Field equations

Gravitation

After Newtonian gravitation was found to be inconsistent with special relativity, Albert Einstein formulated a new theory of gravitation called general relativity. This treats gravitation as a geometric phenomenon ('curved spacetime') caused by masses and represents the gravitational field mathematically by a tensor field called the metric tensor. The Einstein field equations describe how this curvature is produced. Newtonian gravitation is now superseded by Einstein's theory of general relativity, in which gravitation is thought of as being due to a curved spacetime, caused by masses. The Einstein field equations, $${\displaystyle G_{ab}=\kappa T_{ab}}$$ describe how this curvature is produced by matter and radiation, where G_ab is the Einstein tensor, $${\displaystyle G_{ab}\,=R_{ab}-{\frac {1}{2}}Rg_{ab}}$$ written in terms of the Ricci tensorR_ab and Ricci scalarR = R_abg, T_ab is the stress–energy tensor and κ = 8πG/c is a constant. In the absence of matter and radiation (including sources) the vacuum field equations, $${\displaystyle G_{ab}=0}$$ can be derived by varying the Einstein–Hilbert action, $${\displaystyle S=\int R{\sqrt {-g}}\,d^{4}x}$$ with respect to the metric, where g is the determinant of the metric tensorg. Solutions of the vacuum field equations are called vacuum solutions. An alternative interpretation, due to Arthur Eddington, is that ${\displaystyle R}$ is fundamental, ${\displaystyle T}$ is merely one aspect of ${\displaystyle R}$, and ${\displaystyle \kappa }$ is forced by the choice of units.