About spacetime coordinate systems

[Dale]

Search for a solution of Einstein Field Equation (EFE) with given symmetry. For instance in Schwarzschild case start with spatial polar coordinates (r,θ,ϕ) and a coordinate time variable t. Which conditions those functions (map) have to comply with ?

The only conditions that the function must comply with is that it is one-to-one and smooth. The functions do not need to respect the underlying symmetry of the manifold.

Symmetries of the manifold are called Killing vectors. Spherical symmetry means that there are three Killing vectors corresponding to rotations about three orthogonal axes. In addition, a static spacetime means that there is one additional Killing vector corresponding to the symmetry under translations in time.

Now, coordinate systems on this spacetime are not required to respect these symmetries. However, the symmetries also imply the existence of coordinate systems that do respect those underlying symmetries. In such coordinates the metric simplifies in specific ways which makes the calculation of the metric easier. This is shown in detail in Sean Carroll’s Lecture Notes On General Relativity. Using the simplifications available for symmetry-respecting coordinates results in a particularly nice form for the metric.

Surely there exist the requirement to map "close" events in time to "close" coordinate time t values

Yes, that is the smoothness criterion.

 

[sweet springs]

For any coordinate you choose, there is a general procedures to measure proper time and space length and thus find all the metric tensor components.
See section 84 of [link removed].

[Mentors' note: a link to what appears to be a pirated version of " The classical theory of fields" by Landau and Lifshitz has been removed. If anyone is aware of a authorized online reproduction, please PM any mentor so that we can post it here]

 

[cianfa72]

Speed is always relative so when we say that something is at rest or moving we have to say what that's with respect to. But proper acceleration is not relative in that sense; we can use an accelerometer to measure acceleration without reference to any external object.

I'm aware of marvelous Mach point of view about inertia as basically due to the entire universe mass distribution. In that context we can conceive proper acceleration as relative, actually, to the overall mass distribution. Which is the current point of view in the context of general relativity ?

 

[Nugatory]

Which is the current point of view in the context of general relativity ?

General relativity is not inherently Machian: its predictions for the proper acceleration measured at the surface of a rotating body are independent of the hypothetical distribution of hypothetical distant masses in the rest of the universe.

Whether GR is actively hostile to Mach's Principle is a different question. With neither a precise statement of exactly what is meant by "Mach's Principle" nor a testable theory based on that precise definition and that makes different predictions than GR, it's easy to start an argument by asking this question, not so easy to end it.
My advice is to learn GR first; after you understand it, you can decide for yourself whether Mach's principle further advances your understanding. However, if you can't wait... googling for "Mach's Principle and GR" will find many links, and even some to older threads here. Just be aware that the Google search rankings do not reliably distinguish between fringe and real science.

 

[PeterDonis]

its predictions for the proper acceleration measured at the surface of a rotating body are independent of the hypothetical distribution of hypothetical distant masses in the rest of the universe.

Not really. The prediction you refer to assumes an asymptotically flat spacetime, which is equivalent to assuming that whatever other stress-energy exists in the universe is distributed spherically symmetrically outside some large distance from the rotating body, with vacuum inside that distance. The prediction will change if that assumption is violated.

The "non-Machian" nature of GR in this case is better described, I think, by the observation that the asymptotically flat assumption is also satisfied by a spacetime in which the rotating body is absolutely alone in the universe, meaning the spacetime becomes Minkowski at very large distances from the body. But the asymptotic flatness in this version is not due to any stress-energy elsewhere in the universe; it's just there. (Another way of putting it would be to observe that there are multiple vacuum solutions to the Einstein Field equation in GR; but since they are all vacuum solutions, their stress-energy content is all the same--i.e., none--so whatever the difference is between them, it can't be due to a difference in stress-energy, which is the only kind of difference that a purely "Machian" theory would allow.)

 

[PeterDonis]

In that context we can conceive proper acceleration as relative, actually, to the overall mass distribution.

No, it isn't. The proper acceleration is a direct observable; you don't even have to know the overall mass distribution to measure it. Nor do you have to make any choice of coordinates to measure it.

 

[PeterDonis]

respect to what is assumed the black hole non-rotating ?

"Non-rotating" is equivalent to "spherically symmetric" in this context.

 

[Dale]

With neither a precise statement of exactly what is meant by "Mach's Principle" nor a testable theory based on that precise definition and that makes different predictions than GR, it's easy to start an argument by asking this question, not so easy to end it.

This is my objection to Mach’s principle also. The one example that I know of where someone tried to precisely define Mach’s principle and make a testable theory out of it is Brans Dicke Gravity. But with that as the operational definition it seems that the universe is non-Machian.