The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is an exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the scale factor of the universe as a function of time. Depending on geographical or historical preferences, the set of the four scientists – Alexander Friedmann, Georges Lemaître, Howard P. Robertson and Arthur Geoffrey Walker – are customarily grouped as Friedmann or Friedmann–Robertson–Walker (FRW) or Robertson–Walker (RW) or Friedmann–Lemaître (FL). This model is sometimes called the Standard Model of modern cosmology, although such a description is also associated with the further developed Lambda-CDM model. The FLRW model was developed independently by the named authors in the 1920s and 1930s.
Wikipedia states the following in their article about the expansion of the universe:
If the cosmological principle was discovered to be false in our universe, i.e. our universe was discovered to be inhomogeneous or anisotropic or both on very large scales and the FLRW metric does not hold for...
Once having converted the FLRW metric from comoving coordinates ##ds^2=-dt^2+a^2(t)(dr^2+r^2d\phi^2)## to "conformal" coordinates ##ds^2=a^2(n)(-dn^2+dr^2+r^2d\phi^2)##, is there a way to facilitate solving for general geodesics that would otherwise be difficult, such as cases with motion in...
I have calculated the Christoffel symbols for the above given metric, but I don't understand how to calculate a photon's four-momentum using this information. I believe it has something to do with the null geodesic equation but I can't understand how to put that information into the problem...
I need the Ricci scalar for the FRW metric with a general lapse function ##N##:
$$ds^2=-N^2(t) dt^2+a^2(t)\Big[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ d\phi^2)\Big]$$
Could someone put this into Mathematica as I don't have it?
Using the Einstein-Hilbert action for a Universe with just the cosmological constant ##\Lambda##:
$$S=\int\Big[\frac{R}{2}-\Lambda\Big]\sqrt{-g}\ d^4x$$
I would like to derive the equations of motion:
$$\Big(\frac{\dot a}{a}\Big)^2+\frac{k}{a^2}=\frac{\Lambda}{3}\tag{1}$$
$$2\frac{\ddot...
My attempt:
Realize we can work in whatever coordinate system we want, therefore we might as well work in the rest frame of the fluid. In this case ##u^a=(c,\vec{0})##.
The conservation law reads ##\nabla^a T_{ab}=0##. Let us pick the Levi-Civita connection so that we don't have to worry about...
Homework Statement
Prove Rijkl= k/R2 * (gik gjl-gil gjk) where gik is the 3 metric for FRW universe and K =0,+1,-1, and i,j=1,2,3, that is, spatial coordinates.
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Homework Equations
The Christoffel symbol definition:
Γμνρ = ½gμσ(∂ρgνσ+∂νgρσ-∂σgνρ)
and the Riemann tensor definition:
Rμνσρ =...
What do people think of Fulvio Melia's argument for the necessity of "zero active mass" in FRW cosmologies? (i.e. the overall equation of state must be ##\rho+3p=0## at all times)
Here is a link to an interesting lecture video:
Here is a recent paper:
https://arxiv.org/abs/1807.07587
I don't understand the reasoning for any of the three constraints imposed.
why would ##dtdx^i## terms indicate a preferred direction? what if there was identical terms for each ##x^i## would there still be a specified or preferred direction? (or is it that in this case we could rename ##t## to...
Homework Statement
Homework Equations
see above
The Attempt at a Solution
Using the conservation equation for ##p=0##
I find: ##\rho =\frac{ \rho_0}{a^3}##; (I am told this is ##\geq0## , is ##a\geq0## so here I can conclude that ##\rho_0 \geq =0 ## or not?)
Plugging this and ##p=0## into...
Homework Statement
Question:
Homework Equations
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Dp=R(t)Dc where R(t) is the scale factor, Dc is the comoving distance
The Attempt at a Solution
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I must have tried solving this starting 10 different ways now, starting with the fact that distance=integral over velocity dt...
So I have been following various derivations of the FRW metric and have a bit of confusion due to varying convention...
Would it be correct to say that curvature K can be expressed as both K = \frac{k}{a(t)^2} and K = \frac{k}{R(t)^2} where k is the curvature parameter?
If so, is it correct to...
So from a killing tensor the FRW metric is known to possess, for a massless particle we find the well known result that as the universe expands the frequency of the photons decreases . But , what does this do for gr ?
Was this known to happen before gr ?
Thanks a lot.
(I know it is used to...
Homework Statement
(a) Find the FRW metric, equations and density parameter. Express the density parameter in terms of a and H.
(b) Express density parameter as a function of a where density dominates and find values of w.
(c) If curvature is negligible, what values must w be to prevent a...
I'm looking at Tod and Hughston Introduction to GR and writing the metric in the two forms;
[1]##ds^{2}=dt^{2}-R^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}))##
[2] ##ds^{2}=dt^{2}-R^{2}(t)g_{ij}dx^{i}dx^{j}##
where...
This is probably a stupid question but does k=1,0,-1 correspond to closed,flat,open refer to space or space-times?
Looking at a derivation what each geometrically represents is only done when talking about the spatial part of the FRW metric.
As space can be flat and space-time still curved...
So in deriving the metric, the space-time can be foliated by homogenous and isotropic spacelike slices.
And the metric must take the form:
##ds^{2}=-dt^{2}+a^{2}(t)\gamma_{ij}(u)du^{i}du^{j}##,
where ## \gamma_{ij} ## is the metric of a spacelike slice at a constant t
QUESTION:
So I've read...
I'm looking at: http://arxiv.org/pdf/gr-qc/9712019.pdf,
deriving the FRW metric, and I don't fully understand how the Ricci Vectors eq 8.5 can be attained from 7.16, by setting ##\partial_{0} \beta ## and ##\alpha=0##
I see that any christoffel symbol with a ##0## vanish and so so do any...
Mod note: OP warned about not using the homework template.
I have read that 'a(t) determines the value of the constant spatial curvature'..
Where a(t) is the scale factor, and we must have constant spatial curvature - this can be deduced from the isotropic at every point assumption.
I'm trying...
When working with light-propagation in the FRW metric
$$ds^2 = - dt^2 + a^2 ( d\chi^2 + S_k(\chi) d\Omega^2)$$
most texts just set $$ds^2 = 0$$ and obtain the equation
$$\frac{d\chi}{dt} = - \frac{1}{a}$$
for a light-ray moving from the emitter to the observer.
Question1: Do we not strictly...
Hello everyone, I already know that the solution to this question is obvious but I can't find it.
Consider an expanding universe following the FRW metric ds^2=-dt^2-a^2(t)dx^2 (1 space dimension for simplicity). We know that the physical spatial distance x_p is related to the comoving spatial...
In an expanding universe that is modeled by the FRW metric we assume that scale factor of the "present epoch" is unity which is equivalent to a zero redshift. Therefore, most observed galaxies with nonzero redshifts are in our past light cone.
But it is unclear to me how much back in time or...
hi
we know that our universe is homogenous and isotropic in large scale.
the metric describe these conditions is FRW metric.
In FRW, we have constant,k, that represent the surveture of space.
it can be 1,0,-1.
but the the Einstan Eq, Ricci scalar is obtained as function of time! and this...
In the context of Friedmann's time, 1922, how did he know to make the metric scale factor, a, a function of time when Hubble's redshifts were not yet published? I understand that he took the assumption that the universe is homogenous and isotropic, but does that naturally imply that the universe...
Hi, I'm new to Physics Forum and wasn't really sure where to post this since its not strictly speaking a homwork question. So if it happens to be in the wrong place I apologise.
I was looking through some lecture notes from when I did my Physics degree years ago and come across a problem...
What are the symmetries determined by FRW spacetime? I guess they include Lorentz symmetry, rotationally and translationally symmetries, but not time symmetry. Is this right?
Thanks