# Is there a valid solution to Einstein's equations for this cosmological model?

The cosmological model below has been developed in order to explain the flatness problem.

At first it's from Newtonian considerations, then a solution of the Friedman equations is looked for

$$\left(\frac{\dot a}{a}\right)^2+\frac{kc^2}{a^2} = \frac{8 \pi G \rho+\Lambda c^2}{3} \tag{1}$$

$$\frac{\ddot a}{a} = \frac{-4 \pi G}{3} \left(\rho + \frac{3p}{c^2}\right) +\frac{\Lambda c^2}{3} \tag{2}$$

The model has a constant rate of expansion $$\frac{\dot a}{a} = H$$ and the expansion happens to all length scales, including the observer as in this link Cosmology - an expansion of all length scales

The flatness problem is explained as follows, (firstly Newtonian considerations)

In the expansion of the type above each physical quantity $$Q$$ varies with time as $$Q=Q_0 e^{nHt}$$ where $$n$$ is the number of length dimensions.

Gravity is produced in nature to allow a scale-symmetric expansion of the universe without violating conservation of energy.

For an isolated mass its energy $$mc^2$$ becomes $$(mc^2)e^{2Ht}$$ and without gravity conservation of energy is violated.

However the total energy due to the mass in the universe of mass $$M$$ and radius $$R$$ is $$\left(mc^2 - \frac{GMm}{R}\right)e^{2Ht} \tag 3$$

and energy can be conserved if

$$\left(mc^2 - \frac{GMm}{R}\right)e^{2Ht} = 0 \tag 4$$

and the strength of gravity is

$$G=\frac{Rc^2}{M} \tag 5$$

Small numerical constants omitted for simplicity.

The interpretation of this is that gravity is caused to allow the universe to have scaling symmetry. The 'expansion' does not slow down or speed up as the universe expands. Energy is conserved during the 'expansion' due to the gain of internal energy of masses being of balanced by the gain of negative gravitational potential energy.

The flatness problem is naturally explained by (5).

Since no expansion is measurable in such a universe we have a stable, apparently static universe, always at critical density.

Now a solution of the Friedman equations (1) and (2) is sought that represents the above cosmology.

The questions is: Is this solution valid?

The assumptions of the Friedman equations are that the universe is spatial homogenous and isotropic, here is the equivalent solution to (1) and (2) for the model above

with $$Λ=0$$, $$k=0$$, constant $$H$$, and scale factor $$a=a_0 e^{Ht}$$, the equations (1) and (2) reduce to

$$3H^2=8πG\rho \tag 6$$

$$3H^2=−4πG\left(\rho+\frac{3p}{c^2}\right) \tag 7$$

leading to the solution

$$\rho = \frac{3H^2}{8 \pi G} \tag 8$$

$$p=-\rho c^2 \tag 9$$

(5) is equivalent to (8) or (10)

$$G = \frac{3H^2}{8 \pi \rho} \tag {10}$$

If the expansion is of the type described, we now have an apparently static, non-empty solution, always at critical density. Is this solution valid?

Please would anyone answering avoid answers like "supernovae data shows...", but address issues strictly to do with the use of the Friedman equations, like the two below.

1. Is there anything in Einstein's equations or the Friedman equations that say $$\frac{\dot a}{a}$$ must only apply to the distance between galaxies and not to all length scales?

2. The model has a variable speed of light and variable $$G$$, but the universe is apparently static and both are apparently constant, so is this ok?

Thanks for any advice on these questions.

• Models with expanding scales are called self-similar. Here is a reference to a well known one: vixra.org/abs/1107.0016 May 9, 2021 at 2:29
• On your #1, it is simply the Newtonian laws. A non-accelerating expansion applies no force to objects (zero acceleration = zero force), so objects bound by any force cannot be pulled apart. May 9, 2021 at 2:35
• I think your concept of the "flatness problem" differs from the common interpretation. I suggest you take a look at en.wikipedia.org/wiki/Flatness_problem .
– Buzz
May 10, 2021 at 21:31
• @Buzz, thanks, had a look, it says that the flatness problem is that the density is very close to critical density and that in mainstream cosmology it must have been even closer in the past - hence inflation theory to solve it...in this model inflation isn't required as the universe is always at critical density...so it's a kind of alternative to inflation. The question is really about whether the Einstein and Friedman equations allow the interpretation that $\frac{\dot a}{a}$ means the kind of expansion described... May 10, 2021 at 21:40
• Before Eq 6 you have H is a constant. Eq 10 says H^2 = (8/3)G rho, bur rho is density and density grows bigger as t approaches 0. Therefore H cannot be a constant.
– Buzz
May 11, 2021 at 21:25