PlanetMath: Ricci tensor

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Ricci tensor

(Definition)

Definition.

The Ricci curvature tensor is a rank

, symmetric tensor that arises naturally in pseudo-Riemannian geometry. Let $(M,g_{ij})$ be a smooth,

-dimensional pseudo-Riemannian manifold, and let $R^i{}_{jkl}$ denote the corresponding Riemann curvature tensor. The Ricci tensor $R_{ij}$ is commonly defined as the following contraction of the full curvature tensor:

$\displaystyle R_{ij} = R^k{}_{ikj}.$

The index symmetry of $R_{ij}$ , so defined, follows from the symmetry properties of the Riemann curvature. To wit,

$\displaystyle R_{ij} = R^k{}_{ikj} = R_{ki}{}^k{}_j = R^k{}_{jki} = R_{ji}.$

It is also convenient to regard the Ricci tensor as a symmetric bilinear form. To that end for vector-fields

we will write

$\displaystyle \operatorname{Ric}(X,Y) = X^i Y^j R_{ij}.$

Related objects.

Contracting the Ricci tensor, we obtain an important scalar invariant

$\displaystyle R=R^i{}_i,$

called the scalar curvature, and sometimes also called the Ricci scalar. Closely related to the Ricci tensor is the tensor

$\displaystyle G_{ij} = R_{ij} - \frac{1}{2} R\, g_{ij},$

called the Einstein tensor. The Einstein tensor is also known as the trace-reversed Ricci tensor owing to the fact that

$\displaystyle G^i{}_i = - R.$

Another related tensor is

$\displaystyle S_{ij} = R_{ij} - \frac{1}{n} R\, g_{ij}.$

This is called the trace-free Ricci tensor, owing to the fact that the above definition implies that

$\displaystyle S^i{}_i=0.$

Geometric interpretation.

In Riemannian geometry, the Ricci tensor represents the average value of the sectional curvature along a particular direction. Let

$\displaystyle K_x(u,v) = \frac{R_x(u,v,v,u)}{g_x(u,u) g_x(v,v) - g_x(u,v)^2}$

denote the sectional curvature of

along the plane spanned by vectors $u,v\in T_x M$ . Fix a point $x\in M$ and a tangent vector $v\in T_xM$ , and let

$\displaystyle S_x(v)=\{ u\in T_xM \colon g_x(u,u) = 1,\; g_x(u,v)=0 \}$

denote the

dimensional sphere of those unit vectors at

that are perpendicular to

. Let $\mu_x$ denote the natural

-dimensional volume measure on

, normalized so that

$\displaystyle \int_{S_x(v)} \mu_x = 1.$

In this way, the quantity

$\displaystyle \int_{S_x(v)}\!\! K_x(\cdot,v) \mu_x,$

describes the average value of the sectional curvature for all planes in

that contain

. It is possible to show that

$\displaystyle \operatorname{Ric}_x(v,v)= (1-n)\int_{S_x(v)}\!\! K_x(\cdot,v) \mu_x,$

thereby giving us the desired geometric interpretation.

Decomposition of the curvature tensor.

For $n\geq 3$ , the Ricci tensor can be characterized in terms of the decomposition of the full curvature tensor into three covariantly defined summands, namely

$\displaystyle F_{ijkl}$	$\displaystyle = \tfrac{1}{n-2} \left( S_{jl}\, g_{ik}+S_{ik}\, g_{jl}-S_{il}\, g_{jk}-S_{jk}\, g_{il}\right),$
$\displaystyle E_{ijkl}$	$\displaystyle = \tfrac{1}{n(n-1)}R \left( g_{jl}\,g_{ik} - g_{il}\,g_{jk}\right),$
$\displaystyle W_{ijkl}$	$\displaystyle = R_{ijkl}-F_{ijkl}-E_{ijkl}.$

The $W_{ijkl}$ is called the Weyl curvature tensor. It is the conformally invariant, trace-free part of the curvature tensor. Indeed, with the above definitions, we have

$\displaystyle W^k{}_{ikj}=0.$

The $E_{ijkl}$ and $F_{ijkl}$ correspond to the trace-free part of the Ricci curvature tensor, and to the Ricci scalar. Indeed, we can recover $S_{ij}$ and

from $E_{ijkl}$ and $F_{ijkl}$ as follows:

$\displaystyle S_{ij}$	$\displaystyle = F^k{}_{ikj},$
$\displaystyle E^{ij}{}_{ij}$	$\displaystyle = R.$

Relativity.

The Ricci tensor also plays an important role in the theory of general relativity. In this keystone application,

is a 4-dimensional pseudo-Riemannian manifold with signature

. The Einstein field equations assert that the energy-momentum tensor is proportional to the Einstein tensor. In particular, the equation

$\displaystyle R_{ij}=0$

is the field equation for a vacuum space-time. In geometry, a pseudo-Riemannian manifold that satisfies this equation is called Ricci-flat. It is possible to prove that a manifold is Ricci flat if and only if locally, the manifold, is conformally equivalent to flat space.

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"Ricci tensor" is owned by rmilson. [ full author list (2) | owner history (1) ]

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Also defines:

scalar curvature, Einstein tensor, ricci scalar, Weyl tensor, Weyl curvature tensor

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This is version 6 of Ricci tensor, born on 2005-02-16, modified 2006-09-07.
Object id is 6758, canonical name is RicciTensor.
Accessed 9961 times total.

Classification:

AMS MSC:

83C05 (Relativity and gravitational theory :: General relativity :: Einstein's equations )

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