We have been struggling to calculate how this will affect the collapse of a uniform dust ball.
In FLRW models, the expansion of the (dust-filled) universe will slow down as if gravity would be newtonian: we can study a small spatially spherical volume of the universe, and we obtain the deceleration from newtonian gravity.
We believe that the dust-filled universe in an FLRW model behaves like an expanding dust ball in an asymptotically Minkowski space.
We have been struggling to incorporate the retardation effect to this newtonian model of expansion (or collapse) of a dust ball.
Let us, once again, try to figure out what happens. We will study a collapse of a uniform dust ball in an asymptotically Minkowski space.
Retardation makes the gravity potential well shallower: a rubber sheet model
We believe that a clock inside a collapsing dust ball is not aware of the acceleration of individual dust particles far away. Therefore, the potential, as defined by the clock rate, is higher than one would calculate if one would assume that the clock knows the configuration "right now" in the coordinate time.
bulge
___ ___ rubber sheet
\•______----------______•/
weight ---> <--- weight
In a rubber sheet model of gravity, a contracting ring of weights will have the rubber sheet bulged upward at the center, because the rubber at the center of the ring does not yet know that the weights in the ring have been accelerated as they fall lower.
As the weights slide lower, potential energy is released. Most of the potential energy will go to the kinetic energy of the weights. Some will go to a "longitudinal" wave in the rubber sheet, that is, to the stretching of the bulge, and the kinetic energy of the rubber sheet.
In this model, the "gravity" simulated by the rubber sheet is somewhat weaker because not all released potential energy goes to the kinetic energy.
Presumably, the weakening effect grows stronger when the ring contracts. But could it entirely cancel the acceleration of the simulated "gravity" at some point? (Dark energy seems to be accelerating the expansion of the universe.)
Then all the potential energy released would at some point go to the deformation of the rubber sheet and the kinetic energy of the rubber sheet.
• -----______----- •
pit
Could it be that the bulge has time to flip into a "pit" at some point? Then the simulated "gravity" would appear stronger.
Modeling the retarded gravity field with a rubber sheet: a singularity appears
Let us have a mass M which is initially static. Then we start to accelerate it at a constant acceleration a.
observers
● ---> a R × r ×
M 2 1
----->
Let observer 1 be such that he is not yet aware of the acceleration of M.
Let us use the standard retardation rule: observer 2 sees the location of M as if M would have moved at the constant speed v, where v is the last observation that observer 2 made.
The gravity potential V(x) seen by observer 2 is continuous in x, but the derivative dV / dx in not continuous. The derivative should be continuous in a rubber sheet model?
____
/ --> v
-----
rubber sheet
If we have a sharp turn moving in the rubber sheet, then the acceleration of the sheet is infinite at the turn. It is a singularity – nonsensical.
An electric charge which is suddenly accelerated
How does the Edward M. Purcell calculation handle the analogous case for the electromagnetic field?
The electric field lines above are continuous, but make sharp turns. Can we produce those turns with a time-varying magnetic field which does not contain a singularity?
We can determine the curl of B from the time-varying field E. Is there a guarantee that B will not contain sources, and that the curl of E satisfies the upper equation?
If we would use the standard retardation formula for the electric potential V, then at the rightmost point of the circle above,
dE / dt
would be infinite. But the requirement that the electric field lines are continuous means that we cannot apply the retardation rule to the potential V at that point.
What about the sharp turns of the electric field lines at other places? There, dE / dt is infinite and the curl of B is infinite. That seems nonsensical. Suppose then that we make the turns smooth.
When the charge q moves to the right, the magnetic field B takes the well known for of field lines circling the trajectory of q.
Let us accelerate q to the right. The second formula above is, at least approximately, satisfied. The same holds for the first formula.
In electromagnetism, the electric potential V does not have any direct effect on anything. It does not make clocks to tick slower. Therefore, the rubber sheet model does not describe electromagnetism. The potential V can change infinitely fast inside a contracting or expanding shell of charges.
We conclude that the electromagnetic analogy is not useful for us.
A new take: a tense rubber sheet
Let us try to prove that the bulge can turn into a pit.
bulge
___ ___ rubber sheet
\•______----------______•/
weight ---> <--- weight
Let the rubber sheet be very tense. We can assume that the weights move much slower than the waves in the rubber sheet.
Let us assume that the weights start moving. A bulge forms. Could it turn into a pit as the rubber sheet tries to straighten itself up at the center area?
No, it is not possible.
The clock on the bulge runs fast: the observer at the bulge sees the contraction of the dust shell to slow down mysteriously?
This could finally explain why the collapse of a dust shell appears to slow down. If we reverse time, then the expansion would speed up mysteriously, just like in dark energy.
*** WORK IN PROGRESS ***
