Ricci and Weyl Curvature

The Riemann tensor separates algebraically into a trace-free Weyl tensor and pieces built from the Ricci tensor and scalar curvature. In four dimensions each side of that split carries ten independent components. Ricci curvature governs volume convergence and is constrained locally by Einstein’s equation; Weyl curvature carries the remaining tidal shape distortion. Those interpretations are useful, but neither “Ricci equals matter” nor “Weyl equals radiation” is universally correct without further hypotheses.

Required background. We use the algebra and contractions of the Riemann tensor together with the physical limits of local gravitational measurement.

Helpful background. Preparation in Minkowski spacetime, tensor operations, orthonormal tetrads, and Jacobi fields helps connect the algebra to measurements.

Algebraic decomposition

In spacetime dimension $D\ge3$, define the Schouten tensor

$$P_{ab}=\frac{1}{D-2} \left(R_{ab}-\frac{R}{2(D-1)}g_{ab}\right).$$

With antisymmetrization of weight one-half, the Riemann tensor decomposes as

$$R_{abcd} =C_{abcd} +2\left(g_{a[c}P_{d]b}-g_{b[c}P_{d]a}\right).$$

The Weyl tensor $C_{abcd}$ has the same index symmetries as Riemann curvature and all of its traces vanish:

$$g^{ac}C_{abcd}=0.$$

Contracting the decomposition on $a$ and $c$ recovers $R_{bd}$; this is an immediate normalization check. Carroll derives the same split and its dimension dependence Carroll 1997, § 3, pp. 84–88, PDF.

In four dimensions a Riemann tensor has twenty independent components. The symmetric Ricci tensor has ten. The trace-free Weyl remainder therefore has ten. In three dimensions the Weyl tensor vanishes identically, so Ricci curvature determines the full Riemann tensor. In two dimensions the Schouten formula is singular because the Riemann tensor has only one independent scalar component and must be treated separately.

Volume change and shape change

Geodesic deviation is controlled by $R^a{}_{bcd}$. For a timelike congruence with tangent $u^a$, the trace of the tidal matrix is

$$\delta^{ij}R_{i0j0}=R_{00}=R_{ab}u^a u^b$$

in an orthonormal frame. This trace contributes to the evolution of the volume of a small cloud of freely falling particles. The trace-free part changes the cloud’s shape through shear without directly changing its infinitesimal volume at that instant.

The Weyl tensor is trace-free, so its electric part supplies precisely a trace-free tidal contribution. Ricci curvature can also contain anisotropic information. In four dimensions the displayed decomposition gives, in an observer’s orthonormal frame,

$$R_{i0j0}=C_{i0j0}-\frac12R_{ij} +\left(\frac12R_{00}+\frac16R\right)\delta_{ij}.$$

Taking the spatial trace recovers $R_{00}$ because $R=-R_{00}+\sum_iR_{ii}$. Taking the spatial trace-free part instead gives $\mathcal E^{\rm TF}_{ij}=C_{i0j0}-\tfrac12R^{\rm TF}_{ij}$. Thus subtracting the measured tidal trace does not generally isolate electric Weyl curvature: anisotropic Ricci curvature may remain. The slogan “Ricci changes volume, Weyl changes shape” is a guide to the congruence equations, not a complete component-by-component classification.

For null bundles, Ricci contraction $R_{ab}k^ak^b$ enters the expansion equation, while screen-projected Weyl curvature drives shear. This distinction becomes central in gravitational lensing and focusing.

Relation to gravitational sources

Einstein’s equation relates the Einstein tensor, not simply $R_{ab}$, to stress-energy:

$$G_{ab}+\Lambda g_{ab}=8\pi G\,T_{ab}.$$

After tracing and rearranging, local matter variables determine Ricci curvature in classical Einstein gravity. That conclusion assumes the field equation; it is not an algebraic property of curvature. In a higher-curvature or nonminimally coupled theory, the relation changes.

Weyl curvature is not fixed pointwise by $T_{ab}$ through Einstein’s equation. It is constrained and propagated by the differential Bianchi identities and by boundary or initial data. For that reason it is often called the locally free gravitational field. “Free” does not mean arbitrary: constraints, global geometry, and the surrounding matter distribution still restrict it.

Discriminating examples

A Ricci-flat region satisfies $R_{ab}=0$ but can have $C_{abcd}\ne0$. The exterior vacuum field of an isolated spherical source has tidal stretching and compression even where the local stress-energy vanishes; there the Riemann tensor equals the Weyl tensor. This refutes the claim that curvature must vanish wherever matter is absent.

Conversely, a constant-curvature spacetime obeys

$$R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc}),$$

so its Weyl tensor vanishes while its Ricci tensor is $R_{ab}=(D-1)K g_{ab}$. Conformal flatness therefore does not imply flatness.

Nor does nonzero Weyl curvature by itself identify gravitational radiation. Static tidal fields can have nonzero Weyl curvature. A radiation interpretation requires additional structure—typically wave propagation, asymptotic conditions, or invariant algebraic and dynamical criteria.

Exercises

Show from the constant-curvature form that $R_{ab}=(D-1)Kg_{ab}$ and $R=D(D-1)K$, then verify that the Schouten decomposition gives $C_{abcd}=0$.

Solution

Contracting $a$ with $c$ gives

$$R_{bd}=g^{ac}R_{abcd} =K(Dg_{bd}-g_{bd})=(D-1)Kg_{bd}.$$

A second contraction gives $R=D(D-1)K$. Substitution yields $P_{ab}=(K/2)g_{ab}$, so the trace-built term is

$$2(g_{a[c}P_{d]b}-g_{b[c}P_{d]a}) =K(g_{ac}g_{bd}-g_{ad}g_{bc}),$$

equal to the full Riemann tensor. The Weyl remainder is zero.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.