Schwarzschild solution without matter

[paweld]

Is it theoretically possible that the metric in the whole universe would be described
by Kruskal extension of Schwarzschild solution of Einstein equation and in the universe
would be no matter at all (vaccum solution everywhere).
What's the interpretation of parameter m characterising Schwarzschild solution in this case?
Does the matter need to be somewhere in the universe in case of this solution
because we don't want to have white hole? Are there also other arguments for
placing matter in some regions.
What about the statement that matter (stress energy tensor) is the source of
gravitational field. Are nonflat global solution of vacuum Einstein equation acceptable
in whole universe (theoretically) or if there was vacuum everywhere in the universe
it would have to be flat.
Thanks.

 

[bcrowell]

Is it theoretically possible that the metric in the whole universe would be described
by Kruskal extension of Schwarzschild solution of Einstein equation and in the universe
would be no matter at all (vaccum solution everywhere).

I wouldn't agree with "no matter at all" as an interpretation. Just because the matter resides at a singular point, that doesn't mean that the matter doesn't exist.

Also, this spacetime is compatible with any desired spherically symmetric boundary condition at infinity. What this would mean for an observer living in this universe (outside the horizon) is that in time t they could verify the absence of matter and radiation out to r=ct, but they would never be able to determine in any finite time whether there was some r beyond which matter and radiation did exist.

What's the interpretation of parameter m characterising Schwarzschild solution in this case?

It's the amount of mass residing at the central singularity. There are lots of different ways that the observer could determine m. For example, he could use Kepler's laws.

Does the matter need to be somewhere in the universe in case of this solution
because we don't want to have white hole? Are there also other arguments for
placing matter in some regions.

I don't follow you here.

What about the statement that matter (stress energy tensor) is the source of
gravitational field.

There is matter at the singularity. It's the source of the curvature in the surrounding vacuum.

Are nonflat global solution of vacuum Einstein equation acceptable
in whole universe (theoretically) or if there was vacuum everywhere in the universe
it would have to be flat.

You can definitely have nonflat global vacuum solutions with no matter in them. The Petrov metric (see references below) is one example. The Petrov metric is a vacuum solution, and it's geodesically complete, i.e., it's not a case like the Schwarzschild metric where some matter is "hiding" in singularities that aren't properly part of the spacetime. Since the Petrov metric can be interpreted as a uniform gravitational field, this example shows that it's not quite right to say that matter is the source of the gravitational field. Really it's better to say that matter is the source of Ricci curvature.

Another example would be that there are lots of exact vacuum solutions known that can be interpreted as plane gravitational waves.

References on the Petrov metric:
T. Lewis, Proc. Roy. Soc. Lond. A136 (1932) 176 -- original discovery
Petrov, in "Recent Developments in General Relativity," 1962, Pergamon, p. 383 -- rediscovery by Petrov
Gibbons and Gielen, "The Petrov and Kaigorodov-Ozsváth Solutions: Spacetime as a Group Manifold," http://arxiv.org/abs/0802.4082 -- a good description that's available free online

 

[paweld]

Thanks for clarifying lots of things.

I wouldn't agree with "no matter at all" as an interpretation. Just because the matter resides at a singular point, that doesn't mean that the matter doesn't exist.

I only wonder if we can say that matter resides at a singular point. Does this point
belong to the universe?

 

[yossell]

I only wonder if we can say that matter resides at a singular point. Does this point
belong to the universe?

I've wondered about this too and would like to know what people think. Mathematically, space-time is a smooth manifold, and the laws of GR hold everywhere - I've always wondered how, if these singular points were literally part of the space-time manifold, the manifold could have the nice mathematical properties it's supposed to have.

 

[bcrowell]

I only wonder if we can say that matter resides at a singular point. Does this point
belong to the universe?

If you look at a book like Hawking and Ellis, the reason they spend such a huge number of pages on mathematical preliminaries is that issues like this with singularities are extremely complex to deal with rigorously (and a lot of the purpose of the book is to develop singularity theorems).

Mathematically, I think it's pretty clearcut. A singularity like a Schwarzschild one is *not* a point that is part of the manifold.

Physically, the question seems fundamentally uninteresting to me. We know that we would really need a theory of quantum gravity to say what is going on at the center of a black hole, so the mathematical formalism of how it's represented in classical gravity has no physical significance. Another way of getting at this physically is that external measurements will never suffice to prove that a particular object is a singularity; all you can do by scattering higher- and higher-energy particles off of it is to put lower and lower bounds on its size.