- Exact solutions in general relativity
general relativity, an exact solution is a Lorentzian manifoldequipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field. These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations). Following a standard recipe which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress-energy tensor. [cite book | author=Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; & Herlt, E. | title=Exact Solutions of Einstein's Field Equations (2nd edn.) | location=Cambridge | publisher=Cambridge University Press | year=2003 | id=ISBN 0-521-46136-7 The definitive resource for exact solutions in general.] (To wit, whenever 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 satisfy the
Einstein field equations(written here in geometrized units)
In the above field equations, the tensor field standing on the left hand side, the
Einstein tensor, is computed uniquely from the metric tensor which is part of the definition of a Lorentzian manifold. Since giving the Einstein tensor does not fully determine the Riemann tensor, but leaves the Weyl tensorunspecified (see the Ricci decomposition), the Einstein equation may be considered a kind of compatibility condition: the spacetime geometry must be consistent with the amount and motion of any matter or nongravitational fields, in the sense that the immediate presence "here and now" of nongravitational energy-momentum causes a proportional amount of Ricci curvature "here and now". Moreover, taking covariant derivatives of the field equations and applying the Bianchi identities, it is found that a suitably varying amount/motion of nongravitational energy-momentum can cause ripples in curvature to propagate as gravitational radiation, even across "vacuum regions", which contain no matter or nongravitational fields.
Difficulties with the definition
Lorentzian manifold, compute its Einstein tensor, which is a purely mathematical operation, divide by , and declare the resulting symmetric second rank tensor field to be the stress-energy tensor. Thus "any" Lorentzian manifold is a solution of the Einstein field equationwith "some" right hand side. Which of course doesn't make general relativityuseless, but only shows that there are two complementary ways to use it. 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 most of cosmologists do for the last 5-10 years: they assume that the Universe is homogenous, 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
electromagnetic field. But ideally we would like to have some "mathematical characterization" that states some purely mathematical test which we can apply to any putative "stress-energy tensor", which passes everything which might arise from a "reasonable" physical scenario, and rejects everything else. Unfortunately, no such characterization is known. Instead, we have crude tests known as the energy conditions, which are similar to placing restrictions on the eigenvaluesand eigenvectorsof a linear operator. But these conditions, it seems, can satisfy no-one. On the one hand, they are far too permissive: they would admit "solutions" which almost no-one believes are physically reasonable. On the other, they may be far too restrictive: the most popular energy conditions are apparently violated by the Casimir effect.
Einstein also recognized another element of the definition of an exact solution: it should be a Lorentzian manifold (meeting additional criteria), i.e. a
smooth manifold. But in working with general relativity, it turns out to be very useful to admit solutions which are not everywhere smooth; examples include many solutions created by matching a perfect fluid interior solution to a vacuum exterior solution, and impulsive plane waves. Once again, the creative tension between elegance and convenience, respectively, has proven difficult to resolve satisfactorily.
In addition to such local objections, we have the far more challenging problem that there are very many exact solutions which are locally unobjectionable, but globally exhibit causally suspect features such as
closed timelike curves. Some of the best known exact solutions, in fact, have this 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: ; these describe regions in which no matter or nongravitational fields are present,
electrovacuum solutions: must arise entirely from an electromagnetic fieldwhich solves the "source-free" Maxwell equationson 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: 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: 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 mass, momentum, and 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: must arise entirely from a scalar field(often a massless scalar field); these can arise in classical field theory treatments of mesonbeams, or as quintessence,
Lambdavacuum solutions (not a standard term, but a standard concept for which no name yet exists): 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
Segre classificationof the possible algebraic symmetries of the Ricci tensor, but for completeness we note the following facts:
* nonnull electrovacuums have Segre type and
isotropy groupSO(1,1) x SO(2),
* null electrovacuums and null dusts have Segre type and isotropy group E(2),
* perfect fluids have Segre type and isotropy group SO(3),
* Lambdavacuums have Segre type 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.
The Einstein field equation, when fully written out as a system of partial differential equations, takes the form of a rather complicated system of coupled,
nonlinearpartial differential equations. As such, in general, it is very hard to solve.
Nonetheless, several effective techniques for obtaining exact solutions are available.
The simplest involves imposing symmetry conditions on the metric tensor, such as stationarity (symmetry under
time translation) or axisymmetry (symmetry under rotation about some symmetry axis). With sufficiently clever assumptions of this sort, it is often possible to reduce the Einstein field equation to a much simpler system of equations, even a single partial differential equation(as happens in the case of "stationary axisymmetric vacuum solutions", which are characterized by the Ernst equation) or a system of "ordinary" differential equations (as happens in the case of 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
Weyl tensor, Ricci tensor, or Riemann tensor. These are often stated in terms of the Petrov classificationof the possible symmetries of the Weyl tensor, or the Segre classificationof 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
conformal groupon Minkowski spacetimeis the symmetry group of the Maxwell equations. Recall too that solutions of the heat equationcan be found by assuming a scaling "Ansatz". These notions are merely special cases of Sophus Lie's notion of the point symmetryof 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. Emmy Noethershowed that a slight but profound generalization of Lie's notion of symmetry can result in an even more powerful method of attack. This turns out to be closely related to the discovery that some equations, which are said to be completely integrable, enjoy an "infinite sequence of conservation laws". Quite remarkably, both the Ernst equation (which arises several ways in the studies of exact solutions) and the NLS turn out to be completely integrable. They are therefore susceptible to solution by techniques resembling the inverse scattering transformwhich was originally developed to solve the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which arises in the theory of solitons, and which is also completely integrable. Unfortunately, the solutions obtained by these methods are often not as nice as one would like. For example, in a manner analogous to the way that one obtains a multiple soliton solution of the KdV from the single soliton solution (which can be found from Lie's notion of point symmetry), one can obtain a multiple Kerr object solution, but unfortunately, this has some features which make it physically implausible. [cite book | author=Belinski, V.; & Verdaguer, E. | title=Gravitational solitons | location=Cambridge | publisher=Cambridge University Press | year=2001 | id=ISBN 0-521-80586-4 A monograph on the use of soliton methods to produce stationary axisymmetric vacuum solutions, colliding gravitational plane waves, and so forth.]
There are also various transformations which can transform (for example) a vacuum solution found by other means into a new vacuum solution, or into an electrovacuum solution, or a fluid solution. These are analogous to the
Bäcklund transformations known from the theory of certain partial differential equations, including some famous examples of solitonequations. 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, or even globally objectionable.
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 countingmethod. 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
perturbation theory, we can start with Minkowksi vacuum (or another very simple solution, such as the de Sitter lambdavacuum), introduce very small metric perturbations, and retain only terms up to some order in a suitable perturbation expansion-- somewhat like evaluating a kind of Taylor series for the geometry of our spacetime. This approach is essentially the idea behind the post-Newtonian approximations used in constructing models of a gravitating system such as a binary pulsar. 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 Christodoulouand Sergiu Klainermanonly 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 was finally proven by
Richard Schoenand Shing-Tung Yauin 1979, who made an additional technical assumption about the nature of the stress-energy tensor.
The original proof is very difficult;
Edward Wittensoon presented a much shorter "physicist's proof", which has been justified by mathematicians—using further very difficult arguments! Roger Penroseand others have also offered alternative arguments for variants of the original positive energy theorem.
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
energy-momentum tensor, due to a specific kind of matter or field. However, there are some notable exact solutions which contain two or three contributions, including:
Kerr-Newman-NUT-de Sitter solutioncontains 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 are:
wormhole metrics(which can serve as a speculative toy modelof a stargateheld open by a hypothetical kind of exotic matter, as in ; also a toy model of hypothetical time machine, see below),
Alcubierre metric(which has been used as a speculative toy model of effectively superluminalspace travel, as in the warp drivefrom Star Trek).
* "Time machines", i.e. initially nice spacetimes in which at some stage of evolution closed causal curves appear.Some doubt has been cast upon whether sufficient quantity of exotic matter needed for wormholes and Alcubierre bubbles can exist. [L. H. Ford and T. A. Roman (1996) "Quantum field theory constrains traversable wormhole geometries" "Phys. Rev." D 53 5496, See also the cite web | title=eprint version | work=arXiv | url=http://www.arxiv.org/abs/gr-qc/9510071] Later, however, these doubts were shown [S. Krasnikov (2003) "The quantum inequalities do notforbid spacetime shortcuts" "Phys. Rev." D 67 104013, See also the cite web | title=eprint version | work=arXiv | url=http://www.arxiv.org/abs/gr-qc/0507079] 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 [S. W. Hawking (1992) "Chronology protectionconjecture" "Phys. Rev." D 46 603] 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. [S. Krasnikov (2002) "No time machines in classical generalrelativity" "Class. and Quantum Grav." 19 4109, See also the cite web | title=eprint version | work=arXiv | url=http://www.arxiv.org/abs/gr-qc/0111054]
Null dust solution
Petrov classification, for algebraic symmetries of the Weyl tensor
Scalar field solution
Solutions of the Einstein field equations
*cite book | author=Krasiński, A. | title=Inhomogeneous Cosmological Models | location=Cambridge | publisher=Cambridge University Press | year=1997 | id=ISBN 0-521-48180-5
*cite web | author=MacCallum, M. A. H. | title=Finding and using exact solutions of the Einstein Equations| work=arXiv eprint server| url=http://www.arxiv.org/abs/gr-qc/0601102| accessmonthday=February 5 | accessyear=2006 An up-to-date review article, but too brief, compared to the review articles by Bičák or Bonnor et al. (see below).
*cite web | author=Rendall, Alan M. | title=Local and Global Existence Theorems for the Einstein Equations| work=Living Reviews in Relativity | url=http://www.livingreviews.org/lrr-2002-6| accessmonthday=August 11 | accessyear=2005 A thorough and up-to-date review article.
*cite web | author=Friedrich, Helmut | title=Is general relativity `essentially understood' ? | work=arXiv eprint server | url=http://www.arxiv.org/abs/gr-qc/0508016 | accessmonthday=August 11 | accessyear=2005 An excellent and more concise review.
*cite journal | author=Bičák, Jiří | title=Selected exact solutions of Einstein's field equations: their role in general relativity and astrophysics | journal=Lect. Notes Phys. | year=2000 | volume=540 | pages=1–126 | doi=10.1007/3-540-46580-4_1 See also the cite web | title=eprint version | work=arXiv | url=http://www.arxiv.org/abs/gr-qc/0004031| accessmonthday=June 23 | accessyear=2005 An excellent modern survey.
*cite journal | author=Bonnor, W. B.; Griffiths, J. B.; & MacCallum, M. A. H. | title=Physical interpretation of vacuum solutions of Einstein's equations. Part II. Time-dependent solutions | journal=Gen. Rel. Grav. | year=1994 | volume=26 | pages=637–729 | doi=10.1007/BF02116958
*cite journal | author=Bonnor, W. B. | title=Physical interpretation of vacuum solutions of Einstein's equations. Part I. Time-independent solutions | journal=Gen. Rel. Grav. | year=1992 | volume=24 | pages=551–573 | doi=10.1007/BF00760137 A wise review, first of two parts.
*cite book | author=Griffiths, J. B. | title=Colliding Plane Waves in General Relativity | location=Oxford | publisher=Clarendon Press | year=1991 | id=ISBN 0-19-853209-1 The definitive resource on colliding plane waves, but also useful to anyone interested in other exact solutions. [http://www-staff.lboro.ac.uk/~majbg/jbg/book.html available online by the author]
*cite book | author=Hoenselaers, C.; & Dietz, W. | title=Solutions of Einstein's Equations: Techniques and Results| location=New York | publisher=Springer | year=1985 |id=ISBN 3-540-13366-6
*cite conference | author=Ehlers, Jürgen; & Kundt, Wolfgang | title=Exact solutions of the gravitational field equations | booktitle=Gravitation: An Introduction to Current Research | editor=Witten, L. | location=New York | publisher=Wiley | year=1962 | pages=49–101 A classic survey, including important original work such as the symmetry classification of vacuum pp-wave spacetimes.
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