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Measure of the curvature of a pseudo-Riemannian manifold
obtained from the Riemann tensor by subtracting a tensor that is a linear expression in the Ricci tensor. In general relativity, the Weyl curvature is the only
Weyl_tensor
Rank-3 tensor in general relativity associated with gauge fields
The Lanczos tensor or Lanczos potential is a rank 3 tensor in general relativity that generates the Weyl tensor. It was first introduced by Cornelius
Lanczos_tensor
Classification used in differential geometry and general relativity
classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold. It is most often applied in studying
Petrov_classification
German mathematician (1885–1955)
symmetry: see Weyl transformation Weyl tensor Weyl transform Weyl transformation Weyl–Schouten theorem Weyl's criterion (disambiguation) Weyl's lemma on hypoellipticity
Hermann_Weyl
Set of scalars in general relativity
the Weyl tensor.) As one might expect from the Ricci decomposition of the Riemann tensor into the Weyl tensor plus a sum of fourth-rank tensors constructed
Curvature invariant (general relativity)
Curvature_invariant_(general_relativity)
The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero: W i j k l = −
List of formulas in Riemannian geometry
List_of_formulas_in_Riemannian_geometry
Universal construction in multilinear algebra
the Weyl algebra and universal enveloping algebras. The tensor algebra has two different coalgebra structures. One is compatible with the tensor product
Tensor_algebra
Notation in general relativity
the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one
Newman–Penrose_formalism
Non-tensorial representation of the spin group
distinguished from the tensor representations given by Weyl's construction by the weights. Whereas the weights of the tensor representations are integer
Spinor
Notion in geometry
the Weyl tensor and Ricci tensor do not in general determine the full curvature tensor, the Riemann curvature tensor can be decomposed into a Weyl part
Curvature of Riemannian manifolds
Curvature_of_Riemannian_manifolds
Tensor that describes the 4D geometry of spacetime
manifold M {\displaystyle M} and the metric tensor is given as a covariant, second-degree, symmetric tensor on M {\displaystyle M} , conventionally denoted
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Tensor describing energy momentum density in spacetime
stress-energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor field quantity
Stress–energy_tensor
Tensor field in Riemannian geometry
mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the
Riemann_curvature_tensor
Local rescaling of a metric tensor
theoretical physics, the Weyl transformation, named after German mathematician Hermann Weyl, is a local rescaling of the metric tensor: g a b → e − 2 ω ( x
Weyl_transformation
Tensor index notation for tensor-based calculations
notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern
Ricci_calculus
vanishing of the Weyl tensor, while the Cotton tensor just becomes a constant times the divergence of the Weyl tensor. For n < 3 the Cotton tensor is identically
Cotton_tensor
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
inertia tensor of a body calculated at its center of mass, and R {\displaystyle \mathbf {R} } be the displacement vector of the body. The inertia tensor of
Moment_of_inertia
harmonic curvature or harmonic Weyl tensor. In fact, existence of Codazzi tensors impose strict conditions on the curvature tensor of the manifold. Also, the
Codazzi_tensor
Tensor in differential geometry
converge. Formally, it is a symmetric rank-two tensor obtained by taking a trace of the Riemann curvature tensor of a Riemannian or pseudo-Riemannian metric
Ricci_curvature
Set of five scalars
_{3},\Psi _{4}\}} which encode the ten independent components of the Weyl tensor of a four-dimensional spacetime. Given a complex null tetrad { l a ,
Weyl_scalar
Theorem in general relativity
existence of a certain type of congruence with algebraic properties of the Weyl tensor. More precisely, the theorem states that a vacuum solution of the Einstein
Goldberg–Sachs_theorem
Tensor invariant under permutations of vectors it acts on
In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T (
Symmetric_tensor
to the Ricci scalar, the trace-removed Ricci tensor, and the Weyl tensor of the Riemann curvature tensor. In particular, R = S + E + C {\displaystyle
Ricci_decomposition
Second-order tensor
dimension of the manifold. The Weyl tensor equals the Riemann curvature tensor minus the Kulkarni–Nomizu product of the Schouten tensor with the metric. In an
Schouten_tensor
Graphical notation for multilinear algebra calculations
essentially the composition of functions. In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting
Penrose_graphical_notation
Mathematical operation on vector spaces
two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense
Tensor_product
Operation that pairs a left and a right R-module into an abelian group
universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and
Tensor_product_of_modules
Tensor used in general relativity
differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature
Einstein_tensor
Generalization of tensor fields
differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing
Tensor_density
Mathematical object that describes the electromagnetic field in spacetime
electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a tensor that describes
Electromagnetic_tensor
Superenergy tensor of gravitational field flux-energy in a vacuum
{\displaystyle C_{abcd}} is the Weyl tensor. It was introduced by Lluís Bel in 1959. The Bel–Robinson tensor is constructed from the Weyl tensor in a manner analogous
Bel–Robinson_tensor
Branch of mathematics
various areas, including: Classical treatment of tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket notation Multilinear subspace learning
Multilinear_algebra
implements the computation of the Riemann curvature tensor and associated objects (Ricci tensor, Weyl tensor). SageManifolds can also deal with generic affine
Sage_Manifolds
Antisymmetric permutation object acting on tensors
independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms
Levi-Civita_symbol
Type of physical quantity
spacetime Tensor – Algebraic object with geometric applications Tensor density – Generalization of tensor fields Tensor field – Assignment of a tensor continuously
Pseudotensor
Tensor equal to the negative of any of its transpositions
tensor is antisymmetric with respect to its first three indices. If a tensor changes sign under exchange of each pair of its indices, then the tensor
Antisymmetric_tensor
energy–momentum tensor and the Petrov classification of the Weyl tensor. There are various methods of classifying these tensors, some of which use tensor invariants
Mathematics of general relativity
Mathematics_of_general_relativity
Branch of mathematics
importance was Hermann Weyl who made important contributions to the foundations of general relativity, introduced the Weyl tensor providing insight into
Differential_geometry
Coordinate-free definition of a tensor
mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear
Tensor_(intrinsic_definition)
Structure defining distance on a manifold
metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite if g ( v , v ) >
Metric_tensor
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
thought of as a tensor, and is written δ j i {\displaystyle \delta _{j}^{i}} . Sometimes the Kronecker delta is called the substitution tensor. When juxtaposition
Kronecker_delta
Shorthand notation for tensor operations
the multiplication. Given a tensor, one can raise an index or lower an index by contracting the tensor with the metric tensor, g μ ν {\displaystyle g_{\mu
Einstein_notation
field. Tensors also have extensive applications in physics: Electromagnetic tensor (or Faraday's tensor) in electromagnetism Finite deformation tensors for
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Gravity theories that are invariant under Weyl transformations
metric tensor and Ω ( x ) {\displaystyle \Omega (x)} is a function on spacetime. The simplest theory in this category has the square of the Weyl tensor as
Conformal_gravity
Assignment of a tensor continuously varying across a region of space
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space
Tensor_field
Quadratic scalar invariant
{\displaystyle C_{abcd}} is the Weyl tensor, the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. In d {\displaystyle
Kretschmann_scalar
Mathematical theorem in representation theory
the Schur–Weyl duality asserts that under the joint action of the groups Sk and GLn, the tensor space decomposes into a direct sum of tensor products of
Schur–Weyl_duality
the Weyl tensor, Ricci tensor, or Riemann tensor. These are often stated in terms of the Petrov classification of the possible symmetries of the Weyl tensor
Exact solutions in general relativity
Exact_solutions_in_general_relativity
Algebraic object with geometric applications
(electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, etc.), and general relativity (stress–energy tensor, curvature tensor, etc.). In
Tensor
Theory of gravitation as curved spacetime
stress–energy tensor, which includes both energy and momentum densities as well as stress: pressure and shear. Using the equivalence principle, this tensor is readily
General_relativity
semimetal Weyl sequence Weyl spinor Weyl representation Weyl sum, a type of exponential sum Weyl symmetry: see Weyl transformation Weyl tensor Weyl transform
List of things named after Hermann Weyl
List_of_things_named_after_Hermann_Weyl
Theorem in differential geometry
condition. In terms of the Riemann curvature tensor, the Ricci tensor, and the scalar curvature, the Weyl tensor of a pseudo-Riemannian metric g of dimension
Weyl–Schouten_theorem
tensor. It is, along with the Weyl tensor, one of the two primitive invariants in conformal differential geometry. Aside from the obstruction tensor,
Ambient_construction
Decomposition in multilinear algebra
multilinear algebra, the tensor rank decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is minimal
Tensor_rank_decomposition
Operation in mathematics
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. This example
Tensor_contraction
Covariant derivative of the metric tensor
In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It can be interpreted as the failure
Nonmetricity_tensor
Exterior algebraic map taking tensors from p forms to n-p forms
space L ( V , V ) {\displaystyle L(V,V)} is naturally isomorphic to the tensor product V ∗ ⊗ V ≅ V ⊗ V {\displaystyle V^{*}\!\!\otimes V\cong V\otimes
Hodge_star_operator
Specification of a derivative along a tangent vector of a manifold
fields) and to arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction). Given
Covariant_derivative
of tensor theory. For expositions of tensor theory from different points of view, see: Tensor Tensor (intrinsic definition) Application of tensor theory
Glossary_of_tensor_theory
Ways of writing certain laws of physics
t^{2}}-\nabla ^{2}.} The signs in the following tensor analysis depend on the convention used for the metric tensor. The convention used here is (+ − − −), corresponding
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
Canonical connection on a pseudo-Riemannian manifold
Arnoldus Schouten obtained analogous results. In the same year, Hermann Weyl generalized Levi-Civita's results. (M, g) denotes a pseudo-Riemannian manifold
Levi-Civita_connection
Object in differential geometry
differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors
Torsion_tensor
Matrix operation which flips a matrix over its diagonal
tensors Mathematics Kronecker delta Levi-Civita symbol Metric tensor Nonmetricity tensor Ricci curvature Riemann curvature tensor Torsion tensor Weyl
Transpose
Isomorphism between the tangent and cotangent bundles of a manifold
index of an ( r , s ) {\displaystyle (r,s)} tensor gives a ( r − 1 , s + 1 ) {\displaystyle (r-1,s+1)} tensor, while raising an index gives a ( r + 1 ,
Musical_isomorphism
Representation of mechanical stress at every point within a deformed 3D object
Cauchy stress tensor (symbol σ {\displaystyle {\boldsymbol {\sigma }}} , named after Augustin-Louis Cauchy), also called true stress tensor or simply stress
Cauchy_stress_tensor
Property of a mathematical space
tensors Mathematics Kronecker delta Levi-Civita symbol Metric tensor Nonmetricity tensor Ricci curvature Riemann curvature tensor Torsion tensor Weyl
Dimension
Abbreviation in the fields of special and general relativity
relativity, a four-tensor is an abbreviation for a tensor in a four-dimensional spacetime. General four-tensors are usually written in tensor index notation
Four-tensor
Law of physics
entropy, the arrow of time and the curvature of spacetime (encoded in the Weyl tensor). Loschmidt's paradox Entropy as an arrow of time Cold Big Bang See Ludwig
Past_hypothesis
An Einstein–Weyl geometry is a smooth conformal manifold, together with a compatible Weyl connection that satisfies an appropriate version of the Einstein
Einstein–Weyl_geometry
Algebraic operation on coordinate vectors
(single-) dot product between a tensor of order n {\displaystyle n} and a tensor of order m {\displaystyle m} is a tensor of order n + m − 2 {\displaystyle
Dot_product
Algebra associated to any vector space
complex Multilinear algebra Symmetric algebra, the symmetric analog Tensor algebra Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic
Exterior_algebra
Tensor having both covariant and contravariant indices
In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed
Mixed_tensor
Second order tensor in vector algebra
mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second-order tensor, written in a notation that fits in with vector algebra. There
Dyadics
Topological space that locally resembles Euclidean space
submanifold of Euclidean space is locally the graph of a function. Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture
Manifold
Conserved physical quantity; rotational analogue of linear momentum
as an anti-symmetric second order tensor, with components ωij. The relation between the two anti-symmetric tensors is given by the moment of inertia which
Angular_momentum
Spinning motion in theoretical physics
theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The spin tensor has application in general
Spin_tensor
Concept in mathematics
In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold
Tensor_bundle
Mathematical notation for tensors and spinors
between tensor factors of type V {\displaystyle V} and those of type V ∗ {\displaystyle V^{*}} . A general homogeneous tensor is an element of a tensor product
Abstract_index_notation
Array of numbers
multiplication can be defined with entries objects of a category equipped with a "tensor product" similar to multiplication in a ring, having coproducts similar
Matrix_(mathematics)
Rank-2 tensor
known conformally invariant tensor that is algebraically independent of the Weyl tensor. In abstract indices the Bach tensor is given by B a b = P c d W
Bach_tensor
Theory of interwoven space and time by Albert Einstein
coordinates are divided by c or factors of c±2 are included in the metric tensor. These numerous conventions can be superseded by using natural units where
Special_relativity
Vector behavior under coordinate changes
consequently a vector is called a contravariant tensor. A vector, which is an example of a contravariant tensor, has components that transform inversely to
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Differential form of degree one or section of a cotangent bundle
forms are also used in differential geometry to encode other tensors, such as the metric tensor; a selected basis of one-forms is called a coframe, and special
One-form
Tensor operator generalizes the notion of operators which are scalars and vectors
graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which
Tensor_operator
Generalization of the Levi-Civita connection
In differential geometry, a Weyl connection (also called a Weyl structure) is a generalization of the Levi-Civita connection that makes sense on a conformal
Weyl_connection
Breakdown of conformal symmetry at the quantum level
of the stress tensor must vanish for a conformally invariant theory. The trace of the stress tensor appears in the divergence of the Weyl current as an
Conformal_anomaly
Soviet mathematician (1910–1972)
Petrov classification. The Petrov classification is related with the Weyl tensor and it was first published by A. Z. Petrov in 1954. A biography of Petrov
Aleksei_Zinovyevich_Petrov
Expression that may be integrated over a region
covariant tensor field of rank k {\displaystyle k} . The differential forms on M {\displaystyle M} are in one-to-one correspondence with such tensor fields
Differential_form
Second-rank tensor in quantum chromodynamics
In theoretical particle physics, the gluon field strength tensor is a second-order tensor field characterizing the gluon interaction between quarks. The
Gluon_field_strength_tensor
Method for specifying point positions
tensors Mathematics Kronecker delta Levi-Civita symbol Metric tensor Nonmetricity tensor Ricci curvature Riemann curvature tensor Torsion tensor Weyl
Coordinate_system
mathematics and physics, a recurrent tensor, with respect to a connection ∇ {\displaystyle \nabla } on a manifold M, is a tensor T for which there is a one-form
Recurrent_tensor
Mathematical Concept
notation is as follows: Write down the second order tensor in matrix form (in the example, the stress tensor) Strike out the diagonal Continue on the third
Voigt_notation
British theoretical physicist (1928-2015)
physical meaning of the curvature tensor, gravitational waves, and the algebraic classification of the Weyl tensor, which he discovered in 1957 independently
Felix_Pirani
practice, the metric tensor g {\displaystyle g} of the manifold M {\displaystyle M} has to be conformal to the flat metric tensor η {\displaystyle \eta
Conformally_flat_manifold
Basis used to express spherical tensors
a basis in a 3-dimensional space is a valid definition for a spherical tensor, it only covers the case for when the rank k {\displaystyle k} is 1. For
Spherical_basis
Continuous surjection satisfying a local triviality condition
tensors Mathematics Kronecker delta Levi-Civita symbol Metric tensor Nonmetricity tensor Ricci curvature Riemann curvature tensor Torsion tensor Weyl
Fiber_bundle
Scalar quantities representing curvature
of different geometries. These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors formed from these by the operations
Curvature_invariant
Electromagnetism in general relativity
inverse of the metric tensor g α β {\displaystyle g_{\alpha \beta }} , and g {\displaystyle g} is the determinant of the metric tensor. Notice that A α {\displaystyle
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
tensor. For this very reason, it is commonly used to express the contribution that the Ricci curvature (or rather, the Schouten tensor) and the Weyl tensor
Kulkarni–Nomizu_product
Function that is invariant under all permutations of its variables
functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k {\displaystyle k} -tensors on a vector
Symmetric_function
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