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The Schwarzchild solution is derived in vacuum, and we find that light cones always move inward when r < 2M. So if a spherically symmetric, non rotating celestial object of mass M has a radius less than 2M, it must be a Schwarzchild black hole with a singularity at r = 0. But we have derived the Schwarzchild metric in vacuum. Unless the black hole is just the point of infinite density at r = 0, the metric will not hold inside the event horizon all the way till r = 0, because we will have to include the stress energy tensor for the matter that lies inside the event horizon. While this is still a good physical model outside the event horizon, how can we be certain that there is a singularity at the centre?

ProfRob
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Panyam Tejas
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  • Are you asking whether it's possible that there is a stable, non-singular distribution of matter inside the event horizon? – J. Murray May 01 '22 at 15:56
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    Does this answer your question? Are black holes really singularities? – John Rennie May 01 '22 at 17:36
  • If you believe that spacetime inside the horizon exists, then a singularity is imposed by the Penrose Singularity Theorem. – safesphere May 04 '22 at 14:26
  • “Unless the black hole is just the point of infinite density at $r = 0$” - Inside a black hole, $r=0$ is not a point that exists in time, but an infinitely long line that happens only momentarily. – safesphere May 04 '22 at 20:21
  • I am asking if we know for sure that there is a point singularity/ ring singularity inside a Schwarzchild/ Kerr Black hole, or if we get the singularity by extending the coordinates beyond the event horizon. I see comments stating singularity theorems, I haven't reached that far in my reading yet. I want to know if they prove, without extension of coordinates, that there is a singularity behind the event horizon. – Panyam Tejas May 05 '22 at 22:29
  • Free falling objects follow the spacetime path called “geodesic” (not the same as the space path called “trajectory”). The singularity theorem states that, if geodesics cross the horizon (also called a “trapped surface”), then they must end somewhere inside. Because a geodesic is a path in spacetime, the end of a geodesic is the end of space and time for a falling object - a singularity. (This is different from the end of a trajectory where the object keeps moving in time at the end.) Thus either no spacetime exists inside the horizon or there must be some sort of a singularity there. – safesphere May 06 '22 at 02:30
  • “a point singularity/ ring singularity inside a Schwarzchild/ Kerr Black hole” - A Schwarzschild singularity is not a “point”, but an infinitely long spacelike line - meaning it is one moment of time along a line in space. So the Schwarzschild singularity does not “exist” (in time), instead it “happens” momentarily when time ends. The Kerr singularity is a timelike ring - meaning it is a ring in space that exists in time, although “time” around it can go in infinite loops. The Kerr singularity is hidden behind the unstable Cauchy horizon and for this reason is believed to be unphysical. – safesphere May 06 '22 at 02:42
  • Consider the $3D$ coordinates $(x,y,z)$ and define $r\equiv\sqrt{x^2+y^2}$. Now notice that the equation $r=0$ describes not a point, but an infinitely long line of the coordinate axis $z$. Similarly, in the $4D$ coordinates $(x,y,z,t)$ where $r\equiv\sqrt{x^2+y^2+z^2}$, the equation $r=0$ does not describe a point, but an infinitely long line of the coordinate axis $t$, which inside the horizon is a direction in space. So the Schwarzschild singularity $r=0$ is not a point, but an infinitely long spacelike line. For details see: https://math.stackexchange.com/questions/2929400 – safesphere May 06 '22 at 02:59

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I was just 3 minutes ago reading Planck Stars by Carlo Rovelli. In it he conjectures that a star does not collapse down to a singularity, but rather it collapses down to Planck density: 10^93 g/cm^3 with a size of approximately 10^-12 meters cubed which is still some 24 orders larger than the Planck size.

He goes on that in its own proper time, the Planck star lives an exceedingly short life before it bounces back. That life is the time required for light to cross the width of a proton, but because of the incredible time dilation would be about 14 billion years of Earth time. Because this happens to match the current age of the universe, perhaps some of these "bounces" will be detectible to us.

Rovelli's concept helps to solve the problem of information loss.

foolishmuse
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how can we be certain that there is a singularity at the centre

You are starting from a Schrwazchild black hole model which is an idealized eternal model and so never changes. This is a useful basic model of a black hole but it's not how real black holes are - they clearly cannot be eternal as the universe didn't exist at one point previous and indeed the originating stellar object did not either. We also know black holes expand as they absorb material so that's another issue. Hawking also showed that they can emit radiation (albeit not a lot) so an eternal black hole model is clearly not a match to real black holes.

While with an idealized Schwarzchild black hole you can state there is a singularity at the center (because that's just the math and it's an ideal), what is inside the event horizon of a real black hole is probably another thing entirely.

Even the next step up from a Schwarzchild black hole (which is non-rotating) to a rotating black hole (the Kerr metric) produces a requirement for a ring singularity and other features not in the Schwarzchild model.

We also don't have a complete and generally accepted theory of gravity that incorporates quantum theory so how matter and energy behave under such extremes as a near-singularity are currently not understood. It's not at all clear a "pure" singularity could exist, nor what could be in it's place.

So with that in mind we don't really know what the interior of a real black hole is like. We probably won't ever know as by definition inside a black hole is beyond the side of the event horizon we want to be on. Unless there is a way around the no hair theorem we might never know.

StephenG - Help Ukraine
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