The variables used in general relativity to describe the shape of spacetime. If your question is about metric units, use the tag "units", and/or "si-units" if it is about the SI system specifically.
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2answers
91 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
1answer
46 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):
$$
\Delta A_{k} ...
5
votes
0answers
156 views
Penrose Conformal diagram for flat 2-dim Lorentz space-time
I have the following metric
$$ds^2 ~=~ Tdv^2 + 2dTdv,$$
defined for
$$(v,T)~\in~ S^1\times \mathbb{R},$$
e.g. $v$ is periodic.
This is the according Penrose diagram:
Question 1) Is the ...
1
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0answers
35 views
Null vector fields given Bondi metric
I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric
$g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$
with $d\Omega$-standard metric ...
1
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0answers
49 views
Singularities in Schwarzchild space-time
Can anyone explain when a co-ordinate and geometric singularity arise in Schwarzschild space-time with the element
$$ ...
1
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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
$$
dl^{2} = R^{2}\left(d \psi^{2} + sin^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})\right),
$$
$$
dl^{2} = R^{2}\left(d ...
1
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0answers
49 views
The interior of a cylinder as an Einstein manifold
The interior of a curved cylinder is an Einstein manifold (the Ricci Curvature Tensor is proportional to the Metric $R_{\mu\nu}=kg_{\mu\nu}$) since it has a constant curvature.
However, I was unable ...
1
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0answers
46 views
When is spacetime homogenous and isotropic?
When is spacetime homogenous and isotropic?
For example, some metric $g_{\mu \nu}$ is homogeneous and isotropic. We now construct effective metric
$$n_{\mu \nu} ~\rightarrow~ g_{\mu \nu} + ...
0
votes
0answers
33 views
metric extension outside the light cone
Could anyone explain what "extending the solution" beyond the past light cone means? Say, for example, if I have a metric (no coordinate singularities), how can I extend it to the outside of the past ...
0
votes
0answers
41 views
Switching from an accelerated frame of reference to a locally inertial reference system
Using the equivalence principle, show that the interval for an accelerated observer ($\textbf{g}$ uniform and constant) has the form
$$
ds^2|_{\text{first order in ...
