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| {{Expert-subject|Physics|date=November 2008}}
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| ''Where appropriate, this article will use the [[abstract index notation]].'' | |
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| '''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 metric]]s. Solutions are broadly classed as ''exact'' or ''non-exact''.
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| The Einstein field equations are
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| :<math>G_{ab} \, = \kappa T_{ab}</math>
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| or more generally
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| :<math>G_{ab} + \Lambda g_{ab} \, = \kappa T_{ab}</math>
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| where '''<math>\kappa</math>''' 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 (general relativity)|metric tensor]] and its partial derivatives; thus, the EFE are a system of ten [[partial differential equation]]s to be solved for the metric.
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| ==Solving the equations==
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| 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]], 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 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 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.
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| 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):
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| :<math>T^{ab}{}_{;b} \, = 0 \,.</math> | |
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| 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). [[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 space-time.
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| 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:
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| * [[Vacuum solution (general relativity)|Vacuum]]:
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| :<math>T_{ab} \, = 0</math>
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| * [[Fluid solution|Perfect fluid]]:
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| :<math>T_{ab} \, = (\rho + p)u_a u_b + p g_{ab}</math> where <math>u^au_a = -1\!</math>
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| 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.
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| * [[Dust solution|Non-interacting dust]] ( a special case of perfect fluid ):
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| :<math>T_{ab} \, = \rho u_a u_b</math>
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| 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.
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| 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]] 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]].
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| A popular choice of gauge is the so-called "De Donder gauge", also known as the [[harmonic coordinate condition|harmonic]] [[coordinate conditions|condition]] or harmonic gauge
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| :<math>g^{\mu\nu} \Gamma^{\sigma}_{\mu\nu} = 0 \,.</math> | |
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| 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
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| :<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>
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| <math>N</math> and <math>N^i</math> can be chosen arbitrarily. The remaining physical degrees of freedom are contained in <math>\gamma_{ij}</math>, which represents the Riemannian metric on 3-hypersurfaces <math>t=const</math>.
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| 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 relativity|numerical methods]]; try to find [[Exact solutions in general relativity|exact solutions]] by imposing [[Spacetime symmetries|symmetries]]; or try middle-ground approaches such as [[Non-exact solutions in general relativity|perturbation methods]] or linear approximations of the [[Einstein tensor]].
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| ==Exact solutions==
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| {{Expand section|date=June 2008}}
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| {{Main|Exact solutions}}
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| Exact solutions are [[Lorentz metric]]s 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]].
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| ==Non-exact solutions==
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| {{Expand section|date=June 2008}}
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| {{Main|Non-exact solutions in general relativity}}
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| 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.
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| ==Applications==
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| {{Expand section|date=June 2008}}
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| There are practical as well as theoretical reasons for studying solutions of the Einstein field equations.
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| 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.
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| ==See also==
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| *[[Ricci calculus]]
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| ==References==
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| * {{cite book | author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}
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| * {{cite book | author=J.A. Wheeler, I. Ciufolini| title=Gravitation and Inertia| publisher=Princeton University Press| year=1995| isbn=978-0-691-03323-5}}
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| * {{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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| {{DEFAULTSORT:Solutions Of The Einstein Field Equations}}
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| [[Category:General relativity]]
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