The Schwarzschild (Exterior) Solution

[evanallmighty]

By the way everyone, I apologize if any of my questions/comments seem elementary at times, I am only 13. So I'm just letting you all know. Although the physics level for my age is pretty low, I do understand a lot, so I hope now that I have told you all you won't give me too much of an elementary explanation. (example-I have understood (with some thinking) mostly everything we have talked about so far)

But yes, Pervect, the reason why I am asking all these questions is because I want to, in the future, create a static, spherically symmetrical non singular black hole model (Schwarzschild) and right now I am just gathering as much information as I can. I plan to find the solutions I need (and the information) and as I learn to read and understand them better, although it might be obsolete by then, create my own non singular model. Yes, I know that sounds like a very specific thing to do and a great undertaking for me to even think about(because I know I sound like I don't know what I'm talking about-sorta), but I have learned a lot in the past year (I am surprised at how elementary I was not to long ago) that I think in the near future I will understand this fairly well.
I know it is not as simple as the article I posted earlier says, but I am just making it a long term goal, so I hope nobody will criticize that. And please, if you have bad criticism, just don't post it, if you want to help me with pursuing my career, or this project, than do. (I hope that didn't sound rude or anything)

Thanks for everyone's help thus far! I am going to read all the links suggested for now... but I probably will have some more questions!

 

[evanallmighty]

I suppose my only questions for now are about patchwork above the event horizon.

 

[yuiop]

This might be a silly question to people who are have more practice in this subject, but how does the Schwarzschild Solution/metric have a singularity at r=0 if it is only valid outside the gravitational mass (black hole)? wouldn't that be in the interior solution, describing the inside of a Schwarzschild black hole, where the gravitational singularity is actually located? I thought the only "singularity" in the exterior was the coordinate singularity at the event horizon.

I think there is a terminology issue here. Most texts seem to refer to the exterior Schwarzschild solution as the vacuum region outside the spherical mass in question. The interior solution covers the non-vacuum region inside the spherical mass and the border between the two regions is not necessarily the event horizon. At the risk of getting the terminology wrong myself because I am going from distant memory, I think you are referring to the external solution (outside the event horizon) and the internal solution (inside the event horizon). These have to be patched together by choosing a suitable integration constant so that world-lines are continuous are continuous across the boundary. There is also the issue of complex numbers and how imaginary numbers are physically interpreted.

Anyway, in conclusion, the exterior Schwarzschild solution for a fully formed black hole includes the region inside and outside the event horizon and all regions that are vacuum which covers infinity>= r >0m because all the mass is considered to be located at r=0m. Hope that makes some sort of sense :P

 

[TrickyDicky]

I think there is a terminology issue here...
infinity>= r >0m because all the mass is considered to be located at r=0m. Hope that makes some sort of sense :P

I have difficulties understanding what you mean by r=0m, does r have mass units? I thought r was the radial coordinate.

 

[yuiop]

I have difficulties understanding what you mean by r=0m, does r have mass units? I thought r was the radial coordinate.

I'm sorry, I was being sloppy and using units such that the gravitational constant G=1 and the speed of light c=1. The event horizon is located at r = 2GM/c^2. When multiplied out, G*M/c^2 has units of length and many texts use 2M as shorthand for 2GM/c^2. Similarly some casual texts the entropy of black hole is S = A/4 but it actually:

$$S_{BH} = \frac{kAc^3}{4G\hbar}$$

when the units are not fully expressed, but that is not so snappy.

 

[bcrowell]

This very interesting discussion motivated me to try to understand more about topology change in GR. I found this paper: http://arxiv.org/abs/gr-qc/9406053

They say that topology change isn't kinematically impossible; you can only prove it doesn't happen by imposing dynamical constraints like the EFE.

"Geroch [5, 6] has shown that topology change may be obtained in these cases only at the price of causality violations, and Tipler [7, 8] has shown that Einstein’s equation cannot hold (with a source with non-negative energy density) on such spacetimes if the spatial topology changes."

They're generalizing the Geroch and Tipler results. They assume something they call "causal compactness," and which they consider a very mild physical condition. Their definition is this. Let I(p) be the set of all points that can be connected to p by a timelike curve; a spacetime is causally compact if the closure of I(p) is compact for all p.

Their interpretation is that causal compactness restricts how you can make arbitrary cuts and holes in spacetime, and that if a spacetime is causally compact, then no point p can 'receive signals from, or send signals to, either regions at infinity or "holes" in the spacetime.'

I don't understand this; doesn't this mean that Minkowski space isn't causally compact? If I pick a point p in Minkowski space, the closure of I(p) is simply its light cone, including the lightlike boundary. This isn't compact.

 

[George Jones]

I have not been following this thread at all, but I have taken a quick look at Borde's paper.

I don't understand this; doesn't this mean that Minkowski space isn't causally compact? If I pick a point p in Minkowski space, the closure of I(p) is simply its light cone, including the lightlike boundary. This isn't compact.

Right, this set isn't compact, but this isn't the type of set that Borde has in mind. Consider any two spacelike hypersurfaces in Minkowski spacetime. The open subset of Minkowski spacetime that lies between these two hypersurfaces is itself a spacetime, and is an example of what Borde calls (an interpolating) spacetime M. Note that M is not all of Minkowski spacetime. Now consider a p in M, and I(p) in M. The closure of I(p) is compact.

In figure 5, spacetime M lies between S1 and S2, and the shaded area, together with the lines that enclose shaded area, is the closure of I(p).

 

[bcrowell]

Ah, I see -- thanks, George!

-Ben

 

[evanallmighty]

I think there is a terminology issue here. Most texts seem to refer to the exterior Schwarzschild solution as the vacuum region outside the spherical mass in question. The interior solution covers the non-vacuum region inside the spherical mass and the border between the two regions is not necessarily the event horizon. At the risk of getting the terminology wrong myself because I am going from distant memory, I think you are referring to the external solution (outside the event horizon) and the internal solution (inside the event horizon). These have to be patched together by choosing a suitable integration constant so that world-lines are continuous are continuous across the boundary. There is also the issue of complex numbers and how imaginary numbers are physically interpreted.

Anyway, in conclusion, the exterior Schwarzschild solution for a fully formed black hole includes the region inside and outside the event horizon and all regions that are vacuum which covers infinity>= r >0m because all the mass is considered to be located at r=0m. Hope that makes some sort of sense :P

So what your saying is the exterior solution describes all of the black hole, and the interior solution describes all of the black hole, sorta? Then if that's true, why would there be an exterior AND an interior solution? According the Brian Koberlein: (http://koberlein.main.ad.rit.edu/journal/2009/12/non-singular-black-holes.php ), "you need the interior (inside the event horizon) and the exterior (outside the event horizon) solutions" to create a non singular black hole model.
In an email I received from him, he said: "Ideally we would like a unique solution that describes both the interior and exterior, but we don't have such a solution yet." So how is the exterior solution describing r=0? I'm not questioning you, just wondering.

Wow, this discussion has been quite a hit!

 

[DrGreg]

So what your saying is the exterior solution describes all of the black hole, and the interior solution describes all of the black hole, sorta? Then if that's true, why would there be an exterior AND an interior solution? According the Brian Koberlein: (http://koberlein.main.ad.rit.edu/journal/2009/12/non-singular-black-holes.php ), "you need the interior (inside the event horizon) and the exterior (outside the event horizon) solutions" to create a non singular black hole model.

If you read carefully what yuiop wrote, he is using "exterior" and "external" with different meanings, and incompatibly with Koberlein's use.