Christoffel Definition and 145 Threads

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...

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...

"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!

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...

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: \begin{equation} \Gamma^{i}_{00} \end{equation} I set my equation up as: \begin{equation} \Gamma^i_{00} = \frac{1}{2} g^{ij} (\partial_0 g_{0j} +...

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}...

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...

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...

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...

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?

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

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...

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...

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...

I was reading chapter 3 of Carroll's book up to page 100, where he mentions that the coefficients of Christoffel symbol in flat spacetime vanish in cartesian coordinates but not in curvilinear coordinate systems. I was thinking on why this happens. My logic tells me that these coefficients...

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...

I have a technical problem. 1. Accordingly to historical E.B. Christoffel’s work (I think year 1869), (Christoffel’s) symbols are symmetric in the two (today writing) lower indices. 2. These symbols have been introduced when studying the preservation of differential forms of degree two. The...

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...

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} )...

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...

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...

I am working from Sean Carroll's Spacetime and Geometry : An Introduction to General Relativity and have got to the geodesic equation. I wanted to test it on the surface of a sphere where I know that great circles are geodesics and is about the simplest non-trivial case I can think of. Carroll...

I hope you can understand my notation. The Christoffel symbol can be defined through the relation$$ \frac{\partial \pmb{Z}_i} {\partial Z^k} = \Gamma_{ik}^j \pmb{Z}_j $$ I can solve for the Christoffel symbol this way: $$ \frac{\partial \pmb{Z}_i} {\partial Z^k} \cdot \pmb{Z}^m = \Gamma_{ik}^j...

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

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...

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...

[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.

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##...

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...

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} +...

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...

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?

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...

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...

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...

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...

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 $$\sigma^k, \sigma_p, \sigma_q$$...

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...

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...

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...

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-...

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...

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...

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??

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...

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...

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?

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...

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...

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...