What is Christoffel: Definition and 145 Discussions

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...

Friends, I wish indicate to me a textbook that teaches you calculate in detail the symbol of Christoffel (undergraduate or graduate level in physics). Thank You for your help!

Homework Statement (a) Find the christoffel symbols (Done). (b) Show that ##\phi## is a solution and find the relation between A and B.[/B] Homework EquationsThe Attempt at a Solution Part(b) \nabla_\mu \nabla^\mu \phi = 0 I suppose for a scalar field, this is simply the normal derivative...

Homework Statement (a) Show acceleration is perpendicular to velocity (b)Show the following relations (c) Show the continuity equation (d) Show if P = 0 geodesics obey: Homework EquationsThe Attempt at a SolutionPart (a) U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v U^{\mu} +...

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} =...

http://www.thephysicsforum.com/vlatex/pics/92_db3a451c7d105432675bef473582556e.png Anyone can help me? I am really stuck at here.

I'm looking at lecture notes on General Relativity by Sean M. Carroll, and after defining the Riemannanian tensor in the usual theorem, the extent to which the partial derivatives of a vector field fail to commute, it says ' having defined curvature tensor as something which characterizes the...

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...

Homework Statement Show U^a \nabla_a U^b = 0 Homework Equations U^a refers to 4-velocity so U^0 =\gamma and U^{1 - 3} = \gamma v^{1 - 3} The Attempt at a Solution I get as far as this: U^a \nabla_a U^b = U^a ( \partial_a U^b + \Gamma^b_{c a} U^c) And I think that the...

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...

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...

We know that the derivative of the general Schwartz - Christoffel map (function) is: f'(z) = λ(z - x_1)^{a_1}...(z-x_n)^{a_n} Question: In various sources around the web, it is mentioned that x_n can be taken to be the "point at infinity", and the last factor can be removed from the above...

Ansatz metric of the four dimensional spacetime: ds^2=a^2 g_{ab}dx^a dx^b - du^2 where: a,b=0,1,2 a(u)=warped factor Christoffel symbol of a three dimensional AdS spacetime: \Gamma^{c}_{ab}= \frac{1}{2} g^{cd}(∂_b g_{da} + ∂_a g_{bd} - ∂_d g_{ba}) Now how to find \Gamma^{a}_{b}?

I'm reading an old article published by Kaluza "On the Unity Problem of Physics" where i encounter an expression for the Ricci tensor given by $$R_{\mu \nu} = \Gamma^\rho_{\ \mu \nu, \rho}$$ where he has used the weak field approximation ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}## where...

Homework Statement ¿Why \Gamma^{k}_{i j} = (1/2) g^{k p} (g_{i p ,j}+ g_{j p ,i}- g_{i j , p}) can't be writed like \Gamma^{k}_{i j} = (1/2) g^{k p} (2 g_{i p ,j}+g_{i j , p}) if i can say that the metric is symmetric? Homework Equations That is the relevant equation The Attempt at a...

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 Find the Christoffel symbols of a surface in the form ##g=f(u,v).## Homework Equations ##f_{u_1u_1} = \Gamma^1_{11} f_{u_1} + \Gamma^2_{11}f_{u_2} + A \vec{N}## ##f_{u_1u_2} = f_{u_2u_1} = \Gamma^1_{12} f_{u_1} + \Gamma^2_{12}f_{u_2} + B \vec{N}## ##f_{u_2u_2} =...

Homework Statement Find the Christoffel symbols of a surface in the form ##g= f(u,v).## Homework Equations ##f_{u_1u_1} = \Gamma^1_{11} f_{u_1} + \Gamma^2_{11}f_{u_2} + A \vec{N}## ##f_{u_1u_2} = f_{u_2u_1} = \Gamma^1_{12} f_{u_1} + \Gamma^2_{12}f_{u_2} + B \vec{N}## ##f_{u_2u_2} =...

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...

In my geometry textbook it is stated that intuitively we can choose a suitable basis of coordinates that the components Christoffel symbol vanishes locally at that point(= 0). However can one obtain a formal proof of it? For example if we use rectification theorem to rectify the geodesics...

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...

The Christoffel symbol is as followed: {\Gamma ^{m}}_{ab}=\frac{1}{2}g^{mk}(g_{ak,b}+g_{bk,a}-g_{ab,k}) where k is a dummy index. What values is it summed over? If I had to guess I'd say 0 to 3, but it seems somewhat counter intuitive. Does the Christoffel symbol become {\Gamma...

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...

Just been reading on Christoffel Symbols and I am having a notational mind block. Say we have: \Gamma^{k}_{ij} v^{i} v^{j} The velocity depends on this expression, but I don't read this term v^{i} v^{j} as a velocity squared do I? It's just the one velocity, are the superscripts here...

Homework Statement Homework Equations The Attempt at a Solution Does the Christoffel symbol \Gamma have a dimension in physics? And if it does, what is its dimension? Thank you!

Hi all, I am trying to find the Christoffel connections of this metric: ds2= -(1+2∅)dt2 +(1-2∅)[dx2+dy2+dz2] where ∅ is a general function of x,y,z,t. I tried to solve this through the least action principle, but some of my results(t-related terms) were different from the answer with a minus...

Homework Statement My teacher solved this in class but I'm not understanding some parts of tis solution. Show that \nabla_i V^i is scalar. Homework Equations \nabla_i V^i = \frac{\partial V^{i}}{\partial q^{i}} + \Gamma^{i}_{ik} V^{k} The Attempt at a Solution To start this...

Is the following true? \Gamma_{\mu \nu \alpha}+\Gamma_{\nu \mu \alpha}=-2\Gamma_{\alpha \mu \nu} where: \Gamma_{\alpha \mu \nu}=g_{\alpha \sigma}\Gamma^{\sigma}_{~\mu \nu} I ask because, while bored in a philosophy lecture, I decided to try to derive the geodesic equation by extremizing...

Hi, I am new to general relativity and as I would like to find out how we could derive the value of christoffel symbol in terms of the metric tensor. I have also heard that it was given as a definition for the christoffel symbol and would like a clarification on that. Regards Bltzmn2012

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...

Homework Statement It's not exactly a homework question. I can find Christoffel Symbols using general definition of Christoffel symbol. But, when I try to find Christoffel Symbols using variational principle, I end up getting zero. I have started with the space-time metric in a weak...

Hi Everyone! Somewhere I read <:uhh:or possibly I think so and am wrong> that christoffel symbols correspond to gravitational field. Is there any physical Quantity corresponding to Christoffel symbols? Could you be more explicit as to how the physical quantity corresponds to Christoffel...

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...

Hi all, I'm reading Sean Carroll's Space Time and Geometry and haven't figure out how equation 4.64 is derived, where he is in the process of deriving Einstein's equation from Hilbert action. Given there is a variation of the metric, g_{\mu\nu} \rightarrow g_{\mu\nu} + \delta g_{\mu\nu}, The...

Is there any book out there or website or something that has a list of christoffel symbols, something analogous to a table of integrals The amount of times I've had to calculate christoffel symbols by hand is unreal and let me tell you, chugging through tensors on a moving bus is hard enough...

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...

Hey everyone, This formula was just provided in a book and I was trying to prove it but I'm having a hard time understanding what it's saying. The formula is attached, along with the definitions given for the Christoffel symbols. In the definitions the i's are the standard basis vectors and...

I haven't learned much of advanced mathematics. It seems that we can use metric tensors to lower or raise index of christoffel symbols. But isn't christoffel symbols made of metric tensors and derivatives of metric tensors? How can we contract indices of a derivative directly with metric tensors...

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...