Event Horizon and Light Exit Cone Question

[noblackhole]

Yes, I'm not entirely clear on this. It's standard to note that R is not really "radial distance", but it's also standard to interpret q (where R=q is the coordinate singularity), by doing q/R=2GM/R, so it seems like R=0 is being placed at the centre of the central body.

Perhaps what's happening is that since the interpretation q=2GM is obtained by matching with Newtonian gravity not in the R=0 limit, but the R=infinity limit, the invalid region R<2GM is not used? Time for me to look at a good text ...

Edit: Take a look at Matthias Blau's Lecture Notes on General Relativity, Section 11.5 "Measuring Length and Time in the Schwarzschild metric": http://www.unine.ch/phys/string/Lecturenotes.html

Dear atyy,

The quantity 'r' or 'R' etc. has a definite geometric meaning. However, it has never been correctly identified by the proponents of the black hole. Consequently, they have never treated it correctly, and have committed fatal mathematical errors. In the line elements subject of this discussion, the quantity 'r', called radial distance or not radial distance or some other vague name or notion, is in fact the inverse square root of the Gaussian curvature of a spherically symmetric geodesic surface in the spatial section of the spacetime manifold. This is easily proved, but completely overlooked by the proponents of black holes.

Concerning q = 2GM you are right - it is inadmissible. It is a post hoc insertion of a Newtonian relation in order to claim a Newtonian approximation - a circular argument, and therefore false. Moreover, Newton's potential function implicitly involves two masses, whereas the empty spacetime associated with the "Schwarzschild solution" excludes such masses by construction. Newton's potential function is therefore incompatible with empty spacetimes such as the Schwarzschild spacetime.

A lucid explanation of the salient facts, using nothing more than the most elementary high school algebra, can be had here:

www.sjcrothers.plasmaresources.com/article-1-1.pdf

 

[atyy]

www.sjcrothers.plasmaresources.com/article-1-1.pdf

Wow! That paper substantiates George Jones's point in post #5 much better than the links he provided: https://www.physicsforums.com/showthread.php?p=1884115&highlight=event+horizon#post1884115.

 

[noblackhole]

The usual claim that the Riemann tensor scalar curvature invariant must be unbounded at some point in the spacetime of Einstein’s gravitational field is spurious.

First, the said scalar curvature invariant is a consequence of a geometry: it does not determine a geometry.

Second, a geometry is fully determined by the form of its line-element (its metric tensor), and the said scalar curvature is determined from the components of the metric tensor.

Third, Einstein’s General Theory of Relativity says nothing about any conditions that the said scalar curvature invariant must satisfy. The proponents of the black hole have never offered a proof that Einstein’s field equations require that the said scalar curvature invariant be unbounded at some point in the associated manifold. They just assume that it is so and then concoct ways to make it so, in violation of the intrinsic geometry of the line-element. M. Kruskal made the very same incorrect assumption in his paper on Kruskal coordinates.

Fourth, requiring that the said scalar curvature invariant have, in a non-Euclidean geometry, the same features that it has in a Euclidean geometry, is an arbitrary imposition that has no valid basis in geometry. The indefinite line-element associated with Einstein’s non-Euclidean (pseudo-Riemannian) geometry causes geometric forms that have no counterparts in Euclidean geometry. For example, null vectors, which are non-zero vectors that have zero magnitude, or equivalently, non-zero vectors that are orthogonal to themselves (e.g. the wave 4-vector of Special Relativity). In the case of a Schwarzschild space it is an inevitable consequence of the form of the line-element (i.e. an intrinsic property of the geometry) that any point in the spatial section of the manifold has the unusual property that it has a zero radius, a zero volume, but a non-zero surface area.

It is also noteworthy that satisfaction of the field equations is a necessary but insufficient condition for a description of Einstein’s gravitational field. For example, one can generate an infinite number of solutions that satisfy Rij = 0 (i,j = 0,1,2,3), but they do not satisfy the boundary conditions required by Einstein (e.g. asymptotically Minkowski).

 

[noblackhole]

These ideas have been thoroughly discredited; see

https://www.physicsforums.com/showthread.php?t=141985&highlight=Abrams

http://arxiv.org/abs/gr-qc/0608033.

The threads you link to do not answer the questions involved. They merely regurgitate the usual demonstrably false arguments. Authority is no substitute for thought. The facts do not support the authorities you cite.

 

[Jonathan Scott]

Wow! That paper substantiates George Jones's point in post #5 much better than the links he provided: https://www.physicsforums.com/showthread.php?p=1884115&highlight=event+horizon#post1884115.

As I had already mentioned in this thread, Stephen J Crothers (who presumably either owns or is connected with the user name noblackhole) is so ready to find fault with so many things that this makes it very difficult to accept anything he says. If I hadn't already agreed with one of the things he said, I'd probably have dismissed him immediately as a crackpot.

For the original point about the Schwarzschild radial coordinate, he makes some very useful mathematical contributions, and I've found his generalization of the radial coordinate very interesting. However, he also claims to find fault with many other aspects of GR and makes some blanket statements which seem very silly to me (for example that SR "doesn't allow infinite energy density"). Although there may be some truth somewhere in these other statements, his readiness to find fault means that I will not be able to accept them unless I can absolutely follow all the logic (especially the hidden assumptions) and prove them to my own satisfaction.

I will however commend him on being open to at least some suggestions, in that he previously insisted on rewriting the term "Euclidean" using a different spelling based on a phonetic spelling of how the equivalent name would be pronounced in modern Greek, but in this latest paper he is now using the traditional spelling.

It is of course very difficult to argue rationally against the establishment position without sounding like a crackpot, but I think that even if Stephen Crothers is right he is going to have a hard time persuading the people that matter with his confrontational approach.

 

[noblackhole]

As I had already mentioned in this thread, Stephen J Crothers (who presumably either owns or is connected with the user name noblackhole) is so ready to find fault with so many things that this makes it very difficult to accept anything he says. If I hadn't already agreed with one of the things he said, I'd probably have dismissed him immediately as a crackpot.

For the original point about the Schwarzschild radial coordinate, he makes some very useful mathematical contributions, and I've found his generalization of the radial coordinate very interesting. However, he also claims to find fault with many other aspects of GR and makes some blanket statements which seem very silly to me (for example that SR "doesn't allow infinite energy density"). Although there may be some truth somewhere in these other statements, his readiness to find fault means that I will not be able to accept them unless I can absolutely follow all the logic (especially the hidden assumptions) and prove them to my own satisfaction.

I will however commend him on being open to at least some suggestions, in that he previously insisted on rewriting the term "Euclidean" using a different spelling based on a phonetic spelling of how the equivalent name would be pronounced in modern Greek, but in this latest paper he is now using the traditional spelling.

It is of course very difficult to argue rationally against the establishment position without sounding like a crackpot, but I think that even if Stephen Crothers is right he is going to have a hard time persuading the people that matter with his confrontational approach.

Crothers did not say anywhere that "SR doesn't allow infinite energy density". He clearly said and proved by simple algebra that SR forbids infinite density, because infinite density implies infinite energy or equivalently that a material body can acquire the speed of light in vacuo. This is in the cited article.

 

[Jonathan Scott]

Crothers did not say anywhere that "SR doesn't allow infinite energy density". He clearly said and proved by simple algebra that SR forbids infinite density, because infinite density implies infinite energy or equivalently that a material body can acquire the speed of light in vacuo. This is in the cited article.

Perhaps I shouldn't have written that in quotes as in the latest paper it wasn't exactly those words, but the statement that "SR forbids infinite density" seems to me to have exactly the same meaning as "SR doesn't allow infinite energy density".

The statement "infinite density implies infinite energy" on its own is extremely naive. SR is perfectly capable of describing a point-like finite mass, which has infinite density. This is like a Dirac delta function (the derivative of a step function). It is outside the scope of SR to describe how such an object could be created, but I'm not aware of any problem with describing how it behaves in SR. It is obviously not possible to achieve infinite density by accelerating a material body, but no-one was suggesting that this method would be used.

As far as I'm concerned, this particular point (and hence section 2 of the article) appears to be completely spurious, and therefore actually undermines the rest of Crothers' argument somewhat, but in any case it seems irrelevant to the specific arguments about the Schwarzschild radial coordinate.

As should already be clear, I find Karl Schwarzschild's original interpretation of the radial coordinate far more plausible than the black hole interpretation. However, that doesn't constitute proof. There is some experimental evidence (relating to the lack of Type I X-ray bursts) which appears to support the idea that neutron stars turn into black holes at some mass threshold, and there is also some experimental evidence (relating to intrinsic magnetic fields and radio-loud quasars) which suggests that supermassive bodies are not in fact black holes.

I don't really care whether Crothers, Mitra, Antoci and others have made mistakes; everyone makes mistakes sometimes. If they do it too often, or the mistakes are too blatant, then that obviously undermines credibility, but that doesn't prove that everything they say can safely be ignored.

As I've said before, what I would really like to hear is a clear rationale from the black hole supporters as to why they use Hilbert's assumptions about the radial coordinate when Schwarzschild's original model is more physically plausible. The mere fact that one CAN (in that there is a coordinate system which gets past the origin) does not seem to me to be a sufficient justification.

 

[atyy]

As I've said before, what I would really like to hear is a clear rationale from the black hole supporters as to why they use Hilbert's assumptions about the radial coordinate when Schwarzschild's original model is more physically plausible.

OK, let's see if I can convey some of Hawking and Ellis without garbling it too much. The two "interpretations" differ only by a coordinate change, so they cannot be different. But in any case, no one uses the interpretation of the Schwarzschild radial coordinate R=0 as the centre of the central mass. Also, the interpretation of the Schwarzschild mass parameter M_o as a Newtonian mass is irrelevant.

The mere fact that one CAN (in that there is a coordinate system which gets past the origin) does not seem to me to be a sufficient justification.

Yes, just because the maximally extended solution exists theoretically doesn't mean it exists in nature. But Birkhoff's theorem states that it is the maximally extended Schwarzschild solution that has to be used outside a spherically symmetric star, so we know that it is the extended coodinate system that is relevant for stars, which we do know exist in nature. The theorem itself doesn't say that we have to use the Schwarzschild part or the extended part, so there is no interpretation yet.

We start with the solution for the interior of the star, with some mass function m(Ri), the interior radial coordinate. Then we figure out where the surface of the star is Ri=Rs (Rs is the stellar surface, not the Schwarzschild radius). This determines which part of the extended Schwarzschild solution we need to use - we need to use it for Schwarzschild coordinate R>Rs - this is the where interpretation of the extended Schwarzschild solution first enters. The interpretation of the Schwarzschild mass parameter is done by setting M_o=m(Rs). Notably, this is not the mass (determined number of atoms, neutrons or other particles) of the star. So it is the interior solution (completely normal physics) which determines the interpretation of the extended Schwarzschild solution.

Then by considering stellar dynamics, it appears that for some stars late in their evolution we need to match to the extended part of the extended Schwarzschild solution. The matching is still determined by the normal physics interior solution, and we are only using part of the extended part of the extended Schwarzschild solution, and there's no singularity. Remarkably, there are a couple of theorems that say that if the normal physics interior solution requires us to match in the extended part, the star cannot be stable, and will collapse until ...? Either a black hole or General Relativity is wrong - the latter could be the case, but I think the usual interpretation of the possibly wrong theory is correct.

So apart from Birkhoff's theorem, Hawking and Ellis rely heavily on theorems that there are no stable normal physics interior solutions if the normal physics interior solution is matched to the extended part of the extended Schwarzschild solution, and that within the framework of General relativity, a singularity is inevitable. These theorems apparently fail if conditions are not sufficiently symmetric, and they are careful to state that.

Also, the M/Ri ~ 2 (M is not the Schwarzschid mass parameter M_o) that I made fun of is actually due to carefulness, because the interior coordinate Ri has different interpretations depending on the density and pressure profile of the star.

I might be totally garbling this, but you should look at Hawking and Ellis, because they really distinguish between all the different mass-like terms, and radius-like terms under consideration, and also talk about effects of temperature and other uncertainties. In short, they seem to have been very careful.

 

[Jonathan Scott]

OK, let's see if I can convey some of Hawking and Ellis without garbling it too much. The two "interpretations" differ only by a coordinate change, so they cannot be different. But in any case, no one uses the interpretation of the Schwarzschild radial coordinate R=0 as the centre of the central mass. Also, the interpretation of the Schwarzschild mass parameter M_o as a Newtonian mass is irrelevant.

I wouldn't say "cannot be different". Outside the radial coordinate of the "event horizon", the interpretation makes no difference to the physics. The physical difference is in the location of the point mass for that hypothetical case, or in the way in which an interior solution for the central mass is joined to the vacuum solution, where the assumption is in the way the radial coordinate for the central mass is assumed to go over into the Schwarzschild radial coordinate (as interpreted by Hilbert), or into some other radial coordinate.

Yes, just because the maximally extended solution exists theoretically doesn't mean it exists in nature. But Birkhoff's theorem states that it is the maximally extended Schwarzschild solution that has to be used outside a spherically symmetric star, so we know that it is the extended coodinate system that is relevant for stars, which we do know exist in nature. The theorem itself doesn't say that we have to use the Schwarzschild part or the extended part, so there is no interpretation yet.

To put it another way, Birkhoff's theorem says that any solution must be a subset of the maximally extended Schwarzschild solution, but this must not be taken to imply that there is necessarily any case where where the part of the solution inside the event horizon applies.

We start with the solution for the interior of the star, with some mass function m(Ri), the interior radial coordinate. Then we figure out where the surface of the star is Ri=Rs (Rs is the stellar surface, not the Schwarzschild radius). This determines which part of the extended Schwarzschild solution we need to use - we need to use it for Schwarzschild coordinate R>Rs - this is the where interpretation of the extended Schwarzschild solution first enters. The interpretation of the Schwarzschild mass parameter is done by setting M_o=m(Rs). Notably, this is not the mass (determined number of atoms, neutrons or other particles) of the star. So it is the interior solution (completely normal physics) which determines the interpretation of the extended Schwarzschild solution.

Yes, I think the implicit assumption about the radial parameter occurs when matching up the radial coordinate for the inner boundary of the vacuum solution with the radial coordinate for the outer boundary of the interior solution. Any specific assumption about how these coordinates join up (given that they are parts of different solutions) has physical implications.