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:''Where appropriate, this article will use the abstract index notation.'' Solutions of the Einstein field equations are metrics of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
s that result from solving the
Einstein field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
(EFE) of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. Solving the field equations gives a Lorentz manifold. Solutions are broadly classed as ''exact'' or ''non-exact''. The Einstein field equations are :G_ + \Lambda g_ \, = \kappa T_ , where G_ is the
Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field ...
, \Lambda is the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
(sometimes taken to be zero for simplicity), g_ is the metric tensor, \kappa is a constant, and T_ is the 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 equations in which the metric tensor can be solved for.


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, 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 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 (to determine evolution of the stress–energy tensor): :T^_ \, = 0 \,. 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 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: *
Vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
: :T_ \, = 0 *
Perfect fluid In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density \rho_m and ''isotropic'' pressure ''p''. Real fluids are "sticky" and contain (and conduct) heat. Perfect fluids are idealized models in whi ...
: :T_ \, = (\rho + p)u_a u_b + p g_ where u^au_a = -1 Here \rho is the mass–energy density measured in a momentary co-moving frame, u_a is the fluid's 4-velocity vector field, and p is the pressure. * Non-interacting dust ( a special case of perfect fluid ): :T_ \, = \rho u_a u_b For a perfect fluid, another equation of state relating density \rho and pressure p 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 T^_ = 0 is a consequence of Einstein's equations. This reflects the fact that the system is
gauge invariant In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
(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
coordinate conditions In general relativity, the laws of physics can be expressed in a generally covariant form. In other words, the description of the world as given by the laws of physics does not depend on our choice of coordinate systems. However, it is often useful ...
. A popular choice of gauge is the so-called "De Donder gauge", also known as the harmonic condition or harmonic gauge :g^ \Gamma^_ = 0 \,. In
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 a ...
, the preferred gauge is the so-called "3+1 decomposition", based on the ADM formalism. In this decomposition, metric is written in the form : ds^2 \, = (-N + N^i N^j \gamma_) dt^2 + 2N^i \gamma_ dt dx^j + \gamma_ dx^i dx^j, where i,j = 1\dots 3 \,. N and N^i are functions of spacetime coordinates and can be chosen arbitrarily in each point. The remaining physical degrees of freedom are contained in \gamma_, which represents the Riemannian metric on 3-hypersurfaces with constant t. For example, a naive choice of N=1, N_i=0, would correspond to a so-called
synchronous Synchronization is the coordination of events to operate a system in unison. For example, the conductor of an orchestra keeps the orchestra synchronized or ''in time''. Systems that operate with all parts in synchrony are said to be synchronou ...
coordinate system: one where t-coordinate coincides with proper time for any comoving observer (particle that moves along a fixed x^i 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 numerical methods, try to find exact solutions by imposing symmetries, or try middle-ground approaches such as
perturbation methods In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
or linear approximations of the
Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field ...
.


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 form.


External reference

Scholarpedia article on the subject
written by Malcolm MacCallum


Non-exact solutions

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 Kurt Gödel's solution to these equations do not describe our universe and are therefore approximations.


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.


See also

* Ricci calculus *
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...


References

* * * {{DEFAULTSORT:Solutions Of The Einstein Field Equations General relativity Albert Einstein