We would assume that a clock at the center of the shell "calculates" the radius of the shell based on the latests contraction velocity v that the clock knows of.
Is this a reasonable assumption?
Let us look at a single mass M and a test mass m (or a test clock) in the field of M. The usual retardation rule is that if M moves at a constant speed v, then the test mass or test clock will know the gravity field of M as if m would know the current position of M in the laboratory coordinates.
It makes a lot of sense to assume that a test clock m can "calculate" its own gravity potential based on the assumption that masses M continue their movement at a constant velocity. The clock then adjusts its rate according to that gravity potential.
Another way to look at this is to assume that M is static in the laboratory coordinates, and the test clock m moves around. An atomic clock on the surface of Earth can adjust its ticking based on what is its distance from the center of Earth.
In our March 15, 2025 blog post we calculated the retardation inside a shell assuming that the clock is not aware of the contraction speed v of the shell. That yields a very large retardation effect. If the ignorance of the clock only concerns the acceleration of the shell, the retardation effect is much smaller. But is the effect still large enough to explain dark energy?
A crude calculation based on the expansion of the universe
The "current" radius of the observable universe is estimated to be 46 billion light-years. The age of the universe is 13.7 billion years.
The universe was expanding significantly faster than now, say, 6.9 billion years ago.
The scale factor in the matter-dominated phase is
a(t) ~ t^⅔.
The time derivative is
da / dt ~ 1 / t^⅓.
When the age of the universe was a half of the current age, the expansion speed was
2^⅓ = 1.26
times the current speed.
If a clock at the center of the observable universe "calculates" its rate by assuming that the universe would still be expanding at that 1.26X speed, then the clock will overestimate its gravity potential and will tick too fast. The speed of light is "too fast" close to the center, which means a repulsive force from the center.
Could it be that the current value of matter and dark matter Ω = 0.3 has something to do with the observed acceleration of the expansion?
Let us try to estimate the repulsion, using the comoving coordinates of the "dust" (= matter and dark matter in the universe). On January 18, 2025 we argued that gravity looks very much newtonian in comoving coordinates. But we did not consider retardation of clocks then.
Conservation of energy in a rubber sheet model of gravity
We can simulate the collapse of a dust ball by letting small slippery weights to slide toward a central depression in the rubber sheet.
In the rubber sheet model, longitudinal, spherically symmetric waves exist.
A rubber sheet model allows the collapse process to "oscillate". The potential energy of the weights flows in a complicated way into the elastic energy of the sheet, as well as to the kinetic energy of the weights.
We want energy conservation in the collapse process. The rubber sheet guarantees energy conservation.
In a simplistic retardation model, conservation of energy probably would be breached.
We conclude that a satisfactory retardation model must involve something similar as the rubber sheet model of gravity.
*** WORK IN PROGRESS ***
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