# A new cosmological model (re are we living in a black hole)

I think my argument is better this way round.

1/ Assume that space is flat and that there is no cosmological constant.

2/ Assume that the size of the universe R is bounded by the distance a light beam travels in the age of the Universe t.

Therefore:

R = c * t

Putting this into the Friedman equations for the evolution of the universe we find the following consequences:

1/ 2 * G * M / R = c^2

where M is the mass of the universe and R is the size of the universe.

If you multiply both sides of equation (1) by the mass of a particle m you get a statement of Mach's principle that half the mass/energy of any particle is equal to its mutual gravitational energy with every other particle in the Universe.

2/ A dark energy exists with the following formula for its mass density rho :

rho = (1 / 8 pi G) * 1 / t^2

where t is the age of the universe.

John

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bcrowell
Staff Emeritus
Gold Member
The Friedmann equations only have one flat-spacetime solution, and that's the Milne universe: http://en.wikipedia.org/wiki/Milne_model As explained in the WP article, the Milne model is not compatible with observation, so it isn't a viable model of the universe we live in. The Milne universe is empty, so your description in terms of a nonvanishing mass M is incorrect. If you're claiming you've found a solution of the Friedmann equations, you might want to post your actual calculations, which I think must be mistaken.

Here are my calculations:

The Friedmann equations are:

(1) (a'/a)^2 + k c^2 / a^2 - gamma c^2 / 3 = 8 pi G rho / 3

(2) a''/a + (a'/ a)^2 + k c^2 / a^2 - gamma c^2 = - 8 pi G p / c^2

where

a = dimensionless scale factor
' = derivative with respect to proper time
c = speed of light
k = curvature of space
gamma = cosmological constant
G = Newton's constant
rho = density
p = pressure

Assumptions:

1/ Curvature of space k = 0

2/ Cosmological constant gamma = 0

3/ R = c * t

where R is the size of the universe and t is the proper time.

Derivation:

(1) R = R_0 * a(t)

where R_0 is the size of the universe at the present time
and a(t_0) = 1 where t_0 is the present time.

Hubble's law:

(2) v = H_0 * r

where v is the recessional velocity of an object, r is its distance
from us and H_0 is the present Hubble constant.

If we set v = c in (2) then we find an expression for the present
size of the universe R_0 in terms of the present Hubble constant H_0:

(3) c = H_0 * R_0

Combining equations (1) and (3) we get an expression for a(t):

(4) a(t) = H_0 * R(t) / c

Using assumption (3), R = c * t, in equation (4) we find:

(5) a(t) = H_0 * t

Therefore:

a'/a = 1/t = c / R

Substituting a'/a = c/R, k=0 and gamma=0 into Friedmann (1) we find:

(6) c^2 / R^2 = 8 pi G rho / 3

Now the density rho can be expressed as:

(7) rho = M / (4/3) pi R^3

where M is the mass of the universe and (4/3) pi R^3 is the volume of
the universe assuming that it is a solid sphere.

Substituting (7) into (6) we get

(8) G M / R = c^2 / 2

This seems to be an expression of Mach's principle that the inertial mass
of a particle is related to the gravitational influence on that particle
from all the other particles in the universe.

To see this simply multiply both sides of (8) by the particle mass m to
give:

(1/2) m c^2 = G M m / R

The above equation implies that half the mass/energy of any particle
is in the form of the mutual gravitational energy between it and
every other particle in the universe.

Substituting a'/a = 1/t, k=0 and gamma=0 into Friedmann (2) we find:

p = - c^2 / 8 pi G t^2

This negative pressure p can be interpreted as a dark energy with
mass density rho_dark given by:

rho_dark = 1 / 8 pi G t^2

thus this dark energy falls off inversely as the square of the age
of the universe.

Using 1/t^2 = c^2 / R^2 = H_0^2 / a^2

and also that a(t_0) = 1 where t_0 is the present time we find that
the current value of the dark energy rho_dark is given by:

rho_dark at present = H_0^2 / 8 pi G

where H_0 is the present Hubble constant.

If we take:

H_0 = 2.3 * 10^-18 s^-1
G = 6.6 * 10^-11 m^3 kg^-1 s^-2

then

rho_dark at present = 3.1 * 10^-27 kg/m^3 = 2 proton masses per m^3

marcus
Gold Member
Dearly Missed
John, are you sure you want to assume "space-time" is flat? It looks to me like you are using spatial flatness (i.e. k = 0 in the Friedmann eqns) but you permit expansion. You let a(t) the scalefactor increase. A U can be spatial flat but not spacetime flat.

Check your assumptions 1 and 2 in post #1
======================
what do you mean by R = "size of universe"? Do you mean actual physical extent?
Why do you assume this increasing at rate c?
That seems out of line---I never heard of a realistic Friedmann model U that where R(t) = ct.
Maybe someone else has, and can explain. Assumption 3 seems completely unjustified.
======================
I have problems with your equation (3) in your second post, post #3.

It seems to me you may have a confused idea of R(t) which you call "size of the universe". You assume without explanation that it is currently equal to c/Ho the Hubble radius!

You have us in a Friedmann model U and it is spatial flat, so without further information one assumes the spatial size is infinite with infinite amount of matter approx uniformly distributed. Anyhow that is how I picture the k=0 case unless someone argues for say a toroidal setup.

You've probably heard R(t) described as "the average distance between galaxies". I like that better although neither is perfect. It is not meant to be taken as the actual physical extent, although it is a useful size-type quantity which can track expansion for us even if (e.g. in the k=0 case) the physical extent is infinite so in the commonlanguage vernacular sense the U has no size. Infinite is not a size.

There is plenty more but Bcrowell is on this. I will stop here.

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Marcus:

1/ You're right about the curvature. I meant the spacial curvature is flat.

2/ I thought that R = c * t seemed a natural measure of the size of the universe. If the universe is 13 billion years old then it is 13 billion light years in size.

3/ Although I quote the Hubble's law in (2) and use it to define the Hubble radius in (3) my argument does not depend on it.

John

marcus
Gold Member
Dearly Missed
Marcus:

1/ You're right about the curvature. I meant the spacial curvature is flat.

2/ I thought that R = c * t seemed a natural measure of the size of the universe. If the universe is 13 billion years old then it is 13 billion light years in size.

3/ Although I quote the Hubble's law in (2) and use it to define the Hubble radius in (3) my argument does not depend on it.

John
3/ Although I quote the Hubble's law in (2) and use it to define the Hubble radius in (3) my argument does not depend on it.​
then how do you justify equation (4)?
Let's remove equation (3). How do you proceed?

2/ I thought that R = c * t seemed a natural measure of the size of the universe. If the universe is 13 billion years old then it is 13 billion light years in size.

Why? This doesn't make sense. This assumption may be the chief flaw in your argument.
It gives everything a kind of fringe or non-mainstream character.
Maybe you should read Lineweaver's "Misconceptions" SciAm article. The link in in my sig.
They explain why the mainstream picture is not that of a bomb exploding outwards into empty space. With debris sailing out at no more than the speed of light. They explain about a lot of popular misconceptions. I don't know how you are picturing things, maybe not as an explosion, maybe some other way, but it sounds like some kind of misconception.

Maybe you think that the speed limit of c in (1905) special rel carries over and applies to the largescale expansion of distances in (1915) general rel. Maybe you are reasoning based on that false assumption about general rel, which is the basis of cosmology.

It's hard to guess. But something is clearly wrong when you just say:
"2/ I thought that R = c * t seemed a natural measure of the size of the universe. If the universe is 13 billion years old then it is 13 billion light years in size."

The assumption R = ct is not too far off - but if one calculates R as ct you can't relate Ho to the same R in the expression c/R since the present value of Ho takes into account the fact that the universe has not expanded at a constant rate ct. However, the approach is refreshing - particulaly since a corrected estimate of R will give very nearly the empirical value of G. There have been some similar posts on this forum before.

I think the op is thinking in terms of the Hubble sphere as the de facto limit of a de Sitter space - flat mathematically, but the geometry and radius of the sphere determines the acceleration factor (c^2)/R

One suggestion re interpretation
- the Friedmann relationship equation 6 in your post 3 is derived on the bases that the expansion flux energy is (1/2)(rho)v^2. If you take the energy as (rho)c^2 then the 1/2 factor disappears - and you can interpret all the energy in all forms as equal to that necessary to make omega 1, specifically equation 8 becomes GM/R = c^2. That this is so within the limits of experimental error has always been a mystery - but resolved if your argument otherwise has merit

One suggestion re interpretation
- the Friedmann relationship equation 6 in your post 3 is derived on the bases that the expansion flux energy is (1/2)(rho)v^2. If you take the energy as (rho)c^2 then the 1/2 factor disappears - and you can interpret all the energy in all forms as equal to that necessary to make omega 1, specifically equation 8 becomes GM/R = c^2. That this is so within the limits of experimental error has always been a mystery - but resolved if your argument otherwise has merit

Hi Yogi,

That's very interesting.

What I know about the Friedman equations I learnt from Wikipedia and the Hyperphysics website. I assumed they were relativistically correct.

You could publish this idea if you think it's interesting as I don't think I'll get around to trying to publish other than putting this post on Physics Forums.

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hi johne

You seem to be doing a great job without formal academic introduction to these concepts. What is amazing - and it happens over and over again, that some upstart crow probing for answers in a new field where he has no expertise, suddenly turns a subject on its head with an observation that the career cosmologists have overlooked - I have been on both sides of the situation - lucky enough to have the opportunity of not knowing that the problem didn't have a simple solution - then found one to everyone's amazement - only to be shattered a few months later by my own confidence in a solution to another part of the puzzle that was far from optimum, as shown to me by a newcomer in the lab.

Another poster on these forums (haven't seem him around lately, but I am not on frequently myself) would be interested as he has published a theory called SCC (Self Creating Colmology). His name is Garth Barber; it has Mach's principle as a cornerstone.

I made a copy of your derivation - it parallels some ideas I published some years ago when everyone thought the universe was decelerating - so some of the relationships were off by as much as 30% - I knew deceleration didn't sound right; all the numbers pointed to an accelerating universe... but I didn't have the courage to buck the reviewers. Have since rewritten most of it based upon the present de Sitter phase - as rewritten both G and the theoretical value of the cosmic mass are now are very close to the experimental values.

Cheers