General Relativity/Riemann tensor

In the mathematical field of differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds. It is one of many things named after Bernhard Riemann. The curvature tensor is given in terms of a Levi-Civita connection by the following formula:

$R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w$

NB. Some authors define the curvature tensor with the opposite sign.

If $u={\frac {\partial }{\partial x^{i}}}$ and $v={\frac {\partial }{\partial x^{j}}}$ are coordinate vector fields then $[u,v]=0$ and therefore the formula simplifies to

$R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w$

i.e. the curvature tensor measures noncommutativity of the covariant derivative.

The Riemann curvature tensor, especially in its coordinate expression (see below), is a central mathematical tool of general relativity, the modern theory of gravity.

Coordinate expression

In local coordinates $x^{\mu }$ the Riemann curvature tensor is given by

${R^{\rho }}_{\sigma \mu \nu }=dx^{\rho }(R(\partial _{\mu },\partial _{\nu })\partial _{\sigma })$

where $\partial _{\mu }=\partial /\partial x^{\mu }$ are the coordinate vector fields. The above expression can be written using Christoffel symbols:

${R^{\rho }}_{\sigma \mu \nu }=\partial _{\mu }\Gamma _{\nu \sigma }^{\rho }-\partial _{\nu }\Gamma _{\mu \sigma }^{\rho }+\Gamma _{\mu \lambda }^{\rho }\Gamma _{\nu \sigma }^{\lambda }-\Gamma _{\nu \lambda }^{\rho }\Gamma _{\mu \sigma }^{\lambda }$

The transformation of a vector $V^{\mu }$ after circling an infinitesimal rectangle $dx^{\nu }dx^{\sigma }$ is: $\delta V^{\mu }=R_{\nu \sigma \tau }^{\mu }dx^{\nu }dx^{\sigma }V^{\tau }$.

Symmetries and identities

The Riemann curvature tensor has the following symmetries:

$R(u,v)=-R(v,u)_{}^{}$

$\langle R(u,v)w,z\rangle =-\langle R(u,v)z,w\rangle _{}^{}$

$R(u,v)w+R(v,w)u+R(w,u)v=0_{}^{}$

The last identity was discovered by Gregorio Ricci-Curbastro, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the Bianchi identity below. These three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has $n^{2}(n^{2}-1)/12$ independent components.

Yet another useful identity follows from these three:

$\langle R(u,v)w,z\rangle =\langle R(w,z)u,v\rangle _{}^{}$

The Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) involves the covariant derivative:

$\nabla _{u}R(v,w)+\nabla _{v}R(w,u)+\nabla _{w}R(u,v)=0$

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

$R_{abcd}^{}=-R_{bacd}=-R_{abdc}$

$R_{abcd}^{}=R_{cdab}$

$R_{a[bcd]}^{}=0$ (first Bianchi identity)

$R_{ab[cd;e]}^{}=0$ (second Bianchi identity)

where the square brackets denote cyclic symmetrisation over the indices and the semi-colon is a covariant derivative.

For surfaces

For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor can be expressed as

$R_{abcd}^{}=K(g_{ac}g_{db}-g_{ad}g_{cb})$

where $g_{ab}$ is the metric tensor and $K$ is a function called the Gaussian curvature and a, b, c and d take values either 1 or 2. As expected we see that the Riemann curvature tensor only has one independent component.

The Gaussian curvature coincides with the sectional curvature of the surface. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by

$\operatorname {Ric} _{ab}=Kg_{ab}.$