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| \paragraph{Definition.} |
The \emph{Ricci tensor} is found in the Einstein field equations and is the contraction of the Riemann tensor $R_{\mu\upsilon\alpha\beta}$ whose components are related to derivatives of the Christoffel symbols $\Gamma^{\mu}_{\upsilon\alpha}$ by |
| The \emph{Ricci curvature tensor} is a rank $2$, symmetric tensor that |
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| arises naturally in pseudo-Riemannian geometry. Let $(M,g_{ij})$ be a |
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| smooth, $n$-dimensional pseudo-Riemannian manifold, and let |
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| $R^i{}_{jkl}$ denote the corresponding Riemann curvature tensor. The |
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| Ricci tensor $R_{ij}$ is commonly defined as the following contraction |
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| of the full curvature tensor: |
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| \[R_{ij} = R^k{}_{ikj}. |
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| \] |
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| The symmetry of $R_{ij}$, so defined, follows from the symmetry |
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| properties of the Riemann curvature. To wit, |
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| \[ R_{ij} = R^k{}_{ikj} = R_{ki}{}^k{}_j = R^k{}_{jki} = R_{ji}.\] |
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| It is also convenient to regard the Ricci tensor as a symmetric bilinear |
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| form. To that end for vector-fields $X,Y$ we will write |
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| \[ \Ric(X,Y) = X^i Y^j R_{ij}.\] |
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| \paragraph{Related objects.} |
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| Contracting the Ricci tensor, we obtain an important scalar invariant |
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| \[R=R^i{}_i,\] called the scalar curvature, and sometimes also called |
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| the Ricci scalar. Closely related to the Ricci tensor is the tensor |
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| \[G_{ij} = R_{ij} - \frac{1}{2} R\, g_{ij},\] called the Einstein |
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| tensor. The Einstein tensor is also known as the trace-reversed Ricci |
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| tensor owing to the fact that |
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| \[ G^i{}_i = - R. \] |
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| Another related tensor is |
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| \[S_{ij} = R_{ij} - \frac{1}{n} R\, g_{ij}.\] |
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| This is called |
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| the |
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| trace-free Ricci tensor, owing to the fact that the above definition |
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| implies that |
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| \[ S^i{}_i=0.\] |
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$$R_{\mu\upsilon\alpha\beta}=g_{\mu\delta}(\partial_{\alpha}\Gamma^{\delta}_{\upsilon\beta}-\partial_{\beta}\Gamma^{\delta}_{\upsilon\alpha}+\Gamma^{\xi}_{\upsilon\beta}\Gamma^{\delta}_{\xi\alpha}-\Gamma^{\xi}_{\upsilon\alpha}\Gamma^{\delta}_{\xi\beta})$$ |
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where the Christoffel symbols are given in terms of the metric tensor by |
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| \paragraph{Geometric interpretation.} |
$$\Gamma^{\mu}_{\upsilon\alpha}=\frac{1}{2}g_{\mu\delta}(\partial_{\upsilon}g_{\delta\upsilon}+\partial_{\alpha}g_{\delta\upsilon}-\partial_{\delta}g_{\mu\alpha})$$ |
| In Riemannian geometry, the Ricci tensor represents the average value |
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| of the sectional curvature along a particular direction. |
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| Let |
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| \[ K_x(u,v) = \frac{R_x(u,v,v,u)}{g_x(u,u) g_x(v,v) - g_x(u,v)^2} |
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| \] |
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| denote the sectional curvature of $M$ along the plane spanned by |
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| vectors $u,v\in T_x M$. Fix a point $x\in M$ and a tangent vector |
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| $v\in T_xM$, and let |
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| \[ |
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| S_x(v)=\{ u\in T_xM \colon g_x(u,u) = 1,\; g_x(u,v)=0 \} |
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| \] denote the $n-2$ dimensional |
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| sphere of those unit vectors at $x$ that are perpendicular to $v$. |
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| Let $\mu_x$ denote the natural |
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| $(n-2)$-dimensional volume measure on $T_xM$, normalized so that |
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| \[ \int_{S_x(v)} \mu_x = 1.\] |
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| In this way, the quantity |
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| \[ \int_{S_x(v)}\!\! K_x(\cdot,v) \mu_x, \] |
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| describes the average value of the sectional curvature for all planes |
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| in $T_x M$ that contain $v$. It is possible to show that |
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| \[ \Ric_x(v,v)= (1-n)\int_{S_x(v)}\!\! K_x(\cdot,v) \mu_x,\] |
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| thereby giving us the desired geometric interpretation. |
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| \paragraph{Decomposition of the curvature tensor.} |
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| For $n\geq 3$, the Ricci tensor can be characterized in terms of the |
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| decomposition of the full curvature tensor into three covariantly |
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| defined summands, namely |
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| \begin{align*} |
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| F_{ijkl} &= \tfrac{1}{n-2} \left( S_{jl}\, g_{ik}+S_{ik}\, |
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| g_{jl}-S_{il}\, g_{jk}-S_{jk}\, g_{il}\right),\\ |
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| E_{ijkl} &= \tfrac{1}{n(n-1)}R \left( g_{jl}\,g_{ik} - |
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| g_{il}\,g_{jk}\right),\\ |
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| W_{ijkl} &= R_{ijkl}-F_{ijkl}-E_{ijkl}. |
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| \end{align*} |
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| The $W_{ijkl}$ is called the Weyl curvature tensor. It is |
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| the conformally invariant, trace-free part of the curvature tensor. |
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| Indeed, with the above definitions, we have |
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| \[ W^k{}_{ikj}=0.\] The $E_{ijkl}$ and $F_{ijkl}$ correspond to the |
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| trace-free part of the Ricci curvature tensor, and to the Ricci |
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| scalar. Indeed, we can recover $S_{ij}$ and $R$ from $E_{ijkl}$ and |
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| $F_{ijkl}$ as follows: |
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| \begin{align*} |
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| S_{ij} &= F^k{}_{ikj},\\ |
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| E^{ij}{}_{ij} &= R. |
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| \end{align*} |
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| \paragraph{Relativity.} The Ricci tensor also plays an important role |
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| in the theory of general relativity. In this keystone application, |
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| $M$ is a 4-dimensional pseudo-Riemannian manifold with signature |
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| $(3,1)$. The Einstein field equations assert that the energy-momentum |
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| tensor is proportional to the Einstein tensor. In particular, the |
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| equation |
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| \[ R_{ij}=0 \] |
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| is the field equation for a vacuum space-time. In geometry, a |
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| pseudo-Riemannian manifold that satisfies this equation is called |
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| Ricci-flat. It is possible to prove that a manifold is Ricci flat if |
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| and only if locally, the manifold, is conformally equivalent to flat space. |
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