Ricci decomposition

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In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties.

1 Mathematical definition
2 The pieces appearing in the decomposition
3 Physical interpretation
4 See also
5 References

Mathematical definition

The decomposition is

$R_{abcd}= \, S_{abcd}+E_{abcd}+C_{abcd}.$

The three pieces are:

the scalar part, the tensor $S a b c d$
the semi-traceless part, the tensor $E a b c d$
the fully traceless part, the Weyl tensor $C a b c d$

Each piece possesses all the algebraic symmetries of the Riemann tensor itself, but has additional properties.

The decomposition works in slightly different ways depending on the signature of the metric tensor $g a b$ , and only makes sense if the dimension satisfies $n > 2$ .

The pieces appearing in the decomposition

The scalar part

$S_{abcd} = \frac{R}{(n-1) \, (n-2)} \, H_{abcd}$

is built using the Ricci scalar $R = {R^m}_m$ , where $R a b$ is the Ricci tensor, and a tensor constructed algebraically from the metric tensor $g a b$ ,

$H_{abcd} = g_{ad} \, g_{cb} - g_{ac} \, g_{db} = 2g_{a[d} \, g_{c]b}.$

The semi-traceless part

$E_{abcd} = \frac{1}{n-2} \, \left( g_{ac} \, R_{bd} - g_{ad} \, R_{bc} + g_{bd} \, R_{ac} - g_{bc} \, R_{ad} \right) = \frac{2}{n-2} \, \left( g_{a[c} \, R_{d]b} - g_{b[c} \, R_{d]a} \right)$

is constructed algebraically using the metric tensor and the traceless part of the Ricci tensor

$S_{ab} = R_{ab} - \frac{1}{n} \, g_{ab} \, R$

where $g a b$ is the metric tensor.

The Weyl tensor or conformal curvature tensor is completely traceless, in the sense that taking the trace, or contraction, over any pair of indices gives zero. Hermann Weyl showed that this tensor measures the deviation of a semi-Riemannian manifold from conformal flatness; if it vanishes, the manifold is (locally) conformally equivalent to a flat manifold.

No additional differentiation is needed anywhere in this construction.

In the case of a Lorentzian manifold, $n = 4$ , the Einstein tensor $G_{ab} = R_{ab} - 1/2 \, g_{ab} R$ has, by design, a trace which is just the negative of the Ricci scalar, so that the traceless part of the Einstein tensor agrees with the traceless part of the Ricci tensor.

$S_{ab} = R_{ab} - \frac{1}{4} \, g_{ab} \, R = G_{ab} - \frac{1}{4} \, g_{ab} \, G$

Terminological note: the notation $R_{abcd}, \, C_{abcd}$ is standard in the modern literature, the notations $S_{ab}, \, E_{abcd}$ are commonly used but not standardized, and there is no standard notation for the scalar part.

Physical interpretation

The Ricci decomposition can be interpreted physically in Einstein's theory of general relativity, where it is sometimes called the Géhéniau-Debever decomposition. In this theory, the Einstein field equation

$G^{ab} = 8 \pi \, T^{ab}$

where $T a b$ is the stress-energy tensor describing the amount and motion of all matter and all nongravitational field energy and momentum, states that the Ricci tensor-- or equivalently, the Einstein tensor-- represents that part of the gravitational field which is due to the immediate presence of nongravitational energy and momentum. The Weyl tensor represents the part of the gravitational field which can propagate as a gravitational wave through a region containing no matter or nongravitational fields. Regions of spacetime in which the Weyl tensor vanishes contain no gravitational radiation and are also conformally flat, which implies for example that light rays passing through such a region exhibit no light bending.

References

Hawking, S. W.; and Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. ISBN 0-521-09906-4. See section 2.6 for the decomposition. This book uses opposite signature but the same Landau-Lifshitz spacelike sign convention used in the Wikipedia.

Weinberg, Steven (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: John Wiley & Sons. ISBN 0-471-92567-5. See section 6.7 for a discussion of the decomposition (but note different sign conventions).

Wald, Robert M. (1984). General Relativity. The University of Chicago Press. ISBN 0-226-87033-2. See section 3.2 for a discussion of the decomposition.

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