UPDATE October 29, 2021: There is an error in the proof of the Theorem. In a phase change, the exponent n changes. The formula (1) is incorrect and the Theorem is incorrect.
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The standard FLRW model of the universe is built on the Friedmann equations:
The formula of the metric has a constant c² factor before the time element dt². A static observer has ds₃ = 0. Then we can define that dt is the proper time interval measured by a static observer anywhere in the universe, and t is his proper time.
The Friedmann equations let pressure only affect the second time derivative of the scale factor a(t):
The time derivative is denoted by the dot over the symbol a (the Newton notation of the derivative).
In principle, we can change the pressure rapidly inside the universe. The FLRW universe should have uniform pressure, but let us relax that requirement, and increase the pressure quickly in a half of the universe, and keep it constant in another half.
Static clocks suddenly tick slower in a half of the universe. Static observers in those zones see the time derivative of the scale factor a(t) to jump suddenly. That is, those observers see the expansion to accelerate rapidly.
The rest of the static observers do not see any jump? Do they see a sudden slowdown?
Let us then raise the pressure also in that half which did not get a pressure lift. Now the pressure is uniform again.
Question. Do static observers in the whole universe see a sudden jump in the expansion rate?
Alexander Friedmann (1888 - 1925). https://commons.wikimedia.org/wiki/File:Aleksandr_Fridman.png#mw-jump-to-license
Two stars approach each other, and we change their internal pressure
star star
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/ \ / \
\____/ \____/
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● external
observer
The question above is related to a thought experiment where two stars approach each other and there are observers inside the stars. It is a kind of a "contracting universe". By changing the pressure inside the stars, we can make the clocks of the internal observers to tick faster or slower.
Clearly, the internal pressure inside the stars cannot affect much the speed seen by an external observer.
However, observers inside the stars see a sudden jump or slowdown in the contraction rate of their mini-universe.
If we put the stars inside a huge pressurized vessel, then increasing the pressure suddenly makes the inertia of the stars to grow, because then they are carrying the energy of the pressure field around. In that case, the an external observer sees the speed of the stars to slow down suddenly.
However, if the stars themselves create the pressure, the pressure cannot affect their inertia much.
The FLRW metric inside a newtonian expanding ball
In literature several sources say that a newtonian expanding ball with uniform density has the Schwarzschild metric outside. Birkhoff's theorem dictates the external metric.
They also claim that the interior metric is the FLRW metric.
Let us suddenly increase the pressure inside the ball.
If we can ignore the acceleration of the matter at the edge of the ball, then the FLRW metric claims that there is no sudden jump in the expansion rate, as seen by internal observers.
The pressure is created by the matter of the ball itself. Can the pressure increase enough the inertia of, say, the left half and the right half of the ball, so that the expansion rate would suddenly slow down in the eyes of an external observer, and internal observers would see the expansion rate to stay constant?
Each half does increase the volume of the other half through its gravity field. Thus, each half does get some extra inertia from the pressure force field of the other half. Does this extra inertia slow down the halves enough, so that observers inside the halves see the expansion to continue at the old rate?
The Friedmann equations do not have a solution at all for a sudden pressure jump
Once we have chosen for the universe some scale factor value a(t), and the time derivative of a(t), at a certain initial time t₀, then the second Friedmann equation determines the scale factor a(t) uniquely for all times t.
However, there is no guarantee that the first equation is satisfied. Maybe in most cases there is no solution at all for the Friedmann equations?
Theorem. If we have a solution for the Friedmann equations, but change the setup so that we add a sudden increase in pressure at a certain time, then there is no solution for the Friedmann equations. We assume here that the time derivative of the scale factor a(t) is not zero at the time of the pressure increase. We assume that a(t) is a continuous function and that its second time derivative is defined.
NOTE October 29, 2021: the proof is incorrect. The value of n changes in the phase change, which spoils the calculation of the perturbation in (1).
Proof. Let us assume that we have a solution a(t) for the Friedmann equations. We assume that the time derivative of a(t) is not zero at t = 0.
We change the original setup so that we suddenly increase the pressure p at the time t = 0 by some fixed amount.
Let us assume that b(t) is the new solution for t > 0.
The second Friedmann equation says that the second time derivative of a(t) jumps up suddenly. Then we can write:
b'(t) = a'(t) - C t
for some constant C, for small t > 0. We denoted the time derivative with the prime mark '.
Then we have
b(t) = a(t) - C t² / 2,
for small t > 0, and
ρ_b(t) = ρ(t) / [ 1 - C t² / 2 * 1 / a(t) ]ⁿ, (1)
where n is a real number in the range [3 ... 4], and we denote by ρ_b(t) the new solution for small t > 0.
We have the square of t in the formulae for b(t) and ρ_b(t). If t > 0 is very small, we may assume that b(t) = a(t) and ρ_b(t) = ρ(t).
Then we immediately see that the first Friedmann equation is not satisfied for very small t > 0. QED.
Corollary. The metric of time has to change in a spherically symmetric universe if there is a sudden uniform change in pressure. We assume here that the Einstein equations have a solution for such a setup - otherwise we can no longer talk about a metric.
Proof. If there is no change in the metric of time, then there is a solution of the Friedmann equations. But we showed above that there is no Friedmann solution. QED.
Corollary. Assume that the universe satisfies the Friedmann equations before a sudden pressure change. After the pressure change, the universe cannot return to a standard massive matter & radiation solution of the Friedmann equations.
Proof. If the universe would return to a standard solution a(t) after a pressure change at a time t₀, then we could continue that solution all the way down to the time t₀, because the universe would obey the standard pressure rules of the matter content.
Then we can show just like in the proof of Theorem that for a very short time interval before t₀, the Friedmann equations cannot be satisfied. But that contradicts the assumption that the equations were satisfied before t₀. QED.
It looks like the Friedmann equations do not have a solution in most cases. Thus, in most universes the metric of time has to change.
The standard cosmological model ΛCDM and changes of pressure
The standard ΛCDM model of cosmology assumes that the universe was radiation-dominated until the age of 380,000 years.
Then it became matter-dominated.
At about 4 billion years ago, it became dominated with dark energy, which is expressed in the cosmological constant Λ.
Does ΛCDM assume that the time of a static observer has the same metric in all these phases? That is probably wrong. Friedmann equations probably have no solution for the pressure changes.
If pressure can affect the metric of time, then increasing pressure from dark matter (dark radiation) might slow down time, and the expansion of the universe would appear to accelerate, even though it keeps decelerating from the viewpoint of an observer outside the universe. It might be that most of dark matter was massive particles until some 4 billion years ago, and then decayed into radiation.
The dark energy field which creates energy from nothing is a very ugly idea in ΛCDM. There is no such field in the physics that we know. All known fields conserve energy.
A much better hypothesis is pressure from dark radiation.
A newtonian ball cannot contain a dark energy field because that would break energy conservation. But there is no problem in assuming an increase of pressure when matter is turned into radiation.
What is the history of dark matter?
From astronomy we can deduce the history of visible matter: it switched from radiation-dominated to matter-dominated about 380,000 years after the Big Bang.
But we do not know the history of dark matter. It might be that dark matter has switched between matter and radiation phases many times in the past 13.5 billion years.
If dark matted switched to radiation 4 billion years ago, that may explain the illusion of an accelerated expansion of the universe.
There may be surprises in the first 380,000 years, too.
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