Christoffel symbols

E58190

mathematical object tensorial connection coefficient

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.

Aliases (1)

Levi-Civita connection coefficients ×1

Statements (49)

Predicate	Object
instanceOf	mathematical object → tensorial connection coefficient →
alsoCalled	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 formulation → computing Ricci tensor → computing Riemann curvature tensor → computing scalar curvature →
vanishIn	local inertial coordinates at a point for Levi-Civita connection →

Referenced by (6)

Subject (surface form when different)	Predicate
Ricci scalar → Riemann curvature tensor →	constructedFrom
Christoffel symbols ("Levi-Civita connection coefficients") →	alsoCalled
Einstein tensor →	dependsOn
Levi-Civita connection →	determinedBy
Theorema Egregium →	usesConcept