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Array of numbers describing a metric connection
- In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface.
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In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface.
Christoffel symbols are one of the most important mathematical objects used in general relativity as well as in Riemannian geometry. They are particularly useful for practical calculations in general relativity, but what do the Christoffel symbols actually represent?
May 16, 2024 · The Christoffel symbols are tensor-like objects derived from a Riemannian metric g. They are used to study the geometry of the metric and appear, for example, in the geodesic equation. There are two closely related kinds of Christoffel symbols, the first kind Gamma_ (i,j,k), and the second kind Gamma_ (i,j)^k.
4 days ago · Christoffel symbols of the second kind are variously denoted as {m; i j} (Walton 1967) or Gamma^m_(ij) (Misner et al. 1973, Arfken 1985). They are also known as affine connections (Weinberg 1972, p. 71) or connection coefficients (Misner et al. 1973, p. 210).
May 23, 2017 · The symbols $\Gamma_{k,ij}$ are called the Christoffel symbols of the first kind, in contrast to the Christoffel symbols of the second kind, $\Gamma^k_{ij}$, defined by \begin{equation*} \Gamma^k_{ij}=\sum_{t=1}^ng^{kt}\Gamma_{t,ij}, \end{equation*} where $g^{kt}$ is defined as follows:
Christoffel Symbols. . Let us consider the metric tensor gij which we know satisfies the transformation law. Define the quantity. (α, β, γ) ∂xa ∂xb. αβ = gab. . ∂xα ∂xβ. ∂gαβ. . ∂gab∂xc ∂xa ∂xb. ∂2xa ∂xb. ∂xa ∂2xb. = . ∂xγ. gab. . gab ∂xc ∂xγ. ∂xα. ∂xβ. . ∂xα∂xγ. ∂xβ. ∂xα ∂xβ∂xγ . and form the combination of terms.
