Recent content by Qwet

Yes, actually. That is the reason for my question. I know, what parallel transport is. I understand that we cannot get the law of energy conservation due to ambiguity of the ways to transport vectors in the space: Yes, I understand, that we cannot integrate 4-vector over a volume element...

No, I just wanted to say, that I don't have references on this point; I did't read any book, where there would be said, that Gauss's theorem cannot be used without metric. Of course, I did't read every book on Earth, so I said that I don't understand, how this two concepts are related.

Well, I would do it like this. Let's write 3-dimensional Gauss's theorem similarly to the form, which Ostrogradskiy got. ## \int_V [\frac{∂A^1}{∂x^1}+\frac{∂A^2}{∂x^2}+\frac{∂A^3}{∂x^3}]dx^1dx^2dx^3= \oint_{∂V} [A^1sgn(n_1)|dx^2dx^3|+A^2sgn(n_2)|dx^1dx^3| + A^3sgn(n_3)|dx^1dx^2|]## If ##A^i##...

I do not understand how partial derivatives (from calculus of function of multiple variables) relate to "space", and Gauss's theorem to metric. In any book in a section about Gauss's theorem, there should not be any word about metric. Why? We only need the "usual" Levi-Civita symbol which is...

What do you mean by "wrong"? Partial derivatives can be used in curved spacetime. Gauss's theorem can be used even in space with undefined metric.

While I was editing that I wasn't right, you posted:biggrin: Thank you!

Only if we define it in static spacetime. If defined in non-static one, it can change with time.

Is that because derivatives are covariant in Gauss's theorem? As I understand, if they would be partial, curvature would not prevent us from defining integral equation corresponding to the law? That means that the spacetime has to be isometric, and there is no another symmetry, energy...

I didn't consider it simple. But it seems to be simpler than in case of Killing vectors. Thank you. Another question - is it possible to use Gauss's theorem to get integral equation for the conservation law in case of no Killing field?

Yes, but in general relativity we can only write the equality of energy-momentum tensor covariant divergence to zero which is the equation of matter motion. And this gives the first law of thermodynamics. But this is not the energy-momentum conservation law itself, because there is no...

Yes, thank you. I did't mention this fact because in this case the problem is trivial. As I understand, it just simplifies the calculations. But I am interested if the law still holds its physical sense in the case of no Killing vector field? As you say, doesn't that mean that there is no...

Right. And that is the reason for my question.

Hello. I have a question about the law of energy conservation in GR. As time is inhmogeneous, we don't have energy-momentum 4-vector which would be preserved during system's dynamical change. It is only possible to define 4-vector locally. And next, the problem regarding how to sum this vectors...