I've stumbled over this article and while reading it I saw the following statement (##\xi## a vectorfield and ##d/d\tau## presumably a covariant derivative***):
$$\begin{align*}\frac{d \xi}{d \tau}&=\frac{d}{d \tau}\left(\xi^{\alpha} \mathbf{e}_{\alpha}\right)=\frac{d \xi^{\alpha}}{d \tau}...
Hello everyone,
I'm sure a lot of you know that the Christoffel symbols are not tensors by themselves but, their variation is a tensor.
I want to revive a post that was made in 2016 about this: The Variation of Christoffel Symbol and ask again "How is that you can calculate ∇ρδgμν if δ{gμν} is...
Let us assume a "toy-metric" of the form
$$ g=-c^2 \mathrm{d}t^2+\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2-\frac{4GJ}{c^3 r^3} (c \mathrm{d}t) \left( \frac{x\mathrm{d}y-y\mathrm{d}x}{r} \right)$$
where ##J## is the angular-momentum vector of the source.
Consider the curve
$$ \gamma(\tau)=(x^\mu...
Homework Statement
My textbook states:
Since the number of particles of dust is conserved we also have the conservation equation
$$\nabla_\mu (\rho u^\mu)=0$$
Where ##\rho=nm=N/(\mathrm{d}x \cdot \mathrm{d}y \cdot \mathrm{d}z) m## is the mass per infinitesimal volume and ## (u^\mu) ## is...
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...
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...
The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as:
$$
R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n}
$$
My question is that it seems that...
Here's what I'm watching:
At about 1:35:00 he leaves it to us to look at a parallel transport issue. Explicitly to caclculate
##D_s D_r T_m - D_r D_s T_m##
And on the last term I'm having some difficulties, the second christoffel symbol.
So we have
##D_s [ \partial_r T_m - \Gamma_{rm}^t T_t]##...
Hi friends,
I'm learning Riemannian geometry. I'm in trouble with understanding the meaning of ##g^{jk}\Gamma^{i}{}_{jk}##. I know it is a contracting relation on the Christoffel symbols and one can show that ##g^{jk}\Gamma^i{}_{jk}=\frac{-1}{\sqrt{g}}\partial_j(\sqrt{g}g^{ij})## using the...
Hi,
Want to know (i) what does Riemannian metric tensor and Christoffel symbols on R2 mean? (ii) How does metric tensor and Christoffel symbols look like on R2? It would be great with an example as I am new to this Differential Geometry.