Stephen J Crothers wrote this biography of Leonard S Abrams, based on information provided by Abrams's widow:

Stephen J Crothers:Dr. Leonard S. Abrams was born in Chicago in 1924 and died on December 28, 2001, in Los Angeles at the age of 77. He received a B.S. in Mathematics from the California Institute of Technology and a Ph.D. in physics from the University of California at Los Angeles at the age of 45. He spent almost all of his career working in the private sector, although he taught at a variety of institutions including California State University at Dominguez Hills and at the University of Southern California. He was a pioneer in applying game theory to business problems and was an expert in noise theory, but his first love always was general relativity. His principle theoretical contributions focused on non-black hole solutions to Einstein’s equations and on the inextendability of the “Schwarzschild” solution. Dr. Abrams is survived by his wife and two children.

Abrams was wrong about the inextendability of the Schwarzschild solution, but he did some good stuff. In what follows, all quotations attributed to Abrams come from

Leonard S Abrams. Black holes: the legacy of Hilbert's error.

Canadian Journal of Physics67, 1989, page 919ff. http://arxiv.org/abs/gr-qc/0102055

The first paragraph of that paper's conclusion, with my editorial comments added in blue:

Leonard S Abrams:We summarize the result of the preceding sections as follows. The K-F (Kruskal-Fronsdal) black hole is the result of a mathematically invalid assumption, explains nothing that is not equally well explained by S

_{S}(Schwarzschild spacetime), cannot be generated by any known process, and is physically unreal. Clearly, it is time to relegate it to the same museum that holds the phlogiston theory of heat, the flat earth, and other will-o’-the-wisps of physics.

That's quite a conclusion.

Although I'm not a physicist myself, it's my impression that many (most?) physicists think the "white-hole" half of Kruskal-Fronsdal spacetime is unlikely to be physically real. The controversial part of Abrams's conclusion is that he regards everything inside the gravitational radius (aka Schwarzschild radius, event horizon) as physically unreal and "the result of a mathematically invalid assumption."

It turns out that the mathematically invalid assumption was made by Abrams in his section 6. The first half of his paper is pretty interesting, however, and most of his Appendix A is okay as well. Before we get to his central error, I'll summarize the good stuff.

**Several spacetime manifolds, all of them equivalent**

In 1916, Karl Schwarzschild found the first nontrivial exact solution to Einstein's field equations, for a static spherically symmetric spacetime around an isolated, non-rotating, neutral star of mass m, regarded as a point mass. This is known as Schwarzschild's exterior solution. (He wrote a second paper that describes a solution for the interior of a star that isn't regarded as a point mass, but that interior solution isn't so relevant here.)

As is now well known, Schwarzschild's exterior solution suffers from a coordinate singularity at what is now called the Schwarzschild radius r=2m. That didn't seem terribly consequential at the time, because the Schwarzschild radius lay well inside all known stars. If you wanted to understand spacetime inside a star, you'd use the interior solution, which didn't have that particular coordinate singularity.

What's less well known is that, using the notational conventions I've been using in other threads, Schwarzschild's line-element (pseudo-metric) was:

That's an algebraically inconvenient way to write the pseudometric, so Abrams uses the following equivalent form, which he attributes to Brillouin:

Abrams defines that spacetime as S_{S}, and refers to it as Schwarzschild's spacetime.

Transforming the r coordinate above by adding the Schwarzschild radius α yields the equation that most of us think of as the Schwarzschild metric:

Abrams refers to that as Flamm spacetime S_{F}. Note well that, in Flamm spacetime, the origin of Schwarzschild spacetime has been transformed from r=0 to r=2m. According to Abrams, that means we must think of the entire 2-sphere at r=2m (and arbitrary t) as representing a single point, which he regards as the point mass.

Note also, however, that the 2-sphere at r=2m lies outside the Flamm spacetime, just as the point mass at r=0 lies outside the Schwarzschild spacetime. Abrams is attaching some extra-mathematical mental baggage to the mathematical manifolds he's defining.

Finally, Abrams defines Hilbert's spacetime as the spacetime that's mathematically identical to the Flamm spacetime, but with different mental baggage: Abrams accuses Hilbert of thinking the point mass lies at r=0, and explains how Abrams thinks Hilbert came to think that way. That's Hilbert's alleged error, as mentioned in the title of the paper.

As a mathematician, I have to ask: So what? Hilbert's mental states are no more relevant than Abrams's. What matters here are the manifolds (spacetimes) themselves. The Flamm and Hilbert spacetimes are obviously isometric. Both end at the Schwarzschild radius. Neither includes any points at the Schwarzschild radius, so their spatial slices are missing either a point (Flamm) or a closed ball (Hilbert) at their centers.

Does it matter whether the part that's missing is a point or a closed ball? Not at all: As noted at the end of my previous post, *the topological space you get by removing a single point from Euclidean 3-space is homeomorphic to the space you get by removing a closed ball*.

In his section 5, Abrams observes that Hilbert spacetime can be extended to a larger manifold that includes the Schwarzschild radius and points inside that radius. That extension bypasses the well-known coordinate singularity at the Schwarzschild radius of the Hilbert/Droste/Weyl metric (which is more popularly known as the Schwarzschild metric). Unfortunately, Abrams thinks the corresponding coordinate singularity at the central point mass of the original Schwarzschild spacetime is an irremovable "quasiregular singularity". Abrams therefore believes he has discovered an important difference between the Schwarzschild and Hilbert spacetimes.

In reality, the "quasiregular singularity" at the central point mass of the original Schwarzschild spacetime can be removed by allowing the radial coordinate r to go negative. You can understand why Abrams never considered that possibility, but he should have: Abrams calls the reader's attention to the fact that the radial coordinates of these spacetimes are not identical to the radial coordinates of Euclidean space. He should have realized that the usual assumptions we make about Euclidean coordinates may not apply.

If you want to see exactly how the original Schwarzschild spacetime can be extended by allowing r to go negative, you can read this recent paper that **DeiRenDopa** cited in another thread:

Christian Corda. A clarification on the debate on "the original Schwarzschild solution". http://arxiv.org/abs/1010.6031

**Where Abrams went badly wrong**

In the original version of Abrams's paper, section 6 begins as follows:

Leonard S Abrams:If two space-times are to be equivalent, it is certainly necessary that they be isometric i.e., that there exists a diffeomorphism from one to the other that carries the metric of one into the metric of the other. And since the presence of singularities of the manifold geometry is unaffected by diffeomorphisms, it is also necessary that equivalent space-times have the same “singularity structure”, i.e., the same singularities as one approaches corresponding boundary points. Now, S

_{S}and S_{H}are isometric under T_{α},

So far, so good. All of that is true. Note, however, that the mathematical definition of a spacetime manifold does not mention any explicit "singularity structure". The singularities and their structure must be inferred from the manifolds themselves. Since S_{S} and S_{H} are mathematically equivalent as manifolds, we must infer the same "singularity structure" for both of them.

Leonard S Abrams:but as shown in the preceding section, S

_{H}has no singularity corresponding to the quasiregular singularity at r = 0 in S_{S}. Consequently, S_{S}and S_{H}are inequivalent. Since it was shown in Sect. 2 that S_{S}is the space-time of a point mass, it follows that S_{H}and its analytic extension (S_{K−F}) are not.

That's where Abrams goes off the rails. He's comparing the extensibility of S_{H} (which is a mathematical fact) to the mental baggage he's attached to the Flamm and Schwarzschild spacetimes, which is a mistake. Abrams thinks the Flamm and Schwarzschild spacetimes are inextensible, but he never actually proved that, and he's wrong.

Abrams must have realized there was something wrong with his section 6, because he published an erratum that replaces his entire section 6 with a paragraph that starts like this:

Leonard S Abrams:By inspection, S

_{S}and S_{H}are isometric via T_{α}and thus equivalent.

Abrams should have stopped right there, and drawn the obvious conclusion.

But he went on:

Leonard S Abrams:However, it was shown above that due to the difference in the topology of their boundaries, they are associated with different singularity structures. Thus, the universes (U

_{S}and U_{H}) corresponding to S_{S}and S_{H}(with their indicated boundaries) are inequivalent...

That's just wrong.

The boundaries aren't part of the spacetime manifolds, so Abrams is discussing his mental baggage. Mathematically, it is more correct to imagine all of those boundaries as 2-spheres (for any t) than to imagine any of those boundaries as points, because all of those manifolds can be extended by attaching a 2-sphere (and points within). None of those manifolds can be extended by attaching a single point.

The "singularity structures" aren't part of the spacetime manifolds either. Singularities must be inferred from the spacetime manifolds. Because these spacetime manifolds are equivalent, they have the same singularities. Full stop. Any differences that Abrams may think he perceives come from the mental baggage he's attached to these manifolds.

In my next post, I'll point out some other assumptions that Abrams should not have made. We'll get to the Crothers papers eventually, which is where the real fun begins.