The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) with metric-selected connection $\mathrm{\nabla }$ (Metric compatibility, torsion freedom and the metric-selected connection class), whose existence and uniqueness are supplied by Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields; that theorem also extends $\mathrm{\nabla }$ to a derivative on tensor fields commuting with contractions and annihilating $g$ and $g^{-1}$. Let $U\subseteq M$ be open and let $X,Y,Z$ be smooth vector fields on $U$; brackets are those of The Lie bracket of smooth vector fields, and contractions those of The contraction of a mixed tensor.

(a) The curvature operator. The curvature operator of $\mathrm{\nabla }$ is the assignment

$R(X,Y)Z:={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z\mathrm{.}$

The right-hand side is a smooth vector field on $U$: ${\mathrm{\nabla }}_{Y}Z$ is smooth because $\mathrm{\nabla }$ is an affine connection (Affine connection on a smooth manifold), $[X,Y]$ is smooth, and ${\mathrm{\nabla }}_X(\cdot )$ and ${\mathrm{\nabla }}_{[X,Y]}(\cdot )$ preserve smoothness. The assignment is defined for the given smooth fields; that its value at a point depends only on the values of $X,Y,Z$ at that point, so that $R$ is a smooth tensor field of type $(1,3)$, is the content of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula ↗ clause (a), and likewise the algebraic symmetries used below are proved in The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors ↗.

(b) Component functions. Let $(V,x)$ be a smooth chart of $M$ and let ${\mathrm{\Gamma }}^a{}_{bc}$ be the Christoffel symbols of $\mathrm{\nabla }$ in this chart, characterized by ${\mathrm{\nabla }}_{{\mathrm{\partial }}_b}{\mathrm{\partial }}_c={\sum }_a{\mathrm{\Gamma }}^a{}_{bc}{\mathrm{\partial }}_a$ (Christoffel symbols of an affine connection); the coordinate fields commute, so ${\mathrm{\nabla }}_{{\mathrm{\partial }}_c}{\mathrm{\partial }}_d$ and ${\mathrm{\nabla }}_{{\mathrm{\partial }}_d}{\mathrm{\partial }}_c$ are obtained from the same symbols. The components of the curvature in the chart are the smooth functions $R^a{}_{bcd}$ on $V$ defined by

$R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b={\sum }_a{R}^a{}_{bcd}{\mathrm{\partial }}_a\mathrm{.}$

The first index is the output slot, the second is the differentiated slot and the last two are the two directions of the loop; indices are $a,b,c,d=0,1,2,3$ and the Einstein summation convention of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model applies.

(c) Lowered tensor, Ricci tensor, scalar curvature, Einstein tensor. On the chart domain define, using the inverse metric $g^{ab}$ of The inverse metric, musical isomorphisms and index raising,

$R_{abcd}:={\sum }_e{g}_{ae}{R}^e{}_{bcd},{\mathrm{Ric}}_{bd}:={\sum }_{a,c}{g}^{ac}{R}_{abcd},S:={\sum }_{b,d}{g}^{bd}{\mathrm{Ric}}_{bd},G_{ab}:={\mathrm{Ric}}_{ab}-\frac{1}{2}S{g}_{ab}\mathrm{.}$

The fields $R_{abcd}$, ${\mathrm{Ric}}_{ab}$, $S$ and $G_{ab}$ are called the Riemann tensor with lowered indices, the Ricci tensor, the scalar curvature and the Einstein tensor of the metric-selected connection. Tensoriality of $R$ is proved in Tensoriality of the curvature, its coordinate formula and its point-normal metric formula ↗ clause (a). Lowering its output slot with the smooth metric and contracting with the smooth inverse metric are intrinsic tensor operations (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components clauses (b)-(c), The contraction of a mixed tensor). Thus these formulas give smooth fields of types $(0,4)$, $(0,2)$, $(0,0)$ and $(0,2)$, respectively, independent of the chart. Symmetry of $\mathrm{Ric}$ and $G$ is proved separately in The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors ↗ clause (d).

• Sign convention. The signs of the curvature operator and of the component order are fixed by the displays of clauses (a) and (b) and are those used by every curvature item on this page and by the curvature consumers on later pages: with these conventions $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$ and $R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b=R^a{}_{bcd}{\mathrm{\partial }}_a$. Other sources order the loop indices differently; no formula of another source is used without checking its convention against this one.

• What is defined where. The operator and components of clauses (a)-(b) are defined from the connection alone; the lowered tensor, Ricci tensor, scalar curvature and Einstein tensor of clause (c) require the metric, and in clause (c) everything is defined chartwise from the components, with the chart-independence supplied by curvature tensoriality and intrinsic metric contractions, as specified there. The Christoffel symbols ${\mathrm{\Gamma }}^a{}_{bc}$ are chart functions and not tensor components (Christoffel symbols of an affine connection); no tensority of $\mathrm{\Gamma }$ is asserted here or later.

• Scope. This definition asserts no relation between curvature and any physical quantity, no equation satisfied by $\mathrm{Ric}$ or $G$, and no vanishing criterion: in particular ${\mathrm{\Gamma }}^a{}_{bc}(p)=0$ at a point does not imply $R^a{}_{bcd}(p)=0$, since curvature is second-order metric information in the precise sense of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula ↗ clause (c). The physical reading of $G_{ab}$ as one side of a field equation is a separate postulate of the framework and is not part of this definition.