Stephen J Crothers has published 17 articles in Progress of Physics. One of those papers was co-authored by Jeremy Dunning-Davies, who edited an issue of The Open Astronomy Journal that's being discussed in the thread that asks "How did crackpot Electric Universe papers get published in a peer-reviewed journal?" Crothers was sole author of the other 16 papers.

Today I'm going to discuss the very first of those papers:

Stephen J Crothers. On the general solution to Einstein's vacuum field and its implications for relativistic degeneracy. Progress in Physics April 2005, Volume 1, pages 68-73.

That paper contains four technical sections, plus a dedication to Leonard S Abrams and an epilogue that explains how Crothers became interested in general relativity through the Abrams paper I discussed previously. The first two technical sections basically repeat what Abrams wrote, adding only some snark and a sloppy reformulation (that I believe to be incorrect) of one relatively unimportant formula.

In section 3, Crothers strikes out on his own. Here's what remains of the first half of section 3 after snark, redundancy, and irrelevancy have been excised:

Stephen J Crothers:

3 Implications for gravitational collapse

As is well known the gravitational potential Φ for an arbitrary metric is


from which it is concluded that gravitational collapse occurs at Φ = 1.


From (21),


so in the weak far field,


and so



According to (23), at r=r0, g00 = 0 and Φ = ½. The weak field potential approaches a finite maximum of ½ (i.e. ½c2), in contrast to Newton's potential. The conventional concept of gravitational collapse at rs = α is therefore meaningless.

Several things:

According to (23), at r=r0, g00 = 0. Equation (23) says nothing about Φ.

If g00 = 0, then equation (21) says Φ = 1, not ½.

I think Crothers got Φ = ½ by using his "weak far field" approximation under the assumption that C(r0)=(2m)2, but that weak field approximation is obviously the wrong formula to use for calculations in the strong field at r=r0. (Crothers takes that value of the radial coordinate r to be the location of the point mass, when it's actually the event horizon, but that doesn't matter here: it's a strong field either way).

The correct calculation of Φ = 1 appears to refute the main point Crothers is trying to make here. (I'm not entirely sure what point Crothers is trying to make here. Perhaps a physicist could explain it to me. Crothers couldn't.)

Finally, g00 is coordinate-dependent. (That's an observation, not a criticism, because the gravitational potential has to be coordinate-dependent.) Crothers thinks he's stuffed all of that coordinate-dependence into C(r), but that's true only for coordinate systems that satisfy a number of assumptions. Crothers made some of those assumptions explicit, but quite a few others are implicit. One of those assumptions is that the coordinates take advantage of spherical symmetry so he can ignore all but the time and radial coordinates; that's fine. Another assumption is that the metric is static. That rules out all coordinate systems that cover the event horizon and its spatial interior.

So I don't even believe equation (21) can be true in general.

The second half of section 3 is constructed atop that pile of fail. Some excerpts:

Stephen J Crothers:

Time stops at the Schwarzschild point-mass. The conventional concepts of the Schwarzschild sphere and its interior are meaningless.


Applying (31) to the Kruskal-Szekeres extension gives rise to the absurdity of an infinite acceleration at r = α where it is conventionally claimed that there is no matter and no singularity. It is plainly evident that gravitational collapse terminates at a Schwarzschild simple point-mass, not in a black hole.


The correct conclusion is that gravitational collapse terminates at the point-mass without the formation of a black hole in all general relativistic circumstances.

That's an extraordinary claim, because it contradicts all mainstream research into the implications of general relativity for black holes. When I can identify obvious errors in the chain of reasoning that's alleged to support the claim, I'm inclined to regard the extraordinary claim as unproven.

When I've done the math myself, and found that correct mathematics refutes most of what Crothers has to say, I'm inclined to regard the extraordinary claim as incorrect.

When I survey a fair sample of the 17 papers Crothers has published in Progress in Physics, and look at the magnificent effort Crothers has put into resisting correction by knowledgeable relativists and mathematicians, I'm inclined to regard Crothers as a crackpot.

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.

For debugging: Click here to validate.