Ngô Quốc Anh

December 13, 2009

R-G: The Einstein Tensor, 2

Filed under: Riemannian geometry — Ngô Quốc Anh @ 20:53

In this topic, we shall give a natural way to construct the Einstein tensor. Let us apply the Ricci contraction to the Bianchi identities

\displaystyle g^{\alpha \mu} \big( R_{\alpha \beta \mu;\lambda} + R_{\alpha \lambda\beta ;\mu} + R_{\alpha \mu\lambda;\beta} \big)= 0.

Since g_{\alpha \beta; \mu}=0 and g^{\alpha \beta; \mu}=0, we can take g_{\alpha \mu} in and out of covariant derivatives at will. We get

\displaystyle g^{\alpha \mu} \big( R^\mu_{\beta \mu \nu ;\lambda} + R^\mu_{\beta\lambda\mu;\nu} + R^\mu_{\beta\nu\lambda;\mu} \big)= 0.

Using the antisymmetry on the indices \mu and \lambda we get

\displaystyle R^\mu_{\beta \mu \nu ;\lambda} - R^\mu_{\beta\mu\lambda;\nu} + R^\mu_{\beta\nu\lambda;\mu} = 0

so

\displaystyle R_{\beta\nu ;\lambda} - R_{\beta\lambda;\nu} + R^\mu_{\beta\nu\lambda;\mu} = 0.

These equations are called the contracted Bianchi identities. Let us now contract a second time on the indices \beta and \nu

\displaystyle g^{\beta\nu} \big( R_{\beta\nu ;\lambda} - R_{\beta\lambda;\nu} + R^\mu_{\beta\nu\lambda;\mu}\big) = 0.

This gives

\displaystyle R^\nu_{\nu ;\lambda} - R^\nu_{\lambda;\nu} + R^{\mu\nu}_{\nu\lambda;\mu} = 0

so

\displaystyle R_{;\lambda} - 2R^\mu_{\lambda;\mu} = 0

or

\displaystyle 2R^\mu_{\lambda;\mu}-R_{;\lambda} = 0.

Since R_{;\lambda} =g^\mu_\lambda R_{;\mu}, we get

\displaystyle \big(R^\mu_{\lambda}-\frac{1}{2}g^\mu_\lambda R\big)_{;\mu }= 0.

Raising the index \lambda with g^{\lambda \nu} we get

\displaystyle \big(R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R\big)_{;\mu }= 0.

Defining

\displaystyle E^{\mu\nu}= R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R

we get

\displaystyle {E^{\mu\nu}}_{;\mu }=0.

Theorem. The tensor E^{\mu\nu} is divergence free in the sense that

\displaystyle {E^{\mu\nu}}_{;\mu }=0.

The tensor E^{\mu\nu} is constructed only from the Riemann tensor and the metric, and it is automatically divergence free as an identity. It is called the Einstein tensor, since its importance for gravity was first understood by Einstein. Some authors denote the Einstein tensor by E_{\mu\nu}. We will see later that Einstein’s field equations for General Relativity in the vacuum case are

\displaystyle E^{\mu\nu}=\frac{8 \pi G}{c^4}T^{\mu\nu}

where T^{\mu\nu} is the stress- energy tensor. The Bianchi identities then imply

\displaystyle {T^{\mu\nu}}_{;\mu}=0

which is the conservation of energy and momentum.

Source: http://www.mth.uct.ac.za/omei/gr/chap6/node14.html

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