Exact solutions in general relativity

In general relativity, an exact solution is a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field. In general relativity, an exact solution is a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational 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 T α β {displaystyle T^{alpha eta }} . (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 be a solution of the Einstein field equations (written here in geometrized units, where speed of light c = Gravitational constant G = 1) In the above field equations, G α β {displaystyle G^{alpha eta }} is the Einstein tensor, 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 non-gravitational fields, in the sense that the immediate presence 'here and now' of non-gravitational energy–momentum causes a proportional amount of Ricci curvature 'here and now'. Moreover, taking covariant derivatives of the field equations and applying the Bianchi identities, it is found that a suitably varying amount/motion of non-gravitational energy–momentum can cause ripples in curvature to propagate as gravitational radiation, even across vacuum regions, which contain no matter or non-gravitational fields. Any Lorentzian manifold is a solution of the Einstein field equation for some right hand side. This is illustrated by the following procedure: