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m Fixed spelling inconsistency ("four dimensional" to "four-dimensional"). |
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* The origins of frames S and S′ are coincident when time ''t'' = 0 for frame S and ''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. 2-3a redraws Fig. 2-2 in a different orientation. Fig. 2-3b illustrates a spacetime diagram from the viewpoint of observer O. Since S and S′ are in standard configuration, their origins coincide at times ''t'' = 0 in frame S and ''t''′ = 0 in frame S′. The ''ct''′ axis passes through the events in frame S′ which have ''x''′ = 0. But the points with ''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 ''ct''′ axis is tilted with respect to the ''ct'' axis by an angle ''θ'' given by<ref name="Kogut_2001">{{cite book |last1=Kogut |first1=John B. |title=Introduction to Relativity |date=2001 |publisher=Harcourt/Academic Press |location=San Diego |isbn=0-12-417561-9}}</ref>{{rp|23–31}}
: <math>\tan(\theta) = v/c.</math>
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The same events P, Q, R are plotted in Fig. 2-3b in the frame of observer O. The light paths have slopes = 1 and −1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to the ''x'' and ''ct'' axes. Since OP = OQ = OR, the angle between ''x''′ and ''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}}
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