In this post, we'll look at the assumptions and constraints that led to what is now known as Schwarzschild's exterior solution of Einstein's field equations for spacetime around a non-rotating, electrically neutral star or black hole.

In the early years, no one gave any thought to the possibility of a black hole. That changed in 1931, when Chadrasekhar realized that large white dwarfs would eventually collapse under their own gravity. In 1939, Oppenheimer predicted that large neutron stars would collapse into black holes. Unfortunately, Oppenheimer misinterpreted the Schwarzschild coordinate singularity to mean time stops at the Schwarzschild radius, and that misinterpretation is still being promoted by the electric universe folk and some others.

Assumptions

In the paper cited above, Abrams set the scene:

Leonard S Abrams:

Consider now the particular U consisting of a single uncharged, nonrotating, nonradiating point mass (whose Newtonian gravitational mass will henceforth be denoted by ‘m’). Historically, the conditions regarded as distinguishing the space-time (MU , gU ) of this U from those of all others where originally formulated by Einstein [8], and together with those implicit in that formulation were enumerated by Finkelstein [9].

Abrams assumed the following conditions:

  1. The spacetime is static.
  2. The spatial part of spacetime is spherically symmetric.
  3. The (pseudo-) metric has Lorentz signature.
  4. One of the coordinates represents global time.
  5. The metric satisfies Einstein's field equations for empty space.
  6. The metric is asymptotically flat (at infinity).
  7. The metric coefficients are analytic functions of the spatial coordinates.
  8. The coefficient for the dΩ2 term is greater than or equal to (2m)2 for all positive values of the radial coordinate.
  9. A single set of coordinates covers all of spacetime.

The first 6 conditions (and possibly the 7th) were assumed by Schwarzschild. Abrams assumed condition 8 in his equation (11); it is not implied by the first 7 conditions. Condition 9 was implicit.

The assumption of static spacetime

I got the definition of static spacetime wrong in another thread, so let me try again. For the purposes of this thread, I think it's enough to say that a static spacetime means none of the metric coefficients depend on time, and dt appears only within the dt2 term (so there are no cross terms involving dt dr, for example).

In 1915, when Einstein formulated this problem, he could not have known that a static solution actually exists. Fortunately, Schwarzschild's exterior solution was static, provided the star's radius was greater than 2m. That was true for all known stars.

For black holes, however, the radius is effectively zero. Schwarzschild's exterior solution didn't go there.

(If you're willing to contemplate a weird interchange of time and radial coordinates inside the Schwarzschild radius, then you can say that Schwarzschild's solution extends inside the Schwarzschild radius, but that interchange of coordinates makes the solution non-static inside the Schwarzschild radius, and you still have to deal with the coordinate singularity at the Schwarzschild radius. To keep things simple, I'm not going to go there.)

Contrary to Abrams's central claim, Schwarzschild's spacetime manifold can be extended to include the event horizon and spatial points inside that horizon, but that extension is non-static inside the event horizon. To understand spacetime in the near vicinity of a black hole, we have to abandon the assumption that spacetime is static.

By 1933, Lemaître already understood that Schwarzschild's coordinate singularity at r=2m could be removed. Lemaître correctly attributed the Schwarzschild singularity to Einstein's (and others') hope that the solution would be static, and demonstrated that removal of that assumption made it possible to eliminate the singularity.

Mental baggage

Many of the electric universe folk appear to be motivated in part by hope that cosmology is static or steady-state or at least stationary (so it repeats itself). As Michael Mozina has demonstrated, some electric universe folk have accused mainstream cosmologists of being creationists, just because mainstream cosmology no longer assumes spacetime is static.

Lemaître was not a creationist, but he was a Catholic priest. It is entirely possible that Lemaître's religious beliefs allowed him to think outside the static box that prevented so many of his contemporaries from finding non-static solutions to Einstein's field equations. So what? Lemaître's religious beliefs have nothing to do with mathematical facts.

It's a mathematical fact that, according to Einstein's field equations, the spacetime manifold surrounding a point mass is non-static. It's a mathematical fact that the only way to regard that spacetime manifold as static is to cut out the part that surrounds the point mass, all the way out to and including the Schwarzschild radius. It's also a mathematical fact that all attempts to justify that surgery on mathematical grounds have failed.

In particular: The attempts by Abrams and by Crothers do not stand up under mathematical scrutiny.

Their failure to prove that the spacetime manifold must end at the Schwarzschild radius does not automatically imply the existence of a spacetime manifold that includes the Schwarzschild radius and its interior. Our next step is to give a mathematical proof of that spacetime manifold's existence.

Mathematical proof of existence

Spacetime manifolds are mathematical objects. We can prove their mathematical existence follows from the laws of logic and the axioms of mathematics, but we cannot use mathematics alone to prove a spacetime manifold accurately describes the physical universe. Whether something exists in a physical sense is a question for science, not mathematics.

On the other hand, science tells us the laws of general relativity do a pretty good job of describing the physical universe as we know it. That's why the mathematical existence of a spacetime manifold should be taken more seriously than electric universe pseudomath and pseudoscience.


This is an HTML translation of something I wrote in March 2012 concerning the mathematics of black hole denialism at the online forum now known as the Internation Skeptics Forum. The original forum posts are still online, but that forum no longer renders equations (LaTeX) correctly. For this translation to HTML, I have replaced LaTeX by images and made a few small corrections and deletions. For context and an index, see my introduction to this material.


Last updated: 7 July 2015.

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