Relativistic Frames of Reference

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Relativistic Frames of Reference

From the Galilean description of motion for a frame of reference moving relative to another frame considered stationary we know that

Figure 2.1: Primed reference frame moving in the z direction with velocity v.

$t= t'$ $x=x'$ $y=y'$ $z=z'+vt$

Using Einstein's Theory of Relativity, we know that the speed of light is a constant, c, for all reference frames. In the unprimed frame, from the definition of speed:

$speed=\frac{\Delta Distance}{\Delta Time}$

$c=\frac{\Delta d}{\Delta t}$

where

$c=3\times 10^8\ m/s$

Using the distance equation in a Cartesian coordinate system, the equation for the speed of light becomes

$c=\frac{\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}}{\Delta t}$

Following the postulate of Special Relativity, this implies for the primed frame

$c=\frac{\sqrt{(\Delta x')^2+(\Delta y')^2+(\Delta z')^2}}{\Delta t'}$

We can rewrite this as

$c^2=\frac{(\Delta x')^2+(\Delta y')^2+(\Delta z')^2}{(\Delta t')^2}\ \ \ \ \ c^2=\frac{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}{(\Delta t)^2}$

$c^2 \Delta t^{'2}=(\Delta x')^2+(\Delta y')^2+(\Delta z')^2\ \ \ \ \ c^2 \Delta t^{2}=(\Delta x)^2+(\Delta y)^2+(\Delta z)^2$

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