Solutions of the Einstein field equations
Encyclopedia
Where appropriate, this article will use the abstract index notation
.
Solutions of the Einstein field equations are spacetime
s that result from solving the Einstein field equations
(EFE) of general relativity
. Solving the field equations actually gives Lorentz metrics. Solutions are broadly classed as exact or non-exact.
The Einstein field equations are
or more generally
where
is a constant, and the Einstein tensor
on the left side of the equation is equated to the stress-energy tensor
representing the energy and momentum present in the spacetime. The Einstein tensor is built up from the metric tensor
and its partial derivatives; thus, the EFE are a system of ten partial differential equation
s to be solved for the metric.
, which depends on the dynamics of matter and energy (such as trajectories of moving particles), which in turn depends on the gravitational field. If one is only interested in the weak field limit of the theory, the dynamics of matter can be computed using special relativity methods and/or Newtonian laws of gravity and then placing the resulting stress-energy tensor into the Einstein field equations. But if the exact solution is required or a solution describing strong fields, the evolution of the metric and the stress-energy tensor must be solved for together.
To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation
(to determine evolution of the stress-energy tensor):
This is clearly not enough, as there are only 14 equations (10 from the field equations and 4 from the continuity equation) for 20 unknowns (10 metric components and 10 stress-energy tensor components). Equations of state
are missing. In the most general case, it's easy to see that at least 6 more equations are required, possibly more if there are internal degrees of freedom (such as temperature) which may vary throughout space-time.
In practice, it is usually possible to simplify the problem by replacing the full set of equations of state with a simple approximation. Some common approximations are:
Here
is the mass-energy density measured in a momentary co-moving frame, 
is the fluid's 4-velocity vector field, and 
is the pressure.
For a perfect fluid, another equation of state relating density
and pressure 
must be added. This equation will often depend on temperature, so a heat transfer equation is required or the postulate that heat transfer can be neglected.
Next, notice that only 10 of the original 14 equations are independent, because the continuity equation
is a consequence of Einstein's equations. This reflects the fact that the system is gauge invariant and a "gauge fixing" is needed, i.e. impose 4 constraints on the system, in order to obtain unequivocal results. These constraints are known as coordinate conditions
.
A popular choice of gauge is the so-called "De Donder gauge", also known as the harmonic
condition
or harmonic gauge
In numerical relativity
, the preferred gauge is the so-called "3+1 decomposition", based on the ADM formalism
. In this decomposition, metric is written in the form
, where 
and 
can be chosen arbitrarily. The remaining physical degrees of freedom are contained in 
, which represents the Riemannian metric on 3-hypersurfaces 
.
Once equations of state are chosen and the gauge is fixed, the complete set of equations can be solved for. Unfortunately, even in the simplest case of gravitational field in the vacuum ( vanishing stress-energy tensor ), the problem turns out too complex to be exactly solvable. To get physical results, we can either turn to numerical methods
; try to find exact solutions
by imposing symmetries
; or try middle-ground approaches such as perturbation methods
or linear approximations of the Einstein tensor
.
.
From a purely mathematical viewpoint, it is interesting to know the set of solutions of the Einstein field equations. Some of these solutions are parametrised by one or more parameters.
Abstract index notation
Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical...
.
Solutions of the Einstein field equations are spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
s that result from solving 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...
(EFE) of general relativity
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...
. Solving the field equations actually gives Lorentz metrics. Solutions are broadly classed as exact or non-exact.
The Einstein field equations are
or more generally
where
is a constant, and the Einstein tensorEinstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
on the left side of the equation is equated 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...
representing the energy and momentum present in the spacetime. The Einstein tensor is built up 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...
and its partial derivatives; thus, the EFE are a system of ten 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 to be solved for the metric.
Solving the equations
It is important to realize that the Einstein field equations alone are not enough to determine the evolution of a gravitational system in many cases. They depend on 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...
, which depends on the dynamics of matter and energy (such as trajectories of moving particles), which in turn depends on the gravitational field. If one is only interested in the weak field limit of the theory, the dynamics of matter can be computed using special relativity methods and/or Newtonian laws of gravity and then placing the resulting stress-energy tensor into the Einstein field equations. But if the exact solution is required or a solution describing strong fields, the evolution of the metric and the stress-energy tensor must be solved for together.
To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation
Continuity equation
A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
(to determine evolution of the stress-energy tensor):
This is clearly not enough, as there are only 14 equations (10 from the field equations and 4 from the continuity equation) for 20 unknowns (10 metric components and 10 stress-energy tensor components). Equations of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
are missing. In the most general case, it's easy to see that at least 6 more equations are required, possibly more if there are internal degrees of freedom (such as temperature) which may vary throughout space-time.
In practice, it is usually possible to simplify the problem by replacing the full set of equations of state with a simple approximation. Some common approximations are:
- VacuumVacuum 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...
:
- Perfect fluidFluid 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....
:
where 
Here
is the mass-energy density measured in a momentary co-moving frame, is the fluid's 4-velocity vector field, and is the pressure.- Non-interacting dustDust solutionIn general relativity, a dust 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 perfect fluid which has positive mass density but vanishing pressure...
( a special case of perfect fluid ):
For a perfect fluid, another equation of state relating density
and pressure must be added. This equation will often depend on temperature, so a heat transfer equation is required or the postulate that heat transfer can be neglected.Next, notice that only 10 of the original 14 equations are independent, because the continuity equation
is a consequence of Einstein's equations. This reflects the fact that the system is gauge invariant and a "gauge fixing" is needed, i.e. impose 4 constraints on the system, in order to obtain unequivocal results. These constraints are known as coordinate conditionsCoordinate conditions
In general relativity, the laws of physics can be expressed in a generally covariant form. In other words, the real world does not care about our coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions...
.
A popular choice of gauge is the so-called "De Donder gauge", also known as the harmonic
Harmonic coordinate condition
The harmonic coordinate condition is one of several coordinate conditions in general relativity, which make it possible to solve the Einstein field equations. A coordinate system is said to satisfy the harmonic coordinate condition if each of the coordinate functions xα satisfies d'Alembert's...
condition
Coordinate conditions
In general relativity, the laws of physics can be expressed in a generally covariant form. In other words, the real world does not care about our coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions...
or harmonic gauge
In numerical relativity
Numerical relativity
Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars and many other phenomena governed by Einstein's Theory...
, the preferred gauge is the so-called "3+1 decomposition", based on the ADM formalism
ADM formalism
The ADM Formalism developed in 1959 by Richard Arnowitt, Stanley Deser and Charles W. Misner is a Hamiltonian formulation of general relativity...
. In this decomposition, metric is written in the form
, where and can be chosen arbitrarily. The remaining physical degrees of freedom are contained in , which represents the Riemannian metric on 3-hypersurfaces .Once equations of state are chosen and the gauge is fixed, the complete set of equations can be solved for. Unfortunately, even in the simplest case of gravitational field in the vacuum ( vanishing stress-energy tensor ), the problem turns out too complex to be exactly solvable. To get physical results, we can either turn to numerical methods
Numerical relativity
Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars and many other phenomena governed by Einstein's Theory...
; try to find exact solutions
Exact solutions in general relativity
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....
by imposing symmetries
Spacetime symmetries
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems, spacetime symmetries finding ample application in the study of exact solutions of Einstein's field...
; or try middle-ground approaches such as perturbation methods
Non-exact solutions in general relativity
Non-exact solutions in general relativity are solutions of Albert Einstein's field equations of general relativity which hold only approximately...
or linear approximations of 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...
.
Exact solutions
Exact solutions are Lorentz metrics that are conformable to a physically realistic stress-energy tensor and which are obtained by solving the EFE exactly in closed formClosed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...
.
Non-exact solutions
Those solutions that are not exact are called non-exact solutions. Such solutions mainly arise due to the difficulty of solving the EFE in closed form and often take the form of approximations to ideal systems. Many non-exact solutions may be devoid of physical content, but serve as useful counterexamples to theoretical conjectures.Applications
There are practical as well as theoretical reasons for studying solutions of the Einstein field equations.From a purely mathematical viewpoint, it is interesting to know the set of solutions of the Einstein field equations. Some of these solutions are parametrised by one or more parameters.