All Questions
131
questions
0
votes
0
answers
86
views
Is there a metric, a solution to Einstein's field equations, for a single body in a space of uniform non-zero density?
The Swarzschild metric describes a single body in an empty space with zero density, while the FLRW metric is presumably for a space with uniform non-zero density but no single body. But is there a ...
- 131
2
votes
1
answer
110
views
Saddle Shaped Universe
The universe, as described by FLRW metric, if $k = -1$ is clearly a 2 sheet 3-hyperboloid described by $x^2+y^2+z^2-w^2=-R^2$. So where does the more common saddle shaped picture of the open universe ...
- 1,161
0
votes
2
answers
68
views
Homogeneous and Isotropic But not Maximally Symmetric Space
Is this statement correct: "In a homogeneous and Isotropic space the sectional curvature is constant, while in a maximally symmetric space the Riemann Curvature Tensor is covariantly constant in ...
- 1,161
0
votes
1
answer
66
views
Name of metric used by Friedmann
In his original paper, Friedmann used the following dynamic and symmetrical metric:
$$\mathrm{d}s^2=a(t)^2\left(\mathrm{d}\chi^2+\sin (\chi)^2\left(\mathrm{d}\theta^2+\sin (\theta)^2 \mathrm{d}\phi^2\...
- 181
1
vote
0
answers
60
views
Robertson-Walker metric exercise [closed]
I'm trying to solve an exercise from my astrophysics and cosmology class, the request is the following, starting from the RW metric expression:
$$ \begin{equation*}
ds^2=c^2 dt^2 - a^2 \left ( \frac{...
- 41
1
vote
0
answers
88
views
Why is $h_{\mu\nu}$ not a tensor in the perturbed Universe in cosmological perturbation theory?
In the cosmological perturbation theory course per Hannu Kurki-Suonio (2022) : https://www.mv.helsinki.fi/home/hkurkisu/CosPer.pdf, there is a remark in the text page 5 that puzzles me. The text goes ...
- 1,109
1
vote
1
answer
148
views
Deriving Klein-Gordon equation in curved spacetime [closed]
I try to drive The Klein-Gordon equation for a massless scalar field in case of FRW metric:
$$
ds^2= a^2(t) [-dt^2 + dx^2]
$$
So I start by:
$$\left(\frac{1}{g^{1/2}}\partial_{\mu}(g^{1/2}g^{\mu\nu}\...
- 395
0
votes
0
answers
230
views
Deriving the Ricci tensor on the flat FLRW metric
I am currently with a difficulty in deriving the space-space components of the Ricci tensor in the flat FLRW metric
$$ds^2 = -c^2dt^2 + a^2(t)[dx^2 + dy^2 + dz^2],$$ to find:
$$R_{ij} = \delta_{ij}[2\...
- 1
0
votes
1
answer
58
views
Making a scale factor invariant *density* in FRW spacetime
For a timelike observer in an FRW spacetime with a perfect fluid, the timelike energy density is given by $T_{\mu\nu}U^\mu U^\nu = \rho(t)$ for a comoving observer.
I want to be able to track changes ...
- 57
2
votes
1
answer
134
views
How do you relate $\Omega_{k}$, the curvature term in the FLRW metric, to the radius of curvature?
I have assumed, for reasons a bit too detailed to go into here, that if $\Omega_{k}$, the curvature term in the FLRW metric, is equal to 1, then the radius of curvature is equal to 13.8 billion light ...
- 371
1
vote
2
answers
245
views
How is the interior Schwarzschild metric derived?
Where does the interior Schwarzschild metric come from? How is it derived and why does it have NOT a singularity? Would it mean that the singularity is only apparent and for those out of the black ...
- 495
2
votes
1
answer
78
views
Why must the components $g_{0i}$ of an isotropic, homogeneous spacetime metric vanish?
In Daniel Baumann's book on Cosmology, it says that the metric of an isotropic, homogeneous spacetime must have the form $$\mathrm{d}s^2=-\mathrm{d}t^2+a^2(t)\gamma_\mathrm{ij}(x)\mathrm{d}x^\mathrm{i}...
- 1,187
1
vote
1
answer
69
views
Divergence of the time component of the stress-energy tensor in FLRW metric
I need some help with the divergence of the time component of the stress energy tensor for dust $\nabla_{\mu}T^{0\mu}$ given the stress energy tensor for dust is
$$T^{\mu\nu}=(\rho+p)u^{\mu}u^{\nu} + ...
0
votes
0
answers
62
views
Spacetimes where symmetries vary from place to place?
Are there spacetimes or metrics where symmetries (like Poincaré, Lorentz, diffeomorphism, translational... invariances) are only local and the symmetries of one local neighbourhood are not, a priori, ...
- 2,462
2
votes
1
answer
196
views
What is the $r$ coordinate in a $\mathbb{S}^3$ FLRW spactime?
I'm having trouble understanding what the $r$ reduced-circumference coordinate really is in a 3-sphere $\mathbb{S}^3$ context.
Let's start with the unit 3-sphere metric in hyperspherical $(\psi, \...
- 163