Kretschmann scalar
E4709
The Kretschmann scalar is a curvature invariant in general relativity that combines components of the Riemann tensor into a single scalar quantity used to characterize the intensity of spacetime curvature, especially near singularities.
Observed surface forms (2)
| Surface form | As subject | As object |
|---|---|---|
| Kretschmann invariant | 0 | 1 |
| Schwarzschild metric | 1 | 0 |
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
curvature invariant
→
scalar quantity in general relativity → |
| alsoKnownAs |
Kretschmann scalar
→
surface form:
Kretschmann invariant
|
| appearsIn |
analysis of cosmological singularities
→
classification of exact solutions in general relativity → study of gravitational collapse → |
| behaviorNear | r = 0 in Schwarzschild spacetime diverges to infinity → |
| category | scalar polynomial curvature invariant → |
| constructedBy | index contraction → |
| constructedFrom | R_{abcd} R^{abcd} → |
| coordinateIndependence | yes → |
| definedAs | full contraction of the Riemann curvature tensor with itself → |
| definedOn | pseudo-Riemannian manifolds → |
| dependsOn | Riemann curvature tensor → |
| dimension | L^{-4} in geometrized units → |
| field |
theory of relativity
→
surface form:
general relativity
|
| finiteAt |
Schwarzschild radius
→
surface form:
Schwarzschild event horizon
|
| hasKretschmannScalar | K = 48 G^2 M^2 / (c^4 r^6) → |
| helpsIdentify | true curvature singularities independent of coordinates → |
| helpsShow |
Schwarzschild radius
→
surface form:
Schwarzschild radius is coordinate singularity
|
| introducedInContextOf | Einstein’s theory of gravitation → |
| invariantType | scalar polynomial invariant of Riemann tensor → |
| isFunctionOf | spacetime point → |
| isLocalQuantity | yes → |
| isRealValued | yes → |
| isScalarInvariantUnder |
Lorentz transformation
→
surface form:
Lorentz transformations
general coordinate transformations → |
| mathematicalExpression | K = R_{abcd} R^{abcd} → |
| namedAfter | Erich Kretschmann → |
| relatedTo |
Ricci tensor invariants
→
Weyl tensor invariants → |
| requires | metric-compatible connection → |
| tensorRank | 0 → |
| usedBy |
gravitational physicists
→
mathematical physicists → relativists → |
| usedIn |
black hole physics
→
cosmology → exact solutions of Einstein field equations → |
| usedTo |
characterize intensity of spacetime curvature
→
compare curvature strength between different spacetimes → detect curvature singularities → distinguish physical singularities from coordinate singularities → |
| valueFor |
Schwarzschild black hole
→
surface form:
Schwarzschild metric
|
| zeroIfAndOnlyIf | spacetime is flat in four-dimensional Lorentzian manifolds with vanishing other curvature invariants (with caveats) → |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Kretschmann invariant