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.
Observed surface forms (1)
| Surface form | As subject | As object |
|---|---|---|
| Ricci curvature scalar | 0 | 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 (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Ricci curvature scalar