RicciTensor 2005-02-16 03:50:27 2006-09-07 10:15:37 Definition scalar curvature Einstein tensor ricci scalar Weyl tensor Weyl curvature tensor % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\Ric}{\operatorname{Ric}} \paragraph{Definition.} The \emph{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 \[ \Ric(X,Y) = X^i Y^j R_{ij}.\] \paragraph{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 \emph{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 \[ S^i{}_i=0.\] \paragraph{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 $v\in T_xM$, and let \[ S_x(v)=\{ u\in T_xM \colon g_x(u,u) = 1,\; g_x(u,v)=0 \} \] 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 $T_xM$, 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 \[ \Ric_x(v,v)= (1-n)\int_{S_x(v)}\!\! K_x(\cdot,v) \mu_x,\] thereby giving us the desired geometric interpretation. \paragraph{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 \begin{align*} F_{ijkl} &= \tfrac{1}{n-2} \left( S_{jl}\, g_{ik}+S_{ik}\, g_{jl}-S_{il}\, g_{jk}-S_{jk}\, g_{il}\right),\\ E_{ijkl} &= \tfrac{1}{n(n-1)}R \left( g_{jl}\,g_{ik} - g_{il}\,g_{jk}\right),\\ W_{ijkl} &= R_{ijkl}-F_{ijkl}-E_{ijkl}. \end{align*} The $W_{ijkl}$ is called the \emph{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: \begin{align*} S_{ij} &= F^k{}_{ikj},\\ E^{ij}{}_{ij} &= R. \end{align*} \paragraph{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.