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  2. Christoffel symbols - Wikipedia

    en.wikipedia.org/wiki/Christoffel_symbols

    The Einstein field equations—which determine the geometry of spacetime in the presence of matter—contain the Ricci tensor, and so calculating the Christoffel symbols is essential. Once the geometry is determined, the paths of particles and light beams are calculated by solving the geodesic equations in which the Christoffel symbols ...

  3. List of formulas in Riemannian geometry - Wikipedia

    en.wikipedia.org/wiki/List_of_formulas_in...

    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

  4. Derivation of the Schwarzschild solution - Wikipedia

    en.wikipedia.org/wiki/Derivation_of_the...

    The static assumption is unneeded, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; the Schwarzschild solution thus follows. Birkhoff's theorem has the consequence that any pulsating star that remains spherically symmetric does not generate gravitational waves , as the ...

  5. Riemann curvature tensor - Wikipedia

    en.wikipedia.org/wiki/Riemann_curvature_tensor

    In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field).

  6. Geodesics in general relativity - Wikipedia

    en.wikipedia.org/wiki/Geodesics_in_general...

    It is important that the last two equations are consistent with each other, when the latter is differentiated with respect to proper time, and the following formula for the Christoffel symbols ensures that consistency: = (+) This last equation does not involve the electromagnetic fields, and it is applicable even in the limit as the ...

  7. Levi-Civita connection - Wikipedia

    en.wikipedia.org/wiki/Levi-Civita_connection

    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.

  8. Differential geometry of surfaces - Wikipedia

    en.wikipedia.org/wiki/Differential_geometry_of...

    These equations can be directly derived from the second definition of Christoffel symbols given above; for instance, the first Codazzi equation is obtained by differentiating the first equation with respect to v, the second equation with respect to u, subtracting the two, and taking the dot product with n. The Gauss equation asserts that [24]

  9. Introduction to the mathematics of general relativity - Wikipedia

    en.wikipedia.org/wiki/Introduction_to_the...

    The Einstein field equations – which determine the geometry of spacetime in the presence of matter – contain the Ricci tensor. Since the Ricci tensor is derived from the Riemann curvature tensor, which can be written in terms of Christoffel symbols, a calculation of the Christoffel symbols is essential.