Non singular black hole solutions

[evanallmighty]

Has anyone read this paper? If so, what is the interior solution it describes?

A non singular solution for spherical configuration with infinite central density
Fuloria, Pratibha; Durgapal, M. C.
Astrophysics and Space Science, Volume 314, Issue 4, pp.249-250
Abstract: A non-singular exact solution with an infinite central density is obtained for the interior of spherically symmetric and static structures. Both the energy density and the pressure are infinite at the center but we have e λ(0)=1 and e ν(0)≠0. The solution admits the possibility of receiving signals from the region of infinite pressure.
Keywords: General relativity, Exact solution, Astrophysics
http://www.springerlink.com/content/up024115727162w5/

 

[bcrowell]

Too bad that only the first page is viewable. Have you checked to see if it's on arxiv? It's hard to tell much from the first page, because they don't seem to be saying much about the type of matter involved. Presumably it's matter that violates an energy condition, because otherwise there'd have to be a singularity because of the Penrose singularity theorem...?

 

[George Jones]

This isn't a black hole solution, and the authors don't claim that it is. This paper matches a non-vacuum interior to a vacuum exterior such that both pressure and density become unbounded as r approaches zero, and that there is no event horizon.

 

[evanallmighty]

Yeah, I assumed that much. No, sadly its not on ArXiv... I was just posting to see if anyone has read it and knows the solution. It really stinks too that it only shows one page because there is only two.

Thanks

 

[evanallmighty]

This isn't a black hole solution, and the authors don't claim that it is. This paper matches a non-vacuum interior to a vacuum exterior such that both pressure and density become unbounded as r approaches zero, and that there is no event horizon.

Ah! Thank you. I suspected this, but since I thought little of it and probably went to fast as to make myself think it was a black hole I missed it. Do you know of any non singular black hole interior solutions?

EDIT---
How is it not a black hole solution, if the are getting rid of the singularity, how could it not be? Only black hole solutions have the gravitational singularity. It says that it admits the possibility if receiving signals from the region of infinite pressure, which if only a problem in black hole solutions, right? I think I am missing part of what your saying.

 

[George Jones]

Ah! Thank you. I suspected this, but since I thought little of it and probably went to fast as to make myself think it was a black hole I missed it. Do you know of any non singular black hole interior solutions?

A non-singular black hole solution would have to violate one of the assumptions of the Penrose-Hawking singularity theorems.

How is it not a black hole solution, if the are getting rid of the singularity, how could it not be? Only black hole solutions have the gravitational singularity. It says that it admits the possibility if receiving signals from the region of infinite pressure, which if only a problem in black hole solutions, right?

I am not sure what you're trying to say.

Inside the event horizon, a black hole spacetime is non-stationary (and thus non-static), but the first paragraph of the Introduction indicates that the paper treats static interior solutions. Their metric (after restoring the accidentally(?) omitted term) is the standard
$$ds^2 = e^{\nu\left(r\right)} dt^2 - e^{\lambda\left(r\right)} dr^2 - r^2 d\theta^2 - r^2 \sin^2 \theta d \phi^2.$$

They match vacuum exterior ton non-vacuum interior at $r = a$, and in equation (13) write
$$e^{\nu\left(a\right)} = e^{-\lambda\left(a\right)} = 1 -2u.$$
In the last paragraph before the discussion, they write "The solution thus remains valid for $u < 0.4$"
Clearly, nothing strange happens to the metric coefficients at the match value $r=a$, i.e., $t$ is timelike there and $r$ is spacelike.

PS Did you try clicking on the "Download PDF" above "Abstract" at the link that you gave?

 

[evanallmighty]

.I am not sure what you're trying to say.

Inside the event horizon, a black hole spacetime is non-stationary (and thus non-static), but the first paragraph of the Introduction indicates that the paper treats static interior solutions. Their metric (after restoring the accidentally(?) omitted term) is the standard
$$ds^2 = e^{\nu\left(r\right)} dt^2 - e^{\lambda\left(r\right)} dr^2 - r^2 d\theta^2 - r^2 \sin^2 \theta d \phi^2.$$

They match vacuum exterior ton non-vacuum interior at $r = a$, and in equation (13) write
$$e^{\nu\left(a\right)} = e^{-\lambda\left(a\right)} = 1 -2u.$$
In the last paragraph before the discussion, they write "The solution thus remains valid for $u < 0.4$"
Clearly, nothing strange happens to the metric coefficients at the match value $r=a$, i.e., $t$ is timelike there and $r$ is spacelike.

PS Did you try clicking on the "Download PDF" above "Abstract" at the link that you gave?

The Schwarzschild metric is a spherical static structure, but it describes a black hole. It is not possible in reality, but it is the most well known and simplest (black hole) metrics out there. If the paper is talking about resolving the singularity and the information paradox at r=0, than it must be talking about a black hole, because no star I know of has those problems.

I did click on the download PDF button but it wants 34 dollars for one page of a PDF.

 

[George Jones]

The Schwarzschild metric is a spherical static structure, but it describes a black hole.

No, the spacetime for a Schwarzschild black hole is static only outside the event horizon. Inside the event horizon, the spacetime for a Schwarzschild black hole is not static (it is not even stationary)

If the paper is talking about resolving the singularity and the information paradox at r=0, than it must be talking about a black hole, because no star I know of has those problems.

Again,

This isn't a black hole solution, and the authors don't claim that it is. This paper matches a non-vacuum interior to a vacuum exterior such that both pressure and density become unbounded as r approaches zero, and that there is no event horizon.

 

[evanallmighty]

No, the spacetime for a Schwarzschild black hole is static only outside the event horizon. Inside the event horizon, the spacetime for a Schwarzschild black hole is not static (it is not even stationary)

I've never heard that before. Papers I have read before on Schwarzschild interior/exterior solutions clearly say that the interior Schwarzschild solution represents a static, spherically symmetric mass of imcompressible fluid at rest:

http://www.google.com/url?sa=t&sour...r0cWmJy8HdefqBFZQ&sig2=0byzsQTHI0wrPHKVkw8ILQ

and:

http://www.jstor.org/pss/78530
(I forgot where you can read the whole PDF)

Yes, I know these sources are only describing interior solutions where a>9/8 2m, which means there is no grav. collapse (meaning that it COULD be different from a black hole because it is describing a star...)BUT:

EVERY single source describes that SCINCE the Schwarzschild black hole HAS NO MOMENTUM/IS STATIC, that it's space time singularity is a point, not A RING as in other black hole solutions that have charge/spin, etc. (such as the Kerr or Kerr Newman metric) So How could the exterior and the interior even ever be matched if they are completely different?

 

[atyy]

I've never heard that before. Papers I have read before on Schwarzschild interior/exterior solutions clearly say that the interior Schwarzschild solution represents a static, spherically symmetric mass of imcompressible fluid at rest:

http://www.google.com/url?sa=t&sour...r0cWmJy8HdefqBFZQ&sig2=0byzsQTHI0wrPHKVkw8ILQ

and:

http://www.jstor.org/pss/78530
(I forgot where you can read the whole PDF)

Yes, I know these sources are only describing interior solutions where a>9/8 2m, which means there is no grav. collapse (meaning that it COULD be different from a black hole because it is describing a star...)BUT:

EVERY single source describes that SCINCE the Schwarzschild black hole HAS NO MOMENTUM/IS STATIC, that it's space time singularity is a point, not A RING as in other black hole solutions that have charge/spin, etc. (such as the Kerr or Kerr Newman metric) So How could the exterior and the interior even ever be matched if they are completely different?

George Jones is correct and so are your sources. The terms they use, though similar, refer to different things. Your sources are not referring the part of a Schwarzschild black hole within its event horizon.

 

[bcrowell]

Evanallmighty, please take a deep breath. The ALL CAPS come across as shouting. George Jones has a great deal of expertise in GR and has been very generous with his time in trying to help you. The angry, hostile tone is uncalled for.

 

[evanallmighty]

George Jones is correct and so are your sources. The terms they use, though similar, refer to different things. Your sources are not referring the part of a Schwarzschild black hole within its event horizon.

So they are describing the exterior solution? Kind of like when the schwarzschild exterior solution describes the singularity at r=0 even though it is the EXTERIOR and not the INTERIOR?


Is this the interior/internal and exterior/external argument?

 

[evanallmighty]

Evanallmighty, please take a deep breath. The ALL CAPS come across as shouting. George Jones has a great deal of expertise in GR and has been very generous with his time in trying to help you. The angry, hostile tone is uncalled for.

Oh, I'm sorry all, I didn't know it was coming off like that. Thanks for letting me know! (it was just supposed to kinda point out certain things that I don't get. I wasn't questioning his authority, just wondering for myself, because I am ignorant in this subject. By the way, if that ever happens again or if any questions/comments seem elementary/rude, it's because I am only 13.-Although that is not an exception)
Thanks everyone.

 

[evanallmighty]

George Jones is correct and so are your sources. The terms they use, though similar, refer to different things. Your sources are not referring the part of a Schwarzschild black hole within its event horizon.

Ok, then... How? I really don't get that. In the paper: A New Interior Schwarzschild Solution by P.S Florides, it states: "The most celebrated exact solution of Einstein's vacuum field equations is the Schwarzschild (exterior) solution...It represents the grav. field in the exterior of a spherically symmetric distribution of matter of grav. mass m. The most familiar of Einstein's equations-(1.2) which is static, spherically symmetric, and forms an extension of the solution (Schwarzschild exterior) into the interior of the mass distribution is the Schwarzschild interior solution."
Isn't is saying that the interior is spherically symmetric/static? Or am I completely missing something big?

 

[atyy]

Ok, then... How? I really don't get that. In the paper: A New Interior Schwarzschild Solution by P.S Florides, it states: "The most celebrated exact solution of Einstein's vacuum field equations is the Schwarzschild (exterior) solution...It represents the grav. field in the exterior of a spherically symmetric distribution of matter of grav. mass m. The most familiar of Einstein's equations-(1.2) which is static, spherically symmetric, and forms an extension of the solution (Schwarzschild exterior) into the interior of the mass distribution is the Schwarzschild interior solution."
Isn't is saying that the interior is spherically symmetric/static? Or am I completely missing something big?

Parts of the exterior can be joined to two sorts of solutions.

One sort is a matter solution, including the "interior Schwarzschild solution". There is no event horizon in this sort of solution. The "interior Schwarzschild solution" is static.

Another sort is a vacuum solution, such as the Kruskal-Szekeres extension of the exterior Schwarzschild. This is a solution with no matter, and it has an event horizon. The "interior Schwarzschild solution" is not the "interior Schwazrschild black hole solution". The latter is the same as the interior of the Kruskal-Szekeres black hole, also known as the maximal extension of the vacuum Schwarzschild solution. The interior of the Schwarzschild black hole is not static.

The jargon is confusing, just watch for it.

 

[evanallmighty]

Parts of the exterior can be joined to two sorts of solutions.

One sort is a matter solution, including the "interior Schwarzschild solution". There is no event horizon in this sort of solution. The "interior Schwarzschild solution" is static.

Another sort is a vacuum solution, such as the Kruskal-Szekeres extension of the exterior Schwarzschild. This is a solution with no matter, and it has an event horizon. The "interior Schwarzschild solution" is not the "interior Schwazrschild black hole solution". The latter is the same as the interior of the Kruskal-Szekeres black hole, also known as the maximal extension of the vacuum Schwarzschild solution. The interior of the Schwarzschild black hole is not static.

The jargon is confusing, just watch for it.

Great! Thank you, and thank you all too! I understand it now. This is what I assumed but I was just clarifying/unsure. Now, not to ask more questions or take up anybody's time, but along with these two solutions the Schwarzschild exterior can always be matched with the spherically symmetric perfect fluid solutions too, right?

EDIT--
Oops! Never mind! Answered my own question. The perfect fluid solutions are part of the matter solutions.

 

[pervect]

Question: if you have infinite pressure at the center, and it's not a black hole solution, can it satisfy the hydrostatic equilibrium equations? (Oppenheier-Volkoff eq's). Do the authors claim it satisfies the equations?

 

[George Jones]

I am only 13

Sorry, I didn't realize this. Since you were asking questions about a technical paper. I thought that you wanted advanced, mathematical answers. The exterior/interior vacuum/non-vacuuum stuff is very confusing. I was more than twice your age before I learned about these things. Keep asking questions.

 

[George Jones]

Question: if you have infinite pressure at the center, and it's not a black hole solution, can it satisfy the hydrostatic equilibrium equations? (Oppenheier-Volkoff eq's).

I think that it is possible. From post #6, the solution is consistent with Buchdahl's theorem, $M/a < 4/9$.

Do the authors claim it satisfies the equations?

They don't seem to claim this explicitly.

 

[evanallmighty]

Sorry, I didn't realize this. Since you were asking questions about a technical paper. I thought that you wanted advanced, mathematical answers. The exterior/interior vacuum/non-vacuuum stuff is very confusing. I was more than twice your age before I learned about these things. Keep asking questions.

Well, not trying to sound smarter than I am or anything, but honestly, I have understood the answers given to me thus far, and the papers I have posted, etc. (With some hard thinking) It might take me awhile but I do feel confident after I get it/have been convinced.

I understand what you and Atyy are saying now, well, I figured that all along, but didn't realize it was the answer to my question. It is confusing, but I feel I have grasped the general concept. Correct me if that is impossible or wrong...

I'll keep taking the answers at the level they are now, haha!

Don't have any more questions for now... gimme a while

Thank you and I hope I am not taking up your time!

 

[atyy]

I understand what you and Atyy are saying now, well, I figured that all along, but didn't realize it was the answer to my question. It is confusing, but I feel I have grasped the general concept.

So are there non-singular black hole solutions?

 

[pervect]

I think that it is possible. From post #6, the solution is consistent with Buchdahl's theorem, $M/a < 4/9$.

They don't seem to claim this explicitly.

My intuition is telling me - if you infinite pressure, in the absence of a coordinate singularity, you must have expansion. But I don't have any proof that this is correct, just a strong hunch at this point.

However, I wouldn't be terribly surprised if the authors have found a "new" solution by not bothering to include the hydrostatic equilibrium equations in their solution process.

 

[TrickyDicky]

No, the spacetime for a Schwarzschild black hole is static only outside the event horizon. Inside the event horizon, the spacetime for a Schwarzschild black hole is not static (it is not even stationary)

The interior of the Schwarzschild black hole is not static.

This is the point where I always get stuck too, I just don't understand how a spacetime can be static only in part, I would have thought that the static property, being an invariant property based on the fact that the Lie derivative of the metric with respect to time vanishes, holds for any coordinate system so it shouldn't be abolished by a coordinate transformation.
So how exactly is the staticity of the Schwarzschild spacetime not a global property of the whole Schwarzschild spacetime manifold ?
Does anyone understand this and can explain it?

 

[George Jones]

A vector field doesn't have to have the same causal character everywhere, i.e., it can be timelike in one part of spacetime, lightlike in another part, and spacelike in another. For example consider Minkowski spacetime and the vector field given $x \mathbf{e}_x + t \mathbf{e}_t$.

Schwarzschild spacetime has a Killing vector field that is timelike outside the event horizon, lightlike on the event horizon, and spacelike inside the event horizon.

 

[TrickyDicky]

A vector field doesn't have to have the same causal character everywhere

Ok, but a static manifold has a unique timelike KV field, do you mean then that a manifold can have regions that fulfill this requirement and other regions that don't? What is that manifold called then, static and not static at the same time?

 

[George Jones]

Ok, but a static manifold has a unique timelike KV field

Not necessarily unique.

do you mean then that a manifold can have regions that fulfill this requirement and other regions that don't?

Yes.

What is that manifold called then, static and not static at the same time?

Yes. More precisely, it is called static on (open) region U if there is a timelike, hypersurface-orthogonal Killing vector field defined throughout U.

 

[TrickyDicky]

Not necessarily unique.

After some research on the web I guess you are referring here to the difference between a static manifold (that can be locally or partially static or as you say static on a region like in the case we are discussing) and a standard static manifold, where there truly exist a unique global orthogonal splitting of the spacetime.

 

[George Jones]

What I meant was that in (parts of) some static spacetimes, it is possible to have linearly independent timelike Killing vectors. The additional requirement of hypersurface-orthogonality still does not seem to pin down the Killing vector. Again, consider Minkowski spacetime in a global inertial coordinate system. Then, $\mathbf{e}_t$ and $\mathbf{e}_x + 2 \mathbf{e}_t$ are both timelike, hypersurface orthogonal Killing vector fields.

 

[TrickyDicky]

What I meant was that in (parts of) some static spacetimes, it is possible to have linearly independent timelike Killing vectors. The additional requirement of hypersurface-orthogonality still does not seem to pin down the Killing vector. Again, consider Minkowski spacetime in a global inertial coordinate system. Then, $\mathbf{e}_t$ and $\mathbf{e}_x + 2 \mathbf{e}_t$ are both timelike, hypersurface orthogonal Killing vector fields.

I have the impression you can only do that with cartesian coordinates in flat spacetime, because points in Minkowski space can be identified with vectors but in GR curved spacetime vectors are only defined locally and $\mathbf{e}_x + 2 \mathbf{e}_t$ may not be a vector.

 

[TrickyDicky]

A vector field doesn't have to have the same causal character everywhere, i.e., it can be timelike in one part of spacetime, lightlike in another part, and spacelike in another.

The WP page geodesic entry after explaining the different null, timelike and spacelike types says: "Note that a geodesic cannot be spacelike at one point and timelike at another". Is there a difference in this respect between vectors and geodesics?

 

[George Jones]

The WP page geodesic entry after explaining the different null, timelike and spacelike types says: "Note that a geodesic cannot be spacelike at one point and timelike at another". Is there a difference in this respect between vectors and geodesics?

Yes, even though vectors and curves are related, there is, in principle, a difference in this respect. In particular, the integral curves associated with the Killing vector field $\partial_t$ (curves of contant r, theta,a and phi) are not geodesics, so your result could possible fail for them. It, however, does not fail, i.e., an individual integral curve for $\partial_t$ does not change its causal character, but different integral curves for $\partial_t$ can different characters. Outside the event horizon (where spactime is static), the integral curves of $\partial_t$ are all timelike; inside the event horizon (where spacetime is not even stationary), the integral curves of $\partial_t$ are all spacelike.

The integral curves for $\partial_t$ are the curved dashed lines on

http://www.google.com/imgres?imgurl...MkRTtjWJfTLsQKpl6i-Cg&ved=0CDcQ9QEwBQ&dur=235

These means that the vectors that make up the vector field $\partial_t$ are tangent vectors for these curves. Note the different causal character outside and inside the event horizon.

 

[TrickyDicky]

Thanks.