Exact Solutions in General Relativity

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. 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 . (To wit, whenever 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 satisfy the Einstein field equations (written here in geometrized units, where speed of light c = Gravitational constant G = 1)

In the above field equations, 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 nongravitational fields, in the sense that the immediate presence "here and now" of nongravitational 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 nongravitational energy-momentum can cause ripples in curvature to propagate as gravitational radiation, even across vacuum regions, which contain no matter or nongravitational fields.