I ponder about interpretation of scalar curvature in Schwarzschild interior solution. It reads:
- Rkk = epsilon - 3p,
with epsilon energy density and p pressure. The minus sign on the left side is due to metric signature definition (+,-,-,-). One can calculate that scalar curvature at the origin is positive when the compactness parameter a=rS/R 5/9. The radial geodesics are converging in all cases but this is normally interpreted as a positive scalar curvature. How is to understand the zero and the negative curvature then?
Jan
https://orcid.org/0000-0003-1774-9985
Something else is meant here, since the Schwarzschild solution is the unique solution for the case of a static, spherically symmetric, asymptotically flat spacetime (with no charges) in vacuum. So there’s neither energy, nor pressure.
The convention about the signature shouldn't matter, either, it's the fact that the mass parameter is positive that matters.
When discussing the ``interior'' solution it's necessary to be careful, when trying to use the coordinate system, that's appropriate for an exterior observer, however. In that convention time and space directions get exchanged, when r goes from values greater than 2M to less than 2M. That's why it's more useful to use Kruskal-Szekeres coordinates, when discussing the interior observer-the discussion is equivalent, of course, the chances of making mistakes are, however, smaller.
Cf. for instance: http://www.physik.uni-regensburg.de/forschung/gebhardt/gebhardt_files/skripten/Lect04KruskalCoord.pdf
The static interior Schwarzschild solution to a collapsing model by applying geometrical methods. The collapsing stellar object contracts asymptotically to its minimum extent and needs an infinitely long time to arrive at the final state. The event horizon of the exterior Schwarzschild solution is not reached or even crossed.
Stam Nicolis, thanks a lot for your replay. There are two Schwarzschild solutions, for the interior see for the example: https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric . The so called exterior Schwarzschild solution that is asymptotically flat starts on the surface of constant energy density star. In true, there is only one spacetime that is divided into two regions because two different physical properties - a sphere of fluid and outer spacetime without matter, i.e. epsilon and p equal zero. Without equations it is difficult to explain. If you like please look into https://physics.stackexchange.com/q/681867/281096 . I have formulated my question there with more details.
Md. Tarek Hossain, thank you very much for your answer but I do not mean a collapsing star. My question refers to static solutions, as for example the second Schwarzschild solution ( https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric ). If you like please look to https://physics.stackexchange.com/q/681867/281096 . I have formulated my question there using equations.
Jan Gogolin There aren't two ``solutions''-the solution of Einstein's equations is unique. There doesn't exist a unique coordinate system for all of spacetime, that's all, which is to be expected for any spacetime that's not flat, not, just Schwarzschild. But the different coordinate systems can all be related by general coordinate transformations, that's how it's possible to understand that they all describe the same solution. But these transformations all contain an inevitable singularity, that describes the fact that the spacetime is incomplete. The interior observers will hit the singularity in finite proper time, the exterior observers have the singularity at infinity (which, however, remains inaccessible to the observer-the exterior observers don't have the black hole singularity in their future-cf the introduction of the paper linked to below).
It's because of the possibility of reallizing general coordinate transformations, that it doesn't make sense to make statements about quantities that do change under them. The sign of the scalar curvature doesn't change, however: Schwarzschild spacetime, for a positive value of the mass parameter has positive curvature, that tends to zero at infinity (that's what ``asymptotically flat'' means and is part of the boundary conditions that define the solution) and diverges at the singularity.
The Schwarzschild spacetime is the vacuum solution, it doesn't depend on the equation of state of the star, whose collapse produces it.
It is possible to study perturbations of the Schwarzschild solution and ask the question of its stability. It has been proven that the Schwarzschild solution is stable: Preprint The non-linear stability of the Schwarzschild family of black holes
The question of the interior is addressed here: Article The interior of dynamical vacuum black holes I: The $C^0$-st...
Stam Nicolis, you speak all the time about a black hole. Maybe you have me misunderstood. I speak about a simple matter sphere, as for example earth, and the solution of Einstein field equations for such spacetime (sphere with matter surrounded by empty space). Could I make me understandable?
Stam Nicolis, by the way, the scalar curvature R and Ricci curvature Rik in vacuum Schwarzschild spacetime is zero (no matter, no stress-energy tensor Tik). Only some Riemann tensor Riklm components are non-zero there.
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