Angles between 4-vectors in special relativity?

[LightPhoton]

TL;DR

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?

 

[PeroK]

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?

It means Minkowski spacetime is manifestly non-Euclidean.

 

[Ibix]

If both $U$ and $V$ are unit and timelike then their inner product is $\cosh\psi$, where $\psi$ is the rapidity. Rapidity has a one-to-one relationship with speed, and is additive (at least in one dimension). But it's not much used except to reframe the Lorentz transforms as manifestly the Minkowski analogue to Euclidean rotations.

If both $U$ and $V$ are unit and spacelike then their inner product is the Euclidean inner product.

If the vectors are different types the inner product remains useful, but I'm not sure it has a meaningful interpretation in terms of angle-type quantities.

 

[Ibix]

I always liked rapidity a.k.a. celerity.

Rapidity and celerity are different things. Rapidity is $\psi$ and $v=c\tanh\psi$, but celerity is $dx/d\tau=\gamma v$

I think rapidity is one of those things that isn't particularly useful once you go on to GR, so it's an interesting but fairly niche concept. Celerity is best forgotten about, IMO.

 

[robphy]

velocity = c tanh (rapidity)
celerity = c sinh (rapidity)
time-dilation-factor = cosh (rapidity)
Doppler-factor = exp (rapidity)

 

[robphy]

Old Possibly useful thread:
https://www.physicsforums.com/threads/angle-between-space-like-and-time-like-vectors.74115/