Editing Spacetime (section)

=== Relativistic composition of velocities ===
{{Main|Velocity addition formula}}
[[File:Relativistic composition of velocities.svg|thumb|upright=1.5|Figure 3–2. Relativistic composition of velocities]]
The composition of velocities is quite different in relativistic spacetime. To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light,
: <math>\beta = v/c</math>

Fig. 3-2a illustrates a red train that is moving forward at a speed given by {{nowrap|1=''v''/''c'' = ''β'' = ''s''/''a''}}. From the primed frame of the train, a passenger shoots a bullet with a speed given by {{nowrap|1={{′|''u''}}/''c'' = {{′|''β''}} = ''n''/''m''}}, where the distance is measured along a line parallel to the red {{′|''x''}} axis rather than parallel to the black ''x'' axis. What is the composite velocity ''u'' of the bullet relative to the platform, as represented by the blue arrow? Referring to Fig.&nbsp;3-2b:
# From the platform, the composite speed of the bullet is given by {{nowrap|1=''u'' = ''c''(''s'' + ''r'')/(''a'' + ''b'')}}.
# The two yellow triangles are similar because they are right triangles that share a common angle ''α''. In the large yellow triangle, the ratio {{nowrap|1=''s''/''a'' = ''v''/''c'' = ''β''}}.
# The ratios of corresponding sides of the two yellow triangles are constant, so that {{nowrap|1=''r''/''a'' = ''b''/''s''}} = {{nowrap|1=''n''/''m'' = {{′|''β''}}}}. So {{nowrap|1=''b'' = {{′|''u''}}''s''/''c''}} and {{nowrap|1=''r'' = {{′|''u''}}''a''/''c''}}.
# Substitute the expressions for ''b'' and ''r'' into the expression for ''u'' in step&nbsp;1 to yield Einstein's formula for the addition of velocities:<ref name="Bais">{{cite book|last1=Bais|first1=Sander|title=Very Special Relativity: An Illustrated Guide|url=https://archive.org/details/veryspecialrelat0000bais|url-access=registration|date=2007|publisher=Harvard University Press|location=Cambridge, Massachusetts|isbn=978-0-674-02611-7}}</ref>{{rp|42–48}}
#: <math> u = {v+u'\over 1+(vu'/c^2)} . </math>

The relativistic formula for addition of velocities presented above exhibits several important features:
* If {{′|''u''}} and ''v'' are both very small compared with the speed of light, then the product {{′|''vu''}}/''c''<sup>2</sup> becomes vanishingly small, and the overall result becomes indistinguishable from the Galilean formula (Newton's formula) for the addition of velocities: ''u''&nbsp;=&nbsp;{{′|''u''}}&nbsp;+&nbsp;''v''. The Galilean formula is a special case of the relativistic formula applicable to low velocities.
* If {{′|''u''}} is set equal to ''c'', then the formula yields ''u''&nbsp;=&nbsp;''c'' regardless of the starting value of ''v''. The velocity of light is the same for all observers regardless their motions relative to the emitting source.<ref name="Bais" />{{rp|49}}

{{anchor|Time dilation and length contraction revisited}}