Editing Einstein tensor (section)

== Use in general relativity ==

The Einstein tensor allows the [[Einstein field equations]] to be written in the concise form:
<math display="block">G_{\mu\nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu\nu} ,</math>
where <math>\Lambda</math> is the [[cosmological constant]] and <math>\kappa</math> is the [[Einstein gravitational constant]].

From the [[#Explicit form|explicit form of the Einstein tensor]], the Einstein tensor is a [[nonlinear]] function of the metric tensor, but is linear in the second [[partial derivative]]s of the metric. As a symmetric order-2 tensor, the Einstein tensor has 10 independent components in a 4-dimensional space. It follows that the Einstein field equations are a set of 10 [[Types of differential equations|quasilinear]] second-order partial differential equations for the metric tensor.

The [[contracted Bianchi identities]] can also be easily expressed with the aid of the Einstein tensor:
<math display="block">\nabla_{\mu} G^{\mu\nu} = 0.</math>

The (contracted) Bianchi identities automatically ensure the covariant conservation of the [[stress–energy tensor]] in curved spacetimes:
<math display="block">\nabla_{\mu} T^{\mu\nu} = 0.</math>

The physical significance of the Einstein tensor is highlighted by this identity. In terms of the densitized stress tensor contracted on a [[Killing vector]] {{tmath|1= \xi^\mu }}, an ordinary conservation law holds:
<math display="block">\partial_{\mu}\left(\sqrt{-g}\ T^\mu{}_\nu \xi^\nu\right) = 0.</math>