Weyl tensor

Weyl tensor. In the mathematical field of differential geometry and the physics of general relativity, the Weyl tensor is a measure of the curvature of a pseudo-Riemannian manifold that is not captured by its Ricci curvature. It represents the traceless component of the Riemann curvature tensor, describing how the shape of a body is distorted by tidal forces as it moves along geodesics, independent of changes in its volume. This tensor is central to understanding conformal geometry and the propagation of gravitational waves in vacuum.

Definition and mathematical expression

The Weyl tensor is defined for a manifold of dimension *n* ≥ 3. In terms of the Riemann curvature tensor *R_abcd*, the Ricci tensor *R_ab*, and the Ricci scalar *R*, its components in a coordinate system are given by a specific algebraic combination that ensures it is traceless. This expression involves the metric tensor *g_ab* and symmetries under the permutation group. In four-dimensional spacetime, a common form utilizes the Levi-Civita connection associated with the Einstein field equations. The definition ensures invariance under Weyl transformations, a property studied extensively by Hermann Weyl in his work on gauge theory.

Properties

A fundamental property is that it is conformally invariant, meaning its form is preserved under conformal maps of the metric, a concept pivotal in conformal field theory. It shares the algebraic symmetries of the Riemann curvature tensor, including antisymmetry and satisfaction of the Bianchi identity. In three dimensions, it vanishes identically, making phenomena like gravitational radiation a feature of higher-dimensional spacetimes. Its tensor contraction with itself produces important invariants like the Kretschmann scalar, used to analyze singularities in solutions like the Schwarzschild metric. The Petrov classification scheme, developed by A. Z. Petrov, categorizes spacetimes based on the eigenvalue structure of this tensor.

Physical interpretation

Physically, it encapsulates the tidal force component of the gravitational field that stretches and squeezes matter without changing its volume, analogous to the free-space effects in Newtonian gravity. This is distinct from the volume-changing effects described by the Ricci tensor, which are linked to the presence of matter and energy via the Einstein field equations. In vacuum regions, where the Ricci tensor vanishes, the entire curvature is described by this tensor, governing the dynamics of gravitational waves as predicted by the linearized gravity approximation. Its behavior near black hole horizons, such as in the Kerr metric, influences phenomena like frame-dragging and the Penrose process.

Relation to other curvature tensors

It is the traceless part of the Riemann curvature tensor, obtained by subtracting combinations of the Ricci tensor and Ricci scalar. This decomposition is formalized in the Ricci decomposition theorem, analogous to splitting the electromagnetic tensor into electric and magnetic parts. In the context of the Einstein field equations, the tensor is independent of the local stress–energy tensor, unlike the Einstein tensor which is directly proportional to it. Its dual, constructed using the Hodge star operator, plays a role in formulations of supergravity and string theory. Relationships with the Weyl–Lewis–Papapetrou coordinates are used in studying stationary spacetimes.

Applications in general relativity

A primary application is in characterizing gravitational waves, where its non-zero components in a vacuum signify radiative solutions, as analyzed in the pp-wave spacetime. It is crucial for understanding the geodesic deviation equation in empty space, relevant for detectors like LIGO. In cosmology, its vanishing in Friedmann–Lemaître–Robertson–Walker metric models implies these universes are conformally flat, impacting studies of the cosmic microwave background. The tensor's behavior is key in Penrose–Hawking singularity theorems and studies of cosmic censorship. Advanced topics involve its role in the AdS/CFT correspondence linking quantum gravity to conformal field theory on the boundary of anti-de Sitter space.

Category:Tensors in general relativity Category:Differential geometry Category:Conformal geometry