Christoffel symbols Definition and 97 Threads

  1. Pencilvester

    I Can covariant derivative tensorialness be derived?

    I'm building a mental framework for the Levi-Civita connection that is intuitive to me. I start by imagining an arbitrary manifold with arbitrary coordinates embedded in a higher dimensional Euclidean space, then if I take the derivative of an arbitrary coordinate basis vector with respect to...
  2. 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!
  3. 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_{...
  4. 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...
  5. 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...
  6. 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!
  7. SH2372 General Relativity (3X): Christoffel symbols in polar coordinates

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

  8. SH2372 General Relativity (3): Christoffel symbols

    SH2372 General Relativity (3): Christoffel symbols

  9. 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...
  10. 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}...
  11. 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...
  12. 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...
  13. 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...
  14. D

    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?
  15. 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
  16. 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...
  17. diazdaiz

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

    in video "Einstein Field Equation - for Beginner!" by "DrPhysicsA" on youtube, in 01:10:56, the christoffel symbol equation is written, then i see in "Physics Videos by Eugene Khutoryansky" video with title "Einstein's Field Equations of General Relativity Explained" in minute 05:02 on how the...
  18. 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...
  19. 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}...
  20. P

    A Christoffel Symbols in terms of a Change in Basis

    Hi All Given that the Riemann Curvature Tensor may be derived from the parallel Transport of a Vector around a closed loop, and if that vector is a covariant vector Having contravariant basis The calculation gives the result Now: Given that the Christoffel Symbols represent the...
  21. Luke Tan

    I Transformation of the Christoffel Symbols

    In Landau Book 2 (Classical Field Theory & Relativity), he mentions that the transformation rules of the christoffel symbols can be gotten by "comparing the laws of transformation of the two sides of the equation governing the covariant derivative" I would believe that by the equations...
  22. 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} )...
  23. 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...
  24. 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...
  25. C

    I Riemann Tensor knowing Christoffel symbols (check my result)

    I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are: \Gamma^x_{xx}=\frac{1}{x} and \Gamma^y_{yy}=\frac{2}{y} knowing that: R^\alpha_{\beta\gamma\delta}=\partial_\gamma...
  26. C

    I Christoffel symbols knowing Line Element (check my result)

    Hi! I'm asked to find all the non-zero Christoffel symbols given the following line element: ds^2=2x^2dx^2+y^4dy^2+2xy^2dxdy The result I have obtained is that the only non-zero component of the Christoffel symbols is: \Gamma^x_{xx}=\frac{1}{x} Is this correct? MY PROCEDURE HAS BEEN: the...
  27. W

    A Connection 1-forms to Christoffel symbols

    Let the metric be defined as ##ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2## Through some calculations, we then see that our connection one forms are ##\omega_{12} = -d \theta## and ##\omega_{21}= d\theta##, ##\omega_{13} = -sin\theta d \phi## and ##\omega_{31} = sin\theta d\phi##...
  28. JuanC97

    I Issues with the variation of Christoffel symbols

    Hello everyone, I'm sure a lot of you know that the Christoffel symbols are not tensors by themselves but, their variation is a tensor. I want to revive a post that was made in 2016 about this: The Variation of Christoffel Symbol and ask again "How is that you can calculate ∇ρδgμν if δ{gμν} is...
  29. Destroxia

    Calculating Christoffel Symbols from a given line element

    Homework Statement Given some 2D line element, ## ds^2 = -dt^2 +x^2 dx^2 ##, find the Christoffel Symbols, ## \Gamma_{\beta \gamma}^{\alpha} ##. Homework Equations ## \Gamma_{\beta \gamma}^{\alpha} = \frac {1}{2} g^{\delta \alpha} (\frac{\partial g_{\alpha \beta}}{\partial x^\gamma} +...
  30. P

    A Lense-Thirring effect - General Relativity

    Let us assume a "toy-metric" of the form $$ g=-c^2 \mathrm{d}t^2+\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2-\frac{4GJ}{c^3 r^3} (c \mathrm{d}t) \left( \frac{x\mathrm{d}y-y\mathrm{d}x}{r} \right)$$ where ##J## is the angular-momentum vector of the source. Consider the curve $$ \gamma(\tau)=(x^\mu...
  31. W

    I Manipulating Christoffel Symbols: Questions & Answers

    I have a couple of questions about how Christoffel symbols work. Why can they just be moved inside the partial derivative, as shown just beneath the first blue box here: https://einsteinrelativelyeasy.com/index.php/general-relativity/61-the-riemann-curvature-tensor And if you had the partial...
  32. P

    Dust in special relativity - conservation of particle number

    Homework Statement My textbook states: Since the number of particles of dust is conserved we also have the conservation equation $$\nabla_\mu (\rho u^\mu)=0$$ Where ##\rho=nm=N/(\mathrm{d}x \cdot \mathrm{d}y \cdot \mathrm{d}z) m## is the mass per infinitesimal volume and ## (u^\mu) ## is...
  33. mertcan

    A Christoffel symbols expansion for second derivatives

    Hi, I really wonder how these second derivatives can be written in terms of christofflel symbols. I have made so many search but could not find on internet What is the derivation of equations related to second derivatives in attachment?
  34. A

    How Calculate Coriolis aceleration from Christoffel Symbols?

    Homework Statement Hi, We are trying to calculate the Coriolis acceleration from the Cristoffel symbols in spherical coordinates for the flat space. I think this problem is interesting because, maybe it's a good way if we want to do the calculations with a computer. We start whit the...
  35. S

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

    Consider a force-free particle moving on a geodesic with four-velocity v^\nu. The formula for the four-acceleration in any coordinate system is \frac{dx^\mu}{d\tau} = - \Gamma^\mu_{\nu\lambda} v^\nu v^\lambda Since the four-acceleration on the left side is orthogonal to the four-velocity, this...
  36. Q

    I Christoffel Symbol vs. Vector Potential

    As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative. The Chirstoffel symbol compensates for changes in curvature and helps define what it means for a tensor to remain...
  37. Stella.Physics

    I Christoffel symbols of Schwarzschild metric with Lagrangian

    So the Schwarzschild metric is given by ds2= -(1-2M/r)dt2 + (1-2M/r)-1dr2+r2dθ2+r2sin2θ dφ2 and the Lagragian is ##{\frac{d}{dσ}}[{\frac{1}{L}}{\frac{dx^α}{dσ}}] + {\frac{∂L}{∂x^α}}=0## with L = dτ/dσ. So for each α=0,1,2,3 we have ##{\frac{d^2 x^1}{dτ^2}}=0## for Minkowski spacetime also...
  38. redtree

    A Riemann Tensor Equation: Simplifying the Riemann-Christoffel Tensor

    The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as: $$ R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n} $$ My question is that it seems that...
  39. 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...
  40. 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-...
  41. BiGyElLoWhAt

    I How Do You Correctly Apply Indices in Tensor Calculus for Curvature?

    Here's what I'm watching: At about 1:35:00 he leaves it to us to look at a parallel transport issue. Explicitly to caclculate ##D_s D_r T_m - D_r D_s T_m## And on the last term I'm having some difficulties, the second christoffel symbol. So we have ##D_s [ \partial_r T_m - \Gamma_{rm}^t T_t]##...
  42. A

    I Geodesics on a sphere and the Christoffel symbols

    Hi, I recently tried to derive the equations for a geodesic path on a sphere of radius 1 (which are supposed to come out to be a great circle) using the formula \dfrac{d^2 x^a}{dt^2}+\Gamma^a_{bc} \dfrac{dx^b}{dt}\dfrac{dx^c}{dt}=0 for the geodesic equation, with the metric...
  43. 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...
  44. 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...
  45. 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.
  46. P

    Deriving the Definition of the Christoffel Symbols

    In Sean Carroll's Lecture Notes on General Relativity (Chapter 3, Page 60), in the chapter on Curvature, he derives the definition of the Christoffels Symbols by assuming the connection is metric compatible and torsion free. He then takes the covariant derivative of the metric and cycles through...
  47. U

    Conditions on Christoffel Symbols?

    Homework Statement Write down the geodesic equation. For ##x^0 = c\tau## and ##x^i = constant##, find the condition on the christoffel symbols ##\Gamma^\mu~_{\alpha \beta}##. Show these conditions always work when the metric is of the form ##ds^2 = -c^2dt^2 +g_{ij}dx^idx^j##.Homework...
  48. U

    Flat Space - Christoffel symbols and Ricci = 0?

    Homework Statement [/B] (a) Find christoffel symbols and ricci tensor (b) Find the transformation to the usual flat space form ## g_{\mu v} = diag (-1,1,1,1)##. Homework EquationsThe Attempt at a Solution Part(a) [/B] I have found the metric to be ## g_{tt} = g^{tt} = -1, g_{xt} = g_{tx} =...
  49. E

    Christoffel symbols in differential geometry

    Homework Statement I'm having trouble figuring out how to use Christoffel symbols. Apart from the first three terms here, I can't understand what's going on between line 3 and 4. What formulas/definitions are being used? How do you find the product of two chirstoffel symbols? Where are all the...
  50. Breo

    Variation of the Christoffel Symbols

    So, it is defined that: Γλμυ = Γλμυ + δΓλμυ This makes obvious to see that the variation of the connection, which is defined as a difference of 2 connections, is indeed a tensor. Therefore we can express it as a sum of covariant derivatives. δΓλμυ = ½gλν(-∇λδgμν + ∇μδgλν + ∇νδgλμ) However...
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