Solutions of the nonlinear Field Equations

1. Oct 13, 2006

notknowing

Einsteins field equations are nonlinear but I guess that nobody has already found solutions to the full nonlinear equations (because of course it is very hard to do so). Nevertheless, such solutions could (I think) hold a number of surprises. Instead of linear gravitational waves, one could think of soliton like waves with strange properties. Solutions near spinning black holes could also be siginificantly modified by taking into account nonlinear terms. Finally, even cosmological models could have to be modified because of these nonlinearities. Maybe that there is no need then to invoke dark energy and other strange stuff.
What do you think ?

2. Oct 14, 2006

George Jones

Staff Emeritus
The Friedmann-Robertson-Walker and de Sitter cosmological solutions, as well as the Schwarzschild, Reissner-Nordstron, Kerr and Kerr-Newman black hole solution are all solutions of the full non-linear Einstein equations.

Last edited: Oct 14, 2006
3. Oct 14, 2006

notknowing

This shows indeed my ignorance on this subject.
But what about nonlinear gravitational waves ?
Or could there maybe exist stable structures in spacetime in which the curvature is itself the source of the curvature ?

4. Oct 14, 2006

George Jones

Staff Emeritus
A project underway, LIGO, hopes to detect gravitational radiation in the next decade or so. In order to be detectable, the gravitational radiation will have to be produced by processes in which there are strong gravitational interactions, like collisions that involve black holes and neutron stars. In analyzing such strong interactions, non-linearity must be included. This is done numerically on computers.

Off the top of my head, don't know if any exact soliton-like solutions erxist, and I only have one GR book (Hartle) with me right now.

I'm not sure what you mean here. What is a stable structure?

5. Oct 14, 2006

notknowing

Take the field equations in vacuum, with no mass whatsoever. Then, try to find a stationary solution of the nonlinear field equations. One could image some kind of peculiar curvature of spacetime (for instance with spherical symmetry) in which this curvature is in fact the source of its own curvature. This could for instance look a bit like the space-curvature around a mass, but then without the presence of the mass. Maybe a far-fetched fantasy but can it really be excluded that such a thing exists ?

6. Nov 21, 2006

Chris Hillman

Exact solutions to the fully nonlinear EFE

Hi again, notknowing,

Waay back in 1916, Einstein did say something gloomy to the effect that he feared exact solutions would never be found, but within a month or so Schwarzschild produced a very simple exact solution of fundamental importance (the static spherically symmetric vacuum solution). Levi-Civita soon found dozens of new exact one or two parameter solutions, and Kottler and Reissner and Nordstrom provided important generalizations of the Schwarzschild vacuum.

Next, in 1918 Weyl found ALL the axisymmetric static vacuum solutions of the EFE (they turn out to correspond to axisymmetric harmonic functions), and over the next few decades many more solutions were found, including many important cosmological models such as the FRW dusts and Kasner dusts, as well as the pp waves discovered by Brinkmann in 1925 and an important special case, the gravitational plane waves found by Baldwin and Jeffery in 1926.

All these were found using some symmetry Ansatz. Then in 1963, Kerr made a sensational discovery, the Kerr vacuum, and within a decade after that zillions of new solutions were found, including the Ernst family of all stationary axisymmetric electrovacuum solutions. Even better, this work, and new developments in the field nonlinear PDEs generally, inspired the introduction of fundamental new techniques for finding exact solutions in gtr, as well as techniques for drawing reliable inferences are the solution space itself.

A comparatively recent and rather spectacular achievement, which may turn out to be almost as important as the Kerr vacuum, is the discovery of an exact rigidly rotating disk of dust found by Neugebauer and Meinel in 1993. (The exterior region of) this solution belongs to the Ernst vacuum family, but their achievement was to single out this particular solution from that rather large family. This has since been generalized in various directions.

(Things called "counter-rotating disks of dust" range from the profoundly unrealistic to the unphysical, and are not very interesting compared to the true rotating disk solutions.)

Aha! One of the developments in PDEs I was thinking of was the introduction of the inverse scattering transform for solving the famous KdV equation, and which was soon adapted to attack the EFE. See Belinsky and Verdaguer, Gravitational Solitons. (I should caution you that most "gravitational solitons" are not true solitons and do not exhibit the properties you would expect from n-soliton solutions of the KdV equation. However, current research may yet uncover true solitons lurking in solutions related to the well-known Robinson-Trautman null dusts.)

To elaborate on what George Jones said, the Brinkmann pp-waves form a large class of exact null dust or vacuum solutions of the EFE, which model exact gravitational waves, possibly accompanied by other massless radiation such as EM radiation. The Baldwin-Jeffery plane waves are a large subclass of pp-waves, which include both the curved space generalization of purely electromagnetic plane waves (with not accompanying gravitational radiation) and their gravitational analogs.

In particular, exact null electrovacuum solutions modeling "sinusoidal" EM plane waves with linear or circular polarization are known and can be written in closed form. (See the version of "Monochromatic electromagnetic plane wave" listed at http://en.wikipedia.org/wiki/User:Hillman/Archive; [Broken] I can't vouch for more recent versions, however!) Similarly, exact vacuum solutions (Petrov type N, to use the jargon) modeling "sinusoidal" gravitational plane waves with linear or circular polarizations are known and can be written in closed form (using standard special functions, to be sure, the "Mathieu cosine functions"). Comparing these solutions is very instructive, particularly for anyone who insists that gtr "unifies" electromagnetism and gravitation! (Gtr is a fine theory which has attained impressive accomplishments, but unification is not one of them.)

You might be groping toward something like Wheeler's notion of a "geon".

Some interesting colliding plane wave (CPW) solutions (and some other types of solutions) have been found using the inverse scattering method. The monograph by Belinsky and Verdaguer provides an exposition of this method and gives many detailed examples.

Definitely sounds like you should read about geons. Unfortunately, I am not sure that I can recommend readable sources. The Wikipedia article "geons" in the version listed at http://en.wikipedia.org/wiki/User:Hillman/Archive is probably the best I can offer right now, but I am familiar with its limitations because I wrote it, in the spirit of "something being better than nothing". More recent versions might be much better or much, much worse--- that's just the nature of Wikipedia, so always be very wary of what you read there.

Chris Hillman

Last edited by a moderator: May 2, 2017
7. Nov 21, 2006

Jheriko

I would imagine the most general solutions of the EFE would be numerical, although, numerical solvers have limited accuracy and require computer programming skill as well as the maths.

Still it is probably possible to write a program which would tend towards an exact solution if allowed infinite running time and infinite precision arithmetic.

8. Nov 22, 2006

Chris Hillman

Hi, jheriko,

Surprisingly enough, the number of papers presenting new exact solutions easily outnumbers the number of papers presenting numerical studies, although the latter group is by no means small.

Chris Hillman

9. Nov 22, 2006

cesiumfrog

This makes me think of the Schwarzschild solution, where there is in no place anything but vacuum.

10. Nov 23, 2006

notknowing

This is not really true. To obtain the Schwarzschild solution you really need a mass (as source of curvature). The Schwarzschild solution is than the solution of the Einstein Field Equations in the region where there is no mass present (in contrast to the region inside the mass).

11. Nov 23, 2006

cesiumfrog

In exactly what sense do you need a mass?

12. Nov 23, 2006

notknowing

The Schwarzschild solution is the most symmetric solution (spherically symmetric) outside a spherically symmetric source of curvature(thus a mass). Some input of the experts would be nice here. When you see the derivation of the Schwarzschild solution, one usually does not start with the full Einstein field equations with an energy-stress tensor on the right side. Instead, one uses arguments based on symmetry to arrive at a solution which leads in the weak field limit to geodesics corresponding to Newtonian motion. So, in this last step the requirement for a mass clearly comes in.

13. Nov 23, 2006

George Jones

Staff Emeritus
An eternal Schwarzschild black hole is a vacuum solution to Einstein's equation, i.e., one for which the stress-energy tensor is zero everywhere. I think this is what cesiumfrog means.

This motivated Chandrasekhar to write the following in the prologue to his book The Mathematical Theory of Black Holes.

(These are not a spacetimes for stellar collapse.)

However, these black hole soultions all have singularities - colloquially, they all have edges, rips, or tears in their spacetimes. These black hole singularites are, again colloquially, regions that have excised from spacetime in which one would like to place infinitely dense matter.

However, when considered less colloquially and more rigorously, the actual spacetimes don't contain any matter.

These black hole solutions are not soliton-like. So, notknowing, I think you're looking for solutions that don't have (strong) curvature singularities, and this rules out black hole solutions.

14. Nov 23, 2006

notknowing

Now, I'm really strongly confused . If the stress-energy tensor is zero everywhere, what is then causing the curvature ?? I would tend to think that the stress-energy tensor becomes singular or so, but not zero. And what does "M" then mean in the Schwarzschild solution ? Is this the mass of an object with zero energy-stress tensor ????

15. Nov 23, 2006

George Jones

Staff Emeritus
For an eternal Schwarzschild black hole, components of the curvatiure tensor diverge as the the singularity is approached, but the stress-energy tensor is congruently zero everywhere.

Yes.

Carroll has a good discusion of this, and maybe pervect and/or Chris will say more.

cesiumfrog was correct when he said

You are also right when you say that M is the mass of Schwarzschild black hole.

16. Nov 23, 2006

robphy

The Weyl tensor.

17. Nov 24, 2006

notknowing

Thanks for this clarification. So it seems that I have learned something and that cesiumfrog was right. I was mislead by the popular view that in a black hole a point of infinite density is obtained. So, I recapitulate things to see if I understand it correctly. First you need a sufficient amount of mass to create a black hole in the first place ("feed" it). Then, through some magic, the matter is crunched together in such a way that it is transformed into pure gravitational energy such that "the curvature itself becomes the source of curvature" (the idea which I had originally to create a kind of "geon"). Of course, in a quantum theory of gravitation, things would probably turn out to be very different.

But what then about a charged black hole ? If the stress-energy tensor is also congruently zero everywhere in this case, such that one can not speek of mass in the usual sense, where is then the charge located ? It can probably not be considered to be "smeared out" in the region inside the horizon ?

18. Nov 24, 2006

George Jones

Staff Emeritus
I wouldn't put it this way.

For a black hole that forms from stellar collapse, there is a region of spacetime in which the stress-energy tensor is non-zero. See the diagrams on pages 28 and 36 of Chapter 7 of http://preposterousuniverse.com/grnotes/".

In posts #13 and #15, I was careful, and I said that eternal Scwharzschild black holes are vacuum solutions, and that eternal black holes are not the blackholes which result from stellar collapse.

Also, for eternal black holes, I don't think I'd say that curvature is the source of curvature. I prefer to talk in the imprecise (and possibly wrong) language that I used in post #13.

In this case of a charged, eternal black hole, the stress-energy tensor is not zero - it has the usual form for an electromagetic field. In fact, E_r = Q/r^2, so, again speaking very loosely, it look like the field for charge "residing" at the singularity r = 0. Again, r = 0 is not actually part of the spacetime manifold. Although the black hole has a charge Q, no actual "physical charges" exist anywhere in the spacetime manifold, either inside or outside the horizons.

Last edited by a moderator: Apr 22, 2017
19. Nov 25, 2006

swimmingtoday

<<Take the field equations in vacuum, with no mass whatsoever. Then, try to find a stationary solution of the nonlinear field equations. One could image some kind of peculiar curvature of spacetime (for instance with spherical symmetry) in which this curvature is in fact the source of its own curvature. >>

You seem to be using "curvature" interchangably with "non-linear terms". That is not correct.

In GR, a curvature quantity is set equal to a quantity which involves the mass-density. The "curvature quantity" is not exactly the curvature--we will get to that later--, and the mass-density quantity is a tensor with 16 components, the most prominent one under normal conditions being the mass density.

The curavature quantity contains a non-linear part, which is small under normal conditions (e.g. conditions found on Earth and in the Solar System) So the equations of GR are of a form

mass density tensor = linear portion of curvature quantity plus non-linear portion of curvature quantity.

This is numerically equivalent to

Linear portion of curvature quantity= mass density tensor + non-limear terms from curvature quantity.

In this ladder form, numerically, the behavior is as if both the mass density and a non-linear thing are drivers of the gravitational field. In a sense the non-linear terms are like the energy (actually "stress-energy" would be more accurate, but don't worry about that) of the gravitational field; and the source of gravitation is thus the mass of the particle plus the mass equaivalent (via E = m (c squared)) of the field.

But the curvature quantity is not acting as a source of curvature, rather it is the non-linear terms in the curvarture quantity that are acting like a source for the linear portion of the curvature quantity.

As for the other part of what yoiu were saying, you can indeed have curvature without mass. To see this you need to understand some of the geometry of GR. There is a quantity known as the curvarture tensor. It is a 4th order tensor--it is of the form Rabcd, where a,b,c,and d are subscripts. From it you can construct another tensor known as the contraction of the curvature tensor. It is formed from the curvature tensor as follows. Rac= Ra1c1 + Ra2c2 + Ra3c3 + Ra4c4. (Some of the plus signs are minus signs, but I'm trying to keep this simple). In a region where the mass density tensor is zero, the contracted Rieman tensor is zero. This comes very directly from the equations of GR. But it is quite possible to chave the contraction of a tensor be zero without the original tensor being zero.

20. Nov 25, 2006

swimmingtoday

<<Now, I'm really strongly confused . If the stress-energy tensor is zero everywhere, what is then causing the curvature >>

The stress-energy is not proportional to the curvation. It is proportional to a quantity involving the contraction of the curvature (and the contraction of the contraction). The lack of stress-energy at a point implies the CONTRACTED curvature tensor is zero. It does not imply that the curvature tensor is zero.

Indeed, in the vacuum region of the Schwarzschild solution the curvature tensor is non-zero. In general, it turns out that a region with tidal forces (e.g, the derivative in the x direction of the x component of the gravitational force) will have a non-vanishing curvature tensor. It's doubtful you can have curvature in a spot without a tidal force, but quite possible to have curvature at a point with no matter.

21. Nov 25, 2006

notknowing

Thank you swimmingtoday for this explanation. I'll need some time to digest it though ...

22. Nov 25, 2006

pervect

Staff Emeritus
Einstein's equation says:

G_uv = 8 pi T_uv, which can also be written as

R_uv = 8 pi T_uv - 4 pi T g_ab

which gives the Ricci tensor R_uv directly as an equation from the stress-energy tensor T_uv.

The important thing to note is that R_uv = 0 does not imply no curvature. The general curvature tensor is the Riemann curvature tensor, R_abcd, not the Ricci tensor, which is a contraction of the Riemann.

The significance of this is that the Riemann curvature can be non-zero even when the Ricci tensor is zero. To expand a bit on robphy's rermarks, the Riemann curvature tensor can be decomposed into the Ricci tensor, and the Weyl tensor. If you have a zero stress-energy tensor, the Ricci will be zero, but the Weyl does not have to be zero, which implies that the Riemann tensor also may be nonzero.

23. Nov 25, 2006

swimmingtoday

<<If the stress-energy tensor is zero everywhere, what is then causing the curvature ??...The Weyl tensor>>

I doubt this is true. Suppose that at some point, R1010= -(R2020 + R3030), with all three of these curvature tensor quantities non-zero. And suppose that the Weyl tensor is zero. Hopefully these conditions are self-consistant. If they are, then in this case, R00 will be zero, yet we have 3 components of the Riemann tensor which are non-zero, and so the Riemann Tensor does not vanish, depite both the Ricci Tensor and the Weyl tensor vanishing. (BTW, I suspect that the hypothetical situation I constructed is the actual situation for the vacuum region of the Schwarzschild Solution.)

24. Nov 25, 2006

robphy

For concreteness, here is the Weyl tensor.
$$C_{abc}{}^d= R_{abc}{}^d - \frac{2!}{n-2}g^{de}\left(\left(g_{a[c}R_{e]b}- g_{b[c}R_{e]a}\right) - \frac{1}{n-1}Rg_{a[c}g_{e]b}\right)$$
Weyl is the traceless part of the Riemann tensor.

Does your proposed Riemann tensor (with its claimed vanishing Weyl and Ricci [and therefore scalar-curvature] satisfy this?

(Does your proposed Riemann tensor satisfy the Bianchi identities?)

25. Nov 25, 2006

swimmingtoday

<<Does your proposed Riemann tensor (with its claimed vanishing Weyl and Ricci [and therefore scalar-curvature] satisfy this?>>

Obviously it does, because I chose only 3 components of the Riemann tensor to be non-zero (actually by symetry conditions some other ones will be non-zero too) and those quantities (and the other quantities implied by the symetry conditions) are not part of the traceless part.

<<(Does your proposed Riemann tensor satisfy the Bianchi identities?)>>

Obviously, because I just specified the Riemann tensor at a single point. The Bianchi requirements deal with *derivatives* of the Riemann components. Had I tried to specify a Riemann field over a region larger than a point or had I tried to specify derivatives of the Riemann tensor at that point, then I would have had to be concerned about Bianchi requirements. For example, if I specify an electric field vector and a magnetic field vector at a single point, I do not need to worry about Maxwell's Equations being satisfied, but if I specified them over a finite region, or if I said something about the derivatives of the electric and magnetic field at that point, I would need to be concerned that my arbitrary specification was not one allowed by Maxwell's Equations.