Understand Linear Perturbations to Stationary Black Holes

[Replusz]

TL;DR Summary

Question is in the title, some context:
https://arxiv.org/pdf/1504.08040.pdf It is in the abstract here.

This is on page 2, and I guess it is the key to understanding what they mean by linearized perturbation to a BH in the abstract.
What is meant by gravitational fields, what is delta(g_ab) and delta\Phi ? A perturbation to the metric, and the 'gravitational field', sure. And where are these added? Not in the action/lagrangian right? Just simply take g_ab of a black hole e.g. Schwarzschild and add delta g_ab to everywhere it comes up? What about Phi, what is Phi in general relativity?

Thanks!

 

[PeterDonis]

@Replusz you labeled this thread as "I" level, but the paper you linked to is an "A" level paper; if you don't already have a graduate-level background in theories of gravity you are not going to be able to follow the paper. Anyone with such a background will have no problem understanding all the things you ask about.

Also note that this paper is not about general relativity; it is about hypothetical theories of gravity that have additional terms in the field equations (or the action) beyond those that are present in general relativity.

 

[PeterDonis]

What is meant by gravitational fields

Here it means the metric ##g_{ab}## and the additional scalars ##\phi## that appear in the gravitational portion ##L_g## of the action.

what is delta(g_ab) and delta\Phi ?

The perturbations to the background values of ##g_{ab}## and ##\phi##. This is standard notation in a variational problem.

where are these added? Not in the action/lagrangian right?

I don't know what you mean by "added". The paper is considering a standard variational problem where the action, or at least the gravitational portion ##L_g## of it, is varied around its background value by varying ##g_{ab}## and ##\phi##. If you are not familiar with variational problems you might want to first study the simpler variational problem for the standard Einstein-Hilbert action in GR, which leads to the vacuum Einstein Field Equation. Most GR textbooks discuss this.

Just simply take g_ab of a black hole e.g. Schwarzschild and add delta g_ab to everywhere it comes up?

No.

What about Phi, what is Phi in general relativity?

There is no ##\phi## in general relativity. See the last paragraph of my post #2.

 

[Replusz]

There is no ϕϕ\phi in general relativity. See the last paragraph of my post #2.

Thank you, I missed this, even though it explicitly states on page 1.

The paper is considering a standard variational problem where the action, or at least the gravitational portion LgLgL_g of it, is varied around its background value by varying gabgabg_{ab} and ϕϕ\phi.

I understand this, that is where the eq. of motions come from on the first page.

The perturbations to the background values of gabgabg_{ab} and ϕϕ\phi. This is standard notation in a variational problem.

I understand the variational part. You take the action and the use the Euler-Lagrange method. But how does perturbation come into place? Is that just a synonym for variation or a completely different thing?

Thank you again!

EDIT. I changed the prefix to A.

 

[PeterDonis]

how does perturbation come into place? Is that just a synonym for variation or a completely different thing?

On a further reading, the paper does appear to be mixing up notation a bit. It is evaluating variational integrals, but it also seems to be using "perturbation" to denote a process something like this: we have a stationary black hole with some mass ##M## at early times, a stationary black hole with some slightly larger mass ##M + \delta M## at late times, and a non-stationary "perturbed" configuration in between. We then evaluate the change in entropy during the entire process to see if we can prove that it must be nonnegative. (None of this is stated explicitly in the paper; I'm reading between the lines based on what I have just described being the general sort of scheme that is involved if you're trying to test the second law.)

 

[Replusz]

On a further reading, the paper does appear to be mixing up notation a bit. It is evaluating variational integrals, but it also seems to be using "perturbation" to denote a process something like this: we have a stationary black hole with some mass ##M## at early times, a stationary black hole with some slightly larger mass ##M + \delta M## at late times, and a non-stationary "perturbed" configuration in between. We then evaluate the change in entropy during the entire process to see if we can prove that it must be nonnegative. (None of this is stated explicitly in the paper; I'm reading between the lines based on what I have just described being the general sort of scheme that is involved if you're trying to test the second law.)

So we have a black hole with ##g_{ab}## and ##\Phi##, then we perturb ##g_{ab}## and ##\Phi## with those delta-s, and then for this perturbational process the second law holds. Correct? Thanks so much!

 

[PeterDonis]

So we have a black hole with ##g_{ab}## and ##\Phi##, then we perturb ##g_{ab}## and ##\Phi## with those delta-s, and then for this perturbational process the second law holds. Correct?

That's basically what the paper appears to be doing, yes.

 

[Replusz]

Thank you!

 

[PeterDonis]

Thank you!

You're welcome!