Editing Spacetime (section)

== Spacetime in special relativity {{anchor|In special relativity}}==
{{further|Minkowski spacetime}}

{{anchor|Spacetime interval}} <!-- This section is linked from [[Lorentz transformation]] -->
=== Spacetime interval ===
{{See also|Causal structure}}
In three dimensions, the ''[[Euclidean distance|distance]]'' <math>\Delta{d}</math> between two points can be defined using the [[Pythagorean theorem]]:
: <math>(\Delta{d})^2 = (\Delta{x})^2 + (\Delta{y})^2 + (\Delta{z})^2</math>

Although two viewers may measure the ''x'', ''y'', and ''z'' position of the two points using different coordinate systems, the distance between the points will be the same for both, assuming that they are measuring using the same units. The distance is "invariant".

In special relativity, however, the distance between two points is no longer the same if measured by two different observers, when one of the observers is moving, because of [[Lorentz contraction]]. The situation is even more complicated if the two points are separated in time as well as in space. For example, if one observer sees two events occur at the same place, but at different times, a person moving with respect to the first observer will see the two events occurring at different places, because the moving point of view sees itself as stationary, and the position of the event as receding or approaching. Thus, a different measure must be used to measure the effective "distance" between two events.<ref name="Kogut_2001">{{cite book |last1=Kogut |first1=John B. |title=Introduction to Relativity |date=2001 |publisher=Harcourt/Academic Press |location=Massachusetts |isbn=0-12-417561-9}}</ref>{{rp|48–50;100–102}}

In four-dimensional spacetime, the analog to distance is the interval. Although time comes in as a fourth dimension, it is treated differently than the spatial dimensions. Minkowski space hence differs in important respects from [[Four-dimensional space|four-dimensional Euclidean space]]. The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two [[event (relativity)|events]] (because of [[time dilation]]) or the distance between the two events (because of [[length contraction]]). Special relativity provides a new invariant, called the '''spacetime interval''', which combines distances in space and in time. All observers who measure the time and distance between any two events will end up computing the same spacetime interval. Suppose an observer measures two events as being separated in time by <math>\Delta t</math> and a spatial distance <math>\Delta x.</math> Then the squared spacetime interval <math>(\Delta{s})^2</math> between the two events that are separated by a distance <math>\Delta{x}</math> in space and by <math>\Delta{ct}= c\Delta t</math> in the <math>ct</math>-coordinate is:<ref name="D'Inverno_1002">{{cite book |title=Introducing Einstein's Relativity: A Deeper Understanding |author1=Ray d'Inverno |author2=James Vickers |edition=illustrated |publisher=Oxford University Press |year=2022 |isbn=978-0-19-886202-4 |pages=26–28 |url=https://books.google.com/books?id=LGxvEAAAQBAJ}} [https://books.google.com/books?id=LGxvEAAAQBAJ&pg=PA27 Extract of page 27]</ref>
: <math>(\Delta s)^2 = (\Delta ct)^2 - (\Delta x)^2,</math>
or for three space dimensions,
: <math>(\Delta s)^2 = (\Delta ct)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2.</math>

The constant <math>c,</math> the speed of light, converts time <math>t</math> units (like seconds) into space units (like meters). The squared interval <math>\Delta s^2</math> is a measure of separation between events A and B that are time separated and in addition space separated either because there are two separate objects undergoing events, or because a single object in space is moving inertially between its events. The separation interval is the difference between the square of the spatial distance separating event B from event A and the square of the spatial distance traveled by a light signal in that same time interval <math>\Delta t</math>. If the event separation is due to a light signal, then this difference vanishes and <math>\Delta s =0</math>.

When the event considered is infinitesimally close to each other, then we may write
: <math>ds^2 = c^2dt^2 - dx^2-dy^2-dz^2.</math>

In a different inertial frame, say with coordinates <math>(t',x',y',z')</math>, the spacetime interval <math>ds'</math> can be written in a same form as above. Because of the constancy of speed of light, the light events in all inertial frames belong to zero interval, <math>ds=ds'=0</math>. For any other infinitesimal event where <math>ds\neq 0</math>, one can prove that <math>ds^2=ds'^2</math>
which in turn upon integration leads to <math>s=s'</math>.<ref>Landau, L. D., and Lifshitz, E. M. (2013). The classical theory of fields (Vol. 2).</ref>{{rp|2}} The invariance of the spacetime interval between the same events for all inertial frames of reference is one of the fundamental results of special theory of relativity.

Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general, <math>x</math> means <math>\Delta{x}</math>, etc. We are always concerned with ''differences'' of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values have no essential meaning.

[[File:Spacetime Diagram of Two Photons and a Slower than Light Object.png|thumb|Figure 2–1. Spacetime diagram illustrating two photons, A and B, originating at the same event, and a slower-than-light-speed object, C]]

The equation above is similar to the Pythagorean theorem, except with a minus sign between the <math>(ct)^2</math> and the <math>x^2</math> terms. The spacetime interval is the quantity <math>s^2,</math> not <math>s</math> itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard <math>s^2</math> as a distinct symbol in itself, rather than the square of something.<ref name="Schutz" />{{rp|217}}
: '''Note:''' There are two sign conventions in use in the relativity literature:
:: <math>s^2 = (ct)^2 - x^2 - y^2 - z^2</math>
: and
:: <math>s^2 = -(ct)^2 + x^2 + y^2 + z^2</math>
: These sign conventions are associated with the [[metric signature]]s {{nowrap|(+−−−)}} and {{nowrap|(−+++).}} A minor variation is to place the time coordinate last rather than first. Both conventions are widely used within the field of study.<ref name="Carroll_2022">{{cite book |last1=Carroll |first1=Sean |title=The Biggest Ideas in the Universe |date=2022 |publisher=Penguin Random House LLC |location=New York |isbn=9780593186589 |pages=155–156}}</ref>
: In the following discussion, we use the first convention.

In general <math>s^2</math> can assume any real number value. If <math>s^2</math> is positive, the spacetime interval is referred to as '''timelike'''. Since spatial distance traversed by any massive object is always less than distance traveled by the light for the same time interval, positive intervals are always timelike. If <math>s^2</math> is negative, the spacetime interval is said to be '''spacelike'''. Spacetime intervals are equal to zero when <math>x = \pm ct.</math> In other words, the spacetime interval between two events on the world line of something moving at the speed of light is zero. Such an interval is termed '''lightlike''' or '''null'''. A photon arriving in our eye from a distant star will not have aged, despite having (from our perspective) spent years in its passage.<ref name="Kogut_2001"/>{{rp|48–50}}

A spacetime diagram is typically drawn with only a single space and a single time coordinate. Fig.&nbsp;2-1 presents a spacetime diagram illustrating the [[world lines]] (i.e. paths in spacetime) of two photons, A and B, originating from the same event and going in opposite directions. In addition, C illustrates the world line of a slower-than-light-speed object. The vertical time coordinate is scaled by <math>c</math> so that it has the same units (meters) as the horizontal space coordinate. Since photons travel at the speed of light, their world lines have a slope of ±1.<ref name="Kogut_2001"/>{{rp|23–25}} In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.

{{anchor|Reference frames}}

=== Reference frames ===
{{More citations needed section|date=March 2024}}
[[File:Standard configuration of coordinate systems.svg|thumb|Figure 2-2. Galilean diagram of two frames of reference in standard configuration]]
[[File:Galilean and Spacetime coordinate transformations.png|thumb|upright=1.5|Figure 2–3. (a) Galilean diagram of two frames of reference in standard configuration, (b) spacetime diagram of two frames of reference, (c) spacetime diagram showing the path of a reflected light pulse]]

To gain insight in how spacetime coordinates measured by observers in different [[Inertial frame of reference|reference frames]] compare with each other, it is useful to work with a simplified setup with frames in a ''standard configuration.'' With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig.&nbsp;2-2, two [[Galilean reference frame]]s (i.e. conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S&nbsp;prime") belongs to a second observer O′.
* The ''x'', ''y'', ''z'' axes of frame S are oriented parallel to the respective primed axes of frame S′.
* Frame S′ moves in the ''x''-direction of frame S with a constant velocity ''v'' as measured in frame S.
* The origins of frames S and S′ are coincident when time {{math|1=''t'' = 0}} for frame S and {{math|1=''t''′ = 0}} for frame S′.<ref name="Collier">{{cite book|title=A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity|last1=Collier|first1=Peter|publisher=Incomprehensible Books|year=2017|isbn=978-0-9573894-6-5|edition=3rd}}</ref>{{rp|107}}

Fig.&nbsp;2-3a redraws Fig.&nbsp;2-2 in a different orientation. Fig.&nbsp;2-3b illustrates a ''relativistic'' spacetime diagram from the viewpoint of observer O. Since S and S′ are in standard configuration, their origins coincide at times {{math|1=''t'' = 0}} in frame S and {{math|1=''t''′ = 0}} in frame S′. The {{mvar|ct′}} axis passes through the events in frame S′ which have {{math|1=''x''′ = 0.}} But the points with {{math|1=''x''′ = 0}} are moving in the ''x''-direction of frame S with velocity ''v'', so that they are not coincident with the ''ct'' axis at any time other than zero. Therefore, the {{mvar|ct′}} axis is tilted with respect to the ''ct'' axis by an angle ''θ'' given by<ref name="Kogut_2001"/>{{rp|23–31}}
: <math>\tan(\theta) = v/c.</math>

The ''x''′ axis is also tilted with respect to the ''x'' axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always&nbsp;±1. Fig.&nbsp;2-3c presents a spacetime diagram from the viewpoint of observer O′. Event P represents the emission of a light pulse at {{math|1=''x''′ = 0,}} {{math|1=''ct''′ = −''a''.}} The pulse is reflected from a mirror situated a distance ''a'' from the light source (event Q), and returns to the light source at {{math|1=''x''′ = 0, ''ct''′ = ''a''}} (event R).

The same events P, Q, R are plotted in Fig.&nbsp;2-3b in the frame of observer O. The light paths have {{nowrap|1=slopes = 1}} and&nbsp;−1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to the ''x'' and ''ct'' axes. Since {{nowrap|1=OP = OQ = OR,}} the angle between {{mvar|x′}} and {{mvar|x}} must also be ''θ''.<ref name="Collier" />{{rp|113–118}}

While the rest frame has space and time axes that meet at right angles, the moving frame is drawn with axes that meet at an acute angle. The frames are actually equivalent.<ref name="Kogut_2001"/>{{rp|23–31}} The asymmetry is due to unavoidable distortions in how spacetime coordinates can map onto a [[Cartesian plane]], and should be considered no stranger than the manner in which, on a [[Mercator projection]] of the Earth, the relative sizes of land masses near the poles (Greenland and Antarctica) are highly exaggerated relative to land masses near the Equator.

{{anchor|Light cone}}

=== Light cone ===
{{Main|Light cone}}
{{anchor|Figure 2-4}}
[[File:ModernPhysicsSpaceTimeA.png|thumb|Figure 2–4. The light cone centered on an event divides the rest of spacetime into the future, the past, and "elsewhere"]]

In Fig. 2–4, event&nbsp;O is at the origin of a spacetime diagram, and the two diagonal lines represent all events that have zero spacetime interval with respect to the origin event. These two lines form what is called the ''light cone'' of the event&nbsp;O, since adding a second spatial dimension (Fig.&nbsp;2-5) makes the appearance that of two [[cone|right circular cones]] meeting with their apices at&nbsp;O. One cone extends into the [[future]] (t>0), the other into the [[past]] (t<0).

[[File:World line.png|thumb|Figure 2–5. Light cone in 2D space plus a time dimension]]
A light (double) cone divides spacetime into separate regions with respect to its apex. The interior of the future light cone consists of all events that are separated from the apex by more ''time'' (temporal distance) than necessary to cross their ''spatial distance'' at lightspeed; these events comprise the ''timelike future'' of the event&nbsp;O. Likewise, the ''timelike past'' comprises the interior events of the past light cone. So in ''timelike intervals'' &Delta;''ct'' is greater than &Delta;''x'', making timelike intervals positive.<ref name="Schutz" />{{rp|220}}

The region exterior to the light cone consists of events that are separated from the event&nbsp;O by more ''space'' than can be crossed at lightspeed in the given ''time''. These events comprise the so-called ''spacelike'' region of the event&nbsp;O, denoted "Elsewhere" in Fig.&nbsp;2-4. Events on the light cone itself are said to be ''lightlike'' (or ''null separated'') from&nbsp;O. Because of the invariance of the spacetime interval, all observers will assign the same light cone to any given event, and thus will agree on this division of spacetime.<ref name="Schutz" />{{rp|220}}

The light cone has an essential role within the concept of [[causality]]. It is possible for a not-faster-than-light-speed signal to travel from the position and time of&nbsp;O to the position and time of D (Fig.&nbsp;2-4). It is hence possible for event&nbsp;O to have a causal influence on event&nbsp;D. The future light cone contains all the events that could be causally influenced by O. Likewise, it is possible for a not-faster-than-light-speed signal to travel from the position and time of A, to the position and time of O. The past light cone contains all the events that could have a causal influence on O. In contrast, assuming that signals cannot travel faster than the speed of light, any event, like e.g.&nbsp;B or&nbsp;C, in the spacelike region (Elsewhere), cannot either affect event&nbsp;O, nor can they be affected by event&nbsp;O employing such signalling. Under this assumption any causal relationship between event&nbsp;O and any events in the spacelike region of a light cone is excluded.<ref>{{cite web|last1=Curiel|first1=Erik|last2=Bokulich|first2=Peter|title=Lightcones and Causal Structure|url=https://plato.stanford.edu/entries/spacetime-singularities/lightcone.html|website=Stanford Encyclopedia of Philosophy|publisher=Metaphysics Research Lab, Stanford University|access-date=26 March 2017|archive-date=17 May 2019|archive-url=https://web.archive.org/web/20190517122738/https://plato.stanford.edu/entries/spacetime-singularities/lightcone.html|url-status=live}}</ref>

{{anchor|Relativity of simultaneity}}

=== Relativity of simultaneity ===
{{Main|Relativity of simultaneity}}
{{anchor|Figure 2-6}}
[[File:Relativity of Simultaneity Animation.gif|thumb|Figure 2–6. Animation illustrating relativity of simultaneity]]

All observers will agree that for any given event, an event within the given event's future light cone occurs ''after'' the given event. Likewise, for any given event, an event within the given event's past light cone occurs ''before'' the given event. The before–after relationship observed for timelike-separated events remains unchanged no matter what the [[Frame of reference|reference frame]] of the observer, i.e. no matter how the observer may be moving. The situation is quite different for spacelike-separated events. [[#Figure 2-4|'''Fig.&nbsp;2-4''']] was drawn from the reference frame of an observer moving at {{nowrap|1=''v'' = 0.}} From this reference frame, event&nbsp;C is observed to occur after event&nbsp;O, and event&nbsp;B is observed to occur before event&nbsp;O.<ref name="plato.stanford.edu">{{cite web|last1=Savitt|first1=Steven|title=Being and Becoming in Modern Physics. 3. The Special Theory of Relativity|url=https://plato.stanford.edu/entries/spacetime-bebecome/#Spec|website=The Stanford Encyclopedia of Philosophy|publisher=Metaphysics Research Lab, Stanford University|access-date=26 March 2017|archive-date=11 March 2017|archive-url=https://web.archive.org/web/20170311015404/https://plato.stanford.edu/entries/spacetime-bebecome/#Spec|url-status=live}}</ref>

From a different reference frame, the orderings of these non-causally-related events can be reversed. In particular, one notes that if two events are simultaneous in a particular reference frame, they are ''necessarily'' separated by a spacelike interval and thus are noncausally related. The observation that simultaneity is not absolute, but depends on the observer's reference frame, is termed the [[relativity of simultaneity]].<ref name="plato.stanford.edu"/>

Fig. 2-6 illustrates the use of spacetime diagrams in the analysis of the relativity of simultaneity. The events in spacetime are invariant, but the coordinate frames transform as discussed above for Fig.&nbsp;2-3. The three events {{nowrap|1=(A, B, C)}} are simultaneous from the reference frame of an observer moving at {{nowrap|1=''v'' = 0.}} From the reference frame of an observer moving at {{nowrap|1=''v'' = 0.3''c'',}} the events appear to occur in the order {{nowrap|1=C, B, A.}} From the reference frame of an observer moving at {{nowrap|1=''v'' = −0.5''c''}}, the events appear to occur in the order {{nowrap|1=A, B, C}}. The white line represents a ''plane of simultaneity'' being moved from the past of the observer to the future of the observer, highlighting events residing on it. The gray area is the light cone of the observer, which remains invariant.

A spacelike spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer. A spacelike spacetime interval hence provides a measure of ''proper distance'', i.e. the true distance = <math>\sqrt{-s^2}.</math> Likewise, a timelike spacetime interval gives the same measure of time as would be presented by the cumulative ticking of a clock that moves along a given world line. A timelike spacetime interval hence provides a measure of the ''proper time'' = <math>\sqrt{s^2}.</math><ref name="Schutz" />{{rp|220–221}}

{{anchor|Invariant hyperbola}}

=== Invariant hyperbola ===
{{More citations needed section|date=March 2024}}
{{anchor|Spacelike and Timelike Invariant Hyperbolas}}
[[File:Spacelike and Timelike Invariant Hyperbolas.png|thumb|upright=1.5|Figure 2–7. (a) Families of invariant hyperbolae, (b) Hyperboloids of two sheets and one sheet]]

In Euclidean space (having spatial dimensions only), the set of points equidistant (using the Euclidean metric) from some point form a circle (in two dimensions) or a sphere (in three dimensions). In {{nowrap|(1+1)-dimensional}} Minkowski spacetime (having one temporal and one spatial dimension), the points at some constant spacetime interval away from the origin (using the Minkowski metric) form curves given by the two equations
: <math>(ct)^2 - x^2 = \pm s^2,</math>
with <math> s^2</math>some positive real constant. These equations describe two families of hyperbolae in an ''x''–''ct'' spacetime diagram, which are termed ''invariant hyperbolae''.

In Fig.&nbsp;2-7a, each magenta hyperbola connects all events having some fixed spacelike separation from the origin, while the green hyperbolae connect events of equal timelike separation.

The magenta hyperbolae, which cross the ''x'' axis, are timelike curves, which is to say that these hyperbolae represent actual paths that can be traversed by (constantly accelerating) particles in spacetime: Between any two events on one hyperbola a causality relation is possible, because the inverse of the slope—representing the necessary speed—for all secants is less than <math>c</math>. On the other hand, the green hyperbolae, which cross the ''ct'' axis, are spacelike curves because all intervals ''along'' these hyperbolae are spacelike intervals: No causality is possible between any two points on one of these hyperbolae, because all secants represent speeds larger than <math>c</math>.

Fig.&nbsp;2-7b reflects the situation in {{nowrap|(1+2)-dimensional}} Minkowski spacetime (one temporal and two spatial dimensions) with the corresponding hyperboloids. The invariant hyperbolae displaced by spacelike intervals from the origin generate [[hyperboloid]]s of one sheet, while the invariant hyperbolae displaced by timelike intervals from the origin generate hyperboloids of two sheets.

The (1+2)-dimensional boundary between space- and time-like hyperboloids, established by the events forming a zero spacetime interval to the origin, is made up by degenerating the hyperboloids to the light cone. In (1+1)-dimensions the hyperbolae degenerate to the two grey 45°-lines depicted in Fig.&nbsp;2-7a.

{{anchor|Time dilation and length contraction}}

=== Time dilation and length contraction ===
[[File:Spacetime diagram of invariant hyperbola.png|thumb|Figure 2–8. The invariant hyperbola comprises the points that can be reached from the origin in a fixed proper time by clocks traveling at different speeds]]
Fig. 2-8 illustrates the invariant hyperbola for all events that can be reached from the origin in a proper time of 5&nbsp;meters (approximately {{val|1.67|e=−8|u=s}}). Different world lines represent clocks moving at different speeds. A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time. For a clock traveling at 0.3&nbsp;''c'', the elapsed time measured by the observer is 5.24&nbsp;meters ({{val|1.75|e=−8|u=s}}), while for a clock traveling at 0.7&nbsp;''c'', the elapsed time measured by the observer is 7.00&nbsp;meters ({{val|2.34|e=−8|u=s}}).<ref name="Schutz" />{{rp|220–221}}

This illustrates the phenomenon known as [[time dilation]]. Clocks that travel faster take longer (in the observer frame) to tick out the same amount of proper time, and they travel further along the x–axis within that proper time than they would have without time dilation.<ref name="Schutz" />{{rp|220–221}} The measurement of time dilation by two observers in different inertial reference frames is mutual. If observer O measures the clocks of observer O′ as running slower in his frame, observer O′ in turn will measure the clocks of observer O as running slower.

{{anchor|Figure 2-9}}
[[File:Animated Spacetime Diagram - Length Contraction.gif|thumb|Figure 2–9. In this spacetime diagram, the 1&nbsp;m length of the moving rod, as measured in the primed frame, is the foreshortened distance OC when projected onto the unprimed frame.]]
[[Length contraction]], like time dilation, is a manifestation of the relativity of simultaneity. Measurement of length requires measurement of the spacetime interval between two events that are simultaneous in one's frame of reference. But events that are simultaneous in one frame of reference are, in general, not simultaneous in other frames of reference.

Fig. 2-9 illustrates the motions of a 1&nbsp;m rod that is traveling at 0.5&nbsp;''c'' along the ''x'' axis. The edges of the blue band represent the world lines of the rod's two endpoints. The invariant hyperbola illustrates events separated from the origin by a spacelike interval of 1&nbsp;m. The endpoints O and B measured when {{′|''t''}}&nbsp;=&nbsp;0 are simultaneous events in the S′ frame. But to an observer in frame S, events O and B are not simultaneous. To measure length, the observer in frame S measures the endpoints of the rod as projected onto the ''x''-axis along their world lines. The projection of the rod's ''world sheet'' onto the ''x'' axis yields the foreshortened length OC.<ref name="Collier" />{{rp|125}}

(not illustrated) Drawing a vertical line through A so that it intersects the ''x''′ axis demonstrates that, even as OB is foreshortened from the point of view of observer O, OA is likewise foreshortened from the point of view of observer O′. In the same way that each observer measures the other's clocks as running slow, each observer measures the other's rulers as being contracted.

In regards to mutual length contraction, [[#Figure 2-9|'''Fig.&nbsp;2-9''']] illustrates that the primed and unprimed frames are mutually [[Lorentz transformation#Coordinate transformation|rotated]] by a [[hyperbolic angle]] (analogous to ordinary angles in Euclidean geometry).<ref group=note>In a [[Cartesian plane]], ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the [[hyperbolic metric]].</ref> Because of this rotation, the projection of a primed meter-stick onto the unprimed ''x''-axis is foreshortened, while the projection of an unprimed meter-stick onto the primed x′-axis is likewise foreshortened.

=== Mutual time dilation and the twin paradox ===
{{Main|Twin paradox}}

==== Mutual time dilation ====
Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts. If an observer in frame S measures a clock, at rest in frame S', as running slower than his', while S' is moving at speed ''v'' in S, then the principle of relativity requires that an observer in frame S' likewise measures a clock in frame S, moving at speed −''v'' in S', as running slower than hers. How two clocks can run ''both slower'' than the other, is an important question that "goes to the heart of understanding special relativity."<ref name="Schutz" />{{rp|198}}

This apparent contradiction stems from not correctly taking into account the different settings of the necessary, related measurements. These settings allow for a consistent explanation of the ''only apparent'' contradiction. It is not about the abstract ticking of two identical clocks, but about how to measure in one frame the temporal distance of two ticks of a moving clock. It turns out that in mutually observing the duration between ticks of clocks, each moving in the respective frame, different sets of clocks must be involved. In order to measure in frame S the tick duration of a moving clock W′ (at rest in S′), one uses ''two'' additional, synchronized clocks W<sub>1</sub> and W<sub>2</sub> at rest in two arbitrarily fixed points in S with the spatial distance ''d''.
: <small>Two events can be defined by the condition "two clocks are simultaneously at one place", i.e., when W′ passes each W<sub>1</sub> and W<sub>2</sub>. For both events the two readings of the collocated clocks are recorded. The difference of the two readings of W<sub>1</sub> and W<sub>2</sub> is the temporal distance of the two events in S, and their spatial distance is ''d''. The difference of the two readings of W′ is the temporal distance of the two events in S′. In S′ these events are only separated in time, they happen at the same place in S′. Because of the invariance of the spacetime interval spanned by these two events, and the nonzero spatial separation ''d'' in S, the temporal distance in S′ must be smaller than the one in S: the ''smaller'' temporal distance between the two events, resulting from the readings of the moving clock W′, belongs to the ''slower'' running clock W′.</small>

Conversely, for judging in frame S′ the temporal distance of two events on a moving clock W (at rest in S), one needs two clocks at rest in S′.
: <small>In this comparison the clock W is moving by with velocity −''v''. Recording again the four readings for the events, defined by "two clocks simultaneously at one place", results in the analogous temporal distances of the two events, now temporally and spatially separated in S′, and only temporally separated but collocated in S. To keep the spacetime interval invariant, the temporal distance in S must be smaller than in S′, because of the spatial separation of the events in S′: now clock W is observed to run slower.</small>

The necessary recordings for the two judgements, with "one moving clock" and "two clocks at rest" in respectively S or S′, involves two different sets, each with three clocks. Since there are different sets of clocks involved in the measurements, there is no inherent necessity that the measurements be reciprocally "consistent" such that, if one observer measures the moving clock to be slow, the other observer measures the other clock to be fast.<ref name="Schutz" />{{rp|198–199}}

{{multiple image|perrow = 1|total_width=250
| image2 = Spacetime Diagrams of Mutual Time Dilation B.png  |width2=300|height2=300
| image4 = Spacetime Diagrams of Mutual Time Dilation D.png  |width4=300|height4=300
| footer = Figure 2-10. Mutual time dilation }}
Fig. 2-10 illustrates the previous discussion of mutual time dilation with [[Minkowski diagram]]s. The upper picture reflects the measurements as seen from frame S "at rest" with unprimed, rectangular axes, and frame S′ "moving with ''v''&nbsp;>&nbsp;0", coordinatized by primed, oblique axes, slanted to the right; the lower picture shows frame S′ "at rest" with primed, rectangular coordinates, and frame S "moving with −''v''&nbsp;<&nbsp;0", with unprimed, oblique axes, slanted to the left.

Each line drawn parallel to a spatial axis (''x'', ''x''′) represents a line of simultaneity. All events on such a line have the same time value (''ct'', ''ct''′). Likewise, each line drawn parallel to a temporal axis (''ct'', ''ct′'') represents a line of equal spatial coordinate values (''x'', ''x''′).
: <small>One may designate in both pictures the origin ''O'' (= {{′|''O''}}) as the event, where the respective "moving clock" is collocated with the "first clock at rest" in both comparisons. Obviously, for this event the readings on both clocks in both comparisons are zero. As a consequence, the worldlines of the moving clocks are the slanted to the right ''ct''′-axis (upper pictures, clock W′) and the slanted to the left ''ct''-axes (lower pictures, clock W). The worldlines of W<sub>1</sub> and W′<sub>1</sub> are the corresponding vertical time axes (''ct'' in the upper pictures, and ''ct''′ in the lower pictures).</small>
: <small>In the upper picture the place for W<sub>2</sub> is taken to be ''A<sub>x</sub>'' > 0, and thus the worldline (not shown in the pictures) of this clock intersects the worldline of the moving clock (the ''ct''′-axis) in the event labelled ''A'', where "two clocks are simultaneously at one place". In the lower picture the place for W′<sub>2</sub> is taken to be ''C''<sub>''x''′</sub>&nbsp;<&nbsp;0, and so in this measurement the moving clock W passes W′<sub>2</sub> in the event ''C''.</small>
: <small>In the upper picture the ''ct''-coordinate ''A<sub>t</sub>'' of the event ''A'' (the reading of W<sub>2</sub>) is labeled ''B'', thus giving the elapsed time between the two events, measured with W<sub>1</sub> and W<sub>2</sub>, as ''OB''. For a comparison, the length of the time interval ''OA'', measured with W′, must be transformed to the scale of the ''ct''-axis. This is done by the invariant hyperbola (see also Fig. 2-8) through ''A'', connecting all events with the same spacetime interval from the origin as ''A''. This yields the event ''C'' on the ''ct''-axis, and obviously: ''OC''&nbsp;<&nbsp;''OB'', the "moving" clock W′ runs slower.</small>

To show the mutual time dilation immediately in the upper picture, the event ''D'' may be constructed as the event at ''x''′&nbsp;=&nbsp;0 (the location of clock W′ in S′), that is simultaneous to ''C'' (''OC'' has equal spacetime interval as ''OA'') in S′. This shows that the time interval ''OD'' is longer than ''OA'', showing that the "moving" clock runs slower.<ref name="Collier" />{{rp|124}}

In the lower picture the frame S is moving with velocity −''v'' in the frame S′ at rest. The worldline of clock W is the ''ct''-axis (slanted to the left), the worldline of W′<sub>1</sub> is the vertical ''ct''′-axis, and the worldline of W′<sub>2</sub> is the vertical through event ''C'', with ''ct''′-coordinate ''D''. The invariant hyperbola through event ''C'' scales the time interval ''OC'' to ''OA'', which is shorter than ''OD''; also, ''B'' is constructed (similar to ''D'' in the upper pictures) as simultaneous to ''A'' in S, at ''x''&nbsp;=&nbsp;0. The result ''OB''&nbsp;>&nbsp;''OC'' corresponds again to above.

The word "measure" is important. In classical physics an observer cannot affect an observed object, but the object's state of motion ''can'' affect the observer's ''observations'' of the object.

==== Twin paradox ====
Many introductions to special relativity illustrate the differences between Galilean relativity and special relativity by posing a series of "paradoxes". These paradoxes are, in fact, ill-posed problems, resulting from our unfamiliarity with velocities comparable to the speed of light. The remedy is to solve many problems in special relativity and to become familiar with its so-called counter-intuitive predictions. The geometrical approach to studying spacetime is considered one of the best methods for developing a modern intuition.<ref name="Schutz1985">{{cite book| last1=Schutz |first1= Bernard F. |title=A first course in general relativity|date=1985|publisher=Cambridge University Press|location=Cambridge, UK|isbn=0-521-27703-5|page=26}}</ref>

The [[twin paradox]] is a [[thought experiment]] involving identical twins, one of whom makes a journey into space in a high-speed rocket, returning home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin observes the other twin as moving, and so at first glance, it would appear that each should find the other to have aged less. The twin paradox sidesteps the justification for mutual time dilation presented above by avoiding the requirement for a third clock.<ref name="Schutz" />{{rp|207}} Nevertheless, the ''twin paradox'' is not a true paradox because it is easily understood within the context of special relativity.

The impression that a paradox exists stems from a misunderstanding of what special relativity states. Special relativity does not declare all frames of reference to be equivalent, only inertial frames. The traveling twin's frame is not inertial during periods when she is accelerating. Furthermore, the difference between the twins is observationally detectable: the traveling twin needs to fire her rockets to be able to return home, while the stay-at-home twin does not.<ref name="Weiss" /><ref group=note>Even with no (de)acceleration i.e. using one inertial frame O for constant, high-velocity outward journey and another inertial frame I for constant, high-velocity inward journey – the sum of the elapsed time in those frames (O and I) is shorter than the elapsed time in the stationary inertial frame S. Thus acceleration and deceleration is not the cause of shorter elapsed time during the outward and inward journey. Instead the use of two different constant, high-velocity inertial frames for outward and inward journey is really the cause of shorter elapsed time total. Granted, if the same twin has to travel outward and inward leg of the journey and safely switch from outward to inward leg of the journey, the acceleration and deceleration is required. If the travelling twin could ride the high-velocity outward inertial frame and instantaneously switch to high-velocity inward inertial frame the example would still work. The point is that real reason should be stated clearly. The asymmetry is because of the comparison of sum of elapsed times in two different inertial frames (O and I) to the elapsed time in a single inertial frame S.</ref>

[[File:Introductory Physics fig 4.9.png|thumb|Figure 2–11. Spacetime explanation of the twin paradox]]
These distinctions should result in a difference in the twins' ages. The spacetime diagram of Fig.&nbsp;2-11 presents the simple case of a twin going straight out along the x axis and immediately turning back. From the standpoint of the stay-at-home twin, there is nothing puzzling about the twin paradox at all. The proper time measured along the traveling twin's world line from O to C, plus the proper time measured from C to B, is less than the stay-at-home twin's proper time measured from O to A to B. More complex trajectories require integrating the proper time between the respective events along the curve (i.e. the [[Line integral|path integral]]) to calculate the total amount of proper time experienced by the traveling twin.<ref name="Weiss">{{cite web |last1=Weiss |first1=Michael |title=The Twin Paradox |url=http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html |website=The Physics and Relativity FAQ |access-date=10 April 2017 |archive-date=27 April 2017 |archive-url=https://web.archive.org/web/20170427202915/http://www.math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html |url-status=live }}</ref>

Complications arise if the twin paradox is analyzed from the traveling twin's point of view.

Weiss's nomenclature, designating the stay-at-home twin as Terence and the traveling twin as Stella, is hereafter used.<ref name="Weiss" />

Stella is not in an inertial frame. Given this fact, it is sometimes incorrectly stated that full resolution of the twin paradox requires general relativity:<ref name="Weiss" />
{{blockquote|A pure SR analysis would be as follows: Analyzed in Stella's rest frame, she is motionless for the entire trip. When she fires her rockets for the turnaround, she experiences a pseudo force which resembles a gravitational force.<ref name="Weiss" /> [[#Figure 2-6|'''Figs.&nbsp;2-6''']] and 2-11 illustrate the concept of lines (planes) of simultaneity: Lines parallel to the observer's ''x''-axis (''xy''-plane) represent sets of events that are simultaneous in the observer frame. In Fig.&nbsp;2-11, the blue lines connect events on Terence's world line which, ''from Stella's point of view'', are simultaneous with events on her world line. (Terence, in turn, would observe a set of horizontal lines of simultaneity.) Throughout both the outbound and the inbound legs of Stella's journey, she measures Terence's clocks as running slower than her own. ''But during the turnaround'' (i.e. between the bold blue lines in the figure), a shift takes place in the angle of her lines of simultaneity, corresponding to a rapid skip-over of the events in Terence's world line that Stella considers to be simultaneous with her own. Therefore, at the end of her trip, Stella finds that Terence has aged more than she has.<ref name="Weiss" />}}

Although general relativity is not required to analyze the twin paradox, application of the [[Equivalence Principle]] of general relativity does provide some additional insight into the subject. Stella is not stationary in an inertial frame. Analyzed in Stella's rest frame, she is motionless for the entire trip. When she is coasting her rest frame is inertial, and Terence's clock will appear to run slow. But when she fires her rockets for the turnaround, her rest frame is an accelerated frame and she experiences a force which is pushing her as if she were in a gravitational field. Terence will appear to be high up in that field and because of [[gravitational time dilation]], his clock will appear to run fast, so much so that the net result will be that Terence has aged more than Stella when they are back together.<ref name="Weiss" /> The theoretical arguments predicting gravitational time dilation are not exclusive to general relativity. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence, including Newton's theory.<ref name="Schutz" />{{rp|16}}
{{anchor|Gravitation}}

=== Gravitation ===
<!-- Note to future editors!!! This section is intended to provide a gentle introduction to spacetime. To the limit of what is feasible, avoid mathematics. 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 introduction understandable by the main target audience, which I have envisioned to be a typical high school science student. -->

This introductory section has focused on the spacetime of special relativity, since it is the easiest to describe. Minkowski spacetime is flat, takes no account of gravity, is uniform throughout, and serves as nothing more than a static background for the events that take place in it. The presence of gravity greatly complicates the description of spacetime. In general relativity, spacetime is no longer a static background, but actively interacts with the physical systems that it contains. Spacetime curves in the presence of matter, can propagate waves, bends light, and exhibits a host of other phenomena.<ref name="Schutz" />{{rp|221}} A few of these phenomena are described in the later sections of this article.