Editing Spacetime (section)

=== 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}}