Relativistic Frames of Reference

[math]\displaystyle{ \textbf{\underline{Navigation}} }[/math]

[math]\displaystyle{ \vartriangleleft }[/math] [math]\displaystyle{ \triangle }[/math] [math]\displaystyle{ \vartriangleright }[/math]

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{Distance}{Time} }[/math]

[math]\displaystyle{ c=\frac{d}{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{x^2+y^2+z^2}}{t} }[/math]

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

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

We can rewrite this as

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

This is possible since the ratios of distance to time are multiples of the same base, i.e. the square of the speed of light [math]\displaystyle{ (\frac{3\times 10^8\ m}{s})^2 }[/math]. Therefore for the relative change in the time in one frame, the distance must change by the same factor to maintain the same constant. With this we can write

[math]\displaystyle{ c^2 t^{'2}=x^{'2}+y^{'2}+z^{'2}\ \ \ \ \ c^2 t^{2}=x^2+y^2+ z^2 }[/math]

[math]\displaystyle{ \Rightarrow c^2 t^{'2}- x^{'2}-y^{'2}-z^{'2}= c^2 t^{2}-x^2-y^2-z^2 }[/math]

This quantity is known as the time space interval [math]\displaystyle{ s^2 }[/math]

[math]\displaystyle{ s^2\equiv c^2 t^{'2}-x^{'2}-y^{'2}-z^{'2}= c^2 t^{2}-x^2-y^2-z^2 }[/math]

Since the speed of light is a constant for all frames of reference, this allows the space time interval to also be invariant for inertial frames.

---

[math]\displaystyle{ \textbf{\underline{Navigation}} }[/math]

[math]\displaystyle{ \vartriangleleft }[/math] [math]\displaystyle{ \triangle }[/math] [math]\displaystyle{ \vartriangleright }[/math]