All Questions
Tagged with tensor-calculus cosmology
14
questions
2
votes
1
answer
100
views
Reducing Tensor-rank by fixing an argument
Assume for example that you are given a (2,0) tensor $T^{\mu\nu}$ and you want to create
a vector, i.e., a (1,0) tensor out of it. Is it possible to just fix an index of
$T^{\mu\nu}$ while keeping the ...
- 56
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
0
answers
64
views
Deriving the change in photon frequency under a change of coordinates
I'm trying to derive the following equation relating the frequency of a photon in a frame $\tilde{O}$ to a frame $O$ with a background perturbation. To be more specific, this is equation 9.10 from ...
- 904
1
vote
2
answers
229
views
Continuity equation for the conservation of energy from the conservation of the energy-momentum tensor
I am working through the book Cosmology by Daniel Baumann, and in the subsection that covers the continuity equation (part of section 2.3.1 on perfect fluids) the author makes a claim that confuses me....
- 293
2
votes
1
answer
199
views
FLRW metric derivation
How do I get from the metric for a 3-space of constant curvature $$d\sigma^2 = \frac {dr^2} {1-Kr^2} + r^2 (d\theta^2 + sin^2\theta d\phi^2)$$ to a conformally flat form
$$d\sigma^2=(1+\frac 1 4 K\...
1
vote
1
answer
203
views
Computing the longitudinal and traceless part of the left hand side of Einstein's equation
I am reading a textbook on cosmology. Consider $G^i_j$, the left hand side of Einstein's equation. If $\Psi$ and $\Phi$ are first order perturbations to the time and spatial components respectively of ...
- 1,222
1
vote
1
answer
230
views
Raising and lowering indices in line elements - why do we raise and lower them in line elements?
My question refers to Piattella's lecture notes on cosmology. On page 15, the Euclidean line element is defined as
$$
ds^2 = \vert d\mathbf{x}\vert^2 = \delta_{ij}dx^idx^j.
$$
My first question is ...
- 201
0
votes
1
answer
223
views
Is the equivalence of axial vectors with skew symmetric tensors in 3 dimensions a coincidence?
We happen to live in the only number of (large space-like) dimensions that permits an equivalence between skew symmetric tensors, like the magnetic field, and vectors. Similarly (equivalently? not ...
- 3
-2
votes
1
answer
74
views
Write down the components of metric tensor correctly [closed]
this is a FLRW metric and I want to write down the metric tensor from this FLRW metric accurately. Can anyone please help me to do this? Thanks in advance.
\begin{equation}\tag{1}
ds^2 = a^2 ( \tau) [...
- 55
5
votes
1
answer
249
views
A question from cosmological perturbation theory
We consider the following scalar perturbation on the FRW metric
$$ds^2=-(1+2\Phi)dt^2+2a(\partial_iB)dx^idt+a^2[(1-2\Psi)\delta_{ij}+2\partial_{ij}E]dx^idx^j,$$
where $\Phi$, $B$, $\Psi$ and $E$ are ...
- 3,691
0
votes
1
answer
127
views
Isotropy of 3-space and spacetime metric
The most general spacetime metric is given by $$ds^2=g_{\mu\nu}dx^\mu dx^\nu=c^2dt^2+g_{0i}dtdx^i+g_{ij}dx^i dx^j.$$
Why is the second term said to violate isotropy of 3-space?
It is true that ...
- 26.8k
1
vote
2
answers
407
views
Why is the cosmological constant a scalar?
Maybe my understanding is just off, but the cosmological constant is just a scalar, right?
What are it's units?
Why a scalar? - was a tensor 'cosmological constant' ever considered or is it just not ...
- 11.7k
3
votes
2
answers
517
views
General expression of the redshift: explanation?
In some papers, authors put the following formula for the cosmological redshift $z$ :
$1+z=\frac{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{S}}{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{O}}$
where :
$S$ ...
- 1,109
2
votes
1
answer
241
views
Tensor perturbation inflation
During inflation the metric is de-Sitter so $dt^2-d\underline{X}^2 $.
I know that the eqn.motion governing GW's from inflation (tensor perturbations) is
$$2H\dot{h}+\ddot{h}-\nabla^{2}_{i}h~=~0,$$ ...
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