All Questions
18
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
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
0
votes
2
answers
239
views
Christoffel symbol with conformal time not equal to with cosmic time one when making a change of coordinates for d'Alembertian
I think I am having a misunderstanding that would be nice to clear up.
The covariant d'Alembertian
$$
\Box \phi = g^{\mu\nu}\nabla_\mu\partial_\nu \phi= \left(\partial^2 + \Gamma^\mu_{\mu\lambda}\...
- 23
1
vote
1
answer
105
views
Area of a sphere in curved 3D space
I'm having trouble finding any information on the derivation of the area of sphere in curved 3D space: $A = 4 \pi S^2_{\kappa}$, where $S_{\kappa} = R_o \sin(r/R_o)$.
How did it come about from $ds^2 =...
- 161
0
votes
0
answers
105
views
Raychaudhuri equation and expansion scalar for constant $a(t)$ FLRW metric
On the Wikipedia page for the Raychaudhuri equation, the expansion scalar $\theta$ is described as the rate of change of volume of a ball of matter with respect to the time of a central, comoving ...
- 323
1
vote
2
answers
923
views
Determining the partial derivative of a metric tensor
Im new to the Tensor Calculus and General Theory of Relativity, and I have one question. I want to determine the Christoffel symbols in FRW metric.
This is the general equation of Christoffel symbols:
...
- 25
1
vote
2
answers
517
views
How is spacetime locally Lorentzian?
Following up on the questions raised here:
If you take all the matter and energy out of a significantly large volume of spacetime, what you'll be left with is a small chunk of spacetime that - in the ...
- 584
2
votes
2
answers
555
views
Cosmic string solution to general relativity
I'm having a difficulty in finalizing a resolution of the Einstein equation for a static cosmic string. I start with the following metric ansatz, for a static straight string oriented along the $z$ ...
- 7,592
1
vote
1
answer
208
views
Embedding manifold equipped with FLRW metric
The Friedmann–Lemaître–Robertson–Walker (FLRW) metric is as follows, in natural units:
$$\mathrm{d}s^2=-\mathrm{d}t^2+a(t)^2\left(\frac{\mathrm{d}r^2}{1-\kappa r^2}+r^2\mathrm{d}\theta^2+r^2\sin^2\...
- 3,967
1
vote
1
answer
132
views
What is this metric's scale factor?
While answering this question about a hypothetical 3-sphere universe $S^3$ expanding with a constant acceleration $\phi$ from a zero initial speed
$$ r=\dfrac{\phi}{2}t^2$$
I started from a generic ...
- 12.7k
0
votes
1
answer
243
views
General relativity with space and time on different footing
Excerpt from the textbook below. It seems ambiguous what the author means and I am unable to proceed.
Imagine that you live in a Universe where Einstein never existed. Instead, he was replaced by ...
- 323
12
votes
1
answer
356
views
What are the allowed topologies for a FRW metric?
Given a spacetime that has the maximal amount of spacelike translations and rotations, what are the possible topologies it may take? I am mostly wondering about the "time" topology since the spatial ...
- 16.5k
2
votes
2
answers
638
views
How do we measure distances in the FLRW metric?
We are in the flat FLRW metric in Cartesian comoving coordinates. The metric is expressed as:
$$ds^2 = d\tau^2 + a(\tau)^2\big(dx^2 + dy^2 + dz^2\big)$$
The fact that the "universe is expanding" is ...
- 5,220
6
votes
3
answers
277
views
Two Robertson-Walker observers, at what time will a light signal be received?
Here is a question I have that is inspired by this question here.
The spacetime metric of a radiation-filled, spatially flat ($k = 0$) Robertson-Walker universe is given by$$ds^2 = - dT^2 + T[dx^2 + ...
- 211