Christoffel symbols Definition and 97 Threads

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...

Homework Statement Consider a particle moving through Minkowski space with worldline x^\mu(\lambda). Here \lambda is a continuous parameter which labels different points on the worldline and x^\mu = (t,x,y,z) denotes the usual Cartesian coordinates. We will denote \partial/\partial \lambda by a...

I was looking up ways to solve the Einstein field equations when I came across a couple of sources. http://www.thescienceforum.com/physics/30059-solving-einstein-field-equations.html https://dl.dropboxusercontent.com/u/14461199/Light%20Deflection%20SM.pdf If you look at these sources...

Homework Statement I need to calculate \square A_\mu + R_{\mu \nu} A^\nu if \square = \nabla_\alpha \nabla^\alpha , and is the covariant derivate SEE THIS PDF arXiv:0807.2528v1 i want to get the equation (5) from (3) Homework Equations A^{i}_{{;}{\alpha}} =...

I've been thinking about this quite a bit. So it is clear that one can determine the Christoffel symbols from the first fundamental form. Is it possible to derive the geodesics of a surface from the Christoffel symbols?

Homework Statement I am learning Christoffel symbols and I want to know how to compute a surface parameterized by ##g(u,v) = (u\cos v, u \sin v, u)## by using the definition. Homework Equations Christoffel symbols The Attempt at a Solution Is this website...

I'm having trouble understanding what Christoffel symbols are. In simple language, what are they? What are they used for?

Homework Statement Consider metric ds2 = dx2 + x3 dy2 for 2D space. Calculate all non-zero christoffel symbols of metric. Homework Equations \Gammajik = \partialei / \partial xk \times ej The Attempt at a Solution Christoffel symbols, by definition, takes the partial of each...

I don't think I've ever seen this discussed in a textbook, this is an attempt to throw some light on the connection between Christoffel symbols and forces. In particular I want to derive the later as an approximation of the former, with some limitations on choices of coordinate systems...

I am pretty much confused with all the algebra of Christoffel symbols: I have an expression for infinitesimal length: F= g_{ij} \frac{dx^i dx^j}{du^2} and by using Euler-Lagrange equation (basically finding the shortest distance between two points) want to find the equation for geodesics...

Homework Statement Find the non zero Christoffel symbols of the following metric ds^2 = -dt^2 + \frac{a(t)^2}{(1+\frac{k}{4}(x^2+y^2+z^2))^2} (dx^2 + dy^2 + dz^2 ) and find the non zero Christoffel symbols and Ricci tensor coefficients when k = 0 Homework Equations The...

In the book Mathematical Methods for Physics and Engineering by Riley, Hobson, and Bence, I came across an equation I just can't seem to understand. In the chapter on tensors, they derive the equation for a Christoffel symbol of the second kind, \Gamma^{m}_{ij}=\frac{1}{2}g^{mk}\left(\frac{...

Hello all, I've been going through Bernard Schutz's A First Course In General Relativity, On Chapter 5 questions atm. Should the Christoffel Symbols for a coordinate system (say polar) be the same for vectors and one-forms in that coordinate system? I would have thought yes, but If you...

I don't know exactly what I'm looking for in this question so I'll ask it in a vague way. What is the connection between a particle's proper acceleration and the christoffel symbol of the second kind (single contravariant and double covariant) ? Is this correct...

To avoid hijacking an existing thread, I wanted to start a new one on how "gravitational forces" are represented in GR. There doesn't seem to be a lot on this in the intro textbooks, alas, which mostly deal with the issue by avoiding it. Which suggests there could be some non-obvious...

Hi All, I am currently reading Menzel's "Mathematical Physics" and one part in particular confuses me. When he is introducing Riemannian Geometry he derives the Christoffel symbols almost out of thin air. He starts by differentiating a vector with respect to a coordinate system...

Another "christoffel symbols from the metric" question Homework Statement Find the Christoffel symbols from the metric: ds^2 = -A(r)dt^2 + B(r)dr^2 Homework Equations \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x^a}} \right) = \frac{\partial L}{\partial x^a} The...

Is it true that in GR the gauge is described by Guv while the potential is the Christoffel symbols just like the gauge in EM is described by phase and the potential by the electric and magnetic scalar and vector potential and the observable the electromagnetic field and the Ricci curvature...

Hey guys I'm a bit new to GR and stuck on this question? :/. So we are given that: d2xi/dλ2+\Gammaijk dxi/dλ dxj/dλ = 0 and asked to show that d/dλ(gijdxi/dλdxj/dλ) = 0 So I expanded using the product rule to get: \Gammaijkd2xi/dλ2 dxj/dλ +\Gammaijk dxi/dλd2 xj/dλ2 Then rearranged the...

Homework Statement Prove that the transformation law \Gamma^{\sigma '}_{\lambda '\rho '}=\frac{\partial x^\nu}{\partial x^{\lambda '}}\frac{\partial x^\rho}{\partial x^{\rho '}}\frac{\partial x^{\sigma '}}{\partial x^{\mu}}\Gamma^{\mu}_{\nu\rho}+\frac{\partial x^{\sigma '}}{\partial...

Hello, I am trying to understand what the differences would be in replacing the symmetry equation: g_mn = g_nm with the Hermitian version: g_mn = (g_nm)* In essence, what would happen if we allowed the metric to contain complex elements but be hermitian? I am not talking about...

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.

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...

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, 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...

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...

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 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: [SIZE="3"]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} \...

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 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...

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...

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...

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?

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...