- #1

Suekdccia

- 175

- 21

- TL;DR Summary
- Spacetimes, metrics and symmetries in the theory of relativity?

I was discussing this paper with a couple of physicists colleagues of mine (https://arxiv.org/abs/2011.12970)

In the paper, the authors describe "spacetimes without symmetries". When I mentioned that, one of my friends said that no spacetime predicted or included in the theory of relativity could break the Lorentz and diffeomorphism symmetries...

However, the other one didn't seem to agree, as he asked what he meant exactly by Lorentz and diffeomorphism symmetry:

"

The thing is that I'm having trouble trying to understand how is the Poincaré/Lorentz and diffeomorphism symmetries crucial for General Relativity while there can be solutions to it corresponding to spacetimes which violate them at the same time...

In the paper, the authors describe "spacetimes without symmetries". When I mentioned that, one of my friends said that no spacetime predicted or included in the theory of relativity could break the Lorentz and diffeomorphism symmetries...

However, the other one didn't seem to agree, as he asked what he meant exactly by Lorentz and diffeomorphism symmetry:

"

*Are you talking about a symmetries of the theory? The theory of general relativity is diffeomorphism (and so Lorentz) invariant.*

Are you talking about a symmetry of a spacetime, solution of the equations of motion of general relativity (the Einstein equations)? Defined how? Is it a change of coordinates and the corresponding change of metric components? In this case, a spacetime is trivially invariant (a change of coordinates cannot have any physical effect).

Or is it an active diffeomorphism in which one keeps using the same coordinate system but moves points to new points and then compares the spacetime metric on the old set of points with the spacetime metric on the new set of points? This is the kind of symmetry the authors mention in their paper. When the authors say "spacetime without symmetries", they mean without this latter kind of symmetries.

For example, Minkowski (flat) spacetime has Poincaré symmetries of this kind (e.g. it is translation invariant, but also rotation and boost invariant). The curvature at any point of the spacetime is the same.

Now, imagine bending Minkowski spacetime in an arbitrary way. The result, in general, will not have translation, rotation and boost symmetry, and the curvature at any 2 arbitrary points will be, in general, different."Are you talking about a symmetry of a spacetime, solution of the equations of motion of general relativity (the Einstein equations)? Defined how? Is it a change of coordinates and the corresponding change of metric components? In this case, a spacetime is trivially invariant (a change of coordinates cannot have any physical effect).

Or is it an active diffeomorphism in which one keeps using the same coordinate system but moves points to new points and then compares the spacetime metric on the old set of points with the spacetime metric on the new set of points? This is the kind of symmetry the authors mention in their paper. When the authors say "spacetime without symmetries", they mean without this latter kind of symmetries.

For example, Minkowski (flat) spacetime has Poincaré symmetries of this kind (e.g. it is translation invariant, but also rotation and boost invariant). The curvature at any point of the spacetime is the same.

Now, imagine bending Minkowski spacetime in an arbitrary way. The result, in general, will not have translation, rotation and boost symmetry, and the curvature at any 2 arbitrary points will be, in general, different.

The thing is that I'm having trouble trying to understand how is the Poincaré/Lorentz and diffeomorphism symmetries crucial for General Relativity while there can be solutions to it corresponding to spacetimes which violate them at the same time...