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In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. [1] 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.
In local coordinates the components of the form are called Christoffel symbols: because of the uniqueness of the Levi-Civita connection, there is a formula for these components in terms of the components of g.
Christoffel symbols satisfy the symmetry relations = or, respectively, =, the second of which is equivalent to the torsion-freeness of the Levi-Civita connection. The contracting relations on the Christoffel symbols are given by
the Christoffel symbols that describe components of a metric connection; the stack alphabet in the formal definition of a pushdown automaton, or the tape-alphabet in the formal definition of a Turing machine; the Feferman–Schütte ordinal Γ 0; represents: the specific weight of substances; the lower incomplete gamma function
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel.Levi-Civita, [1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols [2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold , one can additionally arrange that the metric tensor is the Kronecker delta at the point p , and that the first ...
The Christoffel symbols of this connection are given in terms of partial derivatives of the metric in local coordinates by the formula = (+) = (, +,,) (where commas indicate partial derivatives). The curvature of spacetime is then given by the Riemann curvature tensor which is defined in terms of the Levi-Civita connection ∇.
4 Calculating the Christoffel symbols. 5 Using the field equations to find A(r) and B(r) 6 Using the weak-field approximation to find K and S.