Editing Spacetime (section)

=== Galilean transformations ===
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{{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}}