Category talk:Exact solutions in general relativity

Work in progress!

Hi all, I created this category and I plan to heavily populate it! As I create more articles I will gradually rearrange things. Eventually, I plan to have articles discussing various coordinate charts and the physical experiences of various observers in the most important solutions (e.g. Kerr, Schwarzschild, FRW), and I plan to write individual articles on two or three dozen specific solutions and a half dozen or more important families of solutions.

How do I distinguish between "a solution" (often with multiple parameters) and a "family" of solutions? It's an informal distinction, but I use "solution" to mean a spacetime given by explicit metric functions (possibly involving several parameters, or perhaps even an arbitrary smoooth function), and I use "family" to mean a solution given in terms of some differential equation. For example, the Weyl family is "the general static axisymmetric vacuum solution", and each Weyl solution is determined by choosing a harmonic function (axisymmetric solution of 3D Laplace equation). Similarly, each Ernst vacuum is determined by choosing an "Ernst potential" (axisymmetric solution of the 3D Ernst equation, which is very similar to the Laplace equation).---CH (talk) 02:26, 1 August 2005 (UTC)

Modest plans for this category

Here is a list of articles which I have written or plan to write and add to this category:

Major families of vacuum solutions:

• the Weyl vacuums (the family of all static vacuum solutions),
• the Beck vacuums (the family of all cylindrically symmetric nonrotating vacuum solutions),
• the Ernst vacuums (the family of all stationary axisymmetric vacuum solutions),
• the Ehlers vacuums (the family of all cylindrically symmetric vacuum solutions),
• the Szekeres vacuums (the family of all colliding gravitational plane wave models),
• the Gowdy vacuums (cosmological models constructed using gravitational waves),
• the Robinson-Trautman vacuums (compact objects radiating gravitatoinal waves),
• the pp-wave vacuums (generalization of plane gravitational waves).

Several of the families mentioned here, members of which are obtained by solving an appropriate linear or nonlinear, real or complex partial differential equation, turn out to be very closely related, in perhaps surprising ways. I plan to explain this and also some of the reasons why the defining equations are interesting in a much wider context, e.g. some of them appear to be completely integrable and are thus related to the theory of solitons, and some also appear in other physical theories.

I also plan to discuss generalizations of these families to electrovacuums and/or null dusts, and to include a possibly nonzero cosmological constant.

Important individual vacuum solutions:

• Minkowski spacetime (which describes empty space with no cosmological constant)
• Schwarzschild vacuum (which describes the spacetime geometry around a spherical mass),
• Kerr vacuum (which describes the geometry around a rotating object),
• NUT vacuum (a famous counterexample describing the exterior gravitational field of an isolated object with strange properties),
• Penrose-Khan vacuum (a simple colliding plane wave model),
• Oszváth-Schücking vacuum (the circularly polarized sinusoidal gravitational wave),
• Kasner vacuum,
• C-metric.

Important individual non-null electrovacuum solutions include:

• Reissner-Nordström electrovacuum (which describes the geometry around a charged spherical mass),
• Kerr-Newman electrovacuum (which describes the geometry around a charged, rotating object),
• Melvin electrovacuum (a model of a cylindrically symmetric magnetostatic field),
• Bertotti-Robinson electrovacuum (a simple spacetime having a remarkable product structure, which arises from a kind of limiting procedure applied to the RN electrovacuum).

Important null electrovacuum solutions include:

• electromagnetic plane wave,
• Bell-Szekeres electrovacuum (a colliding plane wave model).

Null dust solutions include:

• pp-wave spacetimes (which generalize the gravitational plane waves and electromagnetic plane waves),
• Robinson-Trautmann spacetimes (which model radiation expanding from a radiating object),
• Kinnersley-Walker photon rocket (special case of RT null dusts),
• Vaidya null dust (special case of KW null dusts).

Fluid solutions include:

• the static spherically symmetric perfect fluid models (the simplest models of stars are constructed by matching a fluid in the interior region to a Schwarzschild vacuum exterior region),
• the Neugebauer-Meinel disk (which models a rotating disk of dust matched to an axisymmetric vacuum exterior; this is perhaps the most remarkable exact solution discovered since the Kerr vacuum),
• FRW models (which provide the most basic cosmological models in general relativity),
• Kasner dusts (the simplest cosmological model exhibiting anisotropic expansion),
• Bianchi dust models (generalizations of FRW and Kasner models, exhibiting various types of Lie algebras of Killing vector fields,
• LTB dusts (some of the simplest inhomogeneous cosmological models),
• Kantowski-Sachs dusts (cosmological models which exhibit perturbations from FRW models),
• van Stockum dust (a cylindrically symmetric rotating dust),
• Kasner dust (a useful cosmological model).

Scalar field solutions include various quintessence solutions and

• the Janis/Newman/Winacour solution (the static spherically symmetric massless scalar field).

Important "Lambdavac solutions" (vacuum with nonzero cosmological constant) include:

• de Sitter "vacuum" or dS model,
• anti-de Sitter "vacuum" or AdS model,
• Nariai "vacuum" (the only other solution with a product structure is Bertotti-Robinson),
• Gödel "dust" (a rotating cosmological model and a famous counterexample which is locally unobjectionable but globally suspect).

I plan to link some of these articles to appropriate articles in the category "Coordinate charts in general relativity":

Charts in the Schwarzchild vacuum and its generalizations (e.g. Vaidya/Reissner/Nordstrom/de Sitter):

• Schwarzschild (static polar spherical),
• Eddington (ingoing and outgoing),
• Painleve,
• LeMaître,
• Kruskal-Szekeres (using a special function useful in many contexts, the Lambert W function),
• Penrose (an explicit chart exhibiting the conformal compactification),
• "isotropic" (spatially conformal to E³),
• Costa (spatially conformal to R x S²).

Charts in the Kerr family (e.g. Kerr/Newman/NUT/de Sitter), using the following charts:

• Eddington (ingoing and outgoing),
• Boyer-Lindquist,
• Kerr-Schild,
• Doran (generalization of Painleve).

I plan an extensive discussion of plane wave spacetimes, using these charts:

• Brinkmann (a global chart),
• Rosen (strictly local charts, but comoving with inertial observers).

I plan to discuss charts in some three-manifolds which are also very important in this subject. In particular:

Charts in three-dimensional euclidean space E³:

• cylindrical,
• paraboloidal,
• polar spherical,
• rational and trigonometric versions of prolate spheroidal,
• ditto, for oblate spheroidal,
• a pair of Cassini charts, just to illustrate largely untapped possibilities.

Charts in three-dimensional hyperbolic space H³:

• polar spherical,
• stereographic,
• upper half space,
• cylindrical,
• horospherical.

Charts in the three-dimensional sphere S³:

• polar spherical,
• stereographic,
• cylindrical (adapted to a family of nested Hopf tori).

I also plan to create and link here (as appropriate) articles in another new category, on "Bianchi groups". These will explain Bianchi's classification of three-dimensional real Lie groups, which is important in constructing cosmological models and also in discussing symmetry groups of various solutions. I plan to give the classification and write articles on the nine individual Bianchi types, discussing their interpretation as Kleinian geometries. I plan to link this discussion to the Lorentz group article.

I also plan to discuss some interesting charts for the Minkowski vacuum, including:

• Boyer/Lindquist chart (see Kerr vacuum),
• Kinnersley/Walker chart (see photon rocket),
• Rindler (comoving with certain accelerating particles),
• Kasner.

These can be used to illustrate many concepts such as apparent horizons.

Using these, I plan an extensive discussion of the de Sitter and AdS spacetimes, using the following charts:

• standard static chart,
• static and spatially conformal to S³,
• comoving with certain observers, with S³ hyperslices,
• comoving with different observers, with H³ hyperslices,
• locally conformal to Minkowski vacuum, with E³ hyperslices,
• locally conformal to R x S³,
• conformal to Minkowski vacuum in a different way,
• Brill chart.

These are important spacetimes and these charts give valuable insight. Some of them are discussed for this reason in the monograph by Hawking and Ellis.

I also plan to deprecate some spacetimes which should not be regarded as "solutions" of the EFE:

• certain wormhole metrics (which can serve as a speculative toy model of a stargate held open by a hypothetical kind of exotic matter, as in 2001: A Space Odyssey; also a toy model of hypothetical time travel),
• Alcubierre metric (which has been used as a speculative toy model of effectively superluminal space travel, as in the warp drive from Star Trek).

Grave doubt has been cast upon whether exotic matter of the kind needed for wormholes can exist. The second of these examples, in particular, is an instructive example of the absurd procedure mentioned above for turning any Lorentzian manifold into a "solution", and there are specific reasons for gravely doubting whether any such matter can exist (the reasons come down to the fact that there appears to be no plausible physical reason for spacetime to behave in the manner required).

I might also add some more articles discussing further solutions with interesting physical interpretations, or unsolved problems, such as:

• Schwarschild/Melvin electrovacuum,
• Garfinkle/Melvin electrovacuum,
• gravitational standing wave problem.

I also plan an article on Dimension counting, which will explain how to answer the question: how many functions (of how many variables) are needed to specify a generic vacuum solution? Dust solution? (Etc.).

Some idea of the slow pace to be expected can be gathered from the fact that I have been working off-line on the proposed Robinson/Trautman article(s) for about a week.---CH (talk) 20:56, 4 August 2005 (UTC)