Incorrect to say Einstein's GR implies Black Holes

Question

Einstein’s GR describes space filled with energy and matter. As Schwarzschild’s empty space solution is physically anomalous, is it not incorrect to state Einstein’s GR predicted black-hole singularities? Has anyone shown that Schwarzschild is the limit to a continuum of plausible solutions, rather than an isolated anomaly?

I agree with comments that the Oppenheimer-Snyder metric is crucial, OS was seriously misquoted by Penrose. OS saw the fixed “horizon” ($2MG/c^2$) divides exterior from interior particles. They showed, a particle falling freely would reach the horizon in a finite proper time, as gravitational time dilation causes both falling particles and light speed to slow to zero speed as they approach it. OS also constructed a metric for the interior region, matched to the exterior. At no point does a particle go inside the horizon. Yet Penrose stated “as measured by local comoving observers, the body passes within its Schwarzschild radius”. Analysis of the OS metric shows that there is a single time coordinate $t$ for both the exterior and interior regions. Contraction continues until $t = +ꚙ$, while in spatial terms contraction stops at the gravitational radius (cf. Einstein 1939). Marshall (2016 DOI:10.3390/e18100363). Weinberg (1972) agreed with OS rather than Penrose. Marshall has also followed his differential geometry to derive a proper interior solution (modifying the OS interior metric).

Similar questions have been posed previously, as Necessity of Singularity in General Relativity the discussion in “chat” did not happen. Of the answers, only the “famous singularity theorems” appears conclusive (collapse beyond horizon implies a singularity). Yet the Penrose singularity theorem was challenged by the “famous” Roy Kerr finding counterexamples in the Kerr metric (2023). Kerr also challenged the Raychaudhuri theorem https://en.wikipedia.org/wiki/Raychaudhuri_equation on which Penrose relied (Roger Penrose on video called Kerr’s challenge “rubbish”, but he appears at 93 to be not up to defending his theorem). The mathematical-topology arguments are unsupported, it seems, by differential geometry solutions - fitting of inner and outer metrics – suggested by one answer. So I am asking a similar question pressing the key issue, in the context of Kerr’s general statement that there are no singularities in physics.

Answer 1

Einstein’s GR describes space filled with energy and matter. As Schwarzschild’s empty space solution is physically anomalous, is it not incorrect to state Einstein’s GR predicted black-hole singularities?

No. Quantum mechanics, for example, predicts things that are best demonstrated by completely-unrealistic particles-in-boxes. That doesn’t mean that QM is incorrect for predicting quantum tunnelling or what have ye.

Karl Schwarszchild made an interesting observation about the Einstein field equations when constructing a spherically-symmetric spacetime. That doesn’t mean GR is wrong because that observation wasn’t necessarily the most realistic.

Has anyone shown that Schwarzschild is the limit to a continuum of plausible solutions, rather than an isolated anomaly?

Yes. The Oppenheimer-Snyder metric is arguably Oppenheimer’s most important contribution to physics (often overshadowed by his other work) that models a collapsing star/other supermassive object which results in an exterior-Schwarzschild metric at the end. It was considered weird when it was released, and sort of still is now, but it works out.

Note also that, from far enough away, any spherically-symmetric spacetime is necessarily Schwarzschild. This is Birkhoff’s theorem, and can be rigorously proved as in Gravitation. Any asymptotically-flat spacetime that is spherically-symmetric, arising from or departing to any other configuration, is necessarily Schwarzschild in geometry at some sufficient distance.

I am asking a similar question pressing the key issue, in the context of Kerr’s general statement that there are no singularities in physics.

There needn’t be singularities for there to be black hole-like objects. Gravastars, for example, are models of spacetimes that are black hole-ish on the exterior but have a nonsingular interior.

Answer 2

The "challenge" of Roy Kerr (repeating well-known and openly admitted caveats of the theorems, misinterpreting the maximal analytical extension of his own solution worked out by other people, and providing unrelated comments and anecdotes from his life), was already discussed in this answer. One should never rely on the weight of a personal authority when considering a science argument. Even Einstein struggled with the interpretation of exact solutions of General relativity, leading to the famously mistaken assertion that gravitational waves do not exist in 1936. The fact that a person's name is attached to a seminal result cannot be taken as a guarantee that they keep in touch with the state of the art and that each of their statements about the topic is relevant or informed by the best current knowledge.

Yes, there may be loopholes to the singularity theorems. Their assumptions as well as conclusions are technical. For example, the notion of geodesic incompleteness may or may not be an adequate equivalent to what we would consider a "true" singularity. There may be some very special setups that manage to find a weird loophole and violate the formation of singularities etc. But the fact that once a horizon forms during collapse, the matter will just keep contracting without end and "something" bad will happen is just proven beyond any doubt.

We have ample evidence for the formation of black holes with continuum models

• The Oppenheimer-Snyder collapse, where a continuum of "dust" (a presureless gas) forms a black hole, as published by Oppenheimer and Snyder in On Continued Gravitational Contraction in 1939.
• The collapse of a scalar field within numerical relativity, the study of which lead to the famous Choptuik phenomenon, discussed by Choptuik in 1993.
• The gravitational core collapse of stars leading to the formation of black holes and supernovae, as simulated for the first time by Schwartz in 1967, and gradually refined with the same conclusion over and over. (See Müller's Living Review from 2020 for references on the hundreds of simulations that predict black hole formation by core collapse.)
• Even things such as binary neutron star mergers (see the Living review by Faber and Rasio from 2012) or even the collapse of colliding gravitational waves themselves (see the Letter of Ledvinka and Khirnov from 2021).

From the mathematical standpoint, the fact that these finite simulations produce things that asymptote to black holes is guaranteed by the black hole stability theorems from a few years back (see the work of Giorgi, Szeftel and Kleinerman (2021) and references therein). In other words, if you have initial data that corresponds to a perturbed subextremal black hole, your field is going to relax to a black hole field and not blow up into something else.

In other words, the existence of black holes is an extremely robust prediction of classical relativity, verified by several generations of the most brilliant physicists and mathematicians from a number of points of view.

Answer 3

is it not incorrect to state Einstein’s GR predicted black-hole singularities?

It certainly is not incorrect, in the sense that it is clear that such spacetimes are valid solutions to the Einstein field equations.

Of course, the actual universe has matter. So it is not expected that a vacuum solution accurately represent the universe. However, it is still useful.

Has anyone shown that Schwarzschild is the limit to a continuum of plausible solutions, rather than an isolated anomaly?

Yes. Others have correctly mentioned Oppenheimer-Snyder, but even more important is Birkhoff. He showed that the Schwarzschild spacetime is the unique spherically symmetric vacuum spacetime. So it applies to a vacuum region outside of any spherically symmetric distribution of matter. As such it is a reasonable model for the vacuum region outside a typical planet or star.

At no point does a particle go inside the horizon.

That is simply untrue. At the point that the OS ball of dust collapses to the Schwarzschild radius there is a standard Schwarzschild horizon.

However, that is not the beginning of the horizon. Starting from that moment, if you follow an outgoing radial null geodesic back in time until it left the center, that is the beginning of the event horizon. The horizon starts in the middle of the dust and propagates out through the remainder of the dust.

So, contrary to your statement, all of the dust particles go into the horizon. It happens before the horizon reaches the Schwarzschild region.

Answer 4

Einstein's GR implies black holes, but not the black hole singularity. The ample evidence for the formation of black holes with the continuum model, the Oppenheimer-Snyder collapse, On Continued Gravitational Contraction in 1939, @Void thanks), is flawed. I quote:

"To investigate this question we will solve the field equations with the limiting form of the energy-momentum tensor in which the pressure is zero. When the pressure vanishes there are no static solutions to the field equations except when all components of T vanish. With P =0 we have the free gravitational collapse of the matter. We believe that the general features of the solution obtained this way give a valid indication even for the case that the pressure is not zero, provided that the mass is great enough to cause collapse ".

It is true that for a static perfect fluid sphere there is no static zero pressure solution. So the assumption of vanishing pressure in a non-static case ensures an infinite gravitational collapse, which the authors wanted to prove. I think such a paper would not be accepted for publication today. But as it happened with Wheeler, they coined the name of gravitational collapse with this paper and have been quoted ever since. I think their reasoning is wrong. For a perfect sphere of fluid in equilibrium, i.e. in a static state, the pressure has its maximum at the center and is zero at the surface. If "the mass is large enough", the central pressure diverges like $1/r^2$ while the energy density is finite. For this star to collapse, the pressure should suddenly collapse to zero everywhere but this is physically not possible.

Trigger warning: this is not a canonical textbook mantra answer ;)