Einstein tensor

Definition

In differential geometry, the Einstein tensor $\mathbf {G}$ is a 2-tensor defined over Riemannian manifolds. In index-free notation it looks like this

$\mathbf {G} =\mathbf {R} -{\frac {1}{2}}\mathbf {g} R$,

where $\mathbf {R}$ is the Ricci tensor, $\mathbf {g}$ is the metric tensor and $R$ is the Ricci scalar (or scalar curvature). In component form, the above equation reads

$G_{\mu \nu }=R_{\mu \nu }-{1 \over 2}g_{\mu \nu }R$.

The Einstein tensor is sometimes referred to as the trace-reversed Ricci tensor.

Trace

The trace of the Einstein tensor can be computed by contracting the equation above with the metric $g^{\mu \nu }$,

$g^{\mu \nu }G_{\mu \nu }=g^{\mu \nu }R_{\mu \nu }-{1 \over 2}g^{\mu \nu }g_{\mu \nu }R$,

$G=R-{1 \over 2}(4R)$,

$G=-R$.

hence the name, trace-reversed.

Relation to Bianchi Identities and General Relativity

The Bianchi identities can be easily expressed with the aid of the Einstein tensor:

$\nabla _{\mu }G^{\mu \nu }=0$.

In general relativity, the Einstein tensor allows a compact expression of the Einstein equations:

$G_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }$,

which, using geometrized units, simplifies to

$G_{\mu \nu }=8\pi \,T_{\mu \nu }$.

The Bianchi identities automatically ensure the conservation of the energy-momentum tensor in curved spacetimes:

$\nabla _{\mu }T^{\mu \nu }=0$.