{#invoke:Coordinates | link }}

]]
function coordinates.link(frame)
return coord_link;
end

--[[
dec2dms

Wrapper to allow templates to call dec2dms directly.

Usage:
{{#invoke:Coordinates | dec2dms | decimal_coordinate | positive_suffix |
negative_suffix | precision }}

decimal_coordinate is converted to DMS format. If positive, the positive_suffix
is appended (typical N or E), if negative, the negative suffix is appended. The
specified precision is one of 'D', 'DM', or 'DMS' to specify the level of detail
to use.
]]
coordinates.dec2dms = makeInvokeFunc('_dec2dms')
function coordinates._dec2dms(args)
local coordinate = args[1]
local firstPostfix = args[2] or ''
local secondPostfix = args[3] or ''
local precision = args[4] or ''

return convert_dec2dms(coordinate, firstPostfix, secondPostfix, precision)
end

--[[
Helper function to determine whether to use D, DM, or DMS
format depending on the precision of the decimal input.
]]
function coordinates.determineMode( value1, value2 )
local precision = math.max( math_mod._precision( value1 ), math_mod._precision( value2 ) );
if precision <= 0 then
return 'd'
elseif precision <= 2 then
return 'dm';
else
return 'dms';
end
end

--[[
dms2dec

Wrapper to allow templates to call dms2dec directly.

Usage:
{{#invoke:Coordinates | dms2dec | direction_flag | degrees |
minutes | seconds }}

Converts DMS values specified as degrees, minutes, seconds too decimal format.
direction_flag is one of N, S, E, W, and determines whether the output is
positive (i.e. N and E) or negative (i.e. S and W).
]]
coordinates.dms2dec = makeInvokeFunc('_dms2dec')
function coordinates._dms2dec(args)
local direction = args[1]
local degrees = args[2]
local minutes = args[3]
local seconds = args[4]

return convert_dms2dec(direction, degrees, minutes, seconds)
end

--[[
coord

Main entry point for Lua function to replace {{coord}}

Usage:
{{#invoke:Coordinates | coord }}
{{#invoke:Coordinates | coord | lat | long }}
{{#invoke:Coordinates | coord | lat | lat_flag | long | long_flag }}
...

Refer to {{coord}} documentation page for many additional parameters and
configuration options.

Note: This function provides the visual display elements of {{coord}}. In
order to load coordinates into the database, the {{#coordinates:}} parser
function must also be called, this is done automatically in the Lua
version of {{coord}}.
]]
coordinates.coord = makeInvokeFunc('_coord')
function coordinates._coord(args)
if (not args[1] or not tonumber(args[1])) and not args[2] and mw.wikibase.getEntityObject() then
args[3] = args[1]; args[1] = nil
local entity = mw.wikibase.getEntityObject()
if entity
and entity.claims
and entity.claims.P625
and entity.claims.P625[1].mainsnak.snaktype == 'value'
then
local precision = entity.claims.P625[1].mainsnak.datavalue.value.precision
args[1]=entity.claims.P625[1].mainsnak.datavalue.value.latitude
args[2]=entity.claims.P625[1].mainsnak.datavalue.value.longitude
if precision then
precision=-math_mod._round(math.log(precision)/math.log(10),0)
args[1]=math_mod._round(args[1],precision)
args[2]=math_mod._round(args[2],precision)
end
end
end

local contents, backward = formatTest(args)
local Notes = args.notes or ''
local Display = args.display and args.display:lower() or 'inline'

local function isInline(s)
-- Finds whether coordinates are displayed inline.
return s:find('inline') ~= nil or s == 'i' or s == 'it' or s == 'ti'
end
local function isInTitle(s)
-- Finds whether coordinates are displayed in the title.
return s:find('title') ~= nil or s == 't' or s == 'it' or s == 'ti'
end

local function coord_wrapper(in_args)
-- Calls the parser function {{#coordinates:}}.
return mw.getCurrentFrame():callParserFunction('#coordinates', in_args) or ''
end

local text = ''
if isInline(Display) then
text = text .. displayinline(contents, Notes)
end
if isInTitle(Display) then
text = text
.. displaytitle(contents, Notes)
.. makeWikidataCategories()
end
if not args.nosave then
local page_title, count = mw.title.getCurrentTitle(), 1
if backward then
local tmp = {}
while not string.find((args[count-1] or ''), '[EW]') do tmp[count] = (args[count] or ''); count = count+1 end
tmp.count = count; count = 2*(count-1)
while count >= tmp.count do table.insert(tmp, 1, (args[count] or '')); count = count-1 end
for i, v in ipairs(tmp) do args[i] = v end
else
while count <= 9 do args[count] = (args[count] or ''); count = count+1 end
end
if isInTitle(Display) and not page_title.isTalkPage and page_title.subpageText ~= 'doc' and page_title.subpageText ~= 'testcases' then args[10] = 'primary' end
args.notes, args.format, args.display = nil
text = text .. coord_wrapper(args)
end
return text
end

--[[
coord2text

Extracts a single value from a transclusion of {{Coord}}.
IF THE GEOHACK LINK SYNTAX CHANGES THIS FUNCTION MUST BE MODIFIED.

Usage:

{{#invoke:Coordinates | coord2text | {{Coord}} | parameter }}

Valid values for the second parameter are: lat (signed integer), long (signed integer), type, scale, dim, region, globe, source

]]
function coordinates.coord2text(frame)
if frame.args[1] == '' or frame.args[2] == '' or not frame.args[2] then return nil end
frame.args[2] = mw.text.trim(frame.args[2])
if frame.args[2] == 'lat' or frame.args[2] == 'long' then
local result, negative = mw.text.split((mw.ustring.match(frame.args[1],'[%.%d]+°[NS] [%.%d]+°[EW]') or ''), ' ')
if frame.args[2] == 'lat' then
result, negative = result[1], 'S'
else
result, negative = result[2], 'W'
end
result = mw.text.split(result, '°')
if result[2] == negative then result[1] = '-'..result[1] end
return result[1]
else
return mw.ustring.match(frame.args[1], 'params=.-_'..frame.args[2]..':(.-)[ _]')
end
end

--[[
coordinsert

Injects some text into the Geohack link of a transclusion of {{Coord}} (if that text isn't already in the transclusion). Outputs the modified transclusion of {{Coord}}.
IF THE GEOHACK LINK SYNTAX CHANGES THIS FUNCTION MUST BE MODIFIED.

Usage:

{{#invoke:Coordinates | coordinsert | {{Coord}} | parameter:value | parameter:value | … }}

Do not make Geohack unhappy by inserting something which isn't mentioned in the {{Coord}} documentation.

]]
function coordinates.coordinsert(frame)
for i, v in ipairs(frame.args) do
if i ~= 1 then
if not mw.ustring.find(frame.args[1], (mw.ustring.match(frame.args[i], '^(.-:)') or '')) then
frame.args[1] = mw.ustring.gsub(frame.args[1], '(params=.-)_? ', '%1_'..frame.args[i]..' ')
end
end
end
if frame.args.name then
if not mw.ustring.find(frame.args[1], '') then
local namestr = frame.args.name
frame.args[1] = mw.ustring.gsub(frame.args[1],
'()(<span[^<>]*>[^<>]*<span[^<>]*>[^<>]*<span[^<>]*>[^<>]*)()',
'%1%2 (' .. namestr .. ')%3')
frame.args[1] = mw.ustring.gsub(frame.args[1], '(&params=[^&"<>%[%] ]*) ', '%1&title=' .. mw.uri.encode(namestr) .. ' ')
end
end
return frame.args[1]
end

return coordinates

Template:Scientists whose names are used as non SI units

2017-07-19T22:24:22Z

NickPercival: 1 revision imported

{{Navbox
| name = Scientists whose names are used as non SI units
| title = [[List of scientists whose names are used as non SI units|Scientists whose names are used as non SI units]]
| state = {{{state<includeonly>|autocollapse</includeonly>}}}
| bodyclass = hlist

| list1 =
* [[Anders Jonas Ångström]]
* [[Alexander Graham Bell]]
* [[Marie Curie]]
* [[Pierre Curie]]
* [[John Dalton]]
* [[Peter Debye]]
* [[Loránd Eötvös]]
* [[Daniel Gabriel Fahrenheit]]
* [[Galileo Galilei]]
* [[Carl Friedrich Gauss|Johann Carl Friedrich Gauss]]
* [[William Gilbert (astronomer)|William Gilbert]]
* [[Heinrich Kayser]]
* [[Johann Heinrich Lambert]]
* [[Samuel Pierpont Langley]]
* [[Heinrich Mache]]
* [[James Clerk Maxwell]]
* [[John Napier]]
* [[Hans Christian Ørsted]]
* [[Jean Léonard Marie Poiseuille]]
* [[William John Macquorn Rankine]]
* [[René Antoine Ferchault de Réaumur]]
* [[Wilhelm Röntgen]]
* [[Sir George Stokes, 1st Baronet]]
* [[John Strutt, 3rd Baron Rayleigh]]
* [[J. J. Thomson|Joseph John Thomson]]
* [[Evangelista Torricelli]]

| below =
* [[List of scientists whose names are used as SI units|Scientists whose names are used as SI units]]
* [[List of scientists whose names are used in chemical element names|Scientists whose names are used in chemical element names]]
* [[List of scientists whose names are used in physical constants|Scientists whose names are used in physical constants]]

}}<noinclude>
{{collapsible option}}
[[Category:Measurement templates]]
[[Category:Scientist navigational boxes]]
</noinclude>

Template:Pp-protect

2017-07-19T22:24:20Z

NickPercival: 1 revision imported

#REDIRECT [[Template:Pp]]
[[Category:Top icon protection templates|{{PAGENAME}}]]

Template:Post-nominals/CAN

2017-07-19T22:24:20Z

NickPercival: 1 revision imported

{{#switch: {{{1}}}
| ADC = [[Aide-de-camp|ADC]]
| ADC(P) = [[Personal Aide-de-Camp|ADC(P)]]
| AE = [[Air Efficiency Award|AE]]
| AFC = [[Air Force Cross (United Kingdom)|AFC]]
| AFM = [[Air Force Medal|AFM]]
| AIH = [[Académie Internationale d'Héraldique|AIH]]
| ALS = [[Alberta Land Surveyor|ALS]]
| AM = [[National Assembly for Wales|AM]]
| AOE = [[Alberta Order of Excellence|AOE]]
| ARCT = [[The Royal Conservatory of Music|ARCT]]
| ARIDO = [[Association of Registered Interior Designers of Ontario|ARIDO]]
| ARRC = [[Royal Red Cross|ARRC]]
| AScT = [[Applied Science Technologist|AScT]]
| BCLS = [[British Columbia Land Surveyor|BCLS]]
| BEM = [[British Empire Medal|BEM]]
| Bt = [[Baronet|Bt]]
| Btss = [[Baronet#Baronetesses|Btss]]
| CA = [[Chartered Accountant|CA]]
| CB = [[Order of the Bath|CB]]
| CBE = [[Order of the British Empire|CBE]]
| CBHF = [[Canadian Business Hall of Fame|CBHF]]
| CC = [[Order of Canada|CC]]
| CD = [[Canadian Forces Decoration|CD]]
| C.D. = [[Canadian Forces Decoration|C.D.]]
| CET = [[Certified Engineering Technologist|CET]]
| CFA = [[Chartered Financial Analyst|CFA]]
| C.F.A. = [[Chartered Financial Analyst|C.F.A.]]
| CGA = [[Certified General Accountant|CGA]]
| CGC = [[Conspicuous Gallantry Cross|CGC]]
| CGM = [[Conspicuous Gallantry Medal|CGM]]
| CH = [[Order of the Companions of Honour|CH]]
| CIE = [[Order of the Indian Empire|CIE]]
| CLS = [[Association of Canada Lands Surveyors|CLS]]
| CM = [[Order of Canada|CM]]
| CMA = [[Certified Management Accountants of Canada|CMA]]
| CMG = [[Companion of the Order of St Michael and St George|CMG]]
| CMM = [[Order of Military Merit (Canada)|CMM]]
| COM = [[Order of Merit of the Police Forces|COM]]
| CPA = [[Chartered Professional Accountant|CPA]]
| CPM = [[Overseas Territories Police Medal|CPM]]
| CPMHN(C) = [[Canadian Nurses Association|CPMHN(C)]]
| CQ = [[National Order of Quebec|CQ]]
| CSI = [[Order of the Star of India|CSI]]
| CTech = [[Certified Technician|CTech]]
| CV = [[Cross of Valour (Canada)|CV]]
| CVO = [[Royal Victorian Order|CVO]]
| DBE = [[Order of the British Empire|DBE]]
| DCB = [[Order of the Bath|DCB]]
| DCL = [[Doctor of Civil Law|DCL]]
| DCM = [[Distinguished Conduct Medal|DCM]]
| DCMG = [[Dame Commander of the Order of St Michael and St George|DCMG]]
| DCVO = [[Royal Victorian Order|DCVO]]
| DFC = [[Distinguished Flying Cross (United Kingdom)|DFC]]
| DFC* = [[Distinguished Flying Cross (United Kingdom)#Description|DFC]]
| DFC** = [[Distinguished Flying Cross (United Kingdom)#Description|DFC]]
| DFC2 = [[Distinguished Flying Cross (United Kingdom)#Description|DFC]]
| DFC2 = [[Distinguished Flying Cross (United Kingdom)|DFC]]
| DFM = [[Distinguished Flying Medal|DFM]]
| DFM2 = [[Distinguished Flying Medal#Description|DFM]]
| DL = [[Deputy Lieutenant|DL]]
| DSC = [[Distinguished Service Cross (United Kingdom)|DSC]]
| DSC* = [[Distinguished Service Cross (United Kingdom)#Description|DSC]]
| DSC** = [[Distinguished Service Cross (United Kingdom)#Description|DSC]]
| DSM = [[Distinguished Service Medal (United Kingdom)|DSM]]
| DSM2 = [[Distinguished Service Medal (United Kingdom)#Description|DSM]]
| DSO = [[Distinguished Service Order|DSO]]
| DSO1 = [[Distinguished Service Order|DSO]] & [[Medal bar|Bar]]
| DSO2 = [[Distinguished Service Order|DSO]] & [[Medal bar|Two Bars]]
| DSO3 = [[Distinguished Service Order|DSO]] & [[Medal bar|Three Bars]]
| DSO* = [[Distinguished Service Order#Description|DSO]]
| DSO** = [[Distinguished Service Order#Description|DSO]]
| DSO*** = [[Distinguished Service Order#Description|DSO]]
| ED = [[Canadian Efficiency Decoration|ED]]
| ERD = [[Emergency Reserve Decoration|ERD]]
| FAcSS = [[Academy of Social Sciences|FAcSS]]
| FBA = [[Fellow of the British Academy|FBA]]
| FCA = [[Chartered Accountant|FCA]]
| FCAE = [[Canadian Academy of Engineering|FCAE]]
| FCASI = [[Canadian Aeronautics and Space Institute|FCASI]]
| FCGA = [[Certified General Accountant|FCGA]]
| FCIC = [[Chemical Institute of Canada|FCIC]]
| FCIM = [[Canadian Institute of Mining, Metallurgy and Petroleum|FCIM]]
| FCMA = [[Certified Management Accountants of Canada|FCMA]]
| FCSI = [[Canadian Securities Institute|FCSI]]
| FMedSci = [[Fellow of the Academy of Medical Sciences|FMedSci]]
| FRAIC = [[Royal Architectural Institute of Canada|FRAIC]]
| FRAIC(hon) = [[Royal Architectural Institute of Canada|FRAIC(''hon'')]]
| FRCA = [[Royal Canadian Academy of Arts|FRCA]]
| FRCCO = [[Royal Canadian College of Organists|FRCCO]]
| FRCD = [[Royal College of Dentists of Canada|FRCD]]
| FRCGS = [[Royal Canadian Geographical Society|FRCGS]]
| FRCPC = [[Royal College of Physicians and Surgeons of Canada|FRCPC]]
| FRCS = [[Royal College of Physicians and Surgeons of Canada|FRCS]]
| FRCSC = [[Royal College of Physicians and Surgeons of Canada|FRCSC]]
| FRCPSC(hon) = [[Royal College of Physicians and Surgeons of Canada|FRCPSC(''hon'')]]
| FRGS = [[Fellow of the Royal Geographical Society|FRGS]]
| FRHSC = [[Royal Heraldry Society of Canada|FRHSC]]
| FRHSC(hon) = [[Royal Heraldry Society of Canada|FRHSC(''hon'')]]
| FRMS = [[Royal Microscopical Society|FRMS]]
| FRS = [[Fellow of the Royal Society|FRS]]
| FRSC = [[Fellow of the Royal Society of Canada|FRSC]]
| FRSC(hon) = [[Fellow of the Royal Society of Canada|FRSC(''hon'')]]
| FRSE = [[Fellow of the Royal Society of Edinburgh|FRSE]]
| FRSL = [[Fellow of the Royal Society of Literature|FRSL]]
| FSTS = [[Fellow of the Scottish Tartans Society|FSTS]]
| GBE = [[Order of the British Empire|GBE]]
| GC = [[George Cross|GC]]
| GCB = [[Order of the Bath|GCB]]
| GCIE = [[Order of the Indian Empire|GCIE]]
| GCMG = [[Knight Grand Cross of the Order of St Michael and St George|GCMG]]
| GCMGf = [[Dame Grand Cross of the Order of St Michael and St George|GCMG]]
| GCMGm = [[Knight Grand Cross of the Order of St Michael and St George|GCMG]]
| GCSI = [[Order of the Star of India|GCSI]]
| GCVO = [[Royal Victorian Order|GCVO]]
| GM = [[George Medal|GM]]
| GOQ = [[National Order of Quebec|GOQ]]
| Hon RA = [[Royal Academy of Arts#Membership|Hon RA]]
| Hon. RA = [[Royal Academy of Arts#Membership|Hon. RA]]
| Hon. R.A. = [[Royal Academy of Arts#Membership|Hon. R.A.]]
| HonRA = [[Royal Academy of Arts#Membership|HonRA]]
| IDSM = [[Indian Distinguished Service Medal|IDSM]]
| IOM = [[Indian Order of Merit|IOM]]
| ISO = [[Imperial Service Order|ISO]]
| ISP = [[Information Systems Professional|ISP]]
| JP = [[Justice of the peace#Canada|JP]]
| KBE = [[Order of the British Empire|KBE]]
| KC = [[Queen's Counsel|KC]]
| KCB = [[Order of the Bath|KCB]]
| KCIE = [[Order of the Indian Empire|KCIE]]
| KCMG = [[Knight Commander of the Order of St Michael and St George|KCMG]]
| KCSI = [[Order of the Star of India|KCSI]]
| KCVO = [[Royal Victorian Order|KCVO]]
| KG = [[Order of the Garter|KG]]
| KP = [[Order of St. Patrick|KP]]
| KT = [[Order of the Thistle|KT]]
| LG = [[Order of the Garter|LG]]
| LT = [[Order of the Thistle|LT]]
| LVO = [[Royal Victorian Order|LVO]]
| MAIBC = [[Architectural Institute of British Columbia|MAIBC]]
| MB = [[Medal of Bravery (Canada)|MB]]
| MBE = [[Order of the British Empire|MBE]]
| MC = [[Military Cross|MC]]
| MC* = [[Military Cross|MC]]
| MCIP = [[Canadian Institute of Planners|MCIP]]
| MEP = [[European Parliament|MEP]]
| MHA = [[Newfoundland and Labrador House of Assembly|MHA]]
| MLA = [[Member of the Legislative Assembly|MLA]]
| MLAAB = [[Legislative Assembly of Alberta|MLA]]
| MLABC = [[Legislative Assembly of British Columbia|MLA]]
| MLAMB = [[Legislative Assembly of Manitoba|MLA]]
| MLANB = [[Legislative Assembly of New Brunswick|MLA]]
| MLANS = [[Nova Scotia House of Assembly|MLA]]
| MLANT = [[Legislative Assembly of the Northwest Territories|MLA]]
| MLANU = [[Legislative Assembly of Nunavut|MLA]]
| MLAPE = [[Legislative Assembly of Prince Edward Island|MLA]]
| MLAPEI = [[Legislative Assembly of Prince Edward Island|MLA]]
| MLASK = [[Legislative Assembly of Saskatchewan|MLA]]
| MLAYT = [[Yukon Legislative Assembly|MLA]]
| MLS = [[Manitoba Land Surveyor|MLS]]
| MM = [[Military Medal|MM]]
| MMM = [[Order of Military Merit (Canada)|MMM]]
| MMV = [[Medal of Military Valour|MMV]]
| MNA = [[National Assembly of Quebec|MNA]]
| MOM = [[Order of Merit of the Police Forces|MOM]]
| MP = [[House of Commons of Canada|MP]]
| MPP = [[Legislative Assembly of Ontario|MPP]]
| MRAIC = [[Royal Architectural Institute of Canada|MRAIC]]
| MSC = [[Meritorious Service Cross|MSC]]
| MSM = [[Meritorious Service Medal (Canada)|MSM]]
| MSP = [[Member of the Scottish Parliament|MSP]]
| MSRC = [[Royal Society of Canada|MSRC]]
| MStJ= [[Venerable Order of Saint John|MStJ]]
| MSYP = [[Scottish Youth Parliament|MSYP]]
| MVO = [[Royal Victorian Order|MVO]]
| MYP = [[UK Youth Parliament|MYP]]
| OAA = [[Ontario Association of Architects|OAA]]
| OBC = [[Order of British Columbia|OBC]]
| OBE = [[Order of the British Empire|OBE]]
| OBHF = [[Canadian Business Hall of Fame|OBHF]]
| OBI = [[Order of British India|OBI]]
| OC = [[Order of Canada|OC]]
| OCT = [[Ontario Certified Teacher|OCT]]
| OLS = [[Ontario Land Surveyor|OLS]]
| OM = [[Order of Manitoba|OM]]
| OMC = [[Ontario Medal for Good Citizenship|OMC]]
| OMM = [[Order of Military Merit (Canada)|OMM]]
| OMt = [[Member of the Order of Merit|OM]]
| ONB = [[Order of New Brunswick|ONB]]
| ONL = [[Order of Newfoundland and Labrador|ONL]]
| ONS = [[Order of Nova Scotia|ONS]]
| OOM = [[Order of Merit of the Police Forces|OOM]]
| OOnt = [[Order of Ontario|OOnt]]
| O.Ont = [[Order of Ontario|OOnt]]
| O.Ont. = [[Order of Ontario|OOnt]]
| OPEI = [[Order of Prince Edward Island|OPEI]]
| O.PEI = [[Order of Prince Edward Island|OPEI]]
| OQ = [[National Order of Quebec|OQ]]
| OStJ = [[Venerable Order of Saint John|OStJ]]
| PAg = [[Professional agrologist|PAg]]
| PC = [[Queen's Privy Council for Canada|PC]]
| PCP = [[Canadian Payroll Association|PCP]]
| PEng = [[Canadian Council of Professional Engineers|PEng]]
| PGeo = [[Professional Geoscientist|PGeo]]
| PGeol = [[Professional Geologist|PGeol]]
| PGeoph = [[Professional Geophysicist|PGeoph]]
| PLog = [[Professional Logistician|PLog]]
| PMP = [[Project Management Professional|PMP]]
| QC = [[Queen's Counsel|QC]]
| QC (Can) = [[Queen's Counsel|QC (Can)]]
| QC (Sask) = [[Queen's Counsel|QC (Sask)]]
| QFSM = [[Queen's Fire Service Medal|QFSM]]
| QGM = [[Queen's Gallantry Medal|QGM]]
| QHC = [[Honorary Chaplain to the Queen|QHC]]
| QHDS = [[Monarchy of Canada#Federal residences and royal household|QHDS]]
| QHNS = [[Medical Household|QHNS]]
| QHP = [[Monarchy of Canada#Federal residences and royal household|QHP]]
| QHS = [[Medical Household|QHS]]
| QPM = [[Queen's Police Medal|QPM]]
| QS = [[Serjeant-at-law|QS]]
| RCA = [[Royal Canadian Academy of Arts|RCA]]
| RD = [[Decoration for Officers of the Royal Naval Reserve|RD]]
| RFA = [[Royal Field Artillery|RFA]]
| RFAx = [[Royal Fleet Auxiliary|RFA]]
| RMC = [[Royal Military College of Canada|RMC]]
| RPBio = [[Registered Professional Biologist|RPBio]]
| RPF = [[Registered Professional Forester|RPF]]
| RPP = [[Registered Professional Planner|RPP]]
| RRC = [[Royal Red Cross|RRC]]
| RVM = [[Royal Victorian Medal|RVM]]
| SC = [[Star of Courage (Canada)|SC]]
| SGM = [[Sea Gallantry Medal|SGM]]
| SL = [[Serjeant-at-law|SL]]
| SLS = [[Saskatchewan Land Surveyor|SLS]]
| SMV = [[Star of Military Valour|SMV]]
| SOM = [[Saskatchewan Order of Merit|SOM]]
| StrucEng = [[Structural engineer|StrucEng]]
| SVM = [[Saskatchewan Volunteer Medal|SVM]]
| TD = [[Territorial Decoration|TD]]
| UD = [[Ulster Defence Regiment#Awards, honours and decorations|UD]]
| UE = [[United Empire Loyalist|UE]]
| VC = [[Victoria Cross|VC]]
| VD = [[Volunteer Reserve Decoration|VD]]
| VRD = [[Decoration for Officers of the Royal Naval Volunteer Reserve|VRD]]
| * = [[Medal bar|*]]


| ComIH = [[Order of Prince Henry|ComIH]]
| Com.I.H. = [[Order of Prince Henry|Com.I.H.]]
| CvIH = [[Order of Prince Henry|CvIH]]
| Cv.I.H. = [[Order of Prince Henry|Cv.I.H.]]
| DmIH = [[Order of Prince Henry|DmIH]]
| Dm.I.H. = [[Order of Prince Henry|Dm.I.H.]]
| GCIH = [[Order of Prince Henry|GCIH]]
| G.C.I.H. = [[Order of Prince Henry|G.C.I.H.]]
| GColIH = [[Order of Prince Henry|GColIH]]
| G.Col.I.H. = [[Order of Prince Henry|G.Col.I.H.]]
| GOIH = [[Order of Prince Henry|GOIH]]
| G.O.I.H. = [[Order of Prince Henry|G.O.I.H.]]
| MedOIH = [[Order of Prince Henry|MedOIH]]
| Med.O.I.H. = [[Order of Prince Henry|Med.O.I.H.]]
| MedPIH = [[Order of Prince Henry|MedPIH]]
| Med.P.I.H. = [[Order of Prince Henry|Med.P.I.H.]]
| OIH = [[Order of Prince Henry|OIH]]
| O.I.H. = [[Order of Prince Henry|O.I.H.]]

}}<noinclude>
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