Important and/or Interesting Spacetimes

[George Jones]

What spacetimes do you consider to be important or interesting? Important or interesting as solutions to the Einstein field equation, or important or interesting geometrically. I'll start the list.

Minkowski

Schwarzschild (including extended)

Kerr(-Newman)

Friedmann-Lemaitre-Robertson-Walker

Schwarzschild constant density spherical solution

Vaidya

Morris-Thorne wormholes (and generalizations)

Alcubierre warp drive

The first five spacetimes on the list are now usually treated even in introductory books. Off the top of my head (not with my books right now), I can only only think of two of my books (that aren't compilations of solutions) that cover Vaidya spacetimes, Poisson's A Relativist's Toolkit: The Mathematics of Black Hole Mechanics and Padmanabhan's new, advanced, comprehensive Gravitation: Foundations and Frontiers. Morris-Thorne wormholes and Alcubierre warp drive are covered in some introductory books.

Rindler spacetime and the Milne universe are already included in Minkowski. Similarly, de Sitter and anti-de Sitter are included in the Friedmann-Lemaitre-Robertson-Walker family. I take a spacetime to be a four-dimensional Lorentzian (differentiable) manifold, so each spacetime includes all compatible coordinate charts. Rindler spacetime and the Milne universe are particular coordinate charts for Minkowski spacetime.

If you add a (family of) spacetimes, and you know of a book that isn't a compilation of spacetimes which treats the spacetime, also identify the book.

The two standard compilations are Exact Space-Times in Einstein's General Relativity by Griffiths (Jerry, not David) and Podolsky (which I find it to be endlessly fascinating), and the second edition of Exact Solutions of Einstein's Field Equations by Stephani, Kramer, MacCullum, Hoenselaers, and Herlt.

 

[PeterDonis]

Is the Kinnersley Photon Rocket solution covered by one of the above? (I don't think it is, but I'm not sure.) If not, I think it should be on the list.

Also I would add the "binary pulsar" spacetime, since that's an important experimental test of GR, but I'm not sure if an exact solution is known for it.

 

[George Jones]

Is the Kinnersley Photon Rocket solution covered by one of the above? (I don't think it is, but I'm not sure.) If not, I think it should be on the list.

I, too, think that Kinnersley's photon rocket is quite interesting. Griffiths and Podolsky devote an interesting two-and-a-bit pages to this spacetime; for an excerpt, see post #25 in the "Gravitational waves due to acceleration" thread,

https://www.physicsforums.com/showthread.php?p=3060426#post3060426.

Also I would add the "binary pulsar" spacetime, since that's an important experimental test of GR, but I'm not sure if an exact solution is known for it.

Very important, but verified using linearized GR. Within 10 years or so, we should have experimental verification of strong-field GR near the event horizon of the black hole at the centre of our galaxy.

 

[sheaf]

Goedel solution maybe ? (Hawking and Ellis)

At first I thought you might be excluding it because of the CTCs, but then you have Alcubierre which I think is exotic in the sense that you need matter which violates the energy conditions, so yes, add Goedel, just for fun !

 

[George Jones]

Goedel solution maybe ? (Hawking and Ellis)

At first I thought you might be excluding it because of the CTCs, but then you have Alcubierre which I think is exotic in the sense that you need matter which violates the energy conditions, so yes, add Goedel, just for fun !

Rindler recently published fascinating article on this in the American Journal of physics,

http://ajp.aapt.org/resource/1/ajpias/v77/i6/p498_s1?isAuthorized=no ,

which, unfortunately, is not on the arXiv.

 

[PeterDonis]

Very important, but verified using linearized GR.

Since it's a binary pulsar, though, what solution is being linearized to provide the model? It can't just be one with a single central mass, can it? (Or do you mean that the linearized solution used to verify the gravitational wave predictions *only* models the combined mass of the pulsars and how that combined mass should radiate gravitational waves, not their detailed orbital parameters? That doesn't seem right, since the orbital parameters themselves are part of the data that's used to verify the predictions?)

 

[PAllen]

Since it's a binary pulsar, though, what solution is being linearized to provide the model? It can't just be one with a single central mass, can it? (Or do you mean that the linearized solution used to verify the gravitational wave predictions *only* models the combined mass of the pulsars and how that combined mass should radiate gravitational waves, not their detailed orbital parameters? That doesn't seem right, since the orbital parameters themselves are part of the data that's used to verify the predictions?)

This is covered in outline in sections 4.1 to 4.4 of:

http://relativity.livingreviews.org/Articles/lrr-2006-3/

with references therein to more complete treatments. My understanding is there remains no exact two body solution when both masses are considered significant for determining the geometry.

 

[AEM]

George,

In your first post you mentioned the "Schwarzschild Constant Density Spherical Solution". Is this an "interior" solution? Could you cite a reference for it? Are there any interesting solutions interior to a spherical distribution of "dust" --this could not be static and would have to be highly idealized, I would expect.

Allan

 

[PeterDonis]

This is covered in outline in sections 4.1 to 4.4 of:

http://relativity.livingreviews.org/Articles/lrr-2006-3/

with references therein to more complete treatments. My understanding is there remains no exact two body solution when both masses are considered significant for determining the geometry.

Thanks, I'd only skimmed this section of living reviews in GR before. My understanding after reading it is the same as yours: there is no exact solution for the "two-body" situation, such as the binary pulsar. But it seems like a good enough approximate solution can be built up by adding successive terms of higher and higher order in the "metric perturbation" $h_{\alpha \beta}$ to compare with the data, and also to distinguish GR's predictions from those of other theories of gravity.

From the perspective of the question in the OP, then, I would still suggest that this "binary pulsar" spacetime belongs on the list, even though there is no exact solution for it.

 

[Mentz114]

LENELLS rotating dust solution

arXiv:1003.1453v1 [nlin.SI] 7 Mar 2010

Rotating dust
Neugebauer & Meinel, Phys. Rev. Lett. 75 (1995) 3046]

and

arXiv:gr-qc/9703077v1 26 Mar 1997

arXiv:gr-qc/9910045v2 8 Dec 1999
"The exact global solution of the Einstein equations [Neugebauer &
Meinel, Phys. Rev. Lett. 75 (1995) 3046] describing a rigidly rotating,
self–gravitating disk is discussed. The underlying matter model is a perfect
fluid in the limit of vanishing pressure. The solution represents the
general–relativistic analogue of the classical Maclaurin disk."

arXiv:gr-qc/0302060v1 14 Feb 2003

"In a recent paper we presented analytic expressions for the axis potential, the
disk metric, and the surface mass density of the global solution to Einstein’s
field equations describing a rigidly rotating disk of dust. Here we add the
complete solution in terms of ultraelliptic functions and quadratures."

arXiv:1007.3360v1 [gr-qc] 20 Jul 2010

"In the present paper,
the way in which the black hole limit is approached is investigated in more
detail by means of a parametric Taylor series expansion of the exact solution
describing a rigidly rotating disc of dust."

 

[George Jones]

George,

In your first post you mentioned the "Schwarzschild Constant Density Spherical Solution". Is this an "interior" solution?

Yes.

Could you cite a reference for it?

There is a nice treatment in Misner, Thorne, and Wheeler, which I don't have with me right now, so I can't give the chapter and section. See also the books I give at the bottom of this post.

Are there any interesting solutions interior to a spherical distribution of "dust" --this could not be static and would have to be highly idealized, I would expect.

Allan

The important and groundbreaking Oppenheimer-Snyder collapse (1939)! See section 3.8 of Eric Poisson's notes,

http://www.physics.uoguelph.ca/poisson/research/agr.pdf,

which evolved into the excellent book, A Relativist's Toolkit: The Mathematics of Black Hole Mechanics.

This is covered in outline in sections 4.1 to 4.4 of:

http://relativity.livingreviews.org/Articles/lrr-2006-3/

with references therein to more complete treatments. My understanding is there remains no exact two body solution when both masses are considered significant for determining the geometry.

No exact solution relevant to physical binaries, but all kinds of bizarre "two-body" solutions. For a few, see

https://www.physicsforums.com/showthread.php?p=2528464#post2528464.

Thanks, I'd only skimmed this section of living reviews in GR before. My understanding after reading it is the same as yours: there is no exact solution for the "two-body" situation, such as the binary pulsar. But it seems like a good enough approximate solution can be built up by adding successive terms of higher and higher order in the "metric perturbation" $h_{\alpha \beta}$ to compare with the data, and also to distinguish GR's predictions from those of other theories of gravity.

From the perspective of the question in the OP, then, I would still suggest that this "binary pulsar" spacetime belongs on the list, even though there is no exact solution for it.

Perturbative and numerical solutions are extremely important in physics, and GR is no exception but, in the original post, I really meant exact geometries and/or solutions.

verified using linearized GR

I am not very familiar with this stuff, and before writing the above, I only looked at one book, General Relativity: An Introduction for Physicists by Hobson, Efstathiou, and Lansenby, which uses the wave equation of linearized GR to give a nice, introductory treatment of this. A more advanced treatment is given in Gravitation: Foundations and Frontiers by Padmanabhan.

 

[PAllen]

Yes.
No exact solution relevant to physical binaries, but all kinds of bizarre "two-body" solutions. For a few, see

https://www.physicsforums.com/showthread.php?p=2528464#post2528464.

Very interesting. These solutions all seem to involve balanced forces leading to static (?) geometry. Do you know if there are any special case exact solutions involving anything resembling orbiting (e.g. two precisely identical black holes with some simplest possible mutual orbit)?

 

[Mentz114]

Very interesting. These solutions all seem to involve balanced forces leading to static (?) geometry. Do you know if there are any special case exact solutions involving anything resembling orbiting (e.g. two precisely identical black holes with some simplest possible mutual orbit)?

This paper has a superposition of 2 BHs in a Weyl vacuum.

arXiv:gr-qc/0502062v1 14 Feb 2005

 

[PAllen]

This paper has a superposition of 2 BHs in a Weyl vacuum.

arXiv:gr-qc/0502062v1 14 Feb 2005

Maybe I misunderstand what this paper is doing. It seems to me that the only exact two body solution described is again a static geometry with two black holes rigidly held in place by a conical singularity. They then model the orbit of a star around such a system by examining geodesics. The orbiting star is not considered to perturb the geometry, i.e. it is considered as effectively test particle.

This is, indeed, interesting, but still not even a special case of an exact two body solution that entails orbiting, where each body is considered a source of gravity.

 

[Mentz114]

This is, indeed, interesting, but still not even a special case of an exact two body solution that entails orbiting, where each body is considered a source of gravity.

You're right. I don't think there is a solution for two bodies in orbit around each other. I've never heard of any solution that radiates GWs.

 

[atyy]

If we are allowed 4+1D, then black saturn.

 

[piareround]

You know speaking of Alcubierre and other exotic metrics, I was wondering has anyone Alcubierre or someoneelse ever tried using Alcubierre metric for something than to suggest warp drive. Like what the metric would mean for smaller objects like the Sun?

Just curious ;3