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 Physics2, April 2007, pages 68-74. http://www.ptep-online.com/index_files/2007/PP-09-14.PDF

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 byE^{3}.

Efcleethean?

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 M

^{3}be a 3-dimensional metric manifold. Let there be a one-to-one correspondence between all points of E^{3}and M^{3}.

Wait a second. Does Crothers really want to let M^{3} 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:

- Yes, he wants to talk about fairly arbitrary 3-manifolds (although he's mainly interested in those that are spherically symmetric).
- No, he doesn't mean one-to-one correspondence. He means homeomorphism.

That's a bad combination. Very bad. For an arbitrary 3-dimensional metric manifold M^{3}, there may not exist any homeomorphism at all between M^{3} and E^{3}. 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.

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 E^{3}, 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 E^{3}. 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.