Weyl tensor

Definition

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. This tensor has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor that is a linear expression in the Ricci tensor.

Related concepts

Abstract index notation Affine connection Albert Einstein Angular momentum Antisymmetric tensor Augustin-Louis Cauchy Basis (linear algebra)Bernhard Riemann Carl Friedrich Gauss Cartan formalism (physics)Cauchy stress tensor Christoffel symbols Computer vision Conformal map Conformally flat Connection form Continuum mechanics Coordinate system Cotton tensor Covariance and contravariance of vectors Covariant derivative Curvature Curvature of Riemannian manifolds Differential form Differential geometry Dimension Direct sum of vector bundles Dyadics Einstein field equation Einstein manifold Einstein notation Einstein tensor Electromagnetic tensor Electromagnetism Elwin Bruno Christoffel Encyclopedia of Mathematics Engineering Euclidean geometry European Mathematical Society Exterior calculus Exterior covariant derivative Exterior derivative Exterior form Exterior product Fiber bundle General relativity Geodesic George Francis Rayner Ellis Glossary of tensor theory Gluon field strength tensor Gravitational waves Gregorio Ricci-Curbastro Hermann Grassmann Hermann Weyl Hodge star operator Index notation Integrability condition Irreducible representation Isadore Singer Isothermal coordinates Jan Arnoldus Schouten Kronecker delta Kulkarni–Nomizu product Lanczos tensor Leonhard Euler Levi-Civita connection Levi-Civita symbol Lie derivative Linear map Local coordinate system Manifold Mathematician Mathematics Matrix (mathematics)Method of characteristics Metric tensor Metric tensor (general relativity)Mixed tensor Moment of inertia Multi-index notation Multilinear algebra Multivector Necessary condition Nonmetricity tensor Nordström's theory of gravitation One-form Operation (mathematics)Orthogonal Orthogonal group Peeling theorem Penrose graphical notation Petrov classification Physics Plebanski tensor Pseudo-Riemannian manifold Pseudotensor Raising and lowering indices Ricci-flat manifold Ricci calculus Ricci curvature Ricci decomposition Ricci tensor Riemann curvature tensor Riemannian manifold Scalar curvature Schouten tensor Spacetime Special orthogonal group Spin tensor Spinor Springer-Verlag Stephen Hawking Stress–energy tensor Sufficient condition Symmetric tensor Symmetrization Tensor Tensor (intrinsic definition)Tensor algebra Tensor bundle Tensor calculus Tensor contraction Tensor density Tensor field Tensor operator Tensor product Tensors in curvilinear coordinates Tetrad (index notation)The Large Scale Structure of Space-Time Tidal force Torsion tensor Trace (linear algebra)Traceless Transport phenomena Transpose Tullio Levi-Civita Two-point tensor Van der Waerden notation Vector (mathematics and physics)Vector space Voigt notation Weyl curvature hypothesis Weyl scalar Woldemar Voigt Élie Cartan Øyvind Grøn

38 concepts already in your glossary