Christoffel symbols
E58190
Christoffel symbols are mathematical objects in differential geometry that represent how coordinate bases change from point to point on a curved space or spacetime, and are used to define covariant derivatives and geodesics.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Christoffel symbols canonical | 11 |
| Christoffel symbol | 1 |
| Levi-Civita connection coefficients | 1 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical object
ⓘ
tensorial connection coefficient ⓘ |
| alsoCalled |
Christoffel symbols
ⓘ
surface form:
Levi-Civita connection coefficients
connection coefficients ⓘ |
| appearIn |
Lagrangian formulation of geodesic motion
ⓘ
equations of motion in curved spacetime ⓘ |
| associatedWith |
Levi-Civita connection
ⓘ
affine connection ⓘ metric-compatible connection ⓘ torsion-free connection ⓘ |
| category | coordinate-dependent quantities ⓘ |
| definedFrom | metric tensor ⓘ |
| definedOn | smooth manifold ⓘ |
| definedRelativeTo | coordinate chart ⓘ |
| dependOn | choice of coordinates ⓘ |
| enterEquation |
covariant derivative ∇_μ V^ν = ∂_μ V^ν + Γ^ν_{μρ} V^ρ
ⓘ
covariant derivative ∇_μ ω_ν = ∂_μ ω_ν − Γ^ρ_{μν} ω_ρ ⓘ geodesic equation d^2x^μ/dτ^2 + Γ^μ_{νρ}(dx^ν/dτ)(dx^ρ/dτ) = 0 ⓘ |
| expressibleInTermsOf | first derivatives of the metric ⓘ |
| field |
Riemannian geometry
ⓘ
differential geometry ⓘ general relativity ⓘ pseudo-Riemannian geometry ⓘ |
| indexNotation | Γ^k_{ij} ⓘ |
| mathematicalNature | collection of functions on the manifold in a given chart ⓘ |
| namedAfter | Elwin Bruno Christoffel ⓘ |
| notTensorUnder | general coordinate transformations ⓘ |
| relatedConcept |
Levi-Civita connection
ⓘ
Riemann curvature tensor ⓘ affine connection ⓘ covariant derivative ⓘ geodesic ⓘ metric tensor ⓘ parallel transport ⓘ |
| satisfyProperty | symmetric in lower indices for Levi-Civita connection ⓘ |
| symbol | Γ ⓘ |
| transformAs | connection coefficients under coordinate changes ⓘ |
| usedFor |
defining covariant derivatives
ⓘ
defining geodesics ⓘ describing change of coordinate bases ⓘ expressing curvature components ⓘ expressing parallel transport ⓘ writing covariant derivative of tensor fields ⓘ writing geodesic equation ⓘ |
| usedIn |
Einstein field equations
ⓘ
surface form:
Einstein field equations formulation
computing Ricci tensor ⓘ computing Riemann curvature tensor ⓘ computing scalar curvature ⓘ |
| vanishIn | local inertial coordinates at a point for Levi-Civita connection ⓘ |
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Levi-Civita connection coefficients
this entity surface form:
Christoffel symbol