UPDATE October 29, 2021: We updated the blog post to reflect that the theorem of the October 16, 2021 blog post is erroneous.
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UPDATE October 24, 2021: Our blog post on October 23, 2021 proved that pressure changes cannot explain dark energy.
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The metric of time does not change in a Friedmann solution. Only the mapping of spatial, comoving, coordinates to proper distances changes.
For massive matter, the pressure p = 0, while for radiation,
p = 1/3 ρ / c².
In the second equation, the acceleration suddenly doubles when massive matter is converted into radiation.
Let us analyze what happens.
Assumptions. 1. We assume that prior to the phase transition, the universe is dominated by uniform massive matter, and the metric obeys the Friedmann solution
a(t) ~ t^(2/3),
that is, a(t) is "almost" linear on t, if we regard 2/3 as almost 1. The expansion slows down at a moderate pace.
2. We assume that k = 0 and Λ = 0.
When the pressure suddenly goes up, we believe that the expansion starts to slow down faster.
A sudden jump in the pressure in an Oppenheimer-Snyder collapse
Robert Oppenheimer and Hartland Snyder in their famous 1939 paper were only able to calculate the pressureless p = 0 case, that is, the collapse of a uniform dust ball.
Valerio Faraoni and Farah Atieh (2020) mention the Mashhoon and Partovi result (1980) that the metric inside a collapsing uniform dust ball must be FLRW.
What would happen if uniform large pressure would suddenly appear at a late stage of the collapse?
Since a Friedmann solution uses comoving coordinates, an observer inside the star knows what are the Friedmann coordinates for each particle. The observer can measure the Hubble constant H = da / dt * 1 / a, as well as the matter density ρ. The first Friedmann equation is strictly about observations.
Large pressure lowers the gravitational potential of an observer inside the star. In the previous blog post we argued that an external observer would see the star to continue contracting at roughly the same pace after the pressure is raised. The time of an internal observer slows down. He observes a sudden rise in the Hubble constant.
In this blog we believe that the "true" geometry of spacetime is the flat Minkowski metric, and curved spacetime is just an illusion created by complex effects of newtonian gravity. The Big Bang might be a reverse Oppenheimer-Snyder collapse.
Calculation of the observed increase of the expansion rate in an expanding ball
The expansion rate of the universe appears to have increased by some 20% relative to the t^(2/3) solution in the past 4 billion years.
Let us assume that we live inside an expanding uniform ball in an asymptotic Minkowski space.
The assumed amount of dark matter in the visible universe is so large that the expanding ball might even be inside its own Schwarzschild radius.
How much does pressure lower the potential of a test mass m inside the ball?
The Schwarzschild metric around the mass m stretches the spatial metric of radial distances. If we consider a sphere of a thickness dr at the distance r, its volume grows by
dV = r_s / r * r² dr
= r_s r dr
= 2 G m / c² * r dr,
where r_s is the Schwarzschild radius of the 1 kg mass. Integrating over r gives:
ΔV = G m / c² * r².
The radius of the observable universe is 46.5 billion light-years, which is 4.65 * 10²⁶ meters. A 1 kg mass increases the volume of the observable universe by
ΔV = 3 * 10²⁵ cubic meters.
That is a cube whose side is 300,000 km.
Let us assume that the observable universe holds a pressure p. The potential of the test mass m is
-p ΔV.
If the potential is -0.2 m c², then time would be slowed down by 20% for the test mass. Let us calculate the required pressure. It is
p = 7 * 10⁻¹⁰ Pa,
which is the opposite number to the assumed negative pressure from dark energy in ΛCDM.
To create the positive pressure p, the radiation must have three times the energy density of the assumed dark energy density in ΛCDM.
In our calculation we assumed that the test mass m only increases the volume of the observable universe. Is this right? Probably not. The test mass affects the metric of the entire expanding ball. If we take the radius r in the above calculation to be much larger, then the required pressure is much lower.
Conclusions
It looks like that we can explain the observed acceleration in the expansion of the universe through massive dark matter turning into dark radiation, but the details are not yet clear.
We do not yet understand what happens if the expanding ball is inside its Schwarzschild radius.
We have not analyzed the effect of large pressure during the first 380,000 years of the universe.
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