# Non singular black hole solutions

1. Jun 27, 2011

### 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

2. Jun 27, 2011

### bcrowell

Staff Emeritus
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...?

3. Jun 27, 2011

### George Jones

Staff Emeritus
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.

4. Jun 27, 2011

### 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

5. Jun 27, 2011

### evanallmighty

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.

6. Jun 27, 2011

### George Jones

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

7. Jun 27, 2011

### evanallmighty

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.

8. Jun 27, 2011

### George Jones

Staff Emeritus
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)
Again,

9. Jun 27, 2011

### evanallmighty

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:

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?

Last edited by a moderator: Apr 26, 2017
10. Jun 27, 2011

### atyy

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.

Last edited by a moderator: Apr 26, 2017
11. Jun 27, 2011

### bcrowell

Staff Emeritus
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.

12. Jun 27, 2011

### evanallmighty

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?

13. Jun 27, 2011

### evanallmighty

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.

Last edited: Jun 27, 2011
14. Jun 27, 2011

### evanallmighty

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?

15. Jun 27, 2011

### atyy

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 Schwarschild black hole is not static.

The jargon is confusing, just watch for it.

Last edited: Jun 27, 2011
16. Jun 27, 2011

### evanallmighty

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.

Last edited: Jun 27, 2011
17. Jun 27, 2011

### pervect

Staff Emeritus
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?

18. Jun 27, 2011

### George Jones

Staff Emeritus
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.

19. Jun 27, 2011

### George Jones

Staff Emeritus
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.

20. Jun 27, 2011

### evanallmighty

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!