About spacetime coordinate systems

[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.