What is Christoffel symbols: Definition and 100 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. 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_{...
  2. Jessie24789

    I Doppler Shift & Christoffel Symbols Issues

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  3. Onyx

    B Calc. Christoffel Symbols of Hiscock Coordinates

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  4. 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!
  5. SH2372 General Relativity (3X): Christoffel symbols in polar coordinates

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    SH2372 General Relativity (3): Christoffel symbols

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  7. 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...
  8. U

    Help with Kaluza Klein Christoffel symbols

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  9. M

    I Calculating Covariant Derivative of Riemann Tensor in Riemann Normal Coordinates

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  10. 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...
  11. D.S.Beyer

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

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  12. docnet

    I Covariant derivatives, connections, metrics, and Christoffel symbols

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  13. AHSAN MUJTABA

    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
  14. JD_PM

    Mathematica Learning how to compute Christoffel symbols using Mathematica

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  15. diazdaiz

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

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  16. C

    I Trying To Calculate Christoffel Symbols

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  17. M

    A Solving Covariant Derivative Notation Confusion

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  18. P

    A Christoffel Symbols in terms of a Change in Basis

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  19. Luke Tan

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  20. snoopies622

    I Equivalent paths to the Christoffel symbols

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  21. D

    I Christoffel Symbols: Difference, Importance & Uses

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  22. P

    I Christoffel symbols and covariant derivative intuition

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  23. C

    I Riemann Tensor knowing Christoffel symbols (check my result)

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  24. C

    I Christoffel symbols knowing Line Element (check my result)

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  25. W

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  26. JuanC97

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  27. Destroxia

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  28. P

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  29. W

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  30. P

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  31. mertcan

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  32. A

    How Calculate Coriolis aceleration from Christoffel Symbols?

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  33. S

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

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  34. W

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  35. Q

    I Christoffel Symbol vs. Vector Potential

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  36. Stella.Physics

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  37. redtree

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  38. C

    A Transformation properties of the Christoffel symbols

    If you want to define a covariant derivative which transforms correctly, you need to define it as ##\nabla_i f_j = \partial_i f_j - f_k \Gamma^k_{ij}##, where ##\Gamma^k_{ij}## has the transformation property ##\bar{\Gamma}^k_{ij} = \frac{\partial \bar{x}_k}{\partial x_c}\frac{\partial...
  39. G

    I Christoffel symbols transformation law

    In Carroll's GR book (pg. 96), the transformation law for Christoffel symbols is derived from the requirement that the covariant derivative be tensorial. I think I understand that, and the derivation Carroll carries out, up until this step (I have a very simple question here, I believe-...
  40. BiGyElLoWhAt

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  41. A

    I Geodesics on a sphere and the Christoffel symbols

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  42. S

    Trouble understanding ##g^{jk}\Gamma^{i}{}_{jk}##

    Hi friends, I'm learning Riemannian geometry. I'm in trouble with understanding the meaning of ##g^{jk}\Gamma^{i}{}_{jk}##. I know it is a contracting relation on the Christoffel symbols and one can show that ##g^{jk}\Gamma^i{}_{jk}=\frac{-1}{\sqrt{g}}\partial_j(\sqrt{g}g^{ij})## using the...
  43. PWiz

    Christoffel symbols derivation

    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...
  44. S

    Riemannian Metric Tensor & Christoffel Symbols: Learn on R2

    Hi, Want to know (i) what does Riemannian metric tensor and Christoffel symbols on R2 mean? (ii) How does metric tensor and Christoffel symbols look like on R2? It would be great with an example as I am new to this Differential Geometry.
  45. P

    Deriving the Definition of the Christoffel Symbols

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  46. U

    Conditions on Christoffel Symbols?

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  47. U

    Flat Space - Christoffel symbols and Ricci = 0?

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  48. E

    Christoffel symbols in differential geometry

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  49. Breo

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  50. K

    Coordinate free Christoffel symbols

    I've been trying to come up with a oordinate free formula of Christoffel symbols. For the Christoffel symbols of the first kind it's really easy. Since \Gamma_{\lambda\mu\nu} = \frac{1}{2}\left( g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu} - g_{\mu\nu,\lambda}\right) we can easily generalize the...
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