WebMar 24, 2024 · Tensor Laplacian The vector Laplacian can be generalized to yield the tensor Laplacian (1) (2) (3) (4) (5) where is a covariant derivative, is the metric tensor , , is the comma derivative (Arfken 1985, p. 165), and (6) is a Christoffel symbol of the second kind . See also Laplacian, Vector Laplacian Explore with Wolfram Alpha More things to try: WebFeb 3, 2024 · Out of all of my time learning General relativity, this is the one identity that I cannot get around. Γααβ = ∂βln√− g where g is the determinant of the metric tensor gαβ. With the Christoffel symbol, we start by contracting Γααβ = 1 2gαγ(∂αgβγ + ∂βgαγ − ∂γgαβ) = 1 2gαα(∂βgαα) = 1 2gαα(∂βgαα) where I took γ → α and gαα = 1 / gαα.

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WebGeneral Relativity: Christoffel symbol identity Ask Question Asked 9 years, 5 months ago Modified 4 years, 2 months ago Viewed 7k times 3 I want to show that Γ μ ν μ = ∂ ν ( ln g ). (Here g denotes the determinant of the metric.) Working out the left hand side: Γ μ ν μ = 1 2 g μ ρ ( ∂ μ g ν ρ + ∂ ν g ρ μ − ∂ ρ g μ ν) = 1 2 g μ ρ ∂ ν g ρ μ Webwhere "ik is the two-dimensional antisymmetric Levi-Civitµa symbol "ik = fl fl fl fl fl –i 1 – i 2 –k 1 – k 2 fl fl fl fl fl = –i 1– k 2 ¡– k 1– i 2; "ik = "ik: 1e„ =@~r=@ u„ is theclassical notation. The modern notation simply calls „ (or even shorter: @u„) canonical local coordinate basis belonging to the ... hurst heating castlebar

Christoffel symbols and metric - Mathematics Stack Exchange

WebNov 11, 2024 · If you now calculate the LHS using the definition and the symmetry of the Christoffel symbols you should get the desired equality (be aware of matching the dummy indices). Share Cite Improve this answer Follow answered Nov 11, 2024 at 10:17 hof_a 101 1 6 blackhole Add a comment Your Answer The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols. See more 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 See more Christoffel symbols of the first kind The Christoffel symbols of the first kind can be derived either from the Christoffel symbols of the second kind and the metric, or from the metric … See more Let X and Y be vector fields with components X and Y . Then the kth component of the covariant derivative of Y with respect to X is given by Here, the Einstein notation is used, so repeated indices indicate summation over indices and … See more • Basic introduction to the mathematics of curved spacetime • Differentiable manifold • List of formulas in Riemannian geometry • Ricci calculus See more The definitions given below are valid for both Riemannian manifolds and pseudo-Riemannian manifolds, such as those of general relativity, with careful distinction being made between upper and lower indices (contra-variant and co-variant indices). The … See more Under a change of variable from $${\displaystyle \left(x^{1},\,\ldots ,\,x^{n}\right)}$$ to $${\displaystyle \left({\bar {x}}^{1},\,\ldots ,\,{\bar {x}}^{n}\right)}$$, Christoffel symbols transform as where the overline … See more In general relativity The Christoffel symbols find frequent use in Einstein's theory of general relativity, where spacetime is represented by a curved 4-dimensional See more WebCylindrical Coordinates. Download Wolfram Notebook. Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height () axis. Unfortunately, there … hursthead primary school stockport