All Questions
22
questions
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:
$$ \begin{equation*}
ds^2=c^2 dt^2 - a^2 \left ( \frac{...
- 41
0
votes
0
answers
230
views
Deriving the Ricci tensor on the flat FLRW metric
I am currently with a difficulty in deriving the space-space components of the Ricci tensor in the flat FLRW metric
$$ds^2 = -c^2dt^2 + a^2(t)[dx^2 + dy^2 + dz^2],$$ to find:
$$R_{ij} = \delta_{ij}[2\...
- 1
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
1
answer
106
views
Confusion about the notation for the tangent vector to the FRLW's metric congruence
I want to solve the following problem (Problem 2.2 on Poisson's book "A Relativist's Toolkit"):
The FLRW metric is
$$\text{d} s^2 = -\text{d} t^2 + a(t)^2 \Bigg(\frac{\text{d} r^2}{1 - kr^2} ...
- 615
0
votes
0
answers
165
views
Killing Vectors from Killing Equations
I need to find the killing vectors of the FLRW metric. However, it seems that they are complicated. Is there a simple/general equation that gives the killing vectors for a given metric? Or do I have ...
- 3,122
1
vote
0
answers
675
views
Ricci tensor for FRW Metric [closed]
I am attempting to prove that the FLRW metric given by
$$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
2
votes
1
answer
157
views
Metric components given by Einstein's equation
In a exterior region without matter to a stationary black hole, spherical symmetric, where the cosmological constant is not zero. From the Cartan's structural equations for space without torsion, we ...
- 1,213
1
vote
1
answer
480
views
Coordinate transformation to find the Minkowski metric
I have the following cosmology exercise:
i) calculate the evolution scale factor for an open empty universe and write down its spacetime metric in terms of coordinate $\chi$
\begin{equation}
d\chi = \...
- 155
0
votes
0
answers
378
views
Continuity equation for the stress-energy tensor in the FLRW metric
I'm trying to compute the continuity equation for the stress-energy tensor $\nabla^\mu T_{\mu\nu}$ in the FLRW metric $$ds^2=-dt^2+a^2(t)ds^2_3$$ where $ds^2_3=g^3_{ij}dx^idx^j$ is the metric for the ...
- 99
1
vote
2
answers
667
views
Derive Christoffel Symbols for FRW
In Weinberg's Cosmology, the FRW metric is
\begin{equation}
d\tau^2=dt^2-a^2\left[d\vec{x}^2+K\dfrac{(\vec{x}\cdot d\vec{x})^2}{1-K\vec{x}^2}\right]
\end{equation}
with $g_{ij}=a^2\left(\delta_{ij}+K\...
- 41
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
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
4
votes
1
answer
705
views
Killing Tensor of Friedman-Robertson-Walker Metric
I would like help showing that the tensor, $$K_{\mu\nu}=a^2(g_{\mu\nu}+u_\mu u_\nu),$$ where $u^\mu =(1,0,0,0)$, is a Killing tensor of the spatially flat FRW metric,
$$ds^2=-dt^2+a(t)^2\left(dr^2+d\...
- 51
1
vote
1
answer
169
views
Spatial part of Robertson-Walker metric
The spatial part of the FRW metric can be written as $$d\Sigma^2=d\rho^2+f^2(\rho)(d\theta^2+{sin}^2\theta d\phi^2)$$ where $f(\rho)$ satisfies $$\frac{df}{d\rho}=\frac{f(2\rho)}{2f(\rho)}.$$ I am ...
- 989
0
votes
1
answer
159
views
How can I split the Klein-Gordon equation into first order ODEs?
The Klein Gordon eqution in conformal time without perturbations is:
I want to solve the equation numerically, but to do so I need to split it into 2 first order differential equations. What would it ...
- 27