All Questions
23
questions
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
4
votes
2
answers
170
views
Why isn't the curvature scale in Robertson-Walker metric dynamic?
$$ds^2=-c^2dt^2+a(t)^2 \left[ {dr^2\over1-k{r^2\over R_0^2}}+r^2d\Omega^2 \right]$$
This is the FRW metric, here k=0 for flat space, k=1 for spherical space, k=-1 for hyperbolic space. $R_0$ is the ...
- 619
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:
$$R_{ij} = \delta_{ij}[2\...
- 1
2
votes
1
answer
134
views
How do you relate $\Omega_{k}$, the curvature term in the FLRW metric, to the radius of curvature?
I have assumed, for reasons a bit too detailed to go into here, that if $\Omega_{k}$, the curvature term in the FLRW metric, is equal to 1, then the radius of curvature is equal to 13.8 billion light ...
- 371
1
vote
1
answer
123
views
What's the physical curvature scale $R_0$ in the FLRW metric?
I'm studing the FLRW metric using Daniel Baumann's book, Cosmology (2022), in this book the author derived the FRLW metric using the following equations:
$$ dl^2 = \textbf{dx}^2 \pm du^2 $$ and $$ \...
1
vote
1
answer
105
views
Area of a sphere in curved 3D space
I'm having trouble finding any information on the derivation of the area of sphere in curved 3D space: $A = 4 \pi S^2_{\kappa}$, where $S_{\kappa} = R_o \sin(r/R_o)$.
How did it come about from $ds^2 =...
- 161
2
votes
0
answers
43
views
How (if) can we connect a 2D "throat" piece of a wormhole to two hyperbolic 2D manifolds?
This question wad closed on the mathematics site, as it lacked clarity. So I try my luck here. My question is cosmology-inspired.
Imagine two 2D hyperbolic manifolds. I connect them by a manifold like ...
1
vote
0
answers
675
views
Ricci tensor for FRW Metric [closed]
I am attempting to prove that the FLRW metric given by
$$ds^2 = -dt^2 + g_{ij}dx^idx^j = -dt^2 + a^2(t)\left(d\vec{x}^2+K\frac{(\vec{x}\cdot d\vec{x})^2}{1-K\vec{x}^2}\right)$$
has
$$R_{ij} = \left[\...
- 201
1
vote
2
answers
517
views
How is spacetime locally Lorentzian?
Following up on the questions raised here:
If you take all the matter and energy out of a significantly large volume of spacetime, what you'll be left with is a small chunk of spacetime that - in the ...
- 584
1
vote
1
answer
59
views
Curvature sign-changing Friedman models
Isotropy and homogeneity of space leads to the spacetime metric of the form
$$
ds^2=-dt^2+d\sigma_k^2,
$$
where $d\sigma_k^2$ is the metric on one of the standard manifolds (the 3-sphere, Euclidean 3-...
- 421
3
votes
2
answers
498
views
How is the first Friedmann equation derived from Einstein's field equations?
I see that Friedmann's first Equation (for flat space) is:
$$\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G}{3}\rho.$$
And I know that Einstein's equation, just considering the time-time component is:
$...
- 584
2
votes
1
answer
63
views
Curvature of spatial 2-section in general form for metric
In this paper (Four-Dimensional Asymptotically AdS Black Holes with Scalar Hair by Gonzalez et. al.), the following standard metric is taken as an ansatz for the hairy black hole:
$$ds^2=-f(r)dt^2+f^{-...
- 2,665
3
votes
2
answers
478
views
I need help computing the effect of curvature on the FRW metric
Apparently there are different forms of the FLRW metric. I'm focusing on Anti-de Sitter space, so I'll just give the hyperbolic version of the function.
$$ds^2=-c^2dt^2+a^2(t)\left[dr^2+R_0\space \...
- 584
1
vote
1
answer
162
views
Are the components of the curvature tensor defined by the Robertson-Walker metric constant?
In the Robertson-Walker solution the curvature is uniform. However the space is "expanding" and so the matter density is decreasing over time (or not because it's modelled as a perfect fluid?). Does ...
- 6,963
2
votes
1
answer
96
views
Negative curved spacetime [closed]
Given a metric of the form $$ ds^2=c^2dt^2-a^2\left[d\chi^2+\frac{\sinh^2(\sqrt{-k}\chi)}{-k}d\Omega^2\right] $$
Where $d\Omega=d\theta^2+\sin^2(\theta)d\phi^2$, $k<0$
and $a=a(t)$.
I came across ...
- 90