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}}{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}}}{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. 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 \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]

[math]\displaystyle{ \Rightarrow 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]

This quantity is known as the time space interval [math]\displaystyle{ ds^2 }[/math] when the change is infinitesimal

[math]\displaystyle{ s^2\equiv c^2 dt^{'2}-dx^{'2}-dy^{'2}-dz^{'2}= c^2 dt^{2}-dx^2-dy^2-dz^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{ s^2\equiv c^2 dt^{'2}-dx^{'2}-dy^{'2}-dz^{'2}= c^2 dt^{2}-dx^2-dy^2-dz^2 }[/math]

[math]\displaystyle{ s^2\equiv c^2 dt^{'2}-dr^{'2}= c^2 dt^{2}-dr^2 }[/math]

[math]\displaystyle{ s^2\equiv (c^2 -v^{'2})dt^{'2}= (c^2 -v^2)dt^{2} }[/math]

From the rest frame of v'=0

[math]\displaystyle{ s^2\equiv c^2 dt^{'2}= (c^2 -v^2)dt^{2} }[/math]

[math]\displaystyle{ \Rightarrow dt^{'2}= (1-\frac{v^2}{c^2 })dt^{2} }[/math]

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