Ricci decomposition

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

The pieces appearing in the decomposition

The decomposition is

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

The three pieces are:

1. the scalar part, the tensor S_abcd
2. the semi-traceless part, the tensor E_abcd
3. the fully traceless part, the Weyl tensor C_abcd

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_ab, and only makes sense if the dimension satisfies n > 2.

The scalar part

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

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

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

The semi-traceless part

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

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_ab 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.

Mathematical definition

Mathematically, the Ricci decomposition is the decomposition of the space of all tensors having the symmetries of the Riemann tensor into its irreducible representations for the action of the orthogonal group (Besse 1987, Chapter 1, §G). Let V be an n-dimensional vector space, equipped with a metric tensor (of possibly mixed signature). Here V is modeled on the cotangent space at a point, so that a curvature tensor R (with all indices lowered) is an element of the tensor product V⊗V⊗V⊗V. The curvature tensor is skew symmetric in its first and last two entries:

$R(x,y,z,w)=-R(y,x,z,w)=-R(x,y,w,z)\,$

and obeys the interchange symmetry

$R(x,y,z,w) = R(z,w,x,y),\,$

for all x,y,z,w ∈ V^∗. As a result R is an element of the subspace S²Λ²V, the second symmetric power of the second exterior power of V. A curvature tensor must also satisfy the Bianchi identity, meaning that it is in the kernel of the linear map

$b(R)(x,y,z,w) = R(x,y,z,w) + R(y,z,x,w) + R(z,x,y,w).\,$

The space RV = ker b in S²Λ²V is the space of algebraic curvature tensors. The Ricci decomposition is the decomposition of this space into irreducible factors. The Ricci contraction mapping

$c : S^2\Lambda^2 V \to S^2V$

is given by

$c(R)(x,y) = \operatorname{tr}R(x,\cdot,y,\cdot).$

This associates a symmetric 2-form to an algebraic curvature tensor. Conversely, given a pair of symmetric 2-forms h and k, the Kulkarni–Nomizu product of h and k

$h\circ k(x,y,z,w) = h(x,z)k(y,w)+h(y,w)k(x,z) -h(x,w)k(y,z)-h(y,z)k(x,w)$

produces an algebraic curvature tensor.

If n ≥ 4, then there is an orthogonal decomposition into (unique) irreducible subspaces

RV = SV ⊕ EV ⊕ CV

where

$\mathbf{S}V = \mathbb{R} g\circ g$

$\mathbf{E}V = g\circ S^2_0V$, where S2
0V is the space of trace-free symmetric 2-forms

$\mathbf{C}V = \ker c \cap \ker b.$

The parts S, E, and C of the Ricci decomposition of a given Riemann tensor R are the orthogonal projections of R onto these invariant factors. In particular,

R = S + E + C

is an orthogonal decomposition in the sense that

| R | ² = | S | ² + | E | ² + | C | ².

This decomposition expresses the space of tensors with Riemann symmetries as a direct sum of the scalar submodule, the Ricci submodule, and Weyl submodule, respectively. Each of these modules is an irreducible representation for the orthogonal group (Singer & Thorpe 1968), and thus the Ricci decomposition is a special case of the splitting of a module for a semisimple Lie group into its irreducible factors. In dimension 4, the Weyl module decomposes further into a pair of irreducible factors for the special orthogonal group: the self-dual and antiself-dual parts W⁺ and W⁻.

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^ab 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.

See also

• Bel decomposition of the Riemann tensor
• Conformal geometry
• Schouten tensor
• Trace-free Ricci tensor

References

• Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: Springer-Verlag, pp. xii+510, ISBN 978-3-540-15279-8 .

• 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.

• Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9 . Section 6.1 discusses the decomposition. Versions of the decomposition also enter into the discussion of conformal and projective geometries, in chapters 7 and 8.

• Singer, I.M.; Thorpe, J.A. (1969), "The curvature of 4-dimensional Einstein spaces", Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, pp. 355–365 .