Definition

---

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

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

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

$G\_\{\backslash mu\backslash nu\}\; =\; R\_\{\backslash mu\backslash nu\}\; -\; \{1\backslash over2\}\; g\_\{\backslash mu\backslash nu\}R$.

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 $g^\{\backslash mu\backslash nu\}$,

$g^\{\backslash mu\backslash nu\}G\_\{\backslash mu\backslash nu\}\; =\; g^\{\backslash mu\backslash nu\}R\_\{\backslash mu\backslash nu\}\; -\; \{1\backslash over2\}\; g^\{\backslash mu\backslash nu\}g\_\{\backslash mu\backslash nu\}R$,

$G\; =\; R\; -\; \{1\backslash over2\}\; (4R)$,

$G\; =\; -R$.

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:

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

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

$G\_\{\backslash mu\backslash nu\}\; =\; \backslash frac\{8\backslash pi\; G\}\{c^4\}\; T\_\{\backslash mu\backslash nu\}$,

which, using geometrized units, simplifies to

$G\_\{\backslash mu\backslash nu\}\; =\; 8\; \backslash pi\; \backslash ,\; T\_\{\backslash mu\backslash nu\}$.

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

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

Tensors in general relativity | Tensors | Albert Einstein