Tagged Questions
1
vote
2answers
47 views
Triple-right triangle experiment: what's the minimum distance?
Among the other ways, one way to prove the Earth is round is the triple-right triangle.
The idea is simple:
Starting from point A you move in a straight line for a certain distance.
At point B, ...
2
votes
1answer
37 views
How can I express the Riemann tensor of the 4-metric in terms of quantities derived from the 3-metric and the normal to it?
I want an expression for the Riemann tensor of the four metric in terms of extrinsic curvature, normal, lie derivative of the normal, etc.
The first Einstein-Codacci eq. gives the Riemann tensor of ...
1
vote
1answer
56 views
Curvature tensor of 2-sphere using exterior differential forms (tetrads)
$ds^2= r^2 (d\theta^2 + \sin^2{\theta}d\phi^2)$
The following is the tetrad basis
$e^{\theta}=r d{\theta} \,\,\,\,\,\,\,\,\,\, e^{\phi}=r \sin{\theta} d{\phi}$
Hence, $de^{\theta}=0 ...
3
votes
1answer
80 views
Problem with calculating the curvature tensor of the $n$ dimensional sphere
I am calculating the Riemann curvature tensor, Ricci curvature tensor, and Ricci scalar of the $n$ sphere $$x_0^2 + x_1^2 + ....+x_n^2=R^2,$$ whose metric is
$$ds^2=R^2(d\phi_1^2 + \sin{\phi_1}^2 ...
3
votes
0answers
76 views
Curvature and spacetime
Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
0
votes
1answer
63 views
Parallel transport of a vector along a closed curve in curvilinear coordinates
There is an expression indicating the change of the vector parallel translation along a closed infinitesimal curve in curvilinear coordinates (one way of introducing curvature tensor):
$$
\Delta A_{k} ...
2
votes
2answers
113 views
Ricci tensor for a 3-sphere without Math packets
Let's have the metric for a 3-sphere:
$$
dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right).
$$
I tried to calculate Riemann or Ricci tensor's ...
0
votes
0answers
39 views
Curved space to flat space calculation
When changing the curved space co-ordinate into a flat space co-ordinate if a cone. I got the result transformation that i cannot get a transformation at the vertex(apex) why?
0
votes
1answer
34 views
What is the Willmore energy of the Earth (or the geoid)?
Wikipedia defines the Willmore energy as:
$$e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\, \mathrm{d}A,$$
where $H$ stands for the mean curvature of the manifold $\mathcal{M}$.
What is the ...
1
vote
2answers
172 views
What is the Riemann curvature tensor contracted with the metric tensor?
Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example
$R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.
2
votes
1answer
193 views
Difference between $\partial$ and $\nabla$ in general relativity
I read a lot in Road to Reality, so I think I might use some general relativity terms where I should only special ones.
In our lectures we just had $\partial_\mu$ which would have the plain partial ...
0
votes
0answers
90 views
Is there any Calculator capable of calculating and displaying differential geometry? [closed]
Is there any Calculator capable of calculating and displaying differential geometry (display curvature of spacetime)?
$$ds^2~=~g_{ab}dx^adx^b.$$
2
votes
1answer
219 views
What bends fabric of space-time?
I know that mass can bend fabric of space-time, which causes gravity by making an object curve around a planet or star but is there anything else that can bend it?
Other energy sources, forces ...
1
vote
1answer
67 views
Is there a formula to work out how much the fabric of spacetime bends?
From my knowledge, a big mass (planet star etc) can bend the fabric of spacetime. Is there a formula that we can use to work out how much it bends?
1
vote
1answer
106 views
What is the curvature scalar $\Psi_{4}$?
What is the curvature scalar $\Psi_{4}$?
Is it related to the scalar curvature $R$?
What does its real and imaginary parts represent?
2
votes
0answers
292 views
de Sitter and anti de Sitter metric
Is the following correct for the distance $d$ from the origin $(0,0)$ to point $(t,x)$ in the 2-dimensional
de-Sitter and anti de-Sitter spaces? Here, $t$ is time and the distance may be called the ...
6
votes
1answer
555 views
What is the stress energy tensor?
I'm trying to understand the Einstein Field equation equipped only with training in Riemannian geometry. My question is very simple although I cant extract the answer from the wikipedia page:
Is the ...
3
votes
2answers
393 views
Where do I start with Non-Euclidean Geometry?
I've been trying to grok General Relativity for a while now, and I've been having some trouble. Many physics textbooks gloss over the subject with an "it's too advanced for this medium", and many ...
11
votes
5answers
3k views
Laplace operator's interpretation
What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
2
votes
1answer
282 views
Superposition of Ricci scalars [closed]
Suppose I have two point/line singularities in spacetime (what is important to me is that they are localized). Also suppose I have some fields in spacetime and that the two singularities interact with ...
