What is Christoffel symbols: Definition and 101 Discussions

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. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group O(p, q). As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold. 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.
In general, there are an infinite number of metric connections for a given metric tensor; however, there is a unique connection that is free of torsion, the Levi-Civita connection. It is common in physics and general relativity to work almost exclusively with the Levi-Civita connection, by working in coordinate frames (called holonomic coordinates) where the torsion vanishes. For example, in Euclidean spaces, the Christoffel symbols describe how the local coordinate bases change from point to point.
At each point of the underlying n-dimensional manifold, for any local coordinate system around that point, the Christoffel symbols are denoted Γijk for i, j, k = 1, 2, ..., n. Each entry of this n × n × n array is a real number. Under linear coordinate transformations on the manifold, the Christoffel symbols transform like the components of a tensor, but under general coordinate transformations (diffeomorphisms) they do not. Most of the algebraic properties of the Christoffel symbols follow from their relationship to the affine connection; only a few follow from the fact that the structure group is the orthogonal group O(m, n) (or the Lorentz group O(3, 1) for general relativity).
Christoffel symbols are used for performing practical calculations. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity, the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γijk are zero.
The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900).

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  1. D

    I Online Christoffel Symbols Calculator

    I would love to hear from you if you have any suggestions, feedback, or criticism. The goal is to build better and more sophisticated software that would push the boundaries of research in astrophysics!
  2. C

    I Linearising Christoffel symbols

    Carroll linearising by perturbation ##g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}## has: (Notes 6.4, Book 7.4) ##\Gamma^{\rho}_{\mu\nu}=\frac{1}{2}g^{\rho\lambda}\left( {\partial_{ \mu}}g_{\nu\lambda}+{\partial_{ \nu}}g_{\lambda\mu}-{\partial_{...
  3. Jessie24789

    I Doppler Shift & Christoffel Symbols Issues

    About a month or two ago I started doing simulations of light physics around black holes and yesterday I got a fast Christoffel symbols function for the Schwarzschild metric in cartesian coordinates, but now the photon ring appears flipped. I feel as though it is wrong. But as I am still pretty...
  4. Onyx

    B Calc. Christoffel Symbols of Hiscock Coordinates

    The Hiscock coordinates read: $$d\tau=(1+\frac{v^2(1-f)}{1-v^2(1-f)^2})dt-\frac{v(1-f)}{1-v^2(1-f)^2}dx$$ ##dr=dx-vdt## Where ##f## is a function of ##r##. Now, in terms of calculating the christoffel symbol ##\Gamma^\tau_{\tau\tau}## of the new metric, where ##g_{\tau\tau}=v^2(1-f)^2-1## and...
  5. G

    I The Christoffel symbols at the origin -- Why zero?

    "the christoffel symbols are all zero at the origin of a local inertial frame" Why must it be at the origin? If it is not?Thanks!
  6. SH2372 General Relativity (3X): Christoffel symbols in polar coordinates

    SH2372 General Relativity (3X): Christoffel symbols in polar coordinates

  7. SH2372 General Relativity (3): Christoffel symbols

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  8. George Keeling

    I Contracted Christoffel symbols in terms determinant(?) of metric

    M. Blennow's book has problem 2.18: Show that the contracted Christoffel symbols ##\Gamma_{ab}^b## can be written in terms of a partial derivative of the logarithm of the square root of the metric tensor $$\Gamma_{ab}^b=\partial_a\ln{\sqrt g}$$I think that means square root of the determinant of...
  9. U

    Help with Kaluza Klein Christoffel symbols

    If I want to calculate ##\tilde{\Gamma}^\lambda_{\mu 5}##, I will write \begin{align} \tilde{\Gamma}^\lambda_{\mu 5} & = \frac{1}{2} \tilde{g}^{\lambda X} \left(\partial_\mu \tilde{g}_{5 X} + \partial_5 \tilde{g}_{\mu X} - \partial_X \tilde{g}_{\mu 5}\right) \\ & =\frac{1}{2}...
  10. M

    I Calculating Covariant Derivative of Riemann Tensor in Riemann Normal Coordinates

    Hello everyone, in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the...
  11. Arman777

    The contracting relations on the Christoffel symbols

    I am trying to find $$\Gamma^{\nu}_{\mu \nu} = \partial_{\mu} log(\sqrt{g})$$ but I cannot. by calculations, I manage to find $$\Gamma^{\nu}_{\mu \nu} = \frac{1}{2}g^{\nu \delta}\partial_{\mu}g_{\nu \delta}$$ and from research I have find that $$det(A) = e^{Tr(log(A))}$$ but still I cannot...
  12. D.S.Beyer

    I Solving GR 2-Body Problem with Summ. of Christoffel Symbols

    Can you approach the GR two body problem through summations of multiple Schwarzschild solutions? Specifically, by using the Schwarzschild metric for each body of mass, then adding the Christoffel symbols together, to arrive at a new geodesic equation. Take point C between bodies A and B...
  13. docnet

    I Covariant derivatives, connections, metrics, and Christoffel symbols

    Is a connection the same thing as a covariant derivative in differential geometry? What Is the difference between a covariant derivative and a regular derivative? If you wanted to explain these concepts to a layperson, what would you tell them?

    General Relativity: How many Christoffel symbols?

    Actually I know there would be some permutations used here. I know how to calculate the symbols but estimating is quite a new thing to me
  15. JD_PM

    Mathematica Learning how to compute Christoffel symbols using Mathematica

    I am using the code provided by Artes here, but I am missing something. The Chrisfoffel-symbol formula is $$\Gamma^{\mu}_{\phantom{\mu}\nu\sigma}=\frac{1}{2}g^{\mu\alpha}\left\{\frac{\partial g_{\alpha\nu}}{\partial x^{\sigma}}+\frac{\partial g_{\alpha\sigma}}{\partial x^{\nu}}-\frac{\partial...
  16. diazdaiz

    Please help me to be able to read the Christoffel symbol in EFE

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  17. C

    I Trying To Calculate Christoffel Symbols

    I am trying to create a function to calculate the Christoffel Symbols of a given metric (in this case the Shwartzchild metric). Calculating the (non zero) Christoffel Symboles for the Shwartzchild connection, I am a double major in Physics and Computer Science so I decided to go the code rout...
  18. M

    A Solving Covariant Derivative Notation Confusion

    I've stumbled over this article and while reading it I saw the following statement (##\xi## a vectorfield and ##d/d\tau## presumably a covariant derivative***): $$\begin{align*}\frac{d \xi}{d \tau}&=\frac{d}{d \tau}\left(\xi^{\alpha} \mathbf{e}_{\alpha}\right)=\frac{d \xi^{\alpha}}{d \tau}...
  19. P

    A Christoffel Symbols in terms of a Change in Basis

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  20. Luke Tan

    I Transformation of the Christoffel Symbols

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  21. snoopies622

    I Equivalent paths to the Christoffel symbols

    I've noticed that for both the surface of a sphere and a paraboloid, one arrives at the same Christoffel symbols whether using \Gamma^i_{kl} = \frac {1}{2} g^{im} ( \frac {\partial g_{mk} }{\partial x^l} + \frac {\partial g_{ml}}{\partial x^k} - \frac {\partial g_{kl}} {\partial x^m} )...
  22. D

    I Christoffel Symbols: Difference, Importance & Uses

    What is the general difference or importance between using christoffel symbols of the first kind and those of the second kind in terms of geometry and their application. The christoffel symbols of the second are identical to those of the first except with the inverse metric tensor in front...
  23. P

    I Christoffel symbols and covariant derivative intuition

    So I'm trying to get sort of an intuitive, geometrical grip on the covariant derivative, and am seeking any input that someone with more experience might have. When I see ##\frac {\partial v^{\alpha}}{\partial x^{\beta}} + v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}##, I pretty easily see a...
  24. C

    I Riemann Tensor knowing Christoffel symbols (check my result)

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  25. C

    I Christoffel symbols knowing Line Element (check my result)

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  26. W

    A Connection 1-forms to Christoffel symbols

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  27. JuanC97

    I Issues with the variation of Christoffel symbols

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  28. Destroxia

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  29. P

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  30. W

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  31. P

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  32. mertcan

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  33. A

    How Calculate Coriolis aceleration from Christoffel Symbols?

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  34. S

    I Is the Christoffel symbol orthogonal to the four-velocity?

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  35. W

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  36. Q

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  37. Stella.Physics

    I Christoffel symbols of Schwarzschild metric with Lagrangian

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  38. redtree

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  39. C

    A Transformation properties of the Christoffel symbols

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  40. G

    I Christoffel symbols transformation law

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  41. BiGyElLoWhAt

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  42. A

    I Geodesics on a sphere and the Christoffel symbols

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  43. S

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  44. PWiz

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    I've attempted to derive an expression for the Christoffel symbols (of the 2nd kind) solely in terms of the covariant and contravariant forms of the metric by only using the definition of the Christoffel symbols. I would like to know if my approach is correct or not. The Christoffel symbols are...
  45. S

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  46. P

    Deriving the Definition of the Christoffel Symbols

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  47. U

    Conditions on Christoffel Symbols?

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  48. U

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  49. E

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  50. Breo

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