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.

[math]\displaystyle{ t= t' }[/math] [math]\displaystyle{ x=x' }[/math] [math]\displaystyle{ y=y' }[/math] [math]\displaystyle{ z=z'+vt }[/math]

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:

[math]\displaystyle{ speed=\frac{\Delta Distance}{\Delta Time} }[/math]

[math]\displaystyle{ c=\frac{\Delta d}{\Delta t} }[/math]

where

[math]\displaystyle{ c=3\times 10^8\ m/s }[/math]

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

[math]\displaystyle{ c=\frac{\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}}{\Delta t} }[/math]

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

[math]\displaystyle{ c=\frac{\sqrt{(\Delta x')^2+(\Delta y')^2+(\Delta z')^2}}{\Delta t'} }[/math]

We can rewrite this as

[math]\displaystyle{ \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} }[/math]

This is possible since the ratios of distance to time are multiples of the same base, i.e. [math]\displaystyle{ \frac{3\times 10^8\ m}{s} }[/math]

[math]\displaystyle{ 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 }[/math]

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