What is Christoffel: Definition and 145 Discussions

Hi everyone! Two question for you (): 1) I know that General relativity may also be seen as a gauge theory, but which kind of gauge group is used there?? 2) In the gauge theory wiew the Christoffel symbols \Gamma^{\alpha}_{\mu\kappa} in the covariant derivative...

Homework Statement Determine the Christoffel symbol \Gamma^{t}_{xx} for the metric ds^2 = -c^2dt^2 + (1+h\sin(\omega t))dx^2 + (1-h\sin(\omega t))dy^2 + dz^2 The answer should be: \frac{h\omega}{2} \cos(\omega t) Homework Equations For the evaluation we have to use...

Hello! Here and there I find that it is possible to make the Christoffel symbols vanish on a curve (e.g. lecture http://www.phys.uu.nl/~thooft/lectures/genrel_2010.pdf" by 't Hooft). The transformation law of the Christoffel symbols is relevant in this case...

My question is just, How can I determinate the Christoffel Symbols? I know that they're given by http://img263.imageshack.us/i/17f2df132717bfc32dc2ce3.png/" but, what does this mean? The subscripts I mean. thank you very much! :)

I am trying to understand everything about general relativity. I know that they have to do with how the Riemann curvature tensor uses parallel transporting a vector around a closed path. I really just don't understand the mathematics behind it. Thank you. I prefer layman's terms.

I'm finding it hard to understand this, does anyone know where I can find worked examples of how to find the components of the christoffel symbols of a metric? Please don't give me one to try, I really need a worked example. Thanks :S

Homework Statement Is the covariant derivative of a Christoffel symbol equal to zero? It seems like it would be since it is composed of nothing but metrics, and the covariant derivative of any metric is zero, right?

Homework Statement If the basic equation for the Christoffel symbol is \Gamma^l_{ki} = \frac{1}{2} g^{lj} (\partial_k g_{ij} + \partial_i g_{jk} - \partial_j g_{ki}) so if you bring multiply the first metric into that equation, won't that turn the first two derivatives into derivatives...

Homework Statement How do I show the following metric have time-like geodesics, if \theta and R are constants ds^{2} = R^{2} (-dt^{2} + (cosh(t))^{2} d\theta^{2}) Homework Equations v^{a}v_{a} = -1 for time-like geodesic, where v^{a} is the tangent vector along the curve The Attempt at a...

Kronicker Delta, Levi-Civita, Christoffel ... and "tensors" For quick reference in grabbing latex equations: http://en.wikipedia.org/wiki/Levi-Civita_symbol http://en.wikipedia.org/wiki/Kronecker_delta http://en.wikipedia.org/wiki/Christoffel_symbols Wiki warns that the Christoffel...

Now I'm just start to study the Kaluza-Klein theory from http://arxiv.org/abs/grqc/9805018. I try to calculate the Einstein Field equations in 5 dimensional vacuum space-time. we start with 5D metric tensor, \hat{g}_{AB}=\begin{pmatrix}...

I am learning about christoffel symbols and there is a pretty standard representation of christoffel symbols as a linear combination of products of the metric tensor and the metric tensors derivative. However when this is derived it is always done in a hoakey manner. Something along the lines of...

Does anyone know where I can find a list of Christoffel Symbols for various metrics? Metrics of general forms, as well as famous ones like Schwarzschild and Robertson-Walker? Yes, I can calculate them all if I really need to, but it's pretty tedious.

Hello:) let make a metric tensor, let it make simple, for random 2-dimentional curved space, ex. g_{jk}=\begin{bmatrix}R&0\\ 0&R\sin\phi\end{bmatrix} where R is constant, and \phi,\varphi are variables. Now I have symbol g_{jk,l}, does it mean just partial derivative \frac{\partial...

hello, i have a question about christoffel symbols . if we have :- http://www.tobikat.com how can I derive these equations :- [PLAIN][PLAIN]http://www.tobikat.com please i want the answer be clear . with very thanks...

Kinda silly question. I would like to know how to pronounce 'Christoffel'. If I were to take a stab at it, I would guess two syllables 'Christ' (as in 'grist', with a ch as in 'Christopher') and 'offel' as in 'awful'. Am I close?

I have that g=L^2 \left( e^{-2U} \left( e^{2A} \left( -dt^2 + d \theta^2 \right) + R^2 dy^2 \right) + e^{2U} dx^2 \right) is the metric on my spacetime. taking \{ t, \theta, x , y \} as a coordinate system for the manifol M, i can write this in matrix form as g_{ab}=L^2 \left( \begin...

The metric of Euclidean \mathbb{R}^3 in spherical coordinates is ds^2=dr^2+r^2(d \theta^2 + \sin^2{\theta} d \phi^2). I am asked to calculate the Christoffel components \Gamma^{\sigma}{}_{\mu \nu} in this coordinate system. i'm not too sure how to go about this. it talks about ds^2 being...

Homework Statement I'm trying (on my own) to derive the geodesic for a sphere of radius a using the geodesic equation \ddot{u}^i + \Gamma^i_{jk}\dot{u}^j\dot{u}^k, where \Gamma^i_{jk} are the Christoffel symbols of the second kind, \dot{u} and \ddot{u} are the the first and second...

http://en.wikipedia.org/wiki/Christoffel_symbols#Definition start with 0=\frac{\partial g_{ik}}{\partial x^l}-g_{mk}\Gamma^m_{il}-g_{im}\Gamma^m_{kl} in wiki it said "By permuting the indices, and resumming, one can solve explicitly for the Christoffel symbols as a function of the metric...

Homework Statement 1) Show that \epsilon_{ijk,m}=0 and (\sqrt{g})_{,k}=0 . Where ' ,k ' , stands for covariant derivative and \epsilon is the epsilon permutation symbol. 2) where the {} is for christoffel symbol of the second kind. Homework Equations The Attempt at a...

Homework Statement This is a problem in General Relativity, where I am trying to find the Christoffel symbols that correspond to a given metric. Any help would be greatly appreciated! OK. I have been given the metric ds^2 = (1+gx)^2 dt^2 - dx^2 - dy^2 - dz^2 and have been...

ROUGH DRAFT I have a beginner's basic question: 1. Schwarzschild Metric components Let \epsilon = rs / r, where rs is the Schwarzschild Radius. Then, as is is well-known: g_{00} = 1 - \epsilon g_{11} = - \left( 1 - \epsilon \right)^{-1} g_{22} = - r^{2} g_{33} = - r^{2} \; sin^{2}(\theta)...

I believe there is a way of calculating Christoffel symbols which is easier and less time-consuming than using the metric formula directly. This involves writing down the Lagrangian in a form that just includes the kinetic energy assuming zero potential energy and then equating the coefficient...

Homework Statement (a) Consider a 2-dimensional manifold M with the following line element ds2=dy2+(1/z2)dz2 For which values of z is this line element well defined. (b) Find the non-vanishing Christoffel symbols (c) Obtain the geodesic equations parameterised by l. (d) Solve...

Hi all! I read about tensor analysis and came about following expressions, where also a questions arose which I cannot explain to me. Perhaps you could help me: I: Consider the following expressions: d\vec v=dc^k e^{(k)} d\vec v=dc^k e_{(k)} where: dc^k=dv^k+v^t\Gamma_{wt}^k dx^w...

Hi everyone -- I have a question about the relation between the spin connection and the Christoffel connection. The spin connection comes from the local (gauge) Lorentz symmetry of how we orient vielbeins at each point in space, it contains a manifold index and two tangent space indices. The...

How can I derive the Christoffel symbol from the vanishing of the covariant derivative of the metric tensor? can somebody write the calculation, I read that I have to do some permutation and resumming but I don't get the result! Thank you!

Hi guys, I'm studying C. symbols for my G.R. class and have some doubts I hope you can clear out. First, I just saw this in the wikipedia article for C.s.: 0 = gik;l= gik;l - gmk \Gammamil - gim \Gammamkl By permuting the indices, and resumming, one can solve explicitly for the Christoffel...

Homework Statement show that the definition of the invariant divergence divA = 1/√g ∂i (√g Ai) is equivalent to the other invariant definition divA = Ai;i Ai;k = ∂Ai/∂xk + ГiklAl Гkij = gkl/2 (∂gil/∂xj+∂glj/∂xi-∂gij/∂xl) Homework Equations g is the metric tensor...

What is the mathematical relationship between Christoffel symbol and connection A in Ashtekar variables? Youngsub

Hey Guys, I'm new here on the forum, and I hope someone can help me out. I'm solving one of my GR homework exercises and I'm asked to find the christoffel symbols corresponding to cylindrical coordinates. I'll post my work and please correct if you see mistakes. I found the metric to be dR^2 +...

Hey Guys, I'm new here on the forum, and I hope someone can help me out. I'm solving one of my GR homework exercises and I'm asked to find the christoffel symbols corresponding to cylindrical coordinates. I'll post my work and please correct if you see mistakes. I found the metric to be dR^2...

I asked this question in the tensor analysis formum but did we did not reach a satisfactory conclusion. Here is the problem: Let \mathbf{x} : U \subset\mathbb{R}^2 \to S be a local parametrization of a regular surface S. Then the coefficients of the second derivatives of x in the basis of...

Hi, Let \mathbf{x}(u,v) be a local parametrization of a regular surface. Then the coefficients of \mathbf{x}_{uu},\mathbf{x}_{uv} etc. in the basis of the tangent space are defined as the Christoffel symbols. On the other hand, if we write the first fundamental form \langle,\rangle in...

Not so much a homework problem, more needing help understanding where something comes from. I've attached a jpg file with what I need help on. I've done a general relativity course at uni but can't seem to work out what should be a very simple problem.

I have Christoffel symbols for a metric and I want to find the connection 1-forms. I have the relation: w(^i j)=Chr(^i j k)*dx(^k) w: conn. 1-form Chr: Christoffel symbols But Christoffel symbols do not share the symmetries of the conn. 1-forms. Do you know any way to make this...

i'm having a hard time computing these so could people show me several examples to help me get a better feel for them before I move on to curvature?

So, I understand in order to evaluate the proper "derivative" of a vector valued function on a curved spacetime manifold, it is necessary to address the fact that the tangent space of the manifold changes as the function moves infinitesimally from one point to another. Therefore, you cannot just...

Has anyone a derivation at hand of the Christoffel symbol by permuting of indices in a free fall system? Roland

Hi, I have met a problem, that is how to prove transformation law for Christoffel symbol of first kind. I have read books about that, but many of them just state: cyclic permutation of the 3 indices and substitution. When I tried to work out, I could not elimate some terms... Can anyone show...

OK, my computer program (GRTensor II) says that \Gamma_{abc} is symmetric in the first two indices. Which leads to the equation \Gamma_{abc} = \frac{1}{2} ( \frac{\partial g_{bc}}{\partial a}+ \frac{\partial g_{ac}}{\partial b} - \frac{\partial g_{ab}}{\partial c} ) And that's...

Well, I think I finally figured out how to get good values for the local values of the Christoffel symbols (aka local gravitational accelerations) in the Schwarzschild metric. Some of the results are moderately interesting, though there is one point that still makes me wonder a bit. If we...

On page 34 of General Relativity, Robert Wald, the author refers to the Christoffel symbol as a tensor of rank (1,2). That whole derivation that led to that is quite involved so I don't quite understand how Wald can say that the Christoffel symbols are tensors since normally one does not refer...

Could someone give me a brief tutorial on the Christoffel equation used in relativity please