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Tensors and Relativity: Chapter 1

Velocity composition law

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Suppose a particle has a velocity tex2html_wrap_inline1472 in tex2html_wrap_inline1118 , then at time t its spacetime coordinates are tex2html_wrap_inline1478 . In a frame tex2html_wrap_inline1124 moving at a speed v relative to tex2html_wrap_inline1124 , its spacetime coordinates will be:


The velocity in tex2html_wrap_inline1124 simply comes from dividing the space parts by the time part i.e.


If tex2html_wrap_inline1488 and tex2html_wrap_inline1490 then


This looks like the Galelian relative velocity if tex2html_wrap_inline1492 . If u=c, tex2html_wrap_inline1496 , so we confirm the universality of the the speed of light.