# What is Christoffel: Definition and 145 Discussions

Christoffel is a Dutch and Afrikaans cognate of the masculine given name Christopher. Short forms include Chris, Christie, Kristof, and Stoffel. Christoffel also occurs as a patronymic surname. People with the name include:

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1. ### 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...
2. ### 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...
3. ### 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!
4. ### 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...
5. ### How do I correctly find the Christoffel symbol for a specific component?

I was not given a formal teaching on christoffel symbols and how to find them so I just need some help. I'm trying to find the cristoffel symbol: $$\Gamma^{i}_{00}$$ I set my equation up as: \Gamma^i_{00} = \frac{1}{2} g^{ij} (\partial_0 g_{0j} +...
6. ### 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}...
7. ### 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...
8. ### 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...
9. E

### Understanding Christoffel Identity and its Application in Differential Geometry

We use ##g_{\alpha \beta} = \vec{e}_{\alpha} \cdot \vec{e}_{\beta}## to show that$$\partial_c g_{ab} = \partial_c (\vec{e}_a \cdot \vec{e}_b) = \vec{e}_a \cdot \partial_c \vec{e}_b + \vec{e}_b \cdot \partial_c \vec{e}_a$$then because ##\partial_{\alpha} \vec{e}_{\beta} := \Gamma_{\alpha...
10. ### 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?
11. ### 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

24. ### I Christoffel symbol and Einstein summation convention

Homework Statement I know that by definition Γijkei=∂ej/∂xk implies that Γmjk=em ⋅ ∂ej/∂xk (e are basis vectors, xk is component of basis vector). Can I write it in the following form? Γjjk=ej ⋅ ∂ej/∂xk Why or why not? Homework EquationsThe Attempt at a Solution
25. ### 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. ### 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. ### I Can the Christoffel connection be observed?

[Moderator's note: Spun off from a previous thread about Maxwell's Equations and QFT.] Perhaps, but Christoffel connection is also an observable. You feel it in your whole body when you accelerate. One usually calls it the inertial force.
28. ### 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##...
29. ### 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...
30. ### 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} +...
31. ### 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. ### 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?
33. 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...
34. ### 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...
35. ### A Understanding Orbital Angular Momentum Coupling to Christoffel Connection

I am trying to understand Wen and Zee's article on topological quantum numbers of Hall fluid on curved space: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.69.953 They passingly mentiond the fact that a spinning particle with orbital angular momentum $s$ moving on a manifold with...
36. ### Find the Riemannian curvature tensor component

Given the metric of the gravitational field of a central gravitational body: ds2 = -ev(r)dt2 + eμ(r)dr2 + r2 (dθ2 + sin2θdΦ2) And the Chritofell connection components: Find the Riemannian curvature tensor component R0110 (which is non-zero). I believe the answer uses the Ricci tensor...
37. ### Levi-Civita connection and Christoffel symbols

Homework Statement Show that g(d \sigma ^k, \sigma _p \wedge \sigma _q) = \Gamma _{ipq} - \Gamma _{iqp}Homework Equations Given $$\omega_{ij}=\hat e_i \cdot d \hat e _j = \Gamma_{ijk} \sigma^k$$, we can also say that $$d \hat e_j = \omega^i_j \hat e_i$$. Where $$\sigma^k, \sigma_p, \sigma_q$$...
38. ### 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...
39. ### 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...
40. ### 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...
41. ### 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-...
42. ### The Variation of Christoffel Symbol

Homework Statement It is shown in Carrol, an Introduction to GR that the variatiom of Christoffel symbols are : https://scontent-sin1-1.xx.fbcdn.net/v/t34.0-12/13535871_1161725257182772_897443562_n.jpg?oh=df1a6d26aa0b199d4684b5f0785bee20&oe=576ECCCA But i have no idea how to derive that, any...
43. ### A Christoffel symbol definition

hi, I have seen that christoffel symbol definition or logic is shown in different ways. For instance, in first attachment ( RED box) you can see a normal vector (n) next to the christoffel symbol, but in second image everything is same except that there is a normal vector. Is there a confusion...
44. ### A Subtraction of Christoffel symbol

hi, How do we prove the torsion tensor is a tensor using the subtraction of christoffel symbols?? I am aware of the fact that subtraction of christoffel symbols equals the torsion. How can we use this fact to prove the tensor?? Could you please give the proof or share the link which prove it??
45. ### A Anti-symmetric Christoffel symbol

hi, I have seen some examples related to christoffel symbol when it was symmetric, but I have not seen any anti symmetric christoffel symbol examples. For instance, in torsion tensor, if we have anti symmetric christoffel symbol, torsion tensor does not vanish. To sum up, in what kind of...
46. ### 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...
47. ### Derivation of the Christoffel symbol

I'm reading Zee's Gravity book, can anyone help me understand the explanation on this part, I understand everything except the last part, he said to use (I.4.14) so that I could solve for the quantity shown in the image, what does he mean by that and how?
48. ### Calculating Covariant Riemann Tensor with Diag Metric gab

Using Ray D'Inverno's Introducing Einstein's Relativity. Ex 6.31 Pg 90. I am trying to calculate the purely covariant Riemann Tensor, Rabcd, for the metric gab=diag(ev,-eλ,-r2,-r2sin2θ) where v=v(t,r) and λ=λ(t,r). I have calculated the Christoffel Symbols and I am now attempting the...
49. ### 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...
50. ### 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...