Editing Special relativity (section)

== Consequences derived from the Lorentz transformation ==
{{See also|Twin paradox|Relativistic mechanics}}

The consequences of special relativity can be derived from the [[Lorentz transformation]] equations.<ref>{{cite book |title=Introduction to special relativity |author=Robert Resnick |publisher=Wiley |date=1968|pages=62–63 |isbn=9780471717249 |url=https://books.google.com/books?id=fsIRAQAAIAAJ}}</ref> These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics at all relative velocities, and most pronounced when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything most humans encounter that some of the effects predicted by relativity are initially [[counterintuitive]].

=== Invariant interval ===
In Galilean relativity, the spatial separation, ({{tmath|1= \Delta r }}), and the temporal separation, ({{tmath|1= \Delta t }}), between two events are independent invariants, the values of which do not change when observed from different frames of reference.
In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an '''invariant interval''', denoted as {{tmath|1= \Delta s^2 }}:
<math display="block"> \Delta s^2 \; \overset\text{def}{=} \; c^2 \Delta t^2 - (\Delta x^2 + \Delta y^2 + \Delta z^2) </math>
In considering the physical significance of {{tmath|1= \Delta s^2 }}, there are three cases to note:<ref name=Morin2007/><ref name="Taylor1992"/>{{rp|25–39}}
* '''Δs<sup>2</sup> > 0:''' In this case, the two events are separated by more time than space, and they are hence said to be ''timelike'' separated. This implies that {{tmath|1= \vert \Delta x / \Delta t \vert < c }}, and given the Lorentz transformation {{tmath|1= \Delta x' = \gamma \ (\Delta x - v \ \Delta t) }}, it is evident that there exists a <math>v</math> less than <math>c</math> for which <math>\Delta x' = 0</math> (in particular, {{tmath|1= v = \Delta x / \Delta t }}). In other words, given two events that are timelike separated, it is possible to find a frame in which the two events happen at the same place. In this frame, the separation in time, {{tmath|1= \Delta s / c }}, is called the ''proper time''.
* '''Δs<sup>2</sup> < 0:''' In this case, the two events are separated by more space than time, and they are hence said to be ''spacelike'' separated. This implies that {{tmath|1= \vert \Delta x / \Delta t \vert > c }}, and given the Lorentz transformation {{tmath|1= \Delta t' = \gamma \ (\Delta t - v \Delta x / c^2) }}, there exists a <math>v</math> less than <math>c</math> for which <math>\Delta t' = 0</math> (in particular, {{tmath|1= v = c^2 \Delta t / \Delta x }}). In other words, given two events that are spacelike separated, it is possible to find a frame in which the two events happen at the same time. In this frame, the separation in space, {{tmath|1= \textstyle \sqrt { - \Delta s^2 } }}, is called the ''proper distance'', or ''proper length''. For values of <math>v</math> greater than and less than {{tmath|1= c^2 \Delta t / \Delta x }}, the sign of <math>\Delta t'</math> changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. But the temporal order of timelike-separated events is absolute, since the only way that <math>v</math> could be greater than <math> c^2 \Delta t / \Delta x</math> would be if {{tmath|1= v > c }}.
* '''Δs<sup>2</sup> = 0:''' In this case, the two events are said to be ''lightlike'' separated. This implies that {{tmath|1= \vert \Delta x / \Delta t \vert = c }}, and this relationship is frame independent due to the invariance of {{tmath|1= s^2 }}. From this, we observe that the speed of light is <math>c</math> in every inertial frame. In other words, starting from the assumption of universal Lorentz covariance, the constant speed of light is a derived result, rather than a postulate as in the two-postulates formulation of the special theory.

The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames.

The form of {{tmath|1= \Delta s^2 }}, being the ''difference'' of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances.{{refn| group=note|The invariance of Δ''s''<sup>2</sup> under standard Lorentz transformation in analogous to the invariance of squared distances Δ''r''<sup>2</sup> under rotations in Euclidean space. Although space and time have an equal ''footing'' in relativity, the minus sign in front of the spatial terms marks space and time as being of essentially different character. They are not the same. Because it treats time differently than it treats the 3 spatial dimensions, [[Minkowski space]] differs from [[four-dimensional Euclidean space]].}} The invariance of this interval is a property of the ''general'' Lorentz transform (also called the [[Poincaré transformation]]), making it an [[isometry]] of spacetime. The general Lorentz transform extends the standard Lorentz transform (which deals with translations without rotation, that is, [[Lorentz boost]]s, in the x-direction) with all other [[translation (geometry)|translations]], [[Reflection (mathematics)|reflections]], and [[rotation (mathematics)|rotations]] between any Cartesian inertial frame.<ref name="Rindler1977">{{cite book |last1=Rindler |first1=Wolfgang |title=Essential Relativity |date=1977 | publisher=Springer-Verlag |location=New York |isbn=978-0-387-10090-6 |edition=2nd}}</ref>{{rp|33–34}}

In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed:
<math display="block">\Delta s^2 \, = \, c^2 \Delta t^2 - \Delta x^2</math>

Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration:<ref name=Morin2007/>
<math display="block">\begin{align}
 c^2 \Delta t^2 - \Delta x^2 &= c^2 \gamma ^2 \left(\Delta t' + \dfrac{v \Delta x'}{c^2} \right)^2 - \gamma ^2 \ (\Delta x' + v \Delta t')^2 \\
&= \gamma ^2 \left( c^2 \Delta t' ^ {\, 2} + 2 v \Delta x' \Delta t' + \dfrac{v^2 \Delta x' ^ {\, 2}}{c^2} \right) - \gamma ^2 \ (\Delta x' ^ {\, 2} + 2 v \Delta x' \Delta t' + v^2 \Delta t' ^ {\, 2}) \\
&= \gamma ^2 c^2 \Delta t' ^ {\, 2} - \gamma ^2 v^2 \Delta t' ^{\, 2} - \gamma ^2 \Delta x' ^ {\, 2} + \gamma ^2 \dfrac{v^2 \Delta x' ^ {\, 2}}{c^2} \\
&= \gamma ^2 c^2 \Delta t' ^ {\, 2} \left( 1 - \dfrac{v^2}{c^2} \right) - \gamma ^2 \Delta x' ^{\, 2} \left( 1 - \dfrac{v^2}{c^2} \right) \\
&= c^2 \Delta t' ^{\, 2} - \Delta x' ^{\, 2}
\end{align}</math>

The value of <math>\Delta s^2</math> is hence independent of the frame in which it is measured.

=== Relativity of simultaneity ===
{{See also|Relativity of simultaneity|Ladder paradox}}
[[File:Relativity of Simultaneity Animation.gif|thumb|Figure 4–1. The three events (A, B, C) are simultaneous in the reference frame of some observer '''O'''. In a reference frame moving at ''v'' = 0.3''c'', as measured by '''O''', the events occur in the order C, B, A. In a reference frame moving at {{nowrap|1=''v'' = −0.5''c''}} with respect to '''O''', the events occur in the order A, B, C. The white lines, the ''lines of simultaneity'', move from the past to the future in the respective frames (green coordinate axes), highlighting events residing on them. They are the locus of all events occurring at the same time in the respective frame. The gray area is the [[light cone]] with respect to the origin of all considered frames.]]

Consider two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer. They may occur non-simultaneously in the reference frame of another inertial observer (lack of [[absolute simultaneity]]).

From {{EquationNote|3|Equation&nbsp;3}} (the forward Lorentz transformation in terms of coordinate differences)
<math display="block">\Delta t' = \gamma \left(\Delta t - \frac{v \,\Delta x}{c^{2}} \right)</math>

It is clear that the two events that are simultaneous in frame ''S'' (satisfying {{nowrap|1=Δ''t'' = 0}}), are not necessarily simultaneous in another inertial frame {{prime|''S''}} (satisfying {{nowrap|1=Δ{{prime|''t''}} = 0}}). Only if these events are additionally co-local in frame ''S'' (satisfying {{nowrap|1=Δ''x'' = 0}}), will they be simultaneous in another frame {{prime|''S''}}.

The [[Sagnac effect]] can be considered a manifestation of the relativity of simultaneity.<ref name="Ashby2003">{{cite journal |last1=Ashby |first1=Neil |title=Relativity in the Global Positioning System |journal=Living Reviews in Relativity |volume=6 |issue=1 |pages=1 |doi=10.12942/lrr-2003-1 |pmid=28163638 |pmc=5253894 |year=2003 |doi-access=free |bibcode=2003LRR.....6....1A }}</ref> Since relativity of simultaneity is a first order effect in {{tmath|1= v }},<ref name=Morin2007/> instruments based on the Sagnac effect for their operation, such as [[ring laser gyroscope]]s and [[fiber optic gyroscope]]s, are capable of extreme levels of sensitivity.<ref name="Lin1979" group="p">{{cite journal |last1=Lin |first1=Shih-Chun |last2=Giallorenzi |first2=Thomas G. |title=Sensitivity analysis of the Sagnac-effect optical-fiber ring interferometer |journal=Applied Optics |date=1979 |volume=18 |issue=6 |pages=915–931 |doi=10.1364/AO.18.000915|pmid=20208844 |bibcode=1979ApOpt..18..915L |s2cid=5343180 }}</ref>

=== Time dilation ===
{{See also|Time dilation}}
The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames.

Suppose a [[clock]] is at rest in the unprimed system ''S''. The location of the clock on two different ticks is then characterized by {{nowrap|1=Δ''x'' = 0}}. To find the relation between the times between these ticks as measured in both systems, {{EquationNote|3|Equation&nbsp;3}} can be used to find:
: <math>\Delta t' = \gamma\, \Delta t </math>{{pad|4}}for events satisfying{{pad|4}}<math>\Delta x = 0 \ .</math>

This shows that the time (Δ{{prime|''t''}}) between the two ticks as seen in the frame in which the clock is moving ({{prime|''S''}}), is ''longer'' than the time (Δ''t'') between these ticks as measured in the rest frame of the clock (''S''). Time dilation explains a number of physical phenomena; for example, the lifetime of high speed [[muon]]s created by the collision of cosmic rays with particles in the Earth's outer atmosphere and moving towards the surface is greater than the lifetime of slowly moving muons, created and decaying in a laboratory.<ref>{{cite book|author1=Daniel Kleppner|author2=David Kolenkow|name-list-style=amp|title=An Introduction to Mechanics|date=1973|pages=[https://archive.org/details/introductiontome00dani/page/468 468–70]|publisher=McGraw-Hill |isbn=978-0-07-035048-9|url=https://archive.org/details/introductiontome00dani/page/468}}</ref>

[[File:Observer_in_special_relativity.svg|thumb|Figure 4–2. Hypothetical infinite array of synchronized clocks associated with an observer's reference frame]]
Whenever one hears a statement to the effect that "moving clocks run slow", one should envision an inertial reference frame thickly populated with identical, synchronized clocks. As a moving clock travels through this array, its reading at any particular point is compared with a stationary clock at the same point.<ref name="French_1968"/>{{rp|149–152}}

The measurements that we would get if we actually ''looked'' at a moving clock would, in general, not at all be the same thing, because the time that we would see would be delayed by the finite speed of light, i.e. the times that we see would be distorted by the [[Doppler effect]]. Measurements of relativistic effects must always be understood as having been made after finite speed-of-light effects have been factored out.<ref name="French_1968">{{cite book |last1=French |first1=A. P. |title=Special Relativity |date=1968 |publisher=W. W. Norton & Company |location=New York |isbn=0-393-09793-5}}</ref>{{rp|149–152}}

==== Langevin's light-clock ====
{{anchor|Langevin's Light-Clock}}
[[File:Langevin Light Clock.gif|thumb|320px|Figure 4–3. Thought experiment using a light-clock to explain time dilation]]
[[Paul Langevin]], an early proponent of the theory of relativity, did much to popularize the theory in the face of resistance by many physicists to Einstein's revolutionary concepts. Among his numerous contributions to the foundations of special relativity were independent work on the mass–energy relationship, a thorough examination of the twin paradox, and investigations into rotating coordinate systems. His name is frequently attached to a hypothetical construct called a "light-clock" (originally developed by Lewis and Tolman in 1909<ref name="Lewis_Tolman_1909">{{cite journal |last1=Lewis |first1=Gilbert Newton |authorlink1=Gilbert N. Lewis|last2=Tolman |first2=Richard Chase| authorlink2=Richard Chase Tolman| title=The Principle of Relativity, and Non-Newtonian Mechanics |journal=Proceedings of the American Academy of Arts and Sciences |date=1909 |volume=44 |issue=25 |pages=709–726 |doi=10.2307/20022495 |jstor=20022495 |url=https://en.wikisource.org/wiki/The_Principle_of_Relativity,_and_Non-Newtonian_Mechanics |access-date=22 August 2023}}</ref>), which he used to perform a novel derivation of the Lorentz transformation.<ref name="Cuvaj_1971">{{cite journal |last1=Cuvaj |first1=Camillo |title=Paul Langeyin and the Theory of Relativity |journal=Japanese Studies in the History of Science |date=1971 |volume=10 |pages=113–142 |url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf |access-date=12 June 2023}}</ref>

A light-clock is imagined to be a box of perfectly reflecting walls wherein a light signal reflects back and forth from opposite faces. The concept of time dilation is frequently taught using a  light-clock that is traveling in uniform inertial motion perpendicular to a line connecting the two mirrors.<ref>{{Cite book |last1=Cassidy |first1=David C. |url=https://books.google.com/books?id=rpQo7f9F1xUC&pg=PA422 |title=Understanding Physics |last2=Holton |first2=Gerald James |last3=Rutherford |first3=Floyd James |publisher=[[Springer-Verlag]] |year=2002 |isbn=978-0-387-98756-9 |pages=422}}</ref><ref>{{Cite book |last=Cutner |first=Mark Leslie |url=https://books.google.com/books?id=2QVmiMW0O0MC&pg=PA128 |title=Astronomy, A Physical Perspective |publisher=[[Cambridge University Press]] |year=2003 |isbn=978-0-521-82196-4 |page=128}}</ref><ref>{{Cite book |last1=Ellis |first1=George F. R. |url=https://books.google.com/books?id=Hos31wty5WIC&pg=PA28 |title=Flat and Curved Space-times |last2=Williams |first2=Ruth M. |publisher=[[Oxford University Press]] |year=2000 |isbn=978-0-19-850657-7 |edition=2n |pages=28–29}}</ref><ref name="Feynman_Lectures_1">{{cite book |last1=Feynman |first1=Richard P. |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew |title=The feynman lectures on physics; vol I: The new millennium edition |date=2011 |publisher=Basic Books |isbn=978-0-465-02414-8 |page=15-5 |url=https://www.feynmanlectures.caltech.edu/I_15.html |access-date=12 June 2023}}</ref> (Langevin himself made use of a light-clock oriented parallel to its line of motion.<ref name="Cuvaj_1971"/>)

Consider the scenario illustrated in {{nowrap|Fig. 4-3A.}} Observer A holds a light-clock of length <math>L</math> as well as an electronic timer with which she measures how long it takes a pulse to make a round trip up and down along the light-clock. Although observer A is traveling rapidly along a train, from her point of view the emission and receipt of the pulse occur at the same place, and she measures the interval using a single clock located at the precise position of these two events. For the interval between these two events, observer A finds {{tmath|1= t_\text{A} = 2 L/c }}. A time interval measured using a single clock that is motionless in a particular reference frame is called a ''[[proper time interval]]''.<ref name="Halliday_1988">{{cite book |last1=Halliday |first1=David |last2=Resnick |first2=Robert |title=Fundamental Physics: Extended Third Edition |date=1988 |publisher=John Wiley & sons |location=New York |isbn=0-471-81995-6 |pages=958–959}}</ref>

Fig. 4-3B illustrates these same two events from the standpoint of observer B, who is parked by the tracks as the train goes by at a speed of {{tmath|1= v }}. Instead of making straight up-and-down motions, observer B sees the pulses moving along a zig-zag line. However, because of the postulate of the constancy of the speed of light, the speed of the pulses along these diagonal lines is the same <math>c</math> that observer A saw for her up-and-down pulses. B measures the speed of the vertical component of these pulses as <math display=inline>\pm \sqrt{c^2 - v^2},</math> so that the total round-trip time of the pulses is <math display=inline>t_\text{B} = 2L \big/ \sqrt{ c^2 - v^2 } = {}</math>{{tmath|1= \textstyle t_\text{A} \big/ \sqrt {1 - v^2 / c^2} }}. Note that for observer B, the emission and receipt of the light pulse occurred at different places, and he measured the interval using two stationary and synchronized clocks located at two different positions in his reference frame. The interval that B measured was therefore ''not'' a proper time interval because he did not measure it with a single resting clock.<ref name="Halliday_1988"/>

==== Reciprocal time dilation ====
In the above description of the Langevin light-clock, the labeling of one observer as stationary and the other as in motion was completely arbitrary. One could just as well have observer B carrying the light-clock and moving at a speed of <math>v</math> to the left, in which case observer A would perceive B's clock as running slower than her local clock.

There is no paradox here, because there is no independent observer C who will agree with both A and B. Observer C necessarily makes his measurements from his own reference frame. If that reference frame coincides with A's reference frame, then C will agree with A's measurement of time. If C's reference frame coincides with B's reference frame, then C will agree with B's measurement of time. If C's reference frame coincides with neither A's frame nor B's frame, then C's measurement of time will disagree with ''both'' A's and B's measurement of time.<ref>{{Cite book |last=Adams |first=Steve |url=https://books.google.com/books?id=1RV0AysEN4oC&pg=PA54 |title=Relativity: An introduction to space-time physics |publisher=[[CRC Press]] |year=1997 |isbn=978-0-7484-0621-0 |page=54}}</ref>

=== Twin paradox ===
{{See also|Twin paradox}}
The reciprocity of time dilation between two observers in separate inertial frames leads to the so-called [[twin paradox]], articulated in its present form by Langevin in 1911.<ref name="Langevin_1911">{{cite journal |last1=Langevin |first1=Paul |title=L'Évolution de l'espace et du temps |journal=Scientia |date=1911 |volume=10 |pages=31–54 |url=https://en.wikisource.org/wiki/Translation:The_Evolution_of_Space_and_Time |access-date=20 June 2023}}</ref> Langevin imagined an adventurer wishing to explore the future of the Earth. This traveler boards a projectile capable of traveling at 99.995% of the speed of light. After making a round-trip journey to and from a nearby star lasting only two years of his own life, he returns to an Earth that is two hundred years older.

This result appears puzzling because both the traveler and an Earthbound observer would see the other as moving, and so, because of the reciprocity of time dilation, one might initially expect that each should have found the other to have aged less. In reality, there is no paradox at all, because in order for the two observers to perform side-by-side comparisons of their elapsed proper times, the symmetry of the situation must be broken: At least one of the two observers must change their state of motion to match that of the other.<ref name='Debs_Redhead'>{{cite journal |author1=Debs, Talal A. |author2=Redhead, Michael L.G. |title=The twin "paradox" and the conventionality of simultaneity |journal=American Journal of Physics |volume=64 |issue=4 |year=1996 |pages=384–392 |doi=10.1119/1.18252 |bibcode=1996AmJPh..64..384D}}</ref>

[[File:Twin paradox Doppler analysis.svg|thumb|Figure 4-4. Doppler analysis of twin paradox]]
Knowing the general resolution of the paradox, however, does not immediately yield the ability to calculate correct quantitative results. Many solutions to this puzzle have been provided in the literature and have been reviewed in the [[Twin paradox]] article. We will examine in the following one such solution to the paradox.

Our basic aim will be to demonstrate that, after the trip, both twins are in perfect agreement about who aged by how much, regardless of their different experiences. {{nowrap|Fig 4-4}} illustrates a scenario where the traveling twin flies at {{nowrap|0.6 c}} to and from a star {{nowrap|3 ly}} distant. During the trip, each twin sends yearly time signals (measured in their own proper times) to the other. After the trip, the cumulative counts are compared. On the outward phase of the trip, each twin receives the other's signals at the lowered rate of {{tmath|1= \textstyle f' = f \sqrt{(1-\beta)/(1+\beta)} }}. Initially, the situation is perfectly symmetric: note that each twin receives the other's one-year signal at two years measured on their own clock. The symmetry is broken when the traveling twin turns around at the four-year mark as measured by her clock. During the remaining four years of her trip, she receives signals at the enhanced rate of {{tmath|1= \textstyle f' ' = f \sqrt{(1+\beta)/(1-\beta)} }}. The situation is quite different with the stationary twin. Because of light-speed delay, he does not see his sister turn around until eight years have passed on his own clock. Thus, he receives enhanced-rate signals from his sister for only a relatively brief period. Although the twins disagree in their respective measures of total time, we see in the following table, as well as by simple observation of the Minkowski diagram, that each twin is in total agreement with the other as to the total number of signals sent from one to the other. There is hence no paradox.<ref name="French_1968"/>{{rp|152–159}}

{| class="wikitable"
|-
! Item!! Measured by the{{br}}stay-at-home !! Fig 4-4 !! Measured by{{br}}the traveler !! Fig 4-4
|-
| Total time of trip
| <math>T = \frac{2L}{v}</math>
| {{nowrap|10 yr}}
| <math>T' =  \frac{2L}{\gamma v} </math>
| {{nowrap|8 yr}}
|-
| Total number of pulses sent
| <math>fT = \frac{2fL}{v}</math>
| 10
| <math>fT' = \frac{2fL}{\gamma v}</math>
| 8
|-
| Time when traveler's turnaround is '''detected'''
| <math>t_1 = \frac{L}{v} + \frac{L}{c}</math>
| {{nowrap|8 yr}}
| <math>t_1' = \frac{L}{\gamma v}</math>
| {{nowrap|4 yr}}
|-
| Number of pulses received at initial <math>f'</math> rate
| <math>f't_1</math> <math>= \frac{fL}{v}(1 + \beta)\left(\frac{1-\beta}{1+\beta}\right)^{1/2}</math>{{br}}<math>= \frac{fL}{v}(1 - \beta ^2)^{1/2}</math>
| 4
| <math>f't_1'</math> <math>= \frac{fL}{v}(1 - \beta^2)^{1/2}\left(\frac{1-\beta}{1+\beta}\right)^{1/2}</math>{{br}}<math>= \frac{fL}{v}(1 - \beta )</math>
| 2
|-
| Time for remainder of trip
| <math>t_2 = \frac{L}{v} - \frac{L}{c}</math>
| {{nowrap|2 yr}}
| <math>t_2' = \frac{L}{\gamma v}</math>
| {{nowrap|4 yr}}
|-
| Number of signals received at final <math>f''</math> rate
| <math>f''t_2</math> <math>= \frac{fL}{v}(1 - \beta)\left( \frac{1 + \beta}{1 - \beta} \right)^{1/2}</math> <math>= \frac{fL}{v}(1 - \beta ^2)^{1/2}</math>
| 4
| <math>f''t_2'</math> <math>= \frac{fL}{v}(1 - \beta^2)^{1/2}\left( \frac{1 + \beta}{1 - \beta} \right)^{1/2}</math> <math>= \frac{fL}{v}(1 + \beta)</math>
| 8
|-
| Total number of received pulses
| <math>\frac{2fL}{v}(1 - \beta ^2)^{1/2}</math> <math>= \frac{2fL}{\gamma v}</math>
| 8
| <math>\frac{2fL}{v}</math>
| 10
|-
| Twin's calculation as to how much the '''''other''''' twin should have aged
| <math>T' = \frac{2L}{\gamma v}</math>
| {{nowrap|8 yr}}
| <math>T = \frac{2L}{v}</math>
| {{nowrap|10 yr}}
|}

=== Length contraction ===
{{See also|Lorentz contraction}}
The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the [[ladder paradox]] involves a long ladder traveling near the speed of light and being contained within a smaller garage).

Similarly, suppose a [[measuring rod]] is at rest and aligned along the ''x''-axis in the unprimed system ''S''. In this system, the length of this rod is written as Δ''x''. To measure the length of this rod in the system {{prime|''S''}}, in which the rod is moving, the distances ''{{prime|x}}'' to the end points of the rod must be measured simultaneously in that system {{prime|''S''}}. In other words, the measurement is characterized by {{nowrap|1=Δ{{prime|''t''}} = 0}}, which can be combined with {{EquationNote|4|Equation&nbsp;4}} to find the relation between the lengths Δ''x'' and Δ{{prime|''x''}}:
: <math>\Delta x' = \frac{\Delta x}{\gamma} </math>{{pad|4}}{{pad|4}}for events satisfying{{pad|4}}<math>\Delta t' = 0 \ .</math>
This shows that the length (Δ{{prime|''x''}}) of the rod as measured in the frame in which it is moving ({{prime|''S''}}), is ''shorter'' than its length (Δ''x'') in its own rest frame (''S'').

Time dilation and length contraction are not merely appearances. Time dilation is explicitly related to our way of measuring ''time intervals'' between events that occur at the same place in a given coordinate system (called "co-local" events). These time intervals (which can be, and are, actually measured experimentally by relevant observers) are ''different'' in another coordinate system moving with respect to the first, unless the events, in addition to being co-local, are also simultaneous. Similarly, length contraction relates to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will ''not'' occur at the same ''spatial distance'' from each other when seen from another moving coordinate system.

=== Lorentz transformation of velocities ===
{{See also|Velocity-addition formula}}

Consider two frames ''S'' and {{prime|''S''}} in standard configuration. A particle in ''S'' moves in the x direction with velocity vector {{tmath|1= \mathbf{u} }}. What is its velocity <math>\mathbf{u'}</math> in frame {{prime|''S''}}?

We can write
{{NumBlk2||<math display="block"> \mathbf{|u|} = u = dx / dt \, . </math>|7}}
{{NumBlk2||<math display="block"> \mathbf{|u'|} = u' = dx' / dt' \, . </math>|8}}

Substituting expressions for <math>dx'</math> and <math>dt'</math> from {{EquationNote|5|Equation&nbsp;5}} into {{EquationNote|8|Equation&nbsp;8}}, followed by straightforward mathematical manipulations and back-substitution from {{EquationNote|7|Equation&nbsp;7}} yields the Lorentz transformation of the speed <math>u</math> to {{tmath|1= u' }}:
{{NumBlk2||<math display="block">u' = \frac{dx'}{dt'}=\frac{\gamma(dx-v\,dt)}{\gamma \left(dt-\dfrac{v\,dx}{c^2} \right)} = \frac{\dfrac{dx}{dt}-v}{1 - \dfrac{v}{c^2} \, \dfrac{dx}{dt} } =\frac{u-v}{1- \dfrac{uv}{c^2}}. </math>|9}}

The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing <math> v </math> with {{tmath|1= -v }}.
{{NumBlk2||<math display="block">u=\frac{u'+v}{1+ u'v / c^2}.</math>|10}}

For <math>\mathbf{u}</math> not aligned along the x-axis, we write:<ref name="Rindler0"/>{{rp|47–49}}
{{NumBlk2||<math display="block"> \mathbf{u} = (u_1, \ u_2,\ u_3 ) = ( dx / dt, \ dy/dt, \ dz/dt) \ . </math>|11}}
{{NumBlk2||<math display="block"> \mathbf{u'} = (u_1', \ u_2', \ u_3') = ( dx' / dt', \ dy'/dt', \ dz'/dt') \ . </math>|12}}

The forward and inverse transformations for this case are:
{{NumBlk2||<math display="block">u_1'=\frac{u_1 -v}{1-u_1 v / c^2 } \ , \qquad u_2'=\frac{u_2}{\gamma \left( 1-u_1 v / c^2 \right) } \ , \qquad u_3'=\frac{u_3}{\gamma \left( 1- u_1 v / c^2 \right) } \ . </math>|13}}
{{NumBlk2||<math display="block">u_1=\frac{u_1' +v}{1+ u_1' v / c^2 } \ , \qquad u_2=\frac{u_2'}{ \gamma \left( 1+ u_1' v / c^2 \right) } \ , \qquad u_3=\frac{u_3'}{\gamma \left( 1+ u_1' v / c^2 \right)} \ . </math>|14}}

{{EquationNote|10|Equation&nbsp;10}} and {{EquationNote|14|Equation&nbsp;14}} can be interpreted as giving the ''resultant'' <math> \mathbf{u} </math> of the two velocities <math> \mathbf{v} </math> and {{tmath|1= \mathbf{u'} }}, and they replace the formula {{tmath|1= \mathbf{u = u' + v} }}. which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the ''relativistic velocity addition (or composition) formulas'', valid for the three axes of ''S'' and {{prime|''S''}} being aligned with each other (although not necessarily in standard configuration).<ref name="Rindler0"/>{{rp|47–49}}

We note the following points:
* If an object (e.g., a [[photon]]) were moving at the speed of light in one frame {{nowrap|1=(i.e., ''u'' = ±''c''}} {{nowrap|1=or {{prime|''u''}} = ±''c'')}}, then it would also be moving at the speed of light in any other frame, moving at {{nowrap|{{abs|''v''}} < ''c''}}.
* The resultant speed of two velocities with magnitude less than ''c'' is always a velocity with magnitude less than ''c''.
* If both {{abs|''u''}} and {{abs|''v''}} (and then also {{abs|{{prime|''u''}}}} and {{abs|{{prime|''v''}}}}) are small with respect to the speed of light (that is, e.g., {{nowrap|{{abs|{{sfrac|''u''|''c''}}}} ≪ {{math|1}})}}, then the intuitive Galilean transformations are recovered from the transformation equations for special relativity
* Attaching a frame to a photon (''riding a light beam'' like Einstein considers) requires special treatment of the transformations.

There is nothing special about the ''x'' direction in the standard configuration. The above [[Formalism (mathematics)|formalism]] applies to any direction; and three orthogonal directions allow dealing with all directions in space by decomposing the velocity vectors to their components in these directions. See ''[[Velocity-addition formula]]'' for details.

=== Thomas rotation ===
{{See also|Thomas rotation}}
{{multiple image
 | direction = vertical
 | width = 220
 | footer = Figure 4-5. Thomas–Wigner rotation
 | image1 = Thomas-Wigner Rotation 1.svg
 | image2 = Thomas-Wigner Rotation 2.svg
}}

The composition of two non-collinear Lorentz boosts (i.e., two non-collinear Lorentz transformations, neither of which involve rotation) results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation.

Thomas rotation results from the relativity of simultaneity. In Fig.&nbsp;4-5a, a rod of length <math>L</math> in its rest frame (i.e., having a [[proper length]] of {{tmath|1= L }}) rises vertically along the y-axis in the ground frame.

In Fig.&nbsp;4-5b, the same rod is observed from the frame of a rocket moving at speed <math>v</math> to the right. If we imagine two clocks situated at the left and right ends of the rod that are synchronized ''in the frame of the rod'', relativity of simultaneity causes the observer in the rocket frame to observe (not [[#Measurement_versus_visual_appearance|''see'']]) the clock at the right end of the rod as being advanced in time by {{tmath|1= Lv/c^2 }}, and the rod is correspondingly observed as tilted.<ref name="Taylor1992"/>{{rp|98–99}}

Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities. For example, this can be seen in the [[Spin–orbit interaction|spin of moving particles]], where [[Thomas precession]] is a relativistic correction that applies to the [[Spin (physics)|spin]] of an elementary particle or the rotation of a macroscopic [[gyroscope]], relating the [[angular velocity]] of the spin of a particle following a [[curvilinear]] orbit to the angular velocity of the orbital motion.<ref name="Taylor1992"/>{{rp|169–174}}

Thomas rotation provides the resolution to the well-known "meter stick and hole paradox".<ref name="Shaw" group=p>{{cite journal |last1=Shaw |first1=R. |title=Length Contraction Paradox |journal=American Journal of Physics |date=1962 |volume=30 |issue=1 |page=72 |doi=10.1119/1.1941907 |bibcode=1962AmJPh..30...72S |s2cid=119855914 }}</ref><ref name="Taylor1992"/>{{rp|98–99}}

=== Causality and prohibition of motion faster than light ===
{{See also|Causality (physics)|Tachyonic antitelephone}}
[[File:Simple light cone diagram.svg|thumb|Figure 4–6. [[Light cone]]]]
In Fig.&nbsp;4-6, the time interval between the events A (the "cause") and B (the "effect") is 'timelike'; that is, there is a frame of reference in which events A and B occur at the ''same location in space'', separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames accessible by a Lorentz transformation. It is possible for matter (or information) to travel (below light speed) from the location of A, starting at the time of A, to the location of B, arriving at the time of B, so there can be a causal relationship (with A the cause and B the effect).

The interval AC in the diagram is 'spacelike'; that is, there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. But no frames are accessible by a Lorentz transformation, in which events A and C occur at the same location. If it were possible for a cause-and-effect relationship to exist between events A and C, paradoxes of causality would result.

For example, if signals could be sent faster than light, then signals could be sent into the sender's past (observer B in the diagrams).<ref>{{cite book | first = Richard C.|last = Tolman|title =The Theory of the Relativity of Motion |location=Berkeley|publisher = University of California Press|date = 1917|page = 54|url = https://books.google.com/books?id=8yodAAAAMAAJ&q=54}}</ref><ref group=p>{{cite journal|author1=G. A. Benford |author2=D. L. Book |author3=W. A. Newcomb |name-list-style=amp |doi=10.1103/PhysRevD.2.263|title=The Tachyonic Antitelephone|date=1970|journal=Physical Review D|volume=2|issue=2|pages=263–265|bibcode = 1970PhRvD...2..263B |s2cid=121124132 }}</ref> A variety of causal paradoxes could then be constructed.

{{multiple image
 | width = 160
 | image_gap = 4
 | image1 = Causality violation 1.svg
 | image2 = Causality violation 2.svg
 | alt1 = Causality violation: Beginning of scenario resulting from use of a fictitious instantaneous communicator   
 | alt2 = Causality violation: B receives the message before having sent it. 
 | footer_align = center
 | footer = Figure 4-7. Causality violation by the use of fictitious<br/>"instantaneous communicators"
}}
Consider the spacetime diagrams in Fig.&nbsp;4-7. A and B stand alongside a railroad track, when a high-speed train passes by, with C riding in the last car of the train and D riding in the leading car. The [[world lines]] of A and B are vertical (''ct''), distinguishing the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards (''{{prime|ct}}''), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground.
# Fig.&nbsp;4-7a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the <math>-x'</math> axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrives ''earlier'' than it was sent.
# Fig.&nbsp;4-7b. The event of "C passing the message to A", who is standing by the railroad tracks, is at the origin of their frames. Now A sends the message along the tracks to B via an "instantaneous communicator". The worldline of this message is the blue fat arrow, along the <math>+x</math> axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, in the primed frames of C and D, B will receive the message before it was sent out, a violation of causality.<ref name="Takeuchi">{{cite web |last1=Takeuchi |first1=Tatsu |title=Special Relativity Lecture Notes – Section 10 |url=https://www1.phys.vt.edu/~takeuchi/relativity/notes/section10.html |publisher=Virginia Tech |access-date=31 October 2018}}</ref>

It is not necessary for signals to be instantaneous to violate causality. Even if the signal from D to C were slightly shallower than the <math>x'</math> axis (and the signal from A to B slightly steeper than the <math>x</math> axis), it would still be possible for B to receive his message before he had sent it. By increasing the speed of the train to near light speeds, the <math>ct'</math> and <math>x'</math> axes can be squeezed very close to the dashed line representing the speed of light. With this modified setup, it can be demonstrated that even signals only ''slightly'' faster than the speed of light will result in causality violation.<ref name="Morin2017">{{cite book|last1=Morin|first1=David|title=Special Relativity for the Enthusiastic Beginner|date=2017|publisher=CreateSpace Independent Publishing Platform|pages=90–92|isbn=9781542323512}}</ref>

Therefore, '''if''' [[causality]] is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel [[faster than light]] in vacuum.

This is not to say that ''all'' faster than light speeds are impossible. Various trivial situations can be described where some "things" (not actual matter or energy) move faster than light.<ref>{{cite web |last1=Gibbs |first1=Philip |title=Is Faster-Than-Light Travel or Communication Possible? |url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/FTL.html |website=Physics FAQ |publisher=Department of Mathematics, University of California, Riverside |access-date=31 October 2018}}</ref> For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly (although this does not violate causality or any other relativistic phenomenon).<ref>{{cite book |title=Applications of Electrodynamics in Theoretical Physics and Astrophysics |edition=illustrated |first1=David |last1=Ginsburg |publisher=CRC Press |year=1989 |isbn=978-2-88124-719-4 |page=206 |url=https://books.google.com/books?id=Lh0tjaBNzg0C|bibcode=1989aetp.book.....G }} [https://books.google.com/books?id=Lh0tjaBNzg0C&pg=PA206 Extract of page 206]</ref><ref>{{cite book |title=Four Decades of Scientific Explanation |author=Wesley C. Salmon |publisher=University of Pittsburgh |date=2006 |isbn=978-0-8229-5926-7 |page=107 |url=https://books.google.com/books?id=FHqOXCd06e8C}}, [https://books.google.com/books?id=FHqOXCd06e8C&pg=PA107 Section 3.7 page 107]</ref><!-- a pair of diagrams, with x–t and x'–t' coordinates would help here -->