Schwarzschild conceived his classical solution as external only, and later provided an internal solution for a stationary spherical body of incompressible fluid with constant density and pressure p=0 at the surface. If applied to a collapsing star, this internal solution is valid up to right before its collapse, because it yields an infinite pressure at r=0 (the center of the sphere) when the radius of the sphere reaches at a particular critical radius before becoming the Schwarzschild radius of the external solution.
But even before reaching that critical radius, and due to the enormous pressure, the scalar curvature S=density-3p of the perfect fluid at r=0 given by the trace of the stress-energy tensor becomes negative. For small bodies such as the Earth, this expression is dominated by the energy density, so it's positive. But for stars close to becoming black holes, it starts to become negative (3p>density) from Rs/R=5/13 to Rs/R=5/9 with Rs the Schwarzschild radius and R the sphere radius. From Rs/R=5/9 until R=Rs, the scalar curvature is always negative for the entire sphere.
What is the meaning and implication of this negative scalar curvature?
Stefan Bernhard Rüster
Dear Bernhard. I posted the question thinking about a star before collapsing into a black hole. This is best described by the interior Schwarzschild metric. As you can see, you get a negative scalar curvature even before the solution becomes unstable (before giving an infinite pressure at the center) and collapse. I dont think a star in this condition is beyond the ultra-relativistic limit nor unrealistic.
Stefan Bernhard Rüster
That's not what is usually being taught. Of course you cannot have an infinite pressure, but why not a contribution from pressure larger than the one from matter? Why can't you have a negative scalar curvature in a stable Schwarzschild interior solution?
What is the meaning? You do not specify what type of meaning you are interested in, mathematical or physical. It is known that the Schwarzschild metric is only physically valid outside the event horizon - and regions close to the event horizon are also dubious. Mathematical solutions to the field equations of GR are often unphysical, and you point to an example. They should always be checked by a good physicist, such as Einstein (who had understood why portions of the Schwarzschild solution are unphysical by 1939). All black hole metrics are (at least partially) unphysical and plagued by a variety of paradoxes. The information paradox is just one of them, but today's academic community cannot even seem to agree that information loss implies they have strayed away from genuine physics.
Dear Stefan Bernhard Rüster
This question was originally posted my friend Jan Gogolin here: https://physics.stackexchange.com/questions/681867/how-to-interpret-the-sign-change-of-scalar-ricci-curvature-in-schwarzschild-inte
I hope he can join here for the discussion
Dear Robin Spivey
It is the exterior Schwarzschild metric that is only physically valid outside the event horizon. We are dealing here with the interior Schwarzschild metric, which is a different metric solution, and nothing to do with black holes.
Hallo Stefan Bernhard Rüster
Yes, I missed to state that the Schwarzschild interior metric considered has homogeneous constant mass density.
Of course this is a simplification and does not represent a real star accurately enough.
Still, one may wonder what does the negative scalar curvature mean/imply, or if there is a physical interpretation for it, even though it is not responsible for the star collapse.
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