Newtonian Friedmann Equation, Referance frame, Homogeneity

[RyanH42]

Hi all
I want to ask a question about NFE(Newtonian Friedmann Equation).I know that NFE is not usefull to describe universe.But we can have a general idea about universe to use that formula.

I know that the only spatial coordinate system is CMB referance frame and NFE is derived from that point(taking RF(Referance Frame) CMB).

My question,
Is Universe homogeneius and isotrophic somewhere else ?
I mean Is there any other point which we can say "Universe homogenic and isotrophic" ?

If there's any other point, and we want to calculate NFE(Referance frame will be our new point) it will be also same as NFE (CMB referance frame taken) isn't it ?

 

[PeterDonis]

I want to ask a question about NFE(Newtonian Friedmann Equation).

Do you have a reference that describes what this is?

 

[RyanH42]

Do you have a reference that describes what this is?

http://hyperphysics.phy-astr.gsu.edu/hbase/astro/fried.html
And
https://en.m.wikipedia.org/wiki/Friedmann_equations
Equations section first equation(I assumed no cosmological constant but I can add no problem for that)
And $p=Ω_m/a^3+Ω_r/a^4+Ω_Λ$

 

[PeterDonis]

These references talk about the Friedmann equation, yes. But you said "Newtonian Friedmann Equation". Where does that term come from? If you are basing it on the fact that the hyperphysics page says "a simplified, non-relativistic version based on Newton's laws", note that the page is incorrect in that statement, as far as I can see: the equations it gives are the correct relativistic ones for cold matter, i.e., matter whose pressure is negligible, and they are derived from Einstein's Field Equation, not from Newton's laws.

 

[RyanH42]

I said like that cause as you said I thought it derived from Newtons law.But I see it wrong what would be the right term to describe that equation NRFE(non-relativistic friedmann equation)?

 

[PeterDonis]

.But I see it wrong what would be the right term to describe that equation NRFE(non-relativistic friedmann equation)?

The right term is "Friedmann equations". There's no such thing as a "non-relativistic" vs. a "relativistic" version. All that changes is what assumptions you make about the energy density $\rho$, the pressure $p$, the curvature constant $k$, and the cosmological constant $\Lambda$ that appear in the equations. If you assume that $p = 0$ and $\Lambda = 0$ (which is what the hyperphysics page you linked to does), that doesn't mean you're using a "non-relativistic" version; it just means you've assumed that those quantities are zero (or at least negligible) for the particular scenario you're analyzing.

 

[RyanH42]

How can $p=0$ ? $p=wcρ$ isn't it so you mean $p_m+p_r=0$ ?
The other things are clear thanks for that.
So what about my questions ?

 

[PeterDonis]

How can ##p=0## ?

If ordinary matter has a sufficiently low temperature, its pressure is negligible in comparison with its energy density, so it can be assumed to be zero as a very good approximation. On cosmological scales, the ordinary matter in our universe meets this criterion.

$p=wcρ$ isn't it

Yes, and if $w = 0$, then $p = 0$. For ordinary "cold" matter, $w = 0$; that's what I was describing above. For radiation, $w = 1/3$, and for dark energy, $w = -1$ (at least in the simplest form of a cosmological constant).

So what about my questions ?

They appeared to me to depend on your belief that the "Newtonian Friedmann equation" was something different from the ordinary one. Since it isn't, I would recommend rethinking your questions and, if necessary, asking them again.

 

[RyanH42]

Ok I understand why $p=0$ thanks again.
The derivation of Friedmann Equation is different.Here the derivation
$1/2mV^2-mMG/r=U$
$V^2-2MG/r=2U/m$
$H^2R^2-8πGR^2/3=-k$
$H^2-8πG/3=-k/R^2$
This derivation is wrong I guess.But this is the way that I learn.The problem is This works only when universe is homogeneius and isotrophic.But this situation exist only CMB referance frame.Is there any other referance frame which universe is still homogeneius and isotrophic.So I can use this equation again.To desribe universe.

I don't know FR works all referance frames or only CMB frame(I think it works only CMB frame)

If my derivation is wrong what's the right one ? The answer will be GR but My derivation is also true isn't it ? It can't be coincidence

 

[PeterDonis]

Here the derivation

Where is this from?

 

[PeterDonis]

Is there any other referance frame which universe is still homogeneius and isotrophic.

Homogeneity and isotropy are properties of spacelike slices of the universe, not the universe as a whole. There is only one family of spacelike slices that have these properties: they are the spacelike slices of constant time in the usual FRW coordinates. But the slices having those properties is not a function of the coordinates; they would have those properties in any coordinates you choose. They just wouldn't be slices of constant time in other coordinates. (The main reason FRW coordinates are chosen is to make those slices be slices of constant coordinate time, to make the math simpler.)

I don't know FR works all referance frames or only CMB frame(I think it works only CMB frame)

The Einstein Field Equations, which is what the Friedmann equations are derived from, work in any frame. But they don't necessarily look equally simple in every frame. You could do cosmology in some other frame besides the usual FRW coordinates, but the math would be more complicated.

 

[RyanH42]

These pics from a book Andrew Little An Introduction to Modern Cosmology and Leonard Susskind Cosmology Lectures

 

[RyanH42]

I understand that we don't need any homogeneity or isotrophic universe to derive Friedmann equation.But the math will be complicated

 

[PeterDonis]

These pics from a book Andrew Little An Introduction to Modern Cosmology and Leonard Susskind Cosmology Lectures

Note the statement early on: "The Newtonian derivation is, however, some way from being completely rigorous." That's a fancy way of saying it's not really valid. As the book notes: "general relativity is required to fully patch it up", i.e., to do a valid derivation. I would guess that the reason the book chose to do things this way is that the author(s) thought Newtonian gravity would be more familiar to the reader than GR, so they gave a derivation that, while not valid, at least would make the Friedmann equation seem plausible (as long as you're willing to accept an invalid derivation as a heuristic argument).

 

[RyanH42]

Note the statement early on: "The Newtonian derivation is, however, some way from being completely rigorous." That's a fancy way of saying it's not really valid. As the book notes: "general relativity is required to fully patch it up", i.e., to do a valid derivation. I would guess that the reason the book chose to do things this way is that the author(s) thought Newtonian gravity would be more familiar to the reader than GR, so they gave a derivation that, while not valid, at least would make the Friedmann equation seem plausible (as long as you're willing to accept an invalid derivation as a heuristic argument).

Ok.I got the idea thanks.I need to learn GR immidiatly.

 

[RyanH42]

We need homogenity and isotrophy but the time slices will be different.So that makes the rquation complicated.Am I right ?

 

[Chalnoth]

I understand that we don't need any homogeneity or isotrophic universe to derive Friedmann equation.But the math will be complicated

Correct, it will be more complicated if you don't have homogeneity or isotropy. In fact, I don't think anybody has figured out how to produce an exact solution to Einstein's equations without using isotropy as an assumption*, and homogeneity adds another symmetry that can be used to reduce the number of necessary parameters further.

Once you have the solution, it's possible to translate any results you want into any coordinate system you choose.

* Edit: Well, I suppose rotating systems such as the Kerr metric may be an exception, depending upon how you define isotropy (the Kerr metric is isotropic far from the origin).

 

[RyanH42]

Thank you Chalnoth and PeterDonis.I get my answers for all questions.

 

[RyanH42]

I am confused about referance frame.Whats the "job" of referance frame in Friedmann Equation ? Why we need it ?

 

[PeterDonis]

Whats the "job" of referance frame in Friedmann Equation ?

"Reference frame" here means the coordinates you choose. We choose a particular set of coordinates (the "comoving" coordinates in which the FRW metric is usually written) in which to write down the Friedmann equation because it makes the equation look as simple as possible.

 

[RyanH42]

We need to write FRW metric to define Universe curvature and solve the Friedmann equation.Thats the reason we pick a referance frame.We need to define FRW metric and FRW metric depends where you pick the referance frame.

Is that true ? If its true then Thank you.

And I want to ask " Whats the general properties of FRW ?".Should I ask this question in new thread ?

 

[PeterDonis]

We need to write FRW metric to define Universe curvature and solve the Friedmann equation.

Sort of. The Friedmann equation is really just the Einstein Field Equation; the FRW metric is a solution of the Einstein Field Equation.

We need to define FRW metric and FRW metric depends where you pick the referance frame.

Not quite. The metric itself--the geometry of spacetime--does not depend on which reference frame you pick. But the mathematical form of how the metric is written--how simple the equations look--does depend on which reference frame you pick. As I said before, the usual FRW coordinates are chosen because they make the equations look as simple as possible. But that's a matter of mathematical convenience, not physics. The physics is the same no matter what coordinates you use.

" Whats the general properties of FRW ?".

The key assumptions that underlie the FRW spacetime geometry--that particular solution of the Einstein Field Equation--are that the universe is homogeneous and isotropic. More precisely, that the spacetime describing the universe can be completely covered by a family of spacelike hypersurfaces that are homogeneous (i.e., they look the same at every point of space) and isotropic (i.e., they look the same in every direction in space).

 

[RyanH42]

I understand the FRW metric.And the referance frame.The FRW metric is just the geometry of spacetime.I mean In Friedmann Equation used to describe geometry of space-time.Lets suppose we have GR eqution.Then we are asuming universe homogenius and isotrophic.Then GR solution turns out Friedmann Equation.Then we can describe our universe space-time.

 

[yogi]

I understand the FRW metric.And the referance frame.The FRW metric is just the geometry of spacetime.I mean In Friedmann Equation used to describe geometry of space-time.Lets suppose we have GR eqution.Then we are asuming universe homogenius and isotrophic.Then GR solution turns out Friedmann Equation.Then we can describe our universe space-time.

In 1934 McCrea and Mill, using only energy relationships and Newtonian mechanics, derived the same gravitational equations as those that had been synthesized from GR by considerable effort and skill. For purposes of understanding, there is nothing wrong with using Friedmann's formulations - surprisingly, nothing is actually left out or mis-stated, a conceptual picture of the universe does not require that the formulaizations be extract from the General Theory, Marcos posted an introductory paragraph to the cosmology category of physics forums to that effect.

 

[Chalnoth]

In 1934 McCrea and Mill, using only energy relationships and Newtonian mechanics, derived the same gravitational equations as those that had been synthesized from GR by considerable effort and skill. For purposes of understanding, there is nothing wrong with using Friedmann's formulations - surprisingly, nothing is actually left out or mis-stated, a conceptual picture of the universe does not require that the formulaizations be extract from the General Theory, Marcos posted an introductory paragraph to the cosmology category of physics forums to that effect.

This is only true for non-relativistic matter. Newtonian gravity doesn't give the right answer for radiation (or relativistic matter).

 

[yogi]

This is only true for non-relativistic matter. Newtonian gravity doesn't give the right answer for radiation (or relativistic matter).

That is true, but for the purposes of modeling the now state of the universe, its mostly matter and very little in the way of relativistic velocities wrt to space ...expansion is relativistic - but this doesn't add to the relativistic energy because the nebula are commoving with recessional space. Moreover, even in GR, the time dilation factor reduces to a simple KE - that of the (1/2)v^2 energy require to escape the gravitational field

Correction - Marcus, not Marcos in the above post

 

[PeterDonis]

even in GR, the time dilation factor reduces to a simple KE - that of the (1/2)v^2 energy require to escape the gravitational field

This is only true in Schwarzschild spacetime, i.e., in the vacuum surrounding an idealized, spherically symmetric massive body that is isolated in otherwise empty space. It does not apply to the universe as a whole, since the universe as a whole is not described by Schwarzschild spacetime.

 

[yogi]

This is only true in Schwarzschild spacetime, i.e., in the vacuum surrounding an idealized, spherically symmetric massive body that is isolated in otherwise empty space. It does not apply to the universe as a whole, since the universe as a whole is not described by Schwarzschild spacetime.

If you model the universe as a black hole - the escape velocity is c - the Schwarzschild radius R* is the Hubble radius R - and all energy in the Hubble universe in any form is expressed in terms of its mc^2 equivalent as M (except the interaction gravitational energy of the Hubble sphere). To model the 3-sphere universe as a black-hole, M is considered to be uniformly spread over the Hubble surface. The Schwarzschild radius for the Hubble sphere is then:

R* = 2GM/c^2

As you will notice, except for the factor 2, this is the well known but somewhat puzzling ratio

GM/Rc^2 = 1

that was studied extensively by Dicke and Brans in their search for a scalar density theory of gravity. The universe as a whole can be described by Schwarzschild formalism

 

[PeterDonis]

If you model the universe as a black hole

Which is not a valid model, since, as I said before, the universe is not described by Schwarzschild spacetime.