Einstein field equations
Special classes of exact solutions are most often studied since they model many gravitational phenomena, such as rotating black holes and the expanding universe.
The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime.
Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions.
The vacuum field equations (obtained when Tμν is everywhere zero) define Einstein manifolds.
, since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner.
When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations.
[10] The above form of the EFE is the standard established by Misner, Thorne, and Wheeler (MTW).
With these definitions Misner, Thorne, and Wheeler classify themselves as (+ + +), whereas Weinberg (1972)[12] is (+ − −), Peebles (1980)[13] and Efstathiou et al. (1990)[14] are (− + +), Rindler (1977),[citation needed] Atwater (1974),[citation needed] Collins Martin & Squires (1989)[15] and Peacock (1999)[16] are (− + −).
The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace
the term containing the cosmological constant Λ was absent from the version in which he originally published them.
Einstein then included the term with the cosmological constant to allow for a universe that is not expanding or contracting.
This effort was unsuccessful because: Einstein then abandoned Λ, remarking to George Gamow "that the introduction of the cosmological term was the biggest blunder of his life".
More recent astronomical observations have shown an accelerating expansion of the universe, and to explain this a positive value of Λ is needed.
Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor:
This tensor describes a vacuum state with an energy density ρvac and isotropic pressure pvac that are fixed constants and given by
This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.
General relativity is consistent with the local conservation of energy and momentum expressed as
which by the symmetry of the bracketed term and the definition of the Einstein tensor, gives, after relabelling the indices,
With his field equations Einstein ensured that general relativity is consistent with this conservation condition.
The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories.
For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of quantum mechanics, which is linear in the wavefunction.
Our simplifying assumptions make the squares of Γ disappear together with the time derivatives
From the latter, it follows by the Poincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential Aα such that
Even so, there are several cases where the field equations have been solved completely, and those are called exact solutions.
[9] The study of exact solutions of Einstein's field equations is one of the activities of cosmology.
It leads to the prediction of black holes and to different models of evolution of the universe.
One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum.
As discussed by Hsu and Wainwright,[23] self-similar solutions to the Einstein field equations are fixed points of the resulting dynamical system.
Substituting this expression of the inverse of the metric into the equations then multiplying both sides by a suitable power of det(g) to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives.
The Einstein-Hilbert action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.
