Ricci tensor

Definition.

The Ricci curvature tensor is a rank $2$, symmetric tensor that arises naturally in pseudo-Riemannian geometry. Let $(M,g_{ij})$ be a smooth, $n$-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:

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

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

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

It is also convenient to regard the Ricci tensor as a symmetric bilinear form. To that end for vector-fields $X,Y$ we will write

$$\mathrm{Ric}(X,Y)=X^i{Y}^j{R}_{ij}.$$

Related objects.

Contracting the Ricci tensor, we obtain an important scalar invariant

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

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

$$G_{ij}=R_{ij}-\frac{1}{2}R{g}_{ij},$$

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

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

Another related tensor is

$$S_{ij}=R_{ij}-\frac{1}{n}R{g}_{ij}.$$

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

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

Geometric interpretation.

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

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

denote the sectional curvature of $M$ along the plane spanned by vectors $u,v\in T_{x}M$. Fix a point $x\in M$ and a tangent vector $v\in T_{x}M$, and let

$$S_x(v)=\{u\in T_{x}M:g_x(u,u)=1,g_x(u,v)=0\}$$

denote the $n-2$ dimensional sphere of those unit vectors at $x$ that are perpendicular to $v$. Let ${\mu }_x$ denote the natural $(n-2)$-dimensional volume measure on $T_{x}M$, normalized so that

$${\int }_{S_x(v)}{\mu }_x=1.$$

In this way, the quantity

$${\int }_{S_x(v)}{K}_x(\cdot ,v){\mu }_x,$$

describes the average value of the sectional curvature for all planes in $T_{x}M$ that contain $v$. It is possible to show that

$${\mathrm{Ric}}_x(v,v)=(1-n){\int }_{S_x(v)}{K}_x(\cdot ,v){\mu }_x,$$

thereby giving us the desired geometric interpretation.

Decomposition of the curvature tensor.

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

$F_{ijkl}$ $=\frac{1}{n-2}\left(S_{jl}{g}_{ik}+S_{ik}{g}_{jl}-S_{il}{g}_{jk}-S_{jk}{g}_{il}\right),$

$E_{ijkl}$ $=\frac{1}{n(n-1)}R\left(g_{jl}{g}_{ik}-g_{il}{g}_{jk}\right),$

$W_{ijkl}$ $=R_{ijkl}-F_{ijkl}-E_{ijkl}.$

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

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

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

$S_{ij}$ $=F^k{}_{ikj},$

$E^{ij}{}_{ij}$ $=R.$

Relativity.

The Ricci tensor also plays an important role in the theory of general relativity. In this keystone application, $M$ is a 4-dimensional pseudo-Riemannian manifold with signature $(3,1)$. The Einstein field equations assert that the energy-momentum tensor is proportional to the Einstein tensor. In particular, the equation

$$R_{ij}=0$$

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.

Title	Ricci tensor
Canonical name	RicciTensor
Date of creation	2013-03-22 15:02:38
Last modified on	2013-03-22 15:02:38
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	9
Author	rmilson (146)
Entry type	Definition
Classification	msc 83C05
Defines	scalar curvature
Defines	Einstein tensor
Defines	ricci scalar
Defines	Weyl tensor
Defines	Weyl curvature tensor