Exact solutions in general relativity: Difference between revisions

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article may be confusing or unclear to readers. Please help clarify the article. There might be a discussion about this on the talk page. (March 2007) (Learn how and when to remove this message) This article contains instructions, advice, or how-to content. Please help rewrite the content so that it is more encyclopedic or move it to Wikiversity, Wikibooks, or Wikivoyage. (March 2017) (Learn how and when to remove this message)

This article needs attention from an expert on the subject. Please add a reason or a talk parameter to this template to explain the issue with the article. When placing this tag, consider associating this request with a WikiProject. (March 2017)

General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
Introduction History Timeline Tests Mathematical formulation
Fundamental concepts Equivalence principle Special relativity World line Pseudo-Riemannian manifold
Phenomena Kepler problem Gravitational lensing Gravitational redshift Gravitational time dilation Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole Spacetime Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge
Equations Formalisms Equations Linearized gravity Einstein field equations Friedmann Geodesics Mathisson–Papapetrou–Dixon Hamilton–Jacobi–Einstein Formalisms ADM BSSN Post-Newtonian Advanced theory Kaluza–Klein theory Quantum gravity
Solutions Schwarzschild (interior) Reissner–Nordström Einstein–Rosen waves Wormhole Gödel Kerr Kerr–Newman Kerr–Newman–de Sitter Kasner Lemaître–Tolman Taub–NUT Milne Robertson–Walker Oppenheimer–Snyder pp-wave van Stockum dust Weyl−Lewis−Papapetrou Hartle–Thorne
Scientists Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedman Milne Zwicky Lemaître Oppenheimer Gödel Wheeler Robertson Bardeen Walker Kerr Chandrasekhar Ehlers Penrose Hawking Raychaudhuri Taylor Hulse van Stockum Taub Newman Yau Thorne others
Physics portal  Category
v t e

In

.

Background and definition

These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy

${\displaystyle T^{\alpha \beta }}$.[1] (A field is described by a Lagrangian, varying with respect to the field should give the field equations and varying with respect to the metric should give the stress-energy contribution due to the field.)

Finally, when all the contributions to the stress–energy tensor are added up, the result must be a

= 1)

${\displaystyle G^{\alpha \beta }=8\pi \,T^{\alpha \beta }.}$

In the above field equations, ${\displaystyle G^{\alpha \beta }}$ is the

, which contain no matter or nongravitational fields.

Difficulties with the definition

Any Lorentzian manifold is a solution of the

Einstein field equation

for some right hand side. This is illustrated by the following procedure:

• take any
  Lorentzian manifold, compute its Einstein tensor
  ${\displaystyle G^{\alpha \beta }}$, which is a purely mathematical operation
• divide by ${\displaystyle 8\pi }$
• declare the resulting symmetric second rank tensor field to be the stress–energy tensor ${\displaystyle T^{\alpha \beta }}$.

This shows that there are two complementary ways to use general relativity:

• One can fix the form of the stress–energy tensor (from some physical reasons, say) and study the solutions of the Einstein equations with such right hand side (for example, if the stress–energy tensor is chosen to be that of the perfect fluid, a spherically symmetric solution can serve as a stellar model)
• Alternatively, one can fix some geometrical properties of a spacetime and look for a matter source that could provide these properties. This is what cosmologists have done since the 2000s: they assume that the Universe is homogeneous, isotropic, and accelerating and try to realize what matter (called dark energy) can support such a structure.

Within the first approach the alleged stress–energy tensor must arise in the standard way from a "reasonable" matter distribution or nongravitational field. In practice, this notion is pretty clear, especially if you restrict the admissible nongravitational fields to the only one known in 1916, the

.

Einstein also recognized another element of the definition of an exact solution: it should be a Lorentzian manifold (meeting additional criteria), i.e. a

. Once again, the creative tension between elegance and convenience, respectively, has proven difficult to resolve satisfactorily.

In addition to such

or structures with points of separation ("trouser worlds"). Some of the best known exact solutions, in fact, have globally a strange character.

Types of exact solution

Many well-known exact solutions belong to one of several types, depending upon the intended physical interpretation of the stress–energy tensor:

• Vacuum solutions: ${\displaystyle T^{\alpha \beta }=0}$; these describe regions in which no matter or nongravitational fields are present,
• Electrovacuum solutions: ${\displaystyle T^{\alpha \beta }}$ must arise entirely from an
  Maxwell equations
  on the given curved Lorentzian manifold; this means that the only source for the gravitational field is the field energy (and momentum) of the electromagnetic field,
• Null dust solutions: ${\displaystyle T^{\alpha \beta }}$ must correspond to a stress–energy tensor which can be interpreted as arising from incoherent electromagnetic radiation, without necessarily solving the Maxwell field equations on the given Lorentzian manifold,
• Fluid solutions: ${\displaystyle T^{\alpha \beta }}$ must arise entirely from the stress–energy tensor of a fluid (often taken to be a perfect fluid); the only source for the gravitational field is the energy, momentum, and stress (pressure and shear stress) of the matter comprising the fluid.

In addition to such well established phenomena as fluids or electromagnetic waves, one can contemplate models in which the gravitational field is produced entirely by the field energy of various exotic hypothetical fields:

• Scalar field solutions: ${\displaystyle T^{\alpha \beta }}$ must arise entirely from a scalar field (often a massless scalar field); these can arise in classical field theory treatments of meson beams, or as quintessence,
• Lambdavacuum solutions (not a standard term, but a standard concept for which no name yet exists): ${\displaystyle T^{\alpha \beta }}$ arises entirely from a nonzero cosmological constant.

One possibility which has received little attention (perhaps because the mathematics is so challenging) is the problem of modeling an elastic solid. Presently, it seems that no exact solutions for this specific type are known.

Below we have sketched a classification by physical interpretation. This is probably more useful for most readers than the

, but for completeness we note the following facts:

• nonnull electrovacuums have Segre type ${\displaystyle \{\,(1,1)(11)\}}$ and
  isotropy group
  SO(1,1) x SO(2),
• null electrovacuums and null dusts have Segre type ${\displaystyle \{\,(2,11)\}}$ and isotropy group E(2),
• perfect fluids have Segre type ${\displaystyle \{\,1,(111)\}}$ and isotropy group SO(3),
• Lambdavacuums have Segre type ${\displaystyle \{\,(1,111)\}}$ and isotropy group SO(1,3).

The remaining Segre types have no particular physical interpretation and most of them cannot correspond to any known type of contribution to the stress–energy tensor.

Examples

Noteworthy examples of vacuum solutions, electrovacuum solutions, and so forth, are listed in specialized articles (see below). These solutions contain at most one contribution to the

, due to a specific kind of matter or field. However, there are some notable exact solutions which contain two or three contributions, including:

• NUT-Kerr–Newman–de Sitter solution contains contributions from an electromagnetic field and a positive vacuum energy, as well as a kind of vacuum perturbation of the Kerr vacuum which is specified by the so-called NUT parameter,
• Gödel dust contains contributions from a pressureless perfect fluid (dust) and from a positive vacuum energy.

Some hypothetical possibilities which don't fit into our rough classification^{[clarification needed]} are:

• certain
  which?
  ],
• Alcubierre metric
  .
• "Time machines", i.e. initially nice spacetimes^{[clarification needed]} in which at some stage of evolution closed causal curves appear^{[clarification needed]}.

Some doubt has been cast^{[according to whom?]} upon whether sufficient quantity of exotic matter needed for wormholes and Alcubierre bubbles can exist.^[2] Later, however, these doubts were shown^[3] to be mostly groundless. The third of these examples, in particular, is an instructive example of the procedure mentioned above for turning any Lorentzian manifold into a "solution". It is along this way that Hawking succeeded in proving^[4] that time machines of a certain type (those with a "compactly generated Cauchy horizon") cannot appear without exotic matter. Such spacetimes are also a good illustration of the fact that unless a spacetime is especially nice ("globally hyperbolic") the Einstein equations do not determine its evolution uniquely. Any spacetime may evolve into a time machine, but it never has to do so.^[5]

Constructing solutions

The Einstein field equations are a system of coupled,

]. Nonetheless, several effective techniques for obtaining exact solutions have been established.

The simplest involves imposing symmetry conditions on the

Schwarzschild vacuum

).

This naive approach usually works best if one uses a frame field rather than a coordinate basis.

A related idea involves imposing algebraic symmetry conditions on the

of the possible symmetries of the Ricci tensor. As will be apparent from the discussion above, such Ansätze often do have some physical content, although this might not be apparent from their mathematical form.

This second kind of symmetry approach has often been used with the Newman–Penrose formalism, which uses spinorial quantities for more efficient bookkeeping.

Even after such symmetry reductions, the reduced system of equations is often difficult to solve. For example, the Ernst equation is a nonlinear partial differential equation somewhat resembling the nonlinear Schrödinger equation (NLS).

But recall that the

of a differential equation (or system of equations), and as Lie showed, this can provide an avenue of attack upon any differential equation which has a nontrivial symmetry group. Indeed, both the Ernst equation and the NLS have nontrivial symmetry groups, and some solutions can be found by taking advantage of their symmetries. These symmetry groups are often infinite dimensional, but this is not always a useful feature.

There are also various transformations (see

equations. This is no coincidence, since this phenomenon is also related to the notions of Noether and Lie regarding symmetry. Unfortunately, even when applied to a "well understood", globally admissible solution, these transformations often yield a solution which is poorly understood and their general interpretation is still unknown.

Existence of solutions

Given the difficulty of constructing explicit small families of solutions, much less presenting something like a "general" solution to the Einstein field equation, or even a "general" solution to the vacuum field equation, a very reasonable approach is to try to find qualitative properties which hold for all solutions, or at least for all vacuum solutions. One of the most basic questions one can ask is: do solutions exist, and if so, how many?

To get started, we should adopt a suitable initial value formulation of the field equation, which gives two new systems of equations, one giving a constraint on the initial data, and the other giving a procedure for evolving this initial data into a solution. Then, one can prove that solutions exist at least locally, using ideas not terribly dissimilar from those encountered in studying other differential equations.

To get some idea of "how many" solutions we might optimistically expect, we can appeal to Einstein's constraint counting method. A typical conclusion from this style of argument is that a generic vacuum solution to the Einstein field equation can be specified by giving four arbitrary functions of three variables and six arbitrary functions of two variables. These functions specify initial data, from which a unique vacuum solution can be evolved. (In contrast, the Ernst vacuums, the family of all stationary axisymmetric vacuum solutions, are specified by giving just two functions of two variables, which are not even arbitrary, but must satisfy a system of two coupled nonlinear partial differential equations. This may give some idea of how just tiny a typical "large" family of exact solutions really is, in the grand scheme of things.)

However, this crude analysis falls far short of the much more difficult question of global existence of solutions. The global existence results which are known so far turn out to involve another idea.

Global stability theorems

We can imagine "disturbing" the gravitational field outside some isolated massive object by "sending in some radiation from infinity". We can ask: what happens as the incoming radiation interacts with the ambient field? In the approach of classical

. However, perturbation expansions are generally not reliable for questions of long-term existence and stability, in the case of nonlinear equations.

The full field equation is highly nonlinear, so we really want to prove that the Minkowski vacuum is stable under small perturbations which are treated using the fully nonlinear field equation. This requires the introduction of many new ideas. The desired result, sometimes expressed by the slogan that the Minkowski vacuum is nonlinearly stable, was finally proven by Demetrios Christodoulou and Sergiu Klainerman only in 1993. Analogous results are known for lambdavac perturbations of the de Sitter lambdavacuum (Helmut Friedrich) and for electrovacuum perturbations of the Minkowski vacuum (Nina Zipser).

The positive energy theorem

Another issue we might worry about is whether the net mass-energy of an isolated concentration of positive mass-energy density (and momentum) always yields a well-defined (and non-negative) net mass. This result, known as the positive energy theorem was finally proven by Richard Schoen and Shing-Tung Yau in 1979, who made an additional technical assumption about the nature of the stress–energy tensor. The original proof is very difficult; Edward Witten soon presented a much shorter "physicist's proof", which has been justified by mathematicians—using further very difficult arguments. Roger Penrose and others have also offered alternative arguments for variants of the original positive energy theorem.

See also

• Friedmann–Lemaître–Robertson–Walker metric
• Petrov classification, for algebraic symmetries of the Weyl tensor

References

Further reading

• Krasiński, A. (1997). Inhomogeneous Cosmological Models. Cambridge: Cambridge University Press.
  ISBN 0-521-48180-5
  .
• MacCallum, M. A. H. (2006). "AIP Conference Proceedings". AIP Conf. Proc. 841: 129–143.
  doi:10.1063/1.2218172. {{cite journal}}: |chapter= ignored (help
  ) An up-to-date review article, but too brief, compared to the review articles by Bičák or Bonnor et al. (see below).
• Exact Solutions of Einstein's equations Malcolm A.H. MacCallum Scholarpedia, 8(12):8584. doi:10.4249/scholarpedia.8584
• Rendall, Alan M. "Local and Global Existence Theorems for the Einstein Equations". Living Reviews in Relativity. Retrieved August 11, 2005. A thorough and up-to-date review article.
• Friedrich, Helmut (2005). "Is general relativity 'essentially understood' ?". Annalen der Physik. 15: 84–108.
  doi:10.1002/andp.200510173
  . An excellent and more concise review.
• Bičák, Jiří (2000). "Selected exact solutions of Einstein's field equations: their role in general relativity and astrophysics". Lect. Notes Phys. Lecture Notes in Physics. 540: 1–126.
  ISBN 978-3-540-67073-5
  . An excellent modern survey.
• Bonnor, W. B.; Griffiths, J. B.; MacCallum, M. A. H. (1994). "Physical interpretation of vacuum solutions of Einstein's equations. Part II. Time-dependent solutions". Gen. Rel. Grav. 26 (7): 637–729.
  doi:10.1007/BF02116958
  .
• Bonnor, W. B. (1992). "Physical interpretation of vacuum solutions of Einstein's equations. Part I. Time-independent solutions". Gen. Rel. Grav. 24 (5): 551–573.
  doi:10.1007/BF00760137
  . A wise review, first of two parts.
• Griffiths, J. B. (1991). Colliding Plane Waves in General Relativity. Oxford:
  ISBN 0-19-853209-1. The definitive resource on colliding plane waves, but also useful to anyone interested in other exact solutions. available online by the author
• Hoenselaers, C.; Dietz, W. (1985). Solutions of Einstein's Equations: Techniques and Results. New York: Springer.
  ISBN 3-540-13366-6
  .
• Ehlers, Jürgen; Kundt, Wolfgang (1962). "Exact solutions of the gravitational field equations". In Witten, L. (ed.). Gravitation: An Introduction to Current Research. New York: Wiley. pp. 49–101. {{cite conference}}: Unknown parameter |booktitle= ignored (|book-title= suggested) (help) A classic survey, including important original work such as the symmetry classification of vacuum pp-wave spacetimes.
• Stephani, Hans; Dietrich Kramer; Malcolm MacCallum; Cornelius Hoenselaers; Eduard Herlt (2009). Exact Solutions of Einstein's Field Equations (2nd ed.). Cambridge: Cambridge University Press.
  ISBN 978-0-521-46702-5
  .

External links