All Questions
Tagged with cosmology metric-tensor
180
questions
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Is the tensor product involved in the metric a symmetric product?
The expression of the FRW metric in Cosmology in usually written as:
$$ds^2=-dt^2+a^2(t)d\vec{x}^2$$
where $c=1$. However, $dt^2$ is a shortening of $dt\otimes dt$, that is, of the tensor product of $...
- 293
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Question coming from Cosmological Perturbation
We consider the following scalar perturbation on the FRW metric:
$$ ds^2 = -(1 + 2\phi)dt^2 +2a\partial_i B dx^i dt + a^2 \left( (1 - 2\psi)\delta_{ij} + 2\partial_{ij}E\right) dx^i dx^j $$
where $\...
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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
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Is the FRW metric, based on spatial homogeneity and isotropy, rotationally and translationally invariant? If so, how?
The spatial part of the Minkowski metric, written in the Cartesian coordinates, $$d\vec{ x}^2=dx^2+dy^2+dz^2,$$ is invariant under spatial translations: $\vec{x}\to \vec{x}+\vec{a}$, where $\vec{a}$ ...
- 11.8k
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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
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68
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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
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199
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Change of variables from FRW metric to Newtonian gauge
My question arises from a physics paper, where they state that if we take the FRW metric as follows, where $t_c$ and $\vec{x}$ are the FRW comoving coordinates:
$$ds^2=-dt_c^2+a^2(t_c)d\vec{x}_c^2$$
...
- 293
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27
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A question about Friedmann cosmological expansion equation
A recent paper, arXiv:2403.01555, gives the equations for homogeneity and isotropy of an expanding 3-space as expressed in the following
distance interval as $x^i = (t, \chi, \theta, \phi)$ and $x^i + ...
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66
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Name of metric used by Friedmann
In his original paper, Friedmann used the following dynamic and symmetrical metric:
$$\mathrm{d}s^2=a(t)^2\left(\mathrm{d}\chi^2+\sin (\chi)^2\left(\mathrm{d}\theta^2+\sin (\theta)^2 \mathrm{d}\phi^2\...
- 181
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120
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Transformation under coordinate transformation of scalar perurbation of FLRW metric
For the past few days I've been studying perturbation in cosmology. More specifically I am now busy with chapter 6 in Dodelson's Modern cosmology. In this book the perturbed FLRW metric which only ...
- 29
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60
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Robertson-Walker metric exercise [closed]
I'm trying to solve an exercise from my astrophysics and cosmology class, the request is the following, starting from the RW metric expression:
$$ \begin{equation*}
ds^2=c^2 dt^2 - a^2 \left ( \frac{...
- 41
1
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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
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1
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148
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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:
$$\left(\frac{1}{g^{1/2}}\partial_{\mu}(g^{1/2}g^{\mu\nu}\...
- 395
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2
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170
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Why isn't the curvature scale in Robertson-Walker metric dynamic?
$$ds^2=-c^2dt^2+a(t)^2 \left[ {dr^2\over1-k{r^2\over R_0^2}}+r^2d\Omega^2 \right]$$
This is the FRW metric, here k=0 for flat space, k=1 for spherical space, k=-1 for hyperbolic space. $R_0$ is the ...
- 619
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230
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Deriving the Ricci tensor on the flat FLRW metric
I am currently with a difficulty in deriving the space-space components of the Ricci tensor in the flat FLRW metric
$$ds^2 = -c^2dt^2 + a^2(t)[dx^2 + dy^2 + dz^2],$$ to find:
$$R_{ij} = \delta_{ij}[2\...
- 1
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1
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58
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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
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1
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134
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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
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2
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245
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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
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1
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78
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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 $$\mathrm{d}s^2=-\mathrm{d}t^2+a^2(t)\gamma_\mathrm{ij}(x)\mathrm{d}x^\mathrm{i}...
- 1,187
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Divergence of the time component of the stress-energy tensor in FLRW metric
I need some help with the divergence of the time component of the stress energy tensor for dust $\nabla_{\mu}T^{0\mu}$ given the stress energy tensor for dust is
$$T^{\mu\nu}=(\rho+p)u^{\mu}u^{\nu} + ...
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52
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Common choice in FLRW between dimensionless of scale factor (and coordinates of r lenght dimension) or the contrary
I have an old cosmology book in which the scale factor is expressed in the Roberston-Walker metric as:
$\mathrm{d} s^2=c^2 \mathrm{~d} t^2-R(t)^2 \mathrm{~d} l^2$
with: $\mathrm{d} l^2=\dfrac{\mathrm{...
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62
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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
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1
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196
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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
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86
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Spacetimes, metrics and symmetries in the theory of relativity?
I was discussing this paper with a couple of physicists colleagues of mine (https://arxiv.org/abs/2011.12970)
In the paper, the authors describe "spacetimes without symmetries". When I ...
- 2,462
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650
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Klein-Gordon equation in FRW spacetime
The metric for FRW spacetime is $$ds^2=a(n)^2(dn^2 - dx^2)$$ where $dn$ is the conformal time differential form. The Klein Gordon equation in curved spacetime is $$\left(\frac{1}{g^{1/2}}\partial_{\mu}...
user310742
1
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123
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What's the physical curvature scale $R_0$ in the FLRW metric?
I'm studing the FLRW metric using Daniel Baumann's book, Cosmology (2022), in this book the author derived the FRLW metric using the following equations:
$$ dl^2 = \textbf{dx}^2 \pm du^2 $$ and $$ \...
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1
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98
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Calculating distance from the FRW metric
My question arises from the book Cosmology by Daniel Baumann, specifically from equation (2.81), where I don't understand how the expression for the distance is calculated. I will start by providing ...
- 293
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2
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239
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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
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73
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Energy conservation, photon, distance
From Weinberg 1972 Gravitation and Cosmology Principles, when discussing the luminosity distance, he talked about due to light wavelength being stretched as well as time interval stretched, in the &...
- 161
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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 =...
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