Editing Spacetime (section)

== Basic mathematics of spacetime {{anchor|Mathematics}} ==
{{anchor|Galilean transformations}}

=== Galilean transformations ===
<!-- Note to future editors!!! The "Basic mathematics of spacetime" section is intended to provide a gentle introduction to the mathematics of spacetime. To the limit of what is feasible, keep to basic algebra. If you are eager to share your knowledge of some highly technical material, put your contribution in one of the later sections of this article and not here. We should endeavor to keep this basic mathematics section understandable by the main target audience, which I have envisioned to be a typical high school science student. -->
{{Main|Galilean group}}

A basic goal is to be able to compare measurements made by observers in relative motion. If there is an observer O in frame S who has measured the time and space coordinates of an event, assigning this event three Cartesian coordinates and the time as measured on his lattice of synchronized clocks {{nowrap|1=(''x'', ''y'', ''z'', ''t'')}} (see [[#Figure 1-1|'''Fig.&nbsp;1-1''']]). A second observer O′ in a different frame S′ measures the same event in her coordinate system and her lattice of synchronized clocks {{nowrap|1=({{′|''x''}}, {{′|''y''}}, {{′|''z''}}, {{′|''t''}})}}. With inertial frames, neither observer is under acceleration, and a simple set of equations allows us to relate coordinates {{nowrap|1=(''x'', ''y'', ''z'', ''t'')}} to {{nowrap|1=({{′|''x''}}, {{′|''y''}}, {{′|''z''}}, {{′|''t''}})}}. Given that the two coordinate systems are in standard configuration, meaning that they are aligned with parallel {{nowrap|1=(''x'', ''y'', ''z'')}} coordinates and that {{nowrap|1=''t'' = 0}} when {{nowrap|1={{′|''t''}} = 0}}, the coordinate transformation is as follows:<ref>{{cite book|last1=Mould|first1=Richard A.|title=Basic Relativity |date=1994 |publisher=Springer |isbn=978-0-387-95210-9 |page=42|edition=1st|access-date=22 April 2017|url=https://books.google.com/books?id=lfGE-wyJYIUC&pg=PA42}}</ref><ref>{{cite book|last1=Lerner|first1=Lawrence S.|title=Physics for Scientists and Engineers, Volume 2|date=1997|publisher=Jones & Bartlett Pub |isbn=978-0-7637-0460-5 |page=1047|edition=1st |access-date=22 April 2017|url=https://books.google.com/books?id=B8K_ym9rS6UC&pg=PA1047}}</ref>
: <math>x' = x - v t </math>
: <math>y' = y </math>
: <math>z' = z </math>
: <math>t' = t .</math>

[[File:Galilean Spacetime and composition of velocities.svg|thumb|Figure 3–1. '''Galilean''' Spacetime and composition of velocities]]
Fig. 3-1 illustrates that in Newton's theory, time is universal, not the velocity of light.<ref name="Bais" />{{rp|36–37}} Consider the following thought experiment: The red arrow illustrates a train that is moving at 0.4&nbsp;c with respect to the platform. Within the train, a passenger shoots a bullet with a speed of 0.4&nbsp;c in the frame of the train. The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0.8&nbsp;c. This is in accordance with our naive expectations.

More generally, assuming that frame S′ is moving at velocity ''v'' with respect to frame S, then within frame S′, observer O′ measures an object moving with velocity {{′|''u''}}. Velocity ''u'' with respect to frame S, since {{nowrap|1=''x'' = ''ut''}}, {{nowrap|1={{′|''x''}} = ''x'' − ''vt''}}, and {{nowrap|1=''t'' = {{′|''t''}}}}, can be written as {{nowrap|1={{′|''x''}} = ''ut'' − ''vt''}} = {{nowrap|1=(''u'' − ''v'')''t''}} = {{nowrap|1=(''u'' − ''v''){{′|''t''}}}}. This leads to {{nowrap|1={{′|''u''}} = {{′|''x''}}/{{′|''t''}}}} and ultimately
: <math>u' = u - v</math>&nbsp;&nbsp;or&nbsp;&nbsp;<math>u = u' + v ,</math>
which is the common-sense '''Galilean law for the addition of velocities'''.
{{anchor|Relativistic composition of velocities}}

=== Relativistic composition of velocities ===
{{Main|Velocity addition formula}}
[[File:Relativistic composition of velocities.svg|thumb|upright=1.5|Figure 3–2. Relativistic composition of velocities]]
The composition of velocities is quite different in relativistic spacetime. To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light,
: <math>\beta = v/c</math>

Fig. 3-2a illustrates a red train that is moving forward at a speed given by {{nowrap|1=''v''/''c'' = ''β'' = ''s''/''a''}}. From the primed frame of the train, a passenger shoots a bullet with a speed given by {{nowrap|1={{′|''u''}}/''c'' = {{′|''β''}} = ''n''/''m''}}, where the distance is measured along a line parallel to the red {{′|''x''}} axis rather than parallel to the black ''x'' axis. What is the composite velocity ''u'' of the bullet relative to the platform, as represented by the blue arrow? Referring to Fig.&nbsp;3-2b:
# From the platform, the composite speed of the bullet is given by {{nowrap|1=''u'' = ''c''(''s'' + ''r'')/(''a'' + ''b'')}}.
# The two yellow triangles are similar because they are right triangles that share a common angle ''α''. In the large yellow triangle, the ratio {{nowrap|1=''s''/''a'' = ''v''/''c'' = ''β''}}.
# The ratios of corresponding sides of the two yellow triangles are constant, so that {{nowrap|1=''r''/''a'' = ''b''/''s''}} = {{nowrap|1=''n''/''m'' = {{′|''β''}}}}. So {{nowrap|1=''b'' = {{′|''u''}}''s''/''c''}} and {{nowrap|1=''r'' = {{′|''u''}}''a''/''c''}}.
# Substitute the expressions for ''b'' and ''r'' into the expression for ''u'' in step&nbsp;1 to yield Einstein's formula for the addition of velocities:<ref name="Bais">{{cite book|last1=Bais|first1=Sander|title=Very Special Relativity: An Illustrated Guide|url=https://archive.org/details/veryspecialrelat0000bais|url-access=registration|date=2007|publisher=Harvard University Press|location=Cambridge, Massachusetts|isbn=978-0-674-02611-7}}</ref>{{rp|42–48}}
#: <math> u = {v+u'\over 1+(vu'/c^2)} . </math>

The relativistic formula for addition of velocities presented above exhibits several important features:
* If {{′|''u''}} and ''v'' are both very small compared with the speed of light, then the product {{′|''vu''}}/''c''<sup>2</sup> becomes vanishingly small, and the overall result becomes indistinguishable from the Galilean formula (Newton's formula) for the addition of velocities: ''u''&nbsp;=&nbsp;{{′|''u''}}&nbsp;+&nbsp;''v''. The Galilean formula is a special case of the relativistic formula applicable to low velocities.
* If {{′|''u''}} is set equal to ''c'', then the formula yields ''u''&nbsp;=&nbsp;''c'' regardless of the starting value of ''v''. The velocity of light is the same for all observers regardless their motions relative to the emitting source.<ref name="Bais" />{{rp|49}}

{{anchor|Time dilation and length contraction revisited}}

=== Time dilation and length contraction revisited ===
{{More citations needed section|date=March 2024}}
{{Main|Time dilation|Length contraction}}
[[File:Spacetime Diagrams Illustrating Time Dilation and Length Contraction.png|thumb|upright=1.5|Figure 3-3. Spacetime diagrams illustrating time dilation and length contraction]]

It is straightforward to obtain quantitative expressions for time dilation and length contraction. Fig.&nbsp;3-3 is a composite image containing individual frames taken from two previous animations, simplified and relabeled for the purposes of this section.

To reduce the complexity of the equations slightly, there are a variety of different shorthand notations for ''ct'':
: <math>\Tau = ct</math> and <math>w = ct</math> are common.
: One also sees very frequently the use of the convention <math>c = 1.</math>

[[File:Lorentz factor.svg|thumb|Figure 3–4. Lorentz factor as a function of velocity]]
In Fig. 3-3a, segments ''OA'' and ''OK'' represent equal spacetime intervals. Time dilation is represented by the ratio ''OB''/''OK''. The invariant hyperbola has the equation {{nowrap|1={{math|''w'' {{=}} {{radical|''x''<sup>2</sup> + ''k''<sup>2</sup>}}}}}} where ''k''&nbsp;=&nbsp;''OK'', and the red line representing the world line of a particle in motion has the equation ''w''&nbsp;=&nbsp;''x''/''β''&nbsp;=&nbsp;''xc''/''v''. A bit of algebraic manipulation yields <math display="inline">OB = OK / \sqrt{1 - v^2/c^2} .</math>

The expression involving the square root symbol appears very frequently in relativity, and one over the expression is called the Lorentz factor, denoted by the Greek letter gamma <math>\gamma</math>:<ref name=Forshaw>{{cite book|last1=Forshaw|first1=Jeffrey|last2=Smith|first2=Gavin|title=Dynamics and Relativity|date=2014|publisher=John Wiley & Sons|isbn=978-1-118-93329-9|page=118|url=https://books.google.com/books?id=5TaiAwAAQBAJ|access-date=24 April 2017|language=en}}</ref>
: <math>\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} </math>

If ''v'' is greater than or equal to ''c'', the expression for <math>\gamma</math> becomes physically meaningless, implying that ''c'' is the maximum possible speed in nature. For any ''v'' greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one.

In Fig. 3-3b, segments ''OA'' and ''OK'' represent equal spacetime intervals. Length contraction is represented by the ratio ''OB''/''OK''. The invariant hyperbola has the equation {{nowrap|1={{math|''x'' {{=}} {{radical|''w''<sup>2</sup> + ''k''<sup>2</sup>}}}}}}, where ''k''&nbsp;=&nbsp;''OK'', and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/''β''&nbsp;=&nbsp;''c''/''v''. Event&nbsp;A has coordinates
(''x'',&nbsp;''w'')&nbsp;=&nbsp;(''&gamma;k'',&nbsp;''&gamma;βk''). Since the tangent line through A and B has the equation ''w''&nbsp;=&nbsp;(''x''&nbsp;−&nbsp;''OB'')/''β'', we have ''&gamma;βk''&nbsp;=&nbsp;(''&gamma;k''&nbsp;−&nbsp;''OB'')/''β'' and
: <math>OB/OK = \gamma (1 - \beta ^ 2) = \frac{1}{\gamma}</math>

{{anchor|Lorentz transformations}}

=== 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}}

=== Doppler effect ===
{{Main|Doppler effect|Relativistic Doppler effect}}

The [[Doppler effect]] is the change in frequency or wavelength of a wave for a receiver and source in relative motion. For simplicity, we consider here two basic scenarios: (1) The motions of the source and/or receiver are exactly along the line connecting them (longitudinal Doppler effect), and (2) the motions are at right angles to the said line ([[transverse Doppler effect]]). We are ignoring scenarios where they move along intermediate angles.

==== Longitudinal Doppler effect ====
The classical Doppler analysis deals with waves that are propagating in a medium, such as sound waves or water ripples, and which are transmitted between sources and receivers that are moving towards or away from each other. The analysis of such waves depends on whether the source, the receiver, or both are moving relative to the medium. Given the scenario where the receiver is stationary with respect to the medium, and the source is moving directly away from the receiver at a speed of ''v<sub>s</sub>'' for a velocity parameter of ''β<sub>s</sub>'', the wavelength is increased, and the observed frequency ''f'' is given by
: <math>f = \frac{1}{1+\beta _s}f_0</math>

On the other hand, given the scenario where source is stationary, and the receiver is moving directly away from the source at a speed of ''v<sub>r</sub>'' for a velocity parameter of ''β<sub>r</sub>'', the wavelength is ''not'' changed, but the transmission velocity of the waves relative to the receiver is decreased, and the observed frequency ''f'' is given by
: <math>f = (1-\beta _r)f_0</math>

[[File:Spacetime Diagram of Relativistic Doppler Effect.svg|thumb|Figure 3–6. Spacetime diagram of relativistic Doppler effect]]
Light, unlike sound or water ripples, does not propagate through a medium, and there is no distinction between a source moving away from the receiver or a receiver moving away from the source. Fig.&nbsp;3-6 illustrates a relativistic spacetime diagram showing a source separating from the receiver with a velocity parameter <math>\beta ,</math> so that the separation between source and receiver at time <math>w</math> is <math>\beta w </math>. Because of time dilation, <math>w = \gamma w' .</math> Since the slope of the green light ray is −1, <math>T =  w + \beta w = \gamma w' (1 + \beta) .</math> Hence, the [[relativistic Doppler effect]] is given by<ref name="Bais" />{{rp|58–59}}
: <math>f = \sqrt{\frac{1 - \beta}{1 + \beta}}\,f_0.</math>

==== Transverse Doppler effect ====
[[File:Transverse Doppler effect scenarios 2.svg|thumb|upright=1.4|Figure 3–7. Transverse Doppler effect scenarios]]
Suppose that a source and a receiver, both approaching each other in uniform inertial motion along non-intersecting lines, are at their closest approach to each other. It would appear that the classical analysis predicts that the receiver detects no Doppler shift. Due to subtleties in the analysis, that expectation is not necessarily true. Nevertheless, when appropriately defined, transverse Doppler shift is a relativistic effect that has no classical analog. The subtleties are these:<ref name="Morin2008">{{cite book |last1=Morin |first1=David |title=Introduction to Classical Mechanics: With Problems and Solutions |date=2008 |publisher=Cambridge University Press |isbn=978-0-521-87622-3 |url=https://archive.org/details/introductiontocl00mori }}</ref>{{rp|541–543}}
{{plainlist|
* Fig. 3-7a. What is the frequency measurement when the receiver is geometrically at its closest approach to the source? This scenario is most easily analyzed from the frame S′ of the source.<ref group=note>The ease of analyzing a relativistic scenario often depends on the frame in which one chooses to perform the analysis. '''[[:File:Transverse Doppler effect scenarios 3.svg|In this linked image]]''', we present alternative views of the transverse Doppler shift scenario where source and receiver are at their closest approach to each other.

(a) If we analyze the scenario in the frame of the receiver, we find that the analysis is more complicated than it should be. The apparent position of a celestial object is displaced from its true position (or geometric position) because of the object's motion during the time it takes its light to reach an observer. The source would be time-dilated relative to the receiver, but the redshift implied by this time dilation would be offset by a blueshift due to the longitudinal component of the relative motion between the receiver and the apparent position of the source.

(b) It is much easier if, instead, we analyze the scenario from the frame of the source. An observer situated at the source knows, from the problem statement, that the receiver is at its closest point to him. That means that the receiver has no longitudinal component of motion to complicate the analysis. Since the receiver's clocks are time-dilated relative to the source, the light that the receiver receives is therefore blue-shifted by a factor of ''gamma''.</ref>
* Fig. 3-7b. What is the frequency measurement when the receiver ''sees'' the source as being closest to it? This scenario is most easily analyzed from the frame S of the receiver.

Two other scenarios are commonly examined in discussions of transverse Doppler shift:
* Fig. 3-7c. If the receiver is moving in a circle around the source, what frequency does the receiver measure?
* Fig. 3-7d. If the source is moving in a circle around the receiver, what frequency does the receiver measure?
}}<!—end plainlist—>
In scenario (a), the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time (i.e. dr/dt&nbsp;=&nbsp;0 where ''r'' is the distance between receiver and source) and hence no longitudinal Doppler shift. The source observes the receiver as being illuminated by light of frequency {{′|''f''}}, but also observes the receiver as having a time-dilated clock. In frame&nbsp;S, the receiver is therefore illuminated by [[blueshifted]] light of frequency
: <math>f = f' \gamma = f' / \sqrt { 1 - \beta ^2  }</math>

In scenario (b) the illustration shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on. Because the source's clocks are time dilated as measured in frame S, and since dr/dt was equal to zero at this point, the light from the source, emitted from this closest point, is [[redshifted]] with frequency
: <math>f = f' / \gamma = f' \sqrt { 1 - \beta ^2  }</math>

Scenarios (c) and (d) can be analyzed by simple time dilation arguments. In (c), the receiver observes light from the source as being blueshifted by a factor of <math>\gamma</math>, and in (d), the light is redshifted. The only seeming complication is that the orbiting objects are in accelerated motion. However, if an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation. (The converse, however, is not true.)<ref name="Morin2008" />{{rp|541–543}} Most reports of transverse Doppler shift refer to the effect as a redshift and analyze the effect in terms of scenarios (b) or (d).<ref group="note">Not all experiments characterize the effect in terms of a redshift. For example, the [[Ives–Stilwell experiment#Relativistic Doppler effect|Kündig experiment]] measures transverse blueshift using a Mössbauer source setup at the center of a centrifuge rotor and an absorber at the rim.</ref>

{{anchor|Energy and momentum}}

=== Energy and momentum ===
{{Main|Four-momentum|Momentum|Mass–energy equivalence}}

==== Extending momentum to four dimensions ====
[[File:Relativistic spacetime momentum vector.svg|thumb|upright=1.5|Figure 3–8. Relativistic spacetime momentum vector. The coordinate axes of the rest frame are: momentum, p, and mass * c. For comparison, we have overlaid a spacetime coordinate system with axes: position, and time * c.]]
In classical mechanics, the state of motion of a particle is characterized by its mass and its velocity. [[Linear momentum]], the product of a particle's mass and velocity, is a [[Euclidean vector|vector]] quantity, possessing the same direction as the velocity: {{math|1='''''p'''''&nbsp;=&nbsp;''m'''v'''''}}. It is a ''conserved'' quantity, meaning that if a [[closed system]] is not affected by external forces, its total linear momentum cannot change.

In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector {{tmath|(x,t)}}. In exploring the properties of the spacetime momentum, we start, in Fig.&nbsp;3-8a, by examining what a particle looks like at rest. In the rest frame, the spatial component of the momentum is zero, i.e. {{math|1=''p''&nbsp;=&nbsp;0}}, but the time component equals ''mc''.

We can obtain the transformed components of this vector in the moving frame by using the Lorentz transformations, or we can read it directly from the figure because we know that {{tmath|1=(m c)^{\prime}=\gamma m c}} and {{tmath|1=p^{\prime}=-\beta \gamma m c}}, since the red axes are rescaled by gamma. Fig.&nbsp;3-8b illustrates the situation as it appears in the moving frame. It is apparent that the space and time components of the four-momentum go to infinity as the velocity of the moving frame approaches ''c''.<ref name="Bais" />{{rp|84–87}}

We will use this information shortly to obtain an expression for the [[four-momentum]].

==== Momentum of light ====
[[File:Calculating the energy of light in different inertial frames.svg|thumb|Figure 3–9. Energy and momentum of light in different inertial frames]]
Light particles, or photons, travel at the speed of ''c'', the constant that is conventionally known as the ''speed of light''. This statement is not a tautology, since many modern formulations of relativity do not start with constant speed of light as a postulate. Photons therefore propagate along a lightlike world line and, in appropriate units, have equal space and time components for every observer.

A consequence of [[Maxwell's theory]] of electromagnetism is that light carries energy and momentum, and that their ratio is a constant: {{tmath|1=E/p = c}}. Rearranging, {{tmath|1=E/c = p}}, and since for photons, the space and time components are equal, ''E''/''c'' must therefore be equated with the time component of the spacetime momentum vector.

Photons travel at the speed of light, yet have finite momentum and energy. For this to be so, the mass term in ''γmc'' must be zero, meaning that photons are [[massless particle]]s. Infinity times zero is an ill-defined quantity, but ''E''/''c'' is well-defined.

By this analysis, if the energy of a photon equals ''E'' in the rest frame, it equals {{tmath|1=E^{\prime}=(1-\beta) \gamma E }} in a moving frame. This result can be derived by inspection of Fig.&nbsp;3-9 or by application of the Lorentz transformations, and is consistent with the analysis of Doppler effect given previously.<ref name="Bais" />{{rp|88}}

==== Mass–energy relationship ====
Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to several important conclusions.

*In the low speed limit as {{math|1=''β'' = ''v''/''c''}} approaches zero, {{mvar|&gamma;}} approaches 1, so the spatial component of the relativistic momentum {{tmath|1=\beta \gamma m c=\gamma m v}} approaches ''mv'', the classical term for momentum. Following this perspective, ''&gamma;m'' can be interpreted as a relativistic generalization of ''m''. Einstein proposed that the ''[[relativistic mass]]'' of an object increases with velocity according to the formula {{tmath|1=m_\text{rel}=\gamma m}}.
*Likewise, comparing the time component of the relativistic momentum with that of the photon, {{tmath|1=\gamma m c=m_\text{rel} c=E / c}}, so that Einstein arrived at the relationship {{tmath|1=E=m_\text{rel} c^{2} }}. Simplified to the case of zero velocity, this is Einstein's equation relating energy and mass.

Another way of looking at the relationship between mass and energy is to consider a series expansion of {{math|1=''&gamma;mc''<sup>2</sup>}} at low velocity:
: <math> E = \gamma m c^2 =\frac{m c^2}{\sqrt{1 - \beta ^ 2}}</math> <math>\approx m c^2 + \frac{1}{2} m v^2 ...</math>
The second term is just an expression for the kinetic energy of the particle. Mass indeed appears to be another form of energy.<ref name="Bais" />{{rp|90–92}}<ref name="Morin" />{{rp|129–130,180}}

The concept of relativistic mass that Einstein introduced in 1905, ''m''<sub>rel</sub>, although amply validated every day in particle accelerators around the globe (or indeed in any instrumentation whose use depends on high velocity particles, such as electron microscopes,<ref>{{cite journal|last1=Rose|first1=H. H.|title=Optics of high-performance electron microscopes|journal=Science and Technology of Advanced Materials|date=21 April 2008|volume=9|issue=1|page=014107|doi=10.1088/0031-8949/9/1/014107|bibcode=2008STAdM...9a4107R|pmc=5099802|pmid=27877933}}</ref> old-fashioned color television sets, etc.), has nevertheless not proven to be a ''fruitful'' concept in physics in the sense that it is not a concept that has served as a basis for other theoretical development. Relativistic mass, for instance, plays no role in general relativity.

For this reason, as well as for pedagogical concerns, most physicists currently prefer a different terminology when referring to the relationship between mass and energy.<ref>{{cite book |last1=Griffiths |first1=David J. |title=Revolutions in Twentieth-Century Physics |date=2013 |publisher=Cambridge University Press |location=Cambridge |isbn=978-1-107-60217-5 |page=60 |url=https://books.google.com/books?id=Tv8cz-kN2z0C&pg=PA60 |access-date=24 May 2017 |language=en}}</ref> "Relativistic mass" is a deprecated term. The term "mass" by itself refers to the rest mass or [[invariant mass]], and is equal to the invariant length of the relativistic momentum vector. Expressed as a formula,
: <math> E^2 - p^2c^2 = m_\text{rest}^2 c^4 </math>
This formula applies to all particles, massless as well as massive. For photons where ''m''<sub>rest</sub> equals zero, it yields, {{tmath|1=E=\pm p c}}.<ref name="Bais" />{{rp|90–92}}

==== Four-momentum ====
Because of the close relationship between mass and energy, the four-momentum (also called 4-momentum) is also called the energy–momentum 4-vector. Using an uppercase ''P'' to represent the four-momentum and a lowercase '''''p''''' to denote the spatial momentum, the four-momentum may be written as
: <math>P \equiv (E/c, \vec{p}) = (E/c, p_x, p_y, p_z)</math> or alternatively,
: <math>P \equiv (E, \vec{p}) = (E, p_x, p_y, p_z) </math> using the convention that <math>c = 1 .</math><ref name="Morin" />{{rp|129–130,180}}

{{anchor|Conservation laws}}

=== Conservation laws ===
{{Main|Conservation law}}

In physics, conservation laws state that certain particular measurable properties of an isolated physical system do not change as the system evolves over time. In 1915, [[Emmy Noether]] discovered that underlying each conservation law is a fundamental symmetry of nature.<ref>{{cite arXiv|last1=Byers|first1=Nina|title=E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws|eprint=physics/9807044|year=1998}}</ref> The fact that physical processes do not care ''where'' in space they take place ([[space translation symmetry]]) yields [[conservation of momentum]], the fact that such processes do not care ''when'' they take place ([[time translation symmetry]]) yields [[conservation of energy]], and so on. In this section, we examine the Newtonian views of conservation of mass, momentum and energy from a relativistic perspective.

==== Total momentum ====
[[File:Relativistic conservation of momentum.png|thumb|Figure 3–10. Relativistic conservation of momentum]]
To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension.

In Newtonian mechanics, two extreme cases of this problem may be distinguished yielding mathematics of minimum complexity:
: (1) The two bodies rebound from each other in a completely elastic collision.
: (2) The two bodies stick together and continue moving as a single particle. This second case is the case of completely inelastic collision.
For both cases (1) and (2), momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat.

In case (2), two masses with momentums {{tmath|1=\boldsymbol{p}_{\boldsymbol{1} }=m_{1} \boldsymbol{v}_{\boldsymbol{1} } }}
and {{tmath|1=\boldsymbol{p}_{\boldsymbol{2} }=m_{2} \boldsymbol{v}_{\boldsymbol{2} } }} collide to produce a single particle of conserved mass {{tmath|1=m=m_{1}+m_{2} }} traveling at the [[center of mass]] velocity of the original system, <math>\boldsymbol{v_{c m}}=\left(m_{1} \boldsymbol{v_1}+m_{2} \boldsymbol{v_2}\right) /\left(m_{1}+m_{2}\right) </math>. The total momentum {{tmath|1=\boldsymbol{p=p_{1}+p_{2} } }} is conserved.

Fig.&nbsp;3-10 illustrates the inelastic collision of two particles from a relativistic perspective. The time components {{tmath|E_{1} / c}} and {{tmath|E_{2} / c}} add up to total ''E/c'' of the resultant vector, meaning that energy is conserved. Likewise, the space components {{tmath|1=\boldsymbol{p_{1} } }} and {{tmath|1=\boldsymbol{p_{2} } }} add up to form ''p'' of the resultant vector. The four-momentum is, as expected, a conserved quantity. However, the invariant mass of the fused particle, given by the point where the invariant hyperbola of the total momentum intersects the energy axis, is not equal to the sum of the invariant masses of the individual particles that collided. Indeed, it is larger than the sum of the individual masses: {{tmath|1=m>m_{1}+m_{2} }}.<ref name="Bais" />{{rp|94–97}}

Looking at the events of this scenario in reverse sequence, we see that non-conservation of mass is a common occurrence: when an unstable [[elementary particle]] spontaneously decays into two lighter particles, total energy is conserved, but the mass is not. Part of the mass is converted into kinetic energy.<ref name="Morin" />{{rp|134–138}}

==== Choice of reference frames ====
{{multiple image|align=right|image1=2-body Particle Decay-Lab.svg|width1=115|image2=2-body Particle Decay-CoM.svg|width2=105|caption1=Figure 3-11. <br />(above) '''Lab Frame'''.<br />(right) '''Center of Momentum Frame'''.|&nbsp;}}
The freedom to choose any frame in which to perform an analysis allows us to pick one which may be particularly convenient. For analysis of momentum and energy problems, the most convenient frame is usually the "[[center-of-momentum frame]]" (also called the zero-momentum frame, or COM frame). This is the frame in which the space component of the system's total momentum is zero. Fig.&nbsp;3-11 illustrates the breakup of a high speed particle into two daughter particles. In the lab frame, the daughter particles are preferentially emitted in a direction oriented along the original particle's trajectory. In the COM frame, however, the two daughter particles are emitted in opposite directions, although their masses and the magnitude of their velocities are generally not the same.<ref name="Idema_2022">{{cite web |last1=Idema |first1=Timon |title=Mechanics and Relativity. Chapter 4.3: Reference Frames |url=https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_(Idema)/04%3A_Momentum/4.03%3A_Reference_Frames |website=LibreTexts Physics  |date=17 April 2019 |publisher=California State University Affordable Learning Solutions Program |access-date=6 July 2024}}</ref>

==== Energy and momentum conservation ====
In a Newtonian analysis of interacting particles, transformation between frames is simple because all that is necessary is to apply the Galilean transformation to all velocities. Since {{tmath|1=v' = v - u}}, the momentum {{tmath|1=p' = p - mu}}. If the total momentum of an interacting system of particles is observed to be conserved in one frame, it will likewise be observed to be conserved in any other frame.<ref name="Morin" />{{rp|241–245}}

Conservation of momentum in the COM frame amounts to the requirement that {{math|1=''p''&nbsp;=&nbsp;0}} both before and after collision. In the Newtonian analysis, conservation of mass dictates that {{tmath|1=m=m_{1}+m_{2} }}. In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition. In the one-dimensional case of a completely elastic collision with no loss of kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities. In the case of a completely inelastic collision with total loss of kinetic energy, the outgoing velocities of the rebounding particles will be zero.<ref name="Morin" />{{rp|241–245}}

Newtonian momenta, calculated as {{tmath|1=p = mv}}, fail to behave properly under Lorentzian transformation. The linear transformation of velocities {{tmath|1=v' = v - u}} is replaced by the highly nonlinear
{{tmath|1= v^{\prime} = (v-u) / (1- {v u}/{ c^{2} } )}} so that a calculation demonstrating conservation of momentum in one frame will be invalid in other frames. Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum. This second option was what he chose.<ref name="Bais" />{{rp|104}}

{{multiple image
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 | image1            = Energy-momentum diagram for pion decay (A).png
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 | caption1          = Figure 3-12a. Energy–momentum diagram for decay of a charged pion.
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 | caption2          = Figure 3-12b. Graphing calculator analysis of charged pion decay.
}}
The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications. Kinetic energy converted into heat or internal potential energy shows up as an increase in mass.<ref name="Morin" />{{rp|127}}

{{smalldiv|1=
'''Example:''' Because of the equivalence of mass and energy, elementary particle masses are customarily stated in energy units, where {{nowrap|1=1 MeV = 10<sup>6</sup>}} electron volts. A charged pion is a particle of mass 139.57&nbsp;MeV (approx. 273 times the electron mass). It is unstable, and decays into a muon of mass 105.66&nbsp;MeV (approx. 207 times the electron mass) and an antineutrino, which has an almost negligible mass. The difference between the pion mass and the muon mass is 33.91&nbsp;MeV.
: {{SubatomicParticle|Pion-}} → {{SubatomicParticle|link=yes|Muon-}} + {{SubatomicParticle|link=yes|Muon antineutrino}}

Fig.&nbsp;3-12a illustrates the energy–momentum diagram for this decay reaction in the rest frame of the pion. Because of its negligible mass, a neutrino travels at very nearly the speed of light. The relativistic expression for its energy, like that of the photon, is {{tmath|1=E_{v}=p c,}} which is also the value of the space component of its momentum. To conserve momentum, the muon has the same value of the space component of the neutrino's momentum, but in the opposite direction.

Algebraic analyses of the energetics of this decay reaction are available online,<ref>{{cite web|last1=Nave|first1=R.|title=Energetics of Charged Pion Decay|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/piondec.html|website=Hyperphysics|publisher=Department of Physics and Astronomy, Georgia State University|access-date=27 May 2017|archive-date=21 May 2017|archive-url=https://web.archive.org/web/20170521075304/http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/piondec.html|url-status=live}}</ref> so Fig.&nbsp;3-12b presents instead a graphing calculator solution. The energy of the neutrino is 29.79&nbsp;MeV, and the energy of the muon is {{nowrap|1=33.91  MeV − 29.79  MeV = 4.12 MeV}}. Most of the energy is carried off by the near-zero-mass neutrino.
}}