Tagged Questions
0
votes
1answer
48 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):
$$
\Delta A_{k} ...
1
vote
2answers
94 views
Ricci tensor for a 3-sphere without Math packets
Let's have the metric for a 3-sphere:
$$
dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right).
$$
I tried to calculate Riemann or Ricci tensor's ...
1
vote
0answers
106 views
How to calculate Riemann and Ricci tensors for a sphere? [closed]
Let's have the metric for a sphere:
$$
dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right).
$$
I tried to calculate Riemann or Ricci tensor's ...
1
vote
0answers
25 views
How to prove the derive the expression for space part of Riemann tensor for homogeneous and isotropic space-time?
It's not a homework!!
For spheric, hyperbolic and flat case
$$
dl^{2} = R^{2}\left(d \psi^{2} + sin^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})\right),
$$
$$
dl^{2} = R^{2}\left(d ...
4
votes
3answers
113 views
How scalar curvature of following spacetime can be equal to zero?
For an interval of this spacetime,
$$
ds^{2} = c^{2}dt^{2} - c^{2}t^{2}(d \psi^{2} + sh^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})),
$$
scalar curvature is equal to zero. Also, Ricci ...
0
votes
2answers
56 views
Changing the scalar curvature (k = 0,+1,-1) with coordinate transformations?
I would like to prove that I can (or can't) change curvature of space, k = 0,+1,-1, via general coordinate transformations, which in principle can mix space and time coordinates together.
4
votes
3answers
139 views
How do you tell if a metric is curved?
I was reading up on the Kerr metric (from Sean Carroll's book) and something that he said confused me.
To start with, the Kerr metric is pretty messy, but importantly, it contains two constants - ...
3
votes
1answer
117 views
Material strain from spacetime curvature
Let's say that you moved an object made of rigid materials into a place with extreme tidal forces. Materials have a modulus of elasticity and a yield strength. Does the corresponding 3D geometric ...
