LightPhoton
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- TL;DR Summary
- How do we define angles in relativity? do we change the usual euclidean definition or extend it?
How is the angle between two 4-vectors defined in special relativity? Consider two 4-velocity vectors:
$$U^\mu=(1,0), \\ V^\mu=\gamma_{rel}(1,v_{rel})$$
Where the vectors are written in the frame of the particle with ##U^\mu##.
The dot product between these is
$$U^\mu V_\mu=\gamma_{rel}$$
If we define angle ##\theta## between two vectors as:
$$\cos\theta=\frac{U^\mu V_\mu}{\vert U\vert\vert V\vert}$$
then since all velocity vectors have magnitude of ##1##, we get ##\cos\theta= \gamma_{rel}\geq 1##.
But what does this mean?
$$U^\mu=(1,0), \\ V^\mu=\gamma_{rel}(1,v_{rel})$$
Where the vectors are written in the frame of the particle with ##U^\mu##.
The dot product between these is
$$U^\mu V_\mu=\gamma_{rel}$$
If we define angle ##\theta## between two vectors as:
$$\cos\theta=\frac{U^\mu V_\mu}{\vert U\vert\vert V\vert}$$
then since all velocity vectors have magnitude of ##1##, we get ##\cos\theta= \gamma_{rel}\geq 1##.
But what does this mean?