Exact solutions in general relativity
Encyclopedia
In general relativity
, an exact solution is a Lorentzian manifold equipped 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
. (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 tensor
unspecified (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 derivative
s 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.
, 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 equation with some right hand side. Which of course doesn't make general relativity
useless, 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 cosmologists have done 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 eigenvalues and eigenvectors of 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 curve
s or structures with points of separation ("trouser worlds"). Some of the best known exact solutions, in fact, have globally a strange character.
depending upon the intended physical interpretation of the stress-energy tensor:
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:
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 classification
of the possible algebraic symmetries of the Ricci tensor, but for completeness we note the following facts:
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.
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 classification
of the possible symmetries of the Weyl tensor, or the Segre classification
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 conformal group
on Minkowski spacetime is the symmetry group of the Maxwell equations. Recall too that solutions of the heat equation
can be found by assuming a scaling Ansatz. These notions are merely special cases of Sophus Lie
's notion of the point symmetry 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.
Emmy Noether
showed 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 transform
which 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.
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 equation
s, including some famous examples of soliton
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, or even globally objectionable.
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.
, 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 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).
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.
Some hypothetical possibilities which don't fit into our rough classification are:
Some doubt has been cast upon whether sufficient quantity of exotic matter needed for wormholes and Alcubierre bubbles can exist. Later, however, these doubts were shown 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 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.
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, an exact solution is a Lorentzian manifold equipped with certain tensor fields
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
which are taken to model states of ordinary matter, such as a fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
, or classical nongravitational fields
Classical field theory
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....
such as the electromagnetic field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
.
These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
). Following a standard recipe which is widely used in mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, these tensor fields should also give rise to specific contributions to the stress-energy tensor
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
. (To wit, whenever a field is described by a LagrangianLagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
, 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
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
(written here in geometrized units)
In the above field equations, the tensor field standing on the left hand side, the Einstein tensor
Einstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
, is computed uniquely from the metric tensor
Metric tensor (general relativity)
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
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 tensor
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...
unspecified (see the Ricci decomposition
Ricci decomposition
In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties...
), 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 derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
s 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
Take any Lorentzian manifold, compute its Einstein tensorEinstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
, which is a purely mathematical operation, divide by , and declare the resulting symmetric second rank tensor field to be the stress-energy tensorStress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
. Thus any Lorentzian manifold is a solution of the Einstein field equation with some right hand side. Which of course doesn't make general relativityGeneral relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
useless, 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
Static spherically symmetric perfect fluid
In metric theories of gravitation, particularly general relativity, a static spherically symmetric perfect fluid solution is a spacetime equipped with suitable tensor fields which models a static round ball of a fluid with isotropic pressure.Such solutions are often used as idealized models of...
). 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 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
Dark energy
In physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...
) 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
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
. 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 eigenvalues and eigenvectors of 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
Casimir effect
In quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field...
.
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
Local spacetime structure
Local spacetime structure refers to the structure of spacetime on a local level, i.e. only considering those points in an open region of a point. This notion is useful in many areas of physics, most notably in Einstein's theory of general relativity....
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 curve
Closed timelike curve
In mathematical physics, a closed timelike curve is a worldline in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point...
s 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 solutionsVacuum solution (general relativity)In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress-energy tensor also vanishes identically, so that no matter or non-gravitational fields are present.More generally, a...
:
; these describe regions in which no matter or nongravitational fields are present,
- Electrovacuum solutionElectrovacuum solutionIn general relativity, an electrovacuum solution is an exact solution of the Einstein field equation in which the only nongravitational mass-energy present is the field energy of an electromagnetic field, which must satisfy the source-free Maxwell equations appropriate to the given geometry...
s:
must arise entirely from an electromagnetic fieldElectromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
which solves the source-free 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 solutionNull dust solutionIn mathematical physics, a null dust solution is a Lorentzian manifold in which the Einstein tensor is null...
s:
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 solutionFluid solutionIn general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid....
s:
must arise entirely from the stress-energy tensor of a fluid (often taken to be a perfect fluidPerfect fluidIn physics, a perfect fluid is a fluid that can be completely characterized by its rest frame energy density ρ and isotropic pressure p....
); 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 solutionScalar field solutionIn general relativity, a scalar field solution is an exact solution of the Einstein field equation in which the gravitational field is due entirely to the field energy and momentum of a scalar field...
s:
must arise entirely from a scalar fieldScalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
(often a massless scalar field); these can arise in classical field theory treatments of mesonMesonIn particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...
beams, or as quintessenceQuintessence (physics)In physics, quintessence is a hypothetical form of dark energy postulated as an explanation of observations of an accelerating universe. It has been proposed by some physicists to be a fifth fundamental force...
,
- Lambdavacuum solutionLambdavacuum solutionIn general relativity, a lambdavacuum solution is an exact solution to the Einstein field equation in which the only term in the stress-energy tensor is a cosmological constant term...
s (not a standard term, but a standard concept for which no name yet exists):
arises entirely from a nonzero cosmological constantCosmological constantIn physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
.
One possibility which has received little attention (perhaps because the mathematics is so challenging) is the problem of modeling an elastic solid
Solid mechanics
Solid mechanics is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions . It is part of a broader study known as continuum mechanics. One of the most common practical applications of solid mechanics is the Euler-Bernoulli beam equation...
. 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 classification
Segre classification
The Segre classification is an algebraic classification of rank two symmetric tensors. The resulting types are then known as Segre types. It is most commonly applied to the energy-momentum tensor and primarily finds application in the classification of exact solutions in general relativity....
of the possible algebraic symmetries of the Ricci tensor, but for completeness we note the following facts:
- nonnull electrovacuums have Segre type
and isotropy group SO(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.
Constructing solutions
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, nonlinear partial 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
Metric tensor (general relativity)
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
, such as stationarity
Stationary spacetime
In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike....
(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
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
(as happens in the case of stationary axisymmetric vacuum solutions, which are characterized by the Ernst equation
Ernst equation
In mathematics, the Ernst equation is the non-linear partial differential equation\displaystyle \Re = ^2+^2.It is used to produce exact solutions of Einstein's equations....
) or a system of ordinary differential equations (as happens in the case of the Schwarzschild vacuum
Deriving the Schwarzschild solution
The Schwarzschild solution is one of the simplest and most useful solutions of theEinstein field equations . It describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object...
).
This naive approach usually works best if one uses a frame field
Frame fields in general relativity
In general relativity, a frame field is a set of four orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime...
rather than a coordinate basis.
A related idea involves imposing algebraic symmetry conditions on the Weyl tensor
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...
, Ricci tensor, or Riemann tensor. These are often stated in terms of the Petrov classification
Petrov classification
In differential geometry and theoretical physics, the Petrov classification describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold....
of the possible symmetries of the Weyl tensor, or the Segre classification
Segre classification
The Segre classification is an algebraic classification of rank two symmetric tensors. The resulting types are then known as Segre types. It is most commonly applied to the energy-momentum tensor and primarily finds application in the classification of exact solutions in general relativity....
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
Newman-Penrose Formalism
The Newman-Penrose Formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for General Relativity. Their notation is an effort to treat General Relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR.The NP formalism is itself a...
, 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
Nonlinear Schrödinger equation
In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state...
(NLS).
But recall that the conformal group
Conformal group
In mathematics, the conformal group is the group of transformations from a space to itself that preserve all angles within the space. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important:*...
on Minkowski spacetime is the symmetry group of the Maxwell equations. Recall too that solutions of the heat equation
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
can be found by assuming a scaling Ansatz. These notions are merely special cases of Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
's notion of the point symmetry 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.
Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
showed 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 transform
Inverse scattering transform
In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years...
which was originally developed to solve the Korteweg-de Vries (KdV) equation
Korteweg–de Vries equation
In mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...
, 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.
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 equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s, including some famous examples of soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
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, 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 counting
Constraint counting
In mathematics, constraint counting is a crude but often useful way of counting the number of free functions needed to specify a solution to a partial differential equation....
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 perturbation theoryPerturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
, 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
Binary pulsar
A binary pulsar is a pulsar with a binary companion, often a white dwarf or neutron star. Binary pulsars are one of the few objects which allow physicists to test general relativity in the case of a strong gravitational field...
. 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
Demetrios Christodoulou
Demetrios Christodoulou is a Greek mathematician and physicist, who first became well known for his proof, together with Sergiu Klainerman, of the nonlinear stability of the Minkowski spacetime...
and Sergiu Klainerman
Sergiu Klainerman
Sergiu Klainerman is a mathematician known for his contributions to the study of hyperbolic differential equations and general relativity. He is a Professor in the Department of Mathematics at Princeton University, a position he held since 1987. From 1980 to 1987 he was a faculty member at New...
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 was finally proven by Richard SchoenRichard Schoen
Richard Melvin Schoen is an American mathematician. Born in Fort Recovery, Ohio, he received his PhD in 1977 from Stanford University where he is currently the Anne T. and Robert M. Bass Professor of Humanities and Sciences...
and Shing-Tung Yau
Shing-Tung Yau
Shing-Tung Yau is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province....
in 1979, who made an additional technical assumption about the nature of the stress-energy tensor.
The original proof is very difficult; Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
soon presented a much shorter "physicist's proof", which has been justified by mathematicians—using further very difficult arguments. Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
and others have also offered alternative arguments for variants of the original positive energy theorem.
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 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:- 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 dustGödel metricThe Gödel metric is an exact solution of the Einstein field equations in which the stress-energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant...
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:
- certain wormhole metrics (which can serve as a speculative toy modelToy modelIn physics, a toy model is a simplified set of objects and equations relating them that can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory....
of a stargateStargateStargate is a adventure military science fiction franchise, initially conceived by Roland Emmerich and Dean Devlin. The first film in the franchise was simply titled Stargate. It was originally released on October 28, 1994, by Metro-Goldwyn-Mayer and Carolco, and became a hit, grossing nearly...
held open by a hypothetical kind of exotic matterExotic matterIn physics, exotic matter is a term which refers to matter which would somehow deviate from the norm and have "exotic" properties. There are several uses of the term....
, as in 2001: A Space Odyssey2001: A Space Odyssey (novel)2001: A Space Odyssey is a science fiction novel by Arthur C. Clarke. It was developed concurrently with Stanley Kubrick's film version and published after the release of the film...
; also a toy model of hypothetical time machineTime travelTime travel is the concept of moving between different points in time in a manner analogous to moving between different points in space. Time travel could hypothetically involve moving backward in time to a moment earlier than the starting point, or forward to the future of that point without the...
, see below), - Alcubierre metric (which has been used as a speculative toy model of effectively superluminal space travel, as in the warp driveWarp drive (Star Trek)Warp drive is a faster-than-light propulsion system in the setting of many science fiction works, most notably Star Trek. A spacecraft equipped with a warp drive may travel at velocities greater than that of light by many orders of magnitude, while circumventing the relativistic problem of time...
from Star TrekStar TrekStar Trek is an American science fiction entertainment franchise created by Gene Roddenberry. The core of Star Trek is its six television series: The Original Series, The Animated Series, The Next Generation, Deep Space Nine, Voyager, and Enterprise...
). - "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. Later, however, these doubts were shown 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 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.
See also
- Electrovacuum solutionElectrovacuum solutionIn general relativity, an electrovacuum solution is an exact solution of the Einstein field equation in which the only nongravitational mass-energy present is the field energy of an electromagnetic field, which must satisfy the source-free Maxwell equations appropriate to the given geometry...
- Fluid solutionFluid solutionIn general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid....
- Friedmann-Lemaître-Robertson-Walker metric
- Lambdavacuum solutionLambdavacuum solutionIn general relativity, a lambdavacuum solution is an exact solution to the Einstein field equation in which the only term in the stress-energy tensor is a cosmological constant term...
- Null dust solutionNull dust solutionIn mathematical physics, a null dust solution is a Lorentzian manifold in which the Einstein tensor is null...
- Petrov classificationPetrov classificationIn differential geometry and theoretical physics, the Petrov classification describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold....
, for algebraic symmetries of the Weyl tensorWeyl tensorIn differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic... - Scalar field solutionScalar field solutionIn general relativity, a scalar field solution is an exact solution of the Einstein field equation in which the gravitational field is due entirely to the field energy and momentum of a scalar field...
- Solutions of the Einstein field equationsSolutions of the Einstein field equationsWhere appropriate, this article will use the abstract index notation.Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations of general relativity. Solving the field equations actually gives Lorentz metrics...
- Vacuum solutionVacuum solution (general relativity)In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress-energy tensor also vanishes identically, so that no matter or non-gravitational fields are present.More generally, a...