Editing Spacetime (section)

=== Lorentz transformations ===
{{Main|Lorentz transformation|Lorentz group}}

The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mid-1800s, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary addition of velocities.

Lorentz transformations are used to transform the coordinates of an event from one frame to another in special relativity.

The Lorentz factor appears in the Lorentz transformations:
: <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>

The inverse Lorentz transformations are:
: <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>

When ''v''&nbsp;≪&nbsp;''c'' and ''x'' is small enough, the ''v''<sup>2</sup>/''c''<sup>2</sup> and ''vx''/''c''<sup>2</sup> terms approach zero, and the Lorentz transformations approximate to the Galilean transformations.

<math>t' = \gamma ( t - v x/c^2),</math> <math>x' = \gamma( x - v t) </math> etc., most often really mean <math>\Delta t' = \gamma (\Delta t - v \Delta x/c^2),</math> <math>\Delta x' = \gamma(\Delta x - v \Delta t) </math> etc. Although for brevity the Lorentz transformation equations are written without deltas, ''x'' means &Delta;''x'', etc. We are, in general, always concerned with the space and time ''differences'' between events.

Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames. Different authors call one or the other set of transformations the "inverse" set. The forwards and inverse transformations are trivially related to each other, since the ''S'' frame can only be moving forwards or reverse with respect to {{′|''S''}}. So inverting the equations simply entails switching the primed and unprimed variables and replacing ''v'' with −''v''.<ref name="Morin" />{{rp|71–79}}

<small>'''Example:''' Terence and Stella are at an Earth-to-Mars space race. Terence is an official at the starting line, while Stella is a participant. At time {{math|1=''t'' = {{′|''t''}} = 0}}, Stella's spaceship accelerates instantaneously to a speed of 0.5&nbsp;''c''. The distance from Earth to Mars is 300 light-seconds (about {{val|90.0|e=6|u=km}}). Terence observes Stella crossing the finish-line clock at {{math|1=''t''&nbsp;=&nbsp;600.00&nbsp;s}}. But Stella observes the time on her ship chronometer to be {{tmath|1=t^{\prime}=\gamma\left(t-v x / c^{2}\right)=519.62\ \text{s} }} as she passes the finish line, and she calculates the distance between the starting and finish lines, as measured in her frame, to be 259.81 light-seconds (about {{val|77.9|e=6|u=km}}).</small>
1).

==== Deriving the Lorentz transformations ====
{{Main|Derivations of the Lorentz transformations}}
[[File:Derivation of Lorentz Transformation.svg|thumb|Figure 3–5. Derivation of Lorentz Transformation]]
There have been many dozens of [[derivations of the Lorentz transformations]] since Einstein's original work in 1905, each with its particular focus. Although Einstein's derivation was based on the invariance of the speed of light, there are other physical principles that may serve as starting points. Ultimately, these alternative starting points can be considered different expressions of the underlying [[principle of locality]], which states that the influence that one particle exerts on another can not be transmitted instantaneously.<ref>{{cite book|last1=Landau|first1=L. D.|last2=Lifshitz|first2=E. M.|title=The Classical Theory of Fields, Course of Theoretical Physics, Volume 2|date=2006|publisher=Elsevier|location=Amsterdam|isbn=978-0-7506-2768-9|pages=1–24|edition=4th}}</ref>

The derivation given here and illustrated in Fig.&nbsp;3-5 is based on one presented by Bais<ref name="Bais" />{{rp|64–66}} and makes use of previous results from the Relativistic Composition of Velocities, Time Dilation, and Length Contraction sections. Event&nbsp;P has coordinates (''w'',&nbsp;''x'') in the black "rest system" and coordinates {{math|1=({{′|''w''}},&nbsp;{{′|''x''}})}} in the red frame that is moving with velocity parameter {{math|1=''β''&nbsp;=&nbsp;''v''/''c''}}. To determine {{′|''w''}} and {{′|''x''}} in terms of ''w'' and ''x'' (or the other way around) it is easier at first to derive the ''inverse'' Lorentz transformation.
#There can be no such thing as length expansion/contraction in the transverse directions. ''y{{'}}'' must equal ''y'' and {{′|''z''}} must equal ''z'', otherwise whether a fast moving 1&nbsp;m ball could fit through a 1&nbsp;m circular hole would depend on the observer. The first postulate of relativity states that all inertial frames are equivalent, and transverse expansion/contraction would violate this law.<ref name="Morin">{{cite book|last1=Morin|first1=David|title=Special Relativity for the Enthusiastic Beginner|date=2017|publisher=CreateSpace Independent Publishing Platform|isbn=978-1-5423-2351-2}}</ref>{{rp|27–28}}
# From the drawing, ''w'' = ''a'' + ''b'' and {{math|1=''x''&nbsp;=&nbsp;''r''&nbsp;+&nbsp;''s''}}
# From previous results using similar triangles, we know that {{math|1=''s''/''a''&nbsp;=&nbsp;''b''/''r'' = ''v''/''c''&nbsp;=&nbsp;''β''}}.
# Because of time dilation, {{math|1=''a''&nbsp;=&nbsp;''&gamma;{{prime|w}}''}}
# Substituting equation (4) into {{math|1=''s''/''a''&nbsp;=&nbsp;''β''}} yields {{math|1=''s''&nbsp;=&nbsp;''&gamma;{{prime|w}}β''}}.
# Length contraction and similar triangles give us {{math|1=''r''&nbsp;=&nbsp;''&gamma;{{prime|x}}''}} and {{math|1=''b''&nbsp;=&nbsp;''βr'' = ''βγ{{prime|x}}''}}
# Substituting the expressions for ''s'', ''a'', ''r'' and ''b'' into the equations in Step&nbsp;2 immediately yield
#: <math>w = \gamma w' + \beta \gamma x' </math>
#: <math>x = \gamma x' + \beta \gamma w' </math>

The above equations are alternate expressions for the t and x equations of the inverse Lorentz transformation, as can be seen by substituting ''ct'' for ''w'', {{′|''ct''}} for {{′|''w''}}, and ''v''/''c'' for ''β''. From the inverse transformation, the equations of the forwards transformation can be derived by solving for {{′|''t''}} and {{′|''x''}}.

==== Linearity of the Lorentz transformations ====
The Lorentz transformations have a mathematical property called linearity, since {{′|''x''}} and {{′|''t''}} are obtained as linear combinations of ''x'' and ''t'', with no higher powers involved. The linearity of the transformation reflects a fundamental property of spacetime that was tacitly assumed in the derivation, namely, that the properties of inertial frames of reference are independent of location and time. In the absence of gravity, spacetime looks the same everywhere.<ref name="Bais" />{{rp|67}} All inertial observers will agree on what constitutes accelerating and non-accelerating motion.<ref name="Morin" />{{rp|72–73}} Any one observer can use her own measurements of space and time, but there is nothing absolute about them. Another observer's conventions will do just as well.<ref name="Schutz" />{{rp|190}}

A result of linearity is that if two Lorentz transformations are applied sequentially, the result is also a Lorentz transformation.

<small>'''Example:''' Terence observes Stella speeding away from him at 0.500&nbsp;''c'', and he can use the Lorentz transformations with {{math|1=''β''&nbsp;=&nbsp;0.500}} to relate Stella's measurements to his own. Stella, in her frame, observes Ursula traveling away from her at 0.250&nbsp;''c'', and she can use the Lorentz transformations with {{math|1=''β''&nbsp;=&nbsp;0.250}} to relate Ursula's measurements with her own. Because of the linearity of the transformations and the relativistic composition of velocities, Terence can use the Lorentz transformations with {{math|1=''β''&nbsp;=&nbsp;0.666}} to relate Ursula's measurements with his own.</small>

{{anchor|Doppler effect}}