Einstein tensor

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[edit] Definition

In physics and 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\over2} g_{\mu\nu}R.

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

[edit] Trace

The trace of the Einstein tensor can be computed by contracting the equation above with the metric gμν,

g^{\mu\nu}G_{\mu\nu} = g^{\mu\nu}R_{\mu\nu} - {1\over2} g^{\mu\nu}g_{\mu\nu}R ,
G = R - {1\over2} (4R) ,
G = -R . \,\!

hence the name, trace-reversed.

[edit] 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.
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