Ricci scalar

E57421

curvature invariant differential geometric quantity geometric invariant scalar curvature scalar field

The Ricci scalar is a curvature invariant in differential geometry and general relativity that summarizes how spacetime is curved at a point by contracting the Ricci tensor into a single scalar quantity.

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All labels observed (2)

Label	Occurrences
Ricci scalar canonical	2
Ricci curvature scalar	1

Statements (49)

Predicate	Object
instanceOf	curvature invariant ⓘ differential geometric quantity ⓘ geometric invariant ⓘ scalar curvature ⓘ scalar field ⓘ
alsoKnownAs	Ricci scalar ⓘ surface form: Ricci curvature scalar scalar curvature ⓘ
appearsIn	Einstein field equations ⓘ Einstein–Hilbert action ⓘ
category	Riemannian geometry concept ⓘ general relativity concept ⓘ
constructedFrom	Christoffel symbols ⓘ inverse metric ⓘ
coordinateExpression	R = g^{\mu\nu} R_{\mu\nu} ⓘ R = g^{ij} R_{ij} ⓘ
definedOn	Riemannian manifolds ⓘ surface form: Riemannian manifold pseudo-Riemannian manifold ⓘ
dependsOn	Levi-Civita connection ⓘ Ricci tensor ⓘ metric tensor ⓘ
dimensionInUnits	inverse length squared ⓘ
equalsZeroFor	Ricci-flat manifolds ⓘ vacuum solutions of Einstein equations without cosmological constant ⓘ
fieldOfStudy	Riemannian manifolds ⓘ surface form: Riemannian geometry differential geometry ⓘ general relativity ⓘ pseudo-Riemannian geometry ⓘ
generalizes	Gaussian curvature in higher dimensions ⓘ
isContractionOf	Ricci curvature tensor ⓘ surface form: Ricci tensor Riemann curvature tensor ⓘ
isInvariantUnder	diffeomorphisms ⓘ
isLocalFunctionOf	metric and its first and second derivatives ⓘ
isScalarUnder	coordinate transformations ⓘ
namedAfter	Gregorio Ricci-Curbastro ⓘ
reducesTo	twice the Gaussian curvature in 2-dimensional Riemannian manifolds (up to conventions) ⓘ
relatedTo	Ricci flow ⓘ Weyl tensor ⓘ sectional curvature ⓘ
roleIn	contributes to gravitational dynamics in general relativity ⓘ encodes volume-averaged curvature at a point ⓘ summarizes trace of Ricci tensor ⓘ
signDependsOn	metric signature convention ⓘ
symbol	R ⓘ R ⓘ
tensorRank	0 ⓘ
usedIn	curvature classification of spacetimes ⓘ definition of scalar curvature invariants ⓘ f(R) gravity ⓘ modified gravity theories ⓘ

Referenced by (3)

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

Einstein tensor → constructedFrom → Ricci scalar ⓘ

Ricci scalar → alsoKnownAs → Ricci scalar ⓘ

this entity surface form: Ricci curvature scalar

Einstein–Hilbert action → dependsOn → Ricci scalar ⓘ