The hard work is over. Let's have some fun.

Let's start with this paper:

Stephen J Crothers. Gravitation on a spherically symmetric metric manifold. Progress in Physics 2, April 2007, pages 68-74.

The third and last paragraph of the introduction to that paper says

Stephen J Crothers:

It is plainly evident, res ipsa locquitur, that the standard claims for black holes and Big Bang cosmology are not consistent with elementary differential geometry and are consequently inconsistent with General Relativity.

That sounds intriguing. I'm always open to learning more differential geometry. If it demolishes the standard theory of black holes and big bang cosmology, so much the better. Breaking things can be fun.

The following section, whose title is "Spherical symmetry of three-dimensional metrics", begins with this sentence:

Stephen J Crothers:

Denote ordinary Efcleethean 3-space by E3.


Crothers provided this helpful footnote:

Stephen J Crothers:

For the geometry due to Efcleethees, usually and abominably rendered as Euclid.

Ah, yes: that Efcleethees. "Efcleethees alone has looked on beauty bare."

According to Wikipedia, Euclid's name was Εὐκλείδης. Crothers had been rendering that name abominably in his previous papers, and went on to render that name abominably in subsequent papers.

"A foolish consistency is the hobgoblin of little minds." Crothers's brain is the size of a planet, with matching personality.

Onward. Here are the next two sentences of that fourth paragraph of the paper:

Stephen J Crothers:

Let M3 be a 3-dimensional metric manifold. Let there be a one-to-one correspondence between all points of E3 and M3.

Wait a second. Does Crothers really want to let M3 be an arbitrary 3-dimensional metric manifold? Does Crothers really mean "one-to-one correspondence", or does he mean homeomorphism?

Reading ahead through the next several paragraphs, the answers to those two questions appear to be:

That's a bad combination. Very bad. For an arbitrary 3-dimensional metric manifold M3, there may not exist any homeomorphism at all between M3 and E3. Doesn't Crothers know that?

Evidently not. In comments posted at the "Dealing with Creationism in Astronomy" blog site, Crothers wrote:

Stephen J Crothers:

It is certainly possible to have M3 and E3 in one-to-one correspondence. My paper on metric manifolds is correct.

Not hardly:

Jason J Sharples:

If it is possible to have M3 and E3 in one-to-one correspondence, as you claim, then please provide us with such a mapping. In fact I'll make it even less onerous - simply provide us with a one-to-one mapping between the 1D circle and 1D Euclidean space.

It appears, however, that Crothers's paper was not discussing an arbitrary 3-dimensional metric manifold after all:

Stephen J Crothers:

The mapping you request is developed in the paper based upon the hypothesis of a one-to-one correspondence between E3 and M3, and associated rigid rotations in one corresponding to rigid motions in the other so that geodesics are mapped into geodesics.

So that was all much ado 'bout nuthin'. The Crothers paper deals only with 3-manifolds that are homeomorphic to E3, and my contrary impression came about only because the paper is poorly written. I guess I'll just have to put up with the poor writing, because I really do hope to learn how the standard theory of big bang cosmology is incompatible with elementary differential geometry.

Wait a minute. In mainstream big bang cosmology, several families of big bang spacetimes involve spatial 3-manifolds that aren't homeomorphic to E3. How can Crothers overthrow big bang cosmology if his paper doesn't even consider those manifolds?

I won't keep you in suspense: He doesn't. The paper fails to deliver on the promises made in its third paragraph. Most of the paper is devoted to Schwarzschild spacetime, and most of that is just a restatement of what Crothers wrote in his very first paper, which was itself mostly a rehash of the Abrams paper I went over in post #8 and post #11. Crothers repeats the errors in the Abrams paper and adds a few more of his own.

Here's the final sentence of section 8 ("That the manifold is inextendable"):

Stephen J Crothers:

Thus, the Schwarzschild manifold described by (20) with (22) (and hence (8)) is inextendable.

Equation (8) is the familiar metric for the Schwarzschild manifold. In post #12 above, I sketched a fairly detailed definition of a spacetime manifold that extends that Schwarzschild manifold.

When Crothers tells me the thing I've done can't be done, I begin to doubt his unsupported claims.

I can't honestly say I've learned nothing about differential geometry from Crothers. Working through his calculations and identifying his errors had some educational value, but the process has been less educational than entertaining.

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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