Weyl TensorEdit
The Weyl tensor is a centerpiece of the modern study of spacetime curvature in differential geometry and general relativity. It encapsulates the part of curvature that cannot be pinned down by local matter alone and, in four dimensions, is the trace-free, conformally invariant remnant of the full Riemann curvature tensor. In practical terms, the Weyl tensor measures how nearby freely falling particles deviate from each other in a way that is not directly sourced by local energy-momentum but rather by the tidal and radiative aspects of the gravitational field. It vanishes for spacetimes that are locally conformally flat, while it carries the information about tidal forces and gravitational waves in vacuum regions.
As a mathematical object, the Weyl tensor sits in the Ricci decomposition of the Riemann curvature tensor. In n dimensions, the Riemann tensor R^a_{bcd} can be decomposed into a portion determined by the Ricci tensor R_{bd} and scalar curvature R, and a trace-free portion that is the Weyl tensor C^a_{bcd}. In four dimensions, this decomposition can be written schematically as
R^a_{bcd} = C^a_{bcd} + a specific combination of R_{bd}, R, and g_{ab} that subtracts out the traces,
where the precise coefficients depend on the dimension. The upshot is that C^a_{bcd} contains the conformal part of curvature: it vanishes exactly when the metric is locally conformally flat, i.e., when g̃{ab} = Ω^2 g{ab} makes the spacetime look flat locally. This makes the Weyl tensor a natural carrier of conformal geometry, the study of properties invariant under angle-preserving rescalings of the metric conformal geometry.
Definition and decomposition
- Riemann curvature tensor: R^a_{bcd}, the full curvature object encoding how spacetime bends.
- Ricci tensor and scalar: R_{bd} and R, obtained by contracting indices of the Riemann tensor.
- Weyl tensor: C^a_{bcd}, the trace-free part of the Riemann tensor that remains after removing the contributions fixed by R_{bd} and R.
In four dimensions (n = 4) the Weyl tensor has 10 independent components. It is conformally invariant: if the metric is rescaled by a smooth, nonvanishing factor Ω^2, then the Weyl tensor remains unchanged up to the scaling performed by the indices, C̃^a_{bcd} = C^a_{bcd} under appropriate index placements. This invariance is one of the reasons the Weyl tensor is central to the study of conformal geometry and conformal structure on spacetimes. By contrast, in three dimensions the Weyl tensor vanishes identically, which means conformal curvature in that case is fully captured by the Ricci part; this highlights the special geometric role played by dimension.
Physical interpretation in general relativity
In the framework of General relativity, Einstein’s field equations link curvature to matter and energy through the stress-energy tensor T_{ab}. The decomposition of curvature into Weyl and Ricci parts clarifies which aspects of curvature are locally determined by matter and which aspects reflect the “free” gravitational field. In vacuum regions where T_{ab} = 0, the Ricci tensor vanishes (R_{ab} = 0), and the Riemann tensor reduces to the Weyl tensor. Consequently, in such regions the Weyl tensor encodes all the curvature: tidal forces, the distortion of shapes, and the propagation of gravitational waves.
A particularly useful viewpoint is to split the Weyl tensor into its electric and magnetic parts, relative to a chosen timelike observer field u^a. The electric part E_{ab} and the magnetic part B_{ab} carry the intuitive content of tidal stretching and frame-dragging–like effects, respectively. Gravitational radiation, measured in astrophysical events, leaves a characteristic imprint in the Weyl tensor components, and this link is central to the modern era of gravitational-wave astronomy. The Weyl tensor’s behavior under conformal transformations also illuminates how gravitational fields propagate through spacetime, independent of how matter clusters locally.
Algebraic structure and notable special cases
- Conformal flatness: If a spacetime is conformally flat, its Weyl tensor vanishes. This criterion is a practical diagnostic for recognizing spacetimes that are, up to a rescaling of the metric, flat. The Friedmann–Lemaître–Robertson–Walker (FLRW) cosmologies, which are used to model a homogeneous and isotropic universe, are conformally flat, so their Weyl tensor is zero.
- Schwarzschild and Kerr spacetimes: In the Schwarzschild solution, which describes a non-rotating black hole, the nonzero curvature is carried by the Weyl part (the Ricci tensor vanishes in the vacuum region outside the mass). In the Kerr solution, representing a rotating black hole, the Weyl tensor remains the dominant curvature carrier in vacuum.
- Petrov classification: The algebraic classification of the Weyl tensor in four dimensions is the Petrov scheme, which distinguishes spacetimes by the multiplicity and alignment of principal null directions. This classification provides deep insights into the geometric structure of gravitational fields and helps organize exact solutions and their physical interpretations. See Petrov classification.
Relations to other curvature objects and equations
- Riemann curvature tensor: The Weyl tensor is obtained by subtracting the parts of curvature fixed by the Ricci tensor and scalar curvature from the full Riemann tensor. This relation makes the Weyl tensor a precise measure of the part of curvature not determined by local matter content alone.
- Einstein field equations: The Einstein tensor G_{ab} combines Ricci curvature and scalar curvature and is directly tied to T_{ab}. The Weyl tensor, by contrast, captures the “free” gravitational field that can exist even when local energy-momentum is constrained, and it carries the information about gravitational radiation and tidal effects that is not algebraically determined by T_{ab}.
- Gravitational waves and tidal forces: The Weyl tensor is the mathematical carrier of gravitational radiation in the vacuum region surrounding sources, and its dynamic components reveal how waves propagate and interact with matter through tidal distortions.
Controversies and debates
Within the broader landscape of gravitational theory, there are ongoing debates about the role of curvature components and alternative theories that emphasize conformal structure. Proponents of conformal gravity, which treats the Weyl tensor as central to the action (for example, via a Weyl-squared term in the action), argue that conformal invariance can address certain cosmological puzzles without invoking dark matter or dark energy. Critics, however, point to empirical challenges: difficulties in fitting galaxy rotation curves, lensing observations, and cosmological data as cleanly as the standard Einstein gravity framework with a dark sector, and tension with solar-system tests. In this sense, the Weyl tensor sits at the heart of a broader conversation about how best to describe gravity at all scales, with the mainstream framework rooted in general relativity and its coupling to matter, and competing approaches emphasizing conformal symmetry and higher-derivative actions facing rigorous scrutiny.
The Weyl tensor also acts as a bridge between classical differential geometry and modern perspectives on spacetime structure, including discussions around conformal compactification, holographic ideas, and the geometry of gravitational radiation. Those debates tend to center on theoretical elegance and compatibility with data, rather than political or cultural critique, reflecting a tradition of prioritizing predictive power and empirical adequacy in physical theories.
See also