View source for Physics:Solutions of the Einstein field equations

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'''Solutions of the Einstein field equations''' are metrics of [[Physics:Spacetime|spacetime]]s that result from solving the [[Physics:Einstein field equations|Einstein field equations]] (EFE) of [[Physics:General relativity|general relativity]]. Solving the field equations gives a Lorentz manifold. Solutions are broadly classed as ''exact'' or ''non-exact''.

The Einstein field equations are

:<math>G_{\mu\nu} + \Lambda g_{\mu\nu} \, = \kappa T_{\mu\nu} ,</math>

where <math>G_{\mu\nu}</math> is the [[Physics:Einstein tensor|Einstein tensor]], <math>\Lambda</math> is the [[Astronomy:Cosmological constant|cosmological constant]] (sometimes taken to be zero for simplicity), <math>g_{\mu\nu}</math> is the [[Physics:Metric tensor (general relativity)|metric tensor]], <math>\kappa</math> is a constant, and <math> T_{\mu\nu} </math> is the [[Stress–energy tensor|stress–energy tensor]].

The Einstein field equations relate the Einstein tensor to the stress–energy tensor, which represents the distribution of energy, momentum and stress in the spacetime manifold. The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, given the stress–energy tensor, the Einstein field equations are a system of ten [[Partial differential equation|partial differential equation]]s in which the metric tensor can be solved for.
:''Where appropriate, this article will use the [[Abstract index notation|abstract index notation]].''

==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 tensor|stress–energy tensor]], 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 [[Physics:Weak-field approximation|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 the resulting stress–energy tensor can be plugged 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]] (to determine evolution of the stress–energy tensor):
:<math>T^{ab}{}_{;b}  \, = 0 \,.</math>

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). [[Physics:Equation of state|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 spacetime.

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:

* [[Physics:Vacuum solution (general relativity)|Vacuum]]:
:<math>T_{ab} \, = 0</math>

* [[Physics:Fluid solution|Perfect fluid]]:
:<math>T_{ab} \, = (\rho + p)u_a u_b + p g_{ab}</math> where <math>u^au_a = -1</math>

Here <math>\rho</math> is the mass–energy density measured in a momentary co-moving frame, <math>u_a</math> is the fluid's 4-velocity vector field, and <math>p</math> is the pressure.

* [[Physics:Dust solution|Non-interacting dust]] ( a special case of perfect fluid ):
:<math>T_{ab} \, = \rho u_a u_b</math>

For a perfect fluid, another equation of state relating density <math>\rho</math> and pressure <math>p</math> 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 <math>T^{ab}{}_{;b} = 0</math> is a consequence of Einstein's equations. This reflects the fact that the system is gauge invariant (in general, absent some symmetry, any choice of a curvilinear coordinate net on the same system would correspond to a numerically different solution.) A "gauge fixing" is needed, i.e. we need to impose 4 (arbitrary) constraints on the coordinate system in order to obtain unequivocal results.  These constraints are known as [[Physics:Coordinate conditions|coordinate conditions]].

A popular choice of gauge is the so-called "De Donder gauge", also known as the [[Physics:Harmonic coordinate condition|harmonic]] [[Physics:Coordinate conditions|condition]] or harmonic gauge
:<math>g^{\mu\nu} \Gamma^{\sigma}{}_{\mu\nu} = 0 \,.</math>

In [[Physics:Numerical relativity|numerical relativity]], the preferred gauge is the so-called "3+1 decomposition", based on the [[Physics:ADM formalism|ADM formalism]]. In this decomposition, metric is written in the form
:<math> ds^2 \, = (-N + N^i N^j \gamma_{ij}) dt^2 + 2N^i \gamma_{ij} dt dx^j + \gamma_{ij} dx^i dx^j</math>, where <math> i,j = 1\dots 3 \,.</math>

<math>N</math> and <math>N^i</math> are functions of spacetime coordinates and can be chosen arbitrarily in each point. The remaining physical degrees of freedom are contained in <math>\gamma_{ij}</math>, which represents the Riemannian metric on 3-hypersurfaces with constant <math>t</math>. For example, a naive choice of <math>N=1</math>, <math>N_i=0</math>, would correspond to a so-called synchronous coordinate system: one where t-coordinate coincides with proper time for any comoving observer (particle that moves along a fixed <math>x^i</math> trajectory.)

Once equations of state are chosen and the gauge is fixed, the complete set of equations can be solved. Unfortunately, even in the simplest case of gravitational field in the vacuum (vanishing stress–energy tensor), the problem is too complex to be exactly solvable. To get physical results, we can either turn to [[Physics:Numerical relativity|numerical methods]], try to find [[Physics:Exact solutions in general relativity|exact solutions]] by imposing [[Physics:Spacetime symmetries|symmetries]], or try middle-ground approaches such as [[Physics:Non-exact solutions in general relativity|perturbation methods]] or linear approximations of the [[Physics:Einstein tensor|Einstein tensor]].

==Exact solutions==
{{Main|Physics:Exact solutions in general relativity}}
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-form expression|closed form]].

===External reference===
[http://www.scholarpedia.org/article/Exact_solutions_of_Einstein%27s_equations Scholarpedia article on the subject] written by Malcolm MacCallum

==Non-exact solutions==
{{Main|Physics:Non-exact solutions in general relativity}}
The 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.

Al Momin argues that [[Biography:Kurt Gödel|Kurt Gödel]]'s solution to these equations do not describe our universe and are therefore approximations.<ref>{{cite web |url=http://www.math.nyu.edu/~momin/stuff/grpaper.pdf |title=The Gödel Solution to the Einstein Field Equations |website=www.math.nyu.edu |author=Al Momin |date=March 24, 2002}}</ref>

==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. From a physical standpoint, knowing the solutions of the Einstein Field Equations allows highly-precise modelling of astrophysical phenomena, including black holes, neutron stars, and stellar systems. Predictions can be made analytically about the system analyzed; such predictions include the [[Astronomy:Tests of general relativity|perihelion precession of Mercury]], the existence of a [[Astronomy:Kerr metric|co-rotating region inside spinning black holes]], and the orbits of objects around massive bodies.

==See also==

*[[Ricci calculus]]
*[[Biography:Albert Einstein|Albert Einstein]]

==References==

{{reflist}}
* {{cite book |author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne | title=Gravitation| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}
* {{cite book |author1=J.A. Wheeler |author2=I. Ciufolini | title=Gravitation and Inertia| publisher=Princeton University Press| year=1995| isbn=978-0-691-03323-5}}
* {{cite book | author=R.J.A. Lambourne| title=Relativity, Gravitation and Cosmology| publisher=The Open University, Cambridge University Press| year=2010| isbn=978-0-521-13138-4}}

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[[Category:General relativity]]

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