I Angles between 4-vectors in special relativity?

LightPhoton
Messages
42
Reaction score
3
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?
 
Physics news on Phys.org
LightPhoton said:
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.
 
  • Like
Likes Hornbein and 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.
 
  • Like
Likes Dale and PeterDonis
Hornbein said:
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.
 
velocity = c tanh (rapidity)
celerity = c sinh (rapidity)
time-dilation-factor = cosh (rapidity)
Doppler-factor = exp (rapidity)
 
Thread 'Can this experiment break Lorentz symmetry?'
1. The Big Idea: According to Einstein’s relativity, all motion is relative. You can’t tell if you’re moving at a constant velocity without looking outside. But what if there is a universal “rest frame” (like the old idea of the “ether”)? This experiment tries to find out by looking for tiny, directional differences in how objects move inside a sealed box. 2. How It Works: The Two-Stage Process Imagine a perfectly isolated spacecraft (our lab) moving through space at some unknown speed V...
Does the speed of light change in a gravitational field depending on whether the direction of travel is parallel to the field, or perpendicular to the field? And is it the same in both directions at each orientation? This question could be answered experimentally to some degree of accuracy. Experiment design: Place two identical clocks A and B on the circumference of a wheel at opposite ends of the diameter of length L. The wheel is positioned upright, i.e., perpendicular to the ground...
According to the General Theory of Relativity, time does not pass on a black hole, which means that processes they don't work either. As the object becomes heavier, the speed of matter falling on it for an observer on Earth will first increase, and then slow down, due to the effect of time dilation. And then it will stop altogether. As a result, we will not get a black hole, since the critical mass will not be reached. Although the object will continue to attract matter, it will not be a...
Back
Top