Weyl tensor

E287410

curvature tensor geometric object mathematical tensor

The Weyl tensor is the traceless part of the Riemann curvature tensor in differential geometry and general relativity, encoding the purely shape-distorting (conformal) aspects of spacetime curvature independent of matter content.

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All labels observed (2)

Label	Occurrences
Weyl tensor canonical	2
conformal curvature tensor	1

Statements (48)

Predicate	Object
instanceOf	curvature tensor ⓘ geometric object ⓘ mathematical tensor ⓘ
alsoKnownAs	Weyl tensor ⓘ surface form: conformal curvature tensor
appearsIn	Petrov classification ⓘ
canBeNonzeroIn	vacuum regions of spacetime ⓘ
captures	nonlocal aspects of gravitational field ⓘ
decomposition	Riemann = Weyl + Ricci-part + scalar-curvature-part ⓘ
definedOn	Riemannian manifold ⓘ pseudo-Riemannian manifold ⓘ
describes	free gravitational field ⓘ tidal gravitational effects in vacuum ⓘ
dimensionRequirement	manifold dimension ≥ 3 ⓘ
doesNotDependOn	local matter content ⓘ
encodes	conformal curvature ⓘ shape-distorting aspects of curvature ⓘ
field	differential geometry ⓘ general relativity ⓘ
hasSymmetry	antisymmetric in first and second index pairs ⓘ satisfies first Bianchi identity ⓘ symmetric under exchange of index pairs ⓘ
isComponentOf	spacetime curvature ⓘ
isConformallyInvariant	in dimension 4 ⓘ
isConstructedFrom	Ricci tensor ⓘ Riemann curvature tensor ⓘ surface form: Riemann tensor scalar curvature ⓘ
isDivergenceFreeIn	vacuum Einstein equations ⓘ
isIndependentOf	Ricci tensor ⓘ
isPartOf	Riemann curvature tensor ⓘ surface form: Riemann curvature tensor decomposition
isTraceless	with respect to any pair of indices ⓘ
isTracelessPartOf	Riemann curvature tensor ⓘ
isUsedIn	conformal geometry ⓘ mathematical relativity ⓘ study of gravitational waves ⓘ
isUsedToDefine	Newman–Penrose formalism ⓘ surface form: Newman–Penrose Weyl scalars
isZeroIfAndOnlyIf	spacetime is locally conformally flat (dimension ≥ 4) ⓘ
isZeroIn	Minkowski space-time ⓘ surface form: Minkowski spacetime
namedAfter	Hermann Weyl ⓘ
rank	(0,4) tensor ⓘ (1,3) tensor ⓘ
relatedTo	gravitational radiation ⓘ
satisfies	Bianchi identities ⓘ same index symmetries as Riemann tensor ⓘ
transformsHomogeneouslyUnder	conformal rescalings of the metric ⓘ
usedFor	classifying algebraic types of spacetime curvature ⓘ
vanishesIdenticallyIn	all 2-dimensional manifolds ⓘ all conformally flat manifolds ⓘ
vanishesIn	FLRW cosmological models ⓘ surface form: Friedmann–Lemaître–Robertson–Walker spacetimes

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Ricci scalar → relatedTo → Weyl tensor ⓘ

Weyl → knownFor → Weyl tensor ⓘ

subject surface form: Hermann Weyl

Weyl tensor → alsoKnownAs → Weyl tensor ⓘ

this entity surface form: conformal curvature tensor