# christoffel

1. ### I Question about geodesics on a sphere

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...
2. ### 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...
3. ### 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...
4. ### A Orbital Angular momentum couples 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...
5. ### 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...
6. ### Riemann Tensor

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...
7. ### How do I differentiate this Scalar Field?

1. Homework Statement (a) Find the christoffel symbols (Done). (b) Show that $\phi$ is a solution and find the relation between A and B. 2. Homework Equations 3. The Attempt at a Solution Part(b) \nabla_\mu \nabla^\mu \phi = 0 I suppose for a scalar field, this is simply the normal...
8. ### Energy-Momentum Tensor Algebra

1. 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: 2. Homework Equations 3. The Attempt at a Solution Part (a) U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v...
9. ### Flat Space - Christoffel symbols and Ricci = 0?

1. Homework Statement (a) Find christoffel symbols and ricci tensor (b) Find the transformation to the usual flat space form $g_{\mu v} = diag (-1,1,1,1)$. 2. Homework Equations 3. The Attempt at a Solution Part(a) I have found the metric to be ## g_{tt} = g^{tt} = -1, g_{xt} =...
10. ### Covariant derivative for four velocity

1. Homework Statement Show U^a \nabla_a U^b = 0 2. Homework Equations U^a refers to 4-velocity so U^0 =\gamma and U^{1 - 3} = \gamma v^{1 - 3} 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...