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

\operatorname{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\colon 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

\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

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:

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

Relativity.

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