Sunday, August 7, 2016

Relativistic composition of velocities in 1D

Adding two velocities v and w (in 1 dimension) in the relativistic mechanics needs to be done with the following formula: $$v\oplus w = \frac{v+w}{1+vw}$$ The speed of light is assumed c=1.  The interval $(-1,1)$ together with $\oplus$ forms a group. Here is an example of such "addition": $$\frac{1}{2}\oplus\frac{1}{2}= \frac{1/2 + 1/2}{ 1+\frac{1}{2}\frac{1}{2}} = \frac{4}{5}$$ The following diagram is sort of a slide rule for space travelers (like Ijon Tichy). The horizontal line is the real axis for the values of velocities. Point labelled "v+w" in the figure represents the relativistic sum of velocities v and w.  Move "v" or "w" to see how the result changes.

You may even experiment with tachyons! Slide velocity v to a value greater than 1 and see how adding a small velocity w slows it down.

For more see this.