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 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...
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...
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...
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...
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...
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} =...
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...