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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.
Learn what Christoffel symbols are, how they represent connection coefficients for the Levi-Civita connection, and how they describe changes in basis vectors. Find out their geometric and physical meaning, methods to calculate and use them, and examples in general relativity.
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 the second type of tensor-like object derived from a Riemannian metric g which is used to study the geometry of the metric. 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).
In this section, which can be skipped at a first reading, we show how the Christoffel symbols can be used to find differential equations that describe geodesics. Characterization of the Geodesic. A geodesic can be defined as a world-line that preserves tangency under parallel transport, Figure \ (\PageIndex {1}\).
Christoffel Symbols and Geodesic Equation. This is a Mathematica program to compute the Christoffel and the geodesic equations, starting from a given metric gab. The Christoffel symbols are calculated from the formula. Gl mn = 1. 2. where gls for the inverse metric is standard [cf (20.17)].
Feb 9, 2018 · Christoffel symbols A vector field in ℝ n can be seen as a differentiable ( C ∞ ) map V : ℝ n → ℝ n . Or as a section ℝ n → V T ( ℝ n ) where T ℝ n ≡ ℝ n × ℝ n is the ℝ n ’s trivial tangent bundle obeying p ↦ ( p , V ( p ) ∈ T p ( ℝ n ) ) with T p ( ℝ n ) ≡ ℝ n being the tangent space at p .
