Novel Geometrical Models of Relativistic Stars: A Physical Analysis

[martinbn]

I haven't read these articles, but it seems that the claim is that there are spherically symmetric static space-times, which can describe compact isolated objects with arbitrarily large masses and arbitrarily small radi without being black holes. That doesn't seem right. What am I missing?



Novel Geometrical Models of Relativistic Stars. I. The General Scheme - P. P. Fiziev

Abstract: In a series of articles we describe a novel class of geometrical models of relativistic stars. Our approach to the static spherically symmetric solutions of Einstein equations is based on a careful physical analysis of radial gauge conditions. It brings us to a two parameter family of relativistic stars without stiff functional dependence between the stelar radius and stelar mass. It turns out that within this family there do exist relativistic stars with arbitrary large mass, which are to have arbitrary small radius and arbitrary small luminosity. In addition, point particle idealization, as a limiting case of bodies with finite dimension, becomes possible in GR, much like in Newton gravity.​

Novel Geometrical Models of Relativistic Stars. II. Incompressible Stars and Heavy Black Dwarfs -https://www.physicsforums.com/find/astro-ph/1/au:+Fiziev_P/0/1/0/all/0/1

Abstract: In a series of articles we describe a novel class of geometrical models of relativistic stars. Our approach to the static spherically symmetric solutions of Einstein equations is based on a careful physical analysis of radial gauge conditions.
It turns out that there exist heavy black dwarfs: relativistic stars with arbitrary large mass, which are to have arbitrary small radius and arbitrary small luminosity. In the present article we mathematically prove this new phenomena, using a detailed consideration of incompressible GR stars. We study the whole two parameter family of solutions of extended TOV equations for incompressible stars. This example is used to illustrate most of the basic features of the new geometrical models of relativistic stars. Comparison with newest observational data is discussed​

Novel Geometrical Models of Relativistic Stars III. The Point Particle Idealization -https://www.physicsforums.com/find/astro-ph/1/au:+Fiziev_P/0/1/0/all/0/1

Abstract: We describe a novel class of geometrical models of relativistic stars. Our approach to the static spherically symmetric solutions of Einstein equations is based on a careful physical analysis of radial gauge conditions. It brings us to a two parameter family of relativistic stars without stiff functional dependence between the stelar radius and stelar mass.
As a result, a point particle idealization -- a limiting case of bodies with finite dimension, becomes possible in GR, much like in Newtonian gravity. We devote this article to detailed mathematical study of this limit.​

 

[Mentz114]

I haven't read these articles, but it seems that the claim is that there are spherically symmetric static space-times, which can describe compact isolated objects with arbitrarily large masses and arbitrarily small radi without being black holes. That doesn't seem right. What am I missing?

That is interesting. The paper below seems to be a necessary precursor to those. I have not had time to digest even this one.

http://arxiv.org/abs/gr-qc/0407088v3.pdf

[edit]
Having read the massive point particles paper once it seems that there is an unexplored dof in the function $\rho(r)$ which appears in the first papers on the spherically symmetric vacuum. The author turns this into two parameters so $M$ is no longer a mass without properties, but a distribution with 2 parameters. Metrics (vacuum solutions ) with $\delta(x)$ functions as sources are solutions of the transformed action Hilbert action.

Looks OK to me but I don't understand how a solution with a Dirac delta as source can be a vacuum solution.

 

[WannabeNewton]

Doesn't this contradict Birkhoff's theorem? The exterior of such a space-time must always be Schwarzschild which obviously has black hole solutions in the $(M,R)$ parameter space.

 

[ShayanJ]

Doesn't this contradict Birkhoff's theorem? The exterior of such a space-time must always be Schwarzschild which obviously has black hole solutions in the $(M,R)$ parameter space.

Birkhoff's theorem applies to Einstein equation without a source. But the paper Mentz mentioned, solves Einstein's equation with a dirac delta as a source. Physically, it seems strange but mathematically its trivial!

 

[Mentz114]

Birkhoff's theorem applies to Einstein equation without a source. But the paper Mentz mentioned, solves Einstein's equation with a dirac delta as a source. Physically, it seems strange but mathematically its trivial!

So it is the solution everywhere ?

Doesn't this contradict Birkhoff's theorem? The exterior of such a space-time must always be Schwarzschild which obviously has black hole solutions in the $(M,R)$ parameter space.

On page 8 the author says

This means that these solutions strictly respect a generalized Birkhoff theorem. Its generalization requires only a justification of the physical domain of variable ρ.

 

[WannabeNewton]

Birkhoff's theorem applies to Einstein equation without a source.

The delta function has compact support so the exterior region solves the Einstein equation without a source and Birkhoff's theorem applies. It's no different from having a thin shell spherical collapse wherein there is no matter outside the collapsing shell but there is matter on the shell itself. However the latter is mathematically well defined because one can easily show that Einstein's equations make sense for a hypersurface distribution i.e. a delta function that has compact support on a hypersurface.

On the other hand it isn't clear to me how the analysis in the paper Mentz linked is even mathematically well-defined. It doesn't make any sense to have a point particle delta function source in the Einstein equation because the equations are non-linear. The paper claims to resolve the issue by using a "novel" approach but at first glance this approach doesn't make sense to me. I'll post a more detailed analysis of their "novel" approach once I finish calculations for my research.

EDIT: For reference, the standard treatment of point particle sources in the Einstein equation are as in http://arxiv.org/pdf/0806.3293v5.pdf and http://arxiv.org/pdf/1102.0529v3.pdf

 

[Mentz114]

The delta function has compact support so the exterior region solves the Einstein equation without a source and Birkhoff's theorem applies. It's no different from having a thin shell spherical collapse wherein there is no matter outside the collapsing shell but there is matter on the shell itself. However the latter is mathematically well defined because one can easily show that Einstein's equations make sense for a hypersurface distribution i.e. a delta function that has compact support on a hypersurface.

On the other hand it isn't clear to me how the analysis in the paper Mentz linked is even mathematically well-defined. It doesn't make any sense to have a point particle delta function source in the Einstein equation because the equations are non-linear. The paper claims to resolve the issue by using a "novel" approach but at first glance this approach doesn't make sense to me. I'll post a more detailed analysis of their "novel" approach once I finish calculations for my research.

EDIT: For reference, the standard treatment of point particle sources in the Einstein equation are as in http://arxiv.org/pdf/0806.3293v5.pdf and http://arxiv.org/pdf/1102.0529v3.pdf

I look forward to it.

(I hope your calculations come out to $0=0$. I do like a happy ending.)

 

[PeterDonis]

It turns out that within this family there do exist relativistic stars with arbitrary large mass, which are to have arbitrary small radius and arbitrary small luminosity

This violates the theorem proved by Einstein in the 1930's, that no object can exist in static equilibrium with a radius of less than 9/4 of its mass (in geometric units--i.e., 9/8 of the Schwarzschild radius for its mass). So I'm immediately skeptical of whatever this approach is doing.

 

[Mentz114]

This violates the theorem proved by Einstein in the 1930's, that no object can exist in static equilibrium with a radius of less than 9/4 of its mass (in geometric units--i.e., 9/8 of the Schwarzschild radius for its mass). So I'm immediately skeptical of whatever this approach is doing.

I share your skepticism about the approach. The author would argue that the theorem assumes $\rho(r)=r$ and so is not true in the 'novel' approach.

Abandoning $\rho(r)=r$ probably wrecks some other symmetry in any case.

 

[PAllen]

This violates the theorem proved by Einstein in the 1930's, that no object can exist in static equilibrium with a radius of less than 9/4 of its mass (in geometric units--i.e., 9/8 of the Schwarzschild radius for its mass). So I'm immediately skeptical of whatever this approach is doing.

Einstein assumed matter can't flow FTL which is equivalent to the dominant energy condition. In fact his proof was sort of a reductio at absurdum, that such a static state would require (locally) FTL matter flow. But, without dominant energy condition, nothing at all prevents FTL matter flow in GR.

I am strongly of the opinion that for macroscopic, classical, predictions GR must assume the dominant energy condition. Otherwise, you do NOT have (even approximately) geodesic motion or that matter follows timelike paths.

 

[PeterDonis]

without dominant energy condition

You're right, the paper's "incompressible" model assumes constant energy density as pressure increases without bound, so it violates this condition.

However, I'm not sure violating the dominant energy condition is sufficient to avoid the conclusion of Einstein's theorem, at least not for the model presented in the paper. See below.

without dominant energy condition, nothing at all prevents FTL matter flow in GR.

But the model in the paper assumes a static solution, so "FTL matter flow" is not present anyway. And IIRC, the key assumption of Einstein's theorem is staticity, not the dominant energy condition. (I don't have a source handy to check.)

 

[PAllen]

But the model in the paper assumes a static solution, so "FTL matter flow" is not present anyway. And IIRC, the key assumption of Einstein's theorem is staticity, not the dominant energy condition. (I don't have a source handy to check.)

We might (or might not) be thinking of different Einstein papers. The one I was thinking of argued that to maintain an overall stationary state, below a radius outside the SC radius, FTL orbits would be needed somewhere.

 

[PeterDonis]

IIRC, the key assumption of Einstein's theorem is staticity, not the dominant energy condition. (I don't have a source handy to check.)

I checked my copy of MTW, and in Box 23.2 they present the solution for a static, spherically symmetric star of uniform density (which, as already noted, obviously violates the dominant energy condition), including the limit I cited, which they give as $2M/R < 8/9$ (they don't credit Einstein with the theorem, though). The derivation makes it clear that no assumption of any energy condition is required.

 

[PeterDonis]

The one I was thinking of argued that to maintain an overall stationary state, below a radius outside the SC radius, FTL orbits would be needed somewhere.

Ah, you're right, I might be mixing up who discovered what theorem. (That would explain why MTW didn't credit Einstein.) The MTW derivation I referred to is for a static solution; that's the one I was thinking of.

 

[PAllen]

I checked my copy of MTW, and in Box 23.2 they present the solution for a static, spherically symmetric star of uniform density (which, as already noted, obviously violates the dominant energy condition), including the limit I cited, which they give as $2M/R < 8/9$ (they don't credit Einstein with the theorem, though). The derivation makes it clear that no assumption of any energy condition is required.

Thanks, I'll look at this after the weekend (I'm off for anniversary celebration), since I have this book.

 

[martinbn]

This violates the theorem proved by Einstein in the 1930's, that no object can exist in static equilibrium with a radius of less than 9/4 of its mass (in geometric units--i.e., 9/8 of the Schwarzschild radius for its mass). So I'm immediately skeptical of whatever this approach is doing.

That's the reason I asked. I've seen this called the Buchdal's theorem.