Enemies or Opponents?

A reply to S. Crothers' recent web postings

Gerhard W. Bruhn, Darmstadt University of Technology


Contributions by S. Crothers in black

As a response of the following private email to Mr. Crothers

Date: Wed, 1 Oct 2008 10:35:14 +0200 (CEST)
From: Ernest S. Gullible
To: Stephen Crothers
Cc: carroll@theory.uchicago.edu, Gerhard Bruhn
Subject: Wrong Way Drivers on the Highway

Mr. Crothers,
I have read your objections against Carroll's chapter 7 already on Evans' blog site. But you know that our relations are not the best, so why should I give you a detailed response? I recommend that you send your remarks to S.M. Carroll. Perhaps he will reply, perhaps not, since meanwhile he will know the Wrong Way Driver scene.
For short let me point to one spot you should correct before bothering Carroll:
You wrote:
At eq. 7.25 he gets exp(2a) = 1 + u/r and then by eqs. 7.27 and 7.28 he gets,
at eq. 7.29,
exp(2a) = 1 + u/r = 1 - 2GM/r
Your remark is wrong: Carroll explicitly assumes r > 2GM. The border of that region is r = 2GM, where the Schwarzschild metric (using the coordinates t,r,theta,phi) (not the manifold) is singular. Using other coordinates (e.g. Kruskal, see Carroll) that border can be crossed without problems.

G.W. Bruhn

Crothers posted a text Dealing with Criminality in the Obsolete Physics on Evans' blog site to be commented on here:

Of course our relations are not the best; we are enemies, by your actions. Owing to your behaviour, I regard you as a fraudster and a criminal. Consequently, I treat you as such.

I ignore your slander above and below since your intention is evident: To prevent me from publically discussing your flaws for being deeply offended. This method does not work here, Mr. Crothers. Be sure, I am not at all impressionable by bad behavior. I am not your enemy but your scientific opponent. We have different views the consistency of which should patiently be discussed. So go on!

My remarks are not wrong. Carroll’s own writings testify clearly and plainly to his claims and I have reported them accurately, directly from his lecture notes. Carroll commits elementary and fatal errors in general logic and in mathematics, just as you do when trying to refute my analysis. Your attempts at rebuttal of my work are laughable, and futile. And like Carroll you superimpose matter into a spacetime that by construction contains no matter (Ric = 0) in a theory where the Principle of Superposition does not apply. Now you take it upon yourself to speak for Carroll, telling us what you say he meant. Contrary to your assertion, nowhere in his analysis does Carroll say that he “assumes” that r > 2GM.

I'm not Carroll's speaker, but OK. I guess that Carroll will not reply to your polemical distortions above. So take this response instead:

When discussing the Schwarzschild metric (7.29) first of all Carroll states at p.9 (p.172 of his L.N.) Chap.7 that there are singularities at r=0 and r=2GM. These singularities mark borders of the validity regions of a metric, since, as the reader has learned in Chap.2, the metric coefficients gμν must be C or at least sufficiently differentiable, and det(gμν) ≠ 0.

And further at p.9 (p.172) (Carroll's contributions in green): Having worried a little about singularities, we should point out that the behavior of Schwarzschild at r ≤ 2GM is of little day-to-day consequence.

Which means a restriction to the case r > 2GM. See also the diagrams at p.19 (p.182) and p.20 (p.183). The discussion of the case r ≤ 2GM is postponed to p.20 (p.183).

Starting at p.20 (p.183), Carroll approaches the problem of passing the border r = 2GM from outside by a step-by-step process of introducing new coordinates.

These considerations were prepared in Chap.2 (Manifolds) where coordinate transformations were discussed in detail: The introduction of new coordinates will in general have the effect that another new region of validity occurs for the transformed metric: Former singularities may disappear and new singularities will appear together with different validity region on the manifold. Such singularities are called ''coordinate singularities''. The entire manifold can be imagined as a patchwork of overlapping validity regions of different coordinates.

At p.22 (p.185) Carroll writes: Let’s consider what we have done. Acting under the suspicion that our coordinates (t,r,θ,φ) may not have been good for the entire manifold, we have changed from our original coordinate t to the new one
. . .
We therefore conclude that our suspicion was correct and our initial coordinate system didn’t do a good job of covering the entire manifold. The region r ≤ 2GM should certainly be included in our spacetime, since physical particles can easily reach there and pass through. . . .

By step-by-step coordinate transforms Carroll shows in concordance with the literature that the Schwarzschild manifold originally given by the Schwarzschild metric (7.29) for the validity region r > 2GM can be extended to a metric (7.80) using the so-called Kruskal-coordinates (v,u,θ,φ) the validity region of which covers also the second validity region 0 < r < 2GM of the Schwarzschild metric (7.29) where no further extensions are possible.

Thus, Mr Crothers, your first claim is evidently wrong.

It's not Carroll's fault that you have ignored his Chap.2.

What else?

His analysis is faulty at the most elementary level. I reiterate Carroll’s argument:

At eq. 7.25 he gets exp(2a) = 1 + μ/r and then by eqs. 7.27 and 7.28 he gets, at eq. 7.29, exp(2a) = 1 + μ/r = 1 - 2GM/r. His subsequent talk of r = 0, r = 2GM, r

Equ.(7.25) for small r > 0 is valid if the ''undetermined constant'' is chosen > 0. However, this is quite irrelevant. Important is that Carroll derived the Schwarzschild metric (7.29) for r > 2GM, and on the following pages 171-183 he doesn't talk about r ≤ 2GM.

As I have pointed out above the case r ≤ 2GM is reached by introducing appropriate new coordinates. To repeat this once more for writing down: When the coordinates are changed then it may happen that the validity regions of old and new coordinates do NOT COINCIDE. Then, if the new validity region has parts outside the old one, you get an extension of the original part of the manifold. This is a well known effect you can study at the 2-sphere, parametrized by x,y or y,z in an Euclidian x,y,z-coordinate system.

And the so-called “ Schwarzschild singularity at r = 2GM ” is easily removed by the simple substitution for ‘r’ in the associated metric by (|r - ro|^n + a^n)^(1/n) where r is any real number, n any positive real number, ro and n entirely arbitrary. The resulting metric is singular at only one place, when r = ro, at which the proper radius is zero and the radius of Gaussian curvature of the spherically symmetric geodesic surface associated with that point in spacetime is ‘a’.

I doubt this miracle. Let us consider the choice n=1, ro=0, a=2GM for the Schwarzschild metric

        ds˛ = (r*−a/r*)dt˛ − (r*/r*−a)dr*˛ − r*˛ (dθ˛ + sin˛θ dφ˛) .

Your proposed transform is

        r* = |r|+a .

The r*-interval [a,∞[ corresponds to the r-intervall [0,∞[. The singularity r*=a of the r*-metric has been mapped to r=0. Since we have r≥0 we can replace |r| with r to obtain the r-metric

        ds˛ = (r/r+a)dt˛ − (r+a/r)dr˛ − (r+a)˛ (dθ˛ + sin˛θ dφ˛) .

which is evidently singular at r=0, i.e.

your transform has not at all removed the singularity under consideration.

You can believe me that - with slight modifications - the step-by-step process of transforms, as described by Carroll, applies to that metric as well. By that way the singularity at r=0 turns out to be a ''coordinate singularity'' which will disappear in the final Kruskal metric.

Thus, the black holers have actually removed the wrong singularity in their corruption of Schwarzschild’s solution, violating geometry in doing so, and violating the Principle of Equivalence and Special Relativity (both of which must manifest in a sufficiently small region of Einstein’s gravitational field, as Einstein stipulated). The Kruskal-Szekeres coordinates are just unmitigated rot as well.

That's your view, not mine. My time is limited, but if you cannot overcome your problems on the way to the Kruskal metric then I offer further patient discussions. I think that Carroll's Lecture Notes on GR should be a good introductory book to discuss your problem along his lines.


G. W. Bruhn

Here is the immediate response of a man who is unable to stand through a scientific debate and instead takes the emergency exit of throwing heavy insults against his opponent, assisted by his master, a notorious proponent of New Math.