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.
4
votes
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263 views
How to prove that Weyl tensor is invariant under conformal transformations?
I need to verify that the solution for vanishing Weyl tensor is conformally flat metric $g_{\mu\nu} = e^{2\varphi}\eta_{\mu\nu}$. The most convenient way to show this is to prove that Weyl tensor is ...
3
votes
0answers
56 views
Squashed 3-sphere?
What is a squashed 3-sphere? In context of quantum gravity. I stumbled upon a term 'squashed 7 sphere' but that's concerning supersymmetry.
Is it just normal 3-sphere metric, that is just 'squashed' ...
3
votes
0answers
92 views
Some hints for special case of metric tensor in GR
Let's have metric
$$
ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - 2f(t - z, x, y)(dt - dz)^2.
$$
I need to prove that it is an exact solution for Einstein equations in vacuum for $\partial_{x}^{2}f + ...
3
votes
0answers
95 views
About Dirac equation in curved spacetime (spherical)
I would like to ask you about the separation of variables of the Dirac equation in curved space-time. The metric is given by $$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+\alpha^{2}r^{2}\sin^{2}\theta ...
2
votes
0answers
87 views
General formula to compute the redshift (first order perturbations)
Consider an expanding universe with the following metric in conformal time/co-moving coordinates:
...
2
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0answers
89 views
Understanding spherically symmetric metric
In these lecture notes the static isotropic metric is treated as follows (p. 71):
Take a spherically symmetric, bounded, static distribution of matter, then we will have a spherically symmetric ...
1
vote
0answers
26 views
Metric to describe an expanding spacetime from coordinates reflecting the perspective of a local observer
The FLRW metric describes the metric expansion of spacetime from the perspective of comoving coordinates. Given the way this metric is usually formulated, comoving distances stay constant, and the ...
1
vote
0answers
84 views
Covariant Derivative with a Torsion Free Metric
Where $\triangledown$ is the covariant derivative and we are to assume that the connection is torsion free (that is, we can exchange the lower indices of the connection coefficients), how can I prove ...
1
vote
0answers
58 views
Ricci scalar higher dimensions
I was wondering if there is a straightforward way to compute the Ricci curvature of a metric that has the form (à la Kaluza-Klein):
$g_{MM}\equiv\begin{pmatrix}g_{\mu\nu}&g_{\mu ...
1
vote
0answers
61 views
The time dilation in an oscillating elevator
Suppose you are in an elevator which oscillates vertically with a frequency $\nu$.
How will we find the time dilation in this oscillating reference frame ?
If the lift is accelerating upward or ...
1
vote
0answers
73 views
Linearized gravity and symmetries
I have naive question.
When we analyzing weak gravity field we introduce expression for metric tensor as
$$
g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}, \quad \eta_{\mu \nu} = diag(1, -1, -1, -1), ...
1
vote
0answers
71 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
59 views
Singularities in Schwarzchild space-time
Can anyone explain when a co-ordinate and geometric singularity arise in Schwarzschild space-time with the element
$$ ...
1
vote
0answers
44 views
How to prove the derive the expression for space part of Riemann tensor for homogeneous and isotropic space-time?
It's not a homework!!
For spheric, hyperbolic and flat case
$$
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
vote
0answers
115 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
vote
0answers
65 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
$$n_{\mu \nu} ~\rightarrow~ g_{\mu \nu} + ...
0
votes
0answers
23 views
How to find solutions to the gravitational potential metric h
I'm working on a problem in which a star of mass M1, radius R1 is surrounded by a shell of mass M2, , radius R2. I want to find the solutions to the gravitational potential h in the regions in between ...
0
votes
0answers
27 views
Why such hypersurface orthogonal vector leading to $g_{0i}=0$ for $i=1,2,3$?
Suppose that the hypersurface orthogonal co-vector $W$ us perpendicular to the family of hypersurface defined by a function $\varphi$ with $\varphi=constant$. If we choose a coordinate in which ...
0
votes
0answers
80 views
Curvature based derivation of Schwarzchild Metric
I'm a third year maths undergrad and I'm trying to find (and follow) a curvature based derivation of the Schwarzchild metric, if there exists such a proof?
0
votes
0answers
42 views
What is the physical meaning of the Eddington - Finkelstein metric?
I want to see a some physical process (experimental) that could explain the many transformations of coordinates into this mathematical procedure.
(really two transformations, but i think that is a ...
0
votes
0answers
67 views
Angular Momentum with Upper Index
I am asked to show $[L^2,L_i] = 0 $, but with the definition :
$L^2 \equiv L_i L^i$
I tried this:
$[L_i L^i,L_i] = L_i [L^i,L_i] + [L_i,L_i]L^i$
We know that : $[L_i,L_i]$ = 0 , so we have,
$[L_i ...
0
votes
0answers
51 views
metric extension outside the light cone
Could anyone explain what "extending the solution" beyond the past light cone means? Say, for example, if I have a metric (no coordinate singularities), how can I extend it to the outside of the past ...
0
votes
0answers
57 views
Switching from an accelerated frame of reference to a locally inertial reference system
Using the equivalence principle, show that the interval for an accelerated observer ($\textbf{g}$ uniform and constant) has the form
$$
ds^2|_{\text{first order in ...
