In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization...

In general relativity and tensor calculus, the contracted Bianchi identities are: ∇ ρ R ρ μ = 1 2 ∇ μ R {\displaystyle \nabla _{\rho }{R^{\rho }}_{\mu...

Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) observed that the Christoffel symbols used to define the curvature could also provide a...

Elwin Bruno Christoffel (German: [kʁɪˈstɔfl̩]; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts...

called Christoffel symbols. The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. Levi-Civita...

{\frac {6\pi G(M+m)}{c^{2}A\left(1-e^{2}\right)}}} The non-vanishing Christoffel symbols for the Schwarzschild-metric are: Γ r t t = − Γ r r r = r s 2 r (...

the covariant derivative can be written in terms of Christoffel symbols. The Christoffel symbols find frequent use in Einstein's theory of general relativity...

except when noted otherwise. In a smooth coordinate chart, the Christoffel symbols of the first kind are given by Γ k i j = 1 2 ( ∂ ∂ x j g k i + ∂...

{\displaystyle \nabla } is the Levi-Civita connection, these symbols agree precisely with the Christoffel symbols from pseudo-Riemannian geometry. The expression for...

{\displaystyle v_{\rm {esc}}={\frac {\sqrt {\gamma ^{2}-1}}{\gamma }}.} The Christoffel symbols Γ j k i = ∑ s = 0 3   g i s 2 ( ∂ g j s ∂ x k + ∂ g s k ∂ x j − ∂...

Bruno Christoffel (1829–1900), German mathematician and physicist Named after him: Christoffel equation, Christoffel symbols, Schwarz–Christoffel mapping...

the Christoffel symbols as coordinates of the second partial derivatives of f. The choice of unit normal has no effect on the Christoffel symbols, since...

{1}{J}}\varepsilon _{ijk}={\cfrac {1}{+{\sqrt {g}}}}\varepsilon _{ijk}} Christoffel symbols of the first kind Γ k i j {\displaystyle \Gamma _{kij}} b i , j =...

the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express...

would be singular at that point. Using the metric above, we find the Christoffel symbols, where the indices are ( 1 , 2 , 3 , 4 ) = ( r , θ , ϕ , t ) {\displaystyle...

and Γ b c a {\displaystyle \Gamma _{bc}^{a}} are the Christoffel symbols. The Christoffel symbols are functions of the metric and are given by: Γ b c a...

derivative, and Rαβγδ is the Riemann curvature tensor, containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation...

map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local...

^{\lambda }{}_{\mu \nu }U^{\mu }U^{\nu }} In inertial coordinates the Christoffel symbols Γ λ μ ν {\displaystyle \Gamma ^{\lambda }{}_{\mu \nu }} are all zero...

described in terms of a set of connection coefficients (also known as Christoffel symbols) specifying what happens to components of basis vectors under infinitesimal...

Publications. p. 52. ISBN 0-486-63612-7. tensor Christoffel symbol. For application of the Christoffel symbols formalism to a rotating coordinate system, see...

where Γ μ ν λ {\displaystyle \Gamma _{\mu \nu }^{\lambda }} are the Christoffel symbols of the metric. This is the geodesic equation, discussed below. Techniques...

derivative by a certain linear operator, whose components are called the Christoffel symbols, which involves no derivatives on the vector field itself. The directional...

the Christoffel symbols. This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are...

used in the Ricci calculus in various calculations involving the Christoffel symbols of the first and second kind. In particular, Cramer's rule can be...

and Γ μ α β {\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }} are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita...

numbers In mathematics, the upper incomplete gamma function The Christoffel symbols in differential geometry In probability theory and statistics, the...

represents the coordinate system's 64 connection coefficients or Christoffel symbols. Note that the Greek subscripts take on four possible values, namely...

pseudo-Riemannian metric. In local smooth coordinates, define the Christoffel symbols Γ i j k := 1 2 g k l ( ∂ i g j l + ∂ j g i l − ∂ l g i j ) R j k...

indices is implied, gmn is the inverse metric tensor and Γl mn are the Christoffel symbols for the selected coordinates. In arbitrary curvilinear coordinates...