Collapse of a dust shell on a cloud of photons: conservation of energy?
• dust
|
v
• ---> ~ ~ ~ <--- •
photons
^
|
•
In newtonian gravity, dust particles will gain some kinetic energy from the decreasing "gravity potential" of the matter inside the shell.
The declining potential drains energy from a photon by making its frequency lower. General relativity says that "time slows down".
In newtonian gravity, we would have a paradox: the photon would keep its frequency, and could escape to space without losing energy.
In a more advanced model of gravity, we have another problem: if the shell starts to contract very quickly, then the photons inside cannot know that they are in a low gravity potential. Their frequency, measured in global coordinates, cannot slow down. Falling dust particles gain kinetic energy, but that energy is not yet subtracted from the photons.
One could claim that the "negative energy" in the gravity field outside the shell grows, and gives the kinetic energy to the dust. But we do not like the concept of a negative energy density.
Suppose that the knowledge about the gravity potential propagates at the speed of light. The photons will soon learn that they are in a low potential, and their frequency will drop. Conservation of energy is restored, but only afterwards.
We may imagine that all matter consists of waves propagating at the speed of light. Then also massive particles lose energy in a low potential through the same mechanism as photons.
We have here a mechanism which may involve very large retardation effects. If energy cannot be "borrowed" in large quantities, then it has to flow from the photons to the kinetic energy of the dust, and that process can take time.
General relativity works around this problem by claiming that there is no clear definition of energy, or energy conservation.
When are clocks slowed down inside the collapsing dust shell?
One could claim that it does not matter when exactly clocks are slowed down inside the shell. That the end result is the same, as long as the slow period has the same length in various alternative histories.
However, the timing of the slow period does affect the end result. We can compare the ticking rates of clocks at various distances from the center, inside the shell.
For example, if the slowing down of clocks would happen instantaneously as the shell falls down, then clocks inside the shell would stay synchronized. But if the slowing down propagates at the speed of light from the shell, then a clock at the center will show a later time than a clock close to the shell.
Lowering two masses on a third one
● ---> ● <--- ●
M M M
Suppose that we have three objects whose mass M is the same when measured alone in the space. We let the masses on the left and the right fall freely on the center mass M. We may imagine that the masses can pass through each other, and completely overlap in the end state.
The total mass-energy of the system remains at 3 M at all times, measured by an observer far away.
But measured locally at the center, the two moving masses M possess more mass-energy than the center one. It makes sense to claim that the moving masses have gained energy from the center mass.
Could there be a retardation effect here? If the masses on both sides are rapidly accelerated to approach the center mass, how does the center mass know that it should give up some energy? This would amount to a breach of Gauss's law for gravity, since Gauss's law states that the energy in a spherically symmetric collapse is immediately available, regardless of an acceleration.
Gauss's law does not hold for gravity?
In electromagnetism, an expanding shell can collect energy from the electric field E at the the immediate vicinity of the shell. There is no obvious reason for any retardation.
The expansion of the universe is the only gravitational collapse/expansion for which we have measured data. The data shows that the expansion does not happen in a way which is compatible with Gauss's law: there seems to be dark energy. Thus, the only empirical data, which we possess, suggests that Gauss's law does not hold for gravity.
We can treat the failure of Gauss's law as the primary hypothesis for gravity.
Do we have empirical data for electromagnetism, such that it would prove that Gauss's law holds? The derivation of the power of electromagnetic waves, by Edward M. Purcell, depends on Gauss's law. The derivation produces the empirically correct formula. We are not aware of any experiments of Gauss's law with static charges, though.
The derivation of the power of gravitational waves is quite different from the derivation in electromagnetism. Also, the power is 16-fold compared to the analogous electromagnetic system. Thus, the derivation might not provide evidence for Gauss's law for gravity. The derivation does provide evidence that the linearized Einstein equations describe the gravity field right for a quadrupole oscillation, in some sense. But a spherically symmetric collapse/expansion is quite a different process from an oscillating quadrupole.
Accelerating a heavy neutron star: Gauss's law probably fails
test mass
m
•
● ---> a acceleration
M
neutron star
Let us use a rope to accelerate a very heavy neutron star M. A test mass m close to its surface will necessarily follow the movement, since the speed of light is very slow at the surface.
The "force" which moves m is distinct from the newtonian gravity force, and distinct from the possible gravitomagnetic force associated with the newtonian force. Thus, there is no reason why the force moving m would satisfy Gauss's law.
Why would Gauss's law hold in the collapse of a dust ball?
Gauss's law in electromagnetism requires that a varying magnetic field attachs the ends of the lines of force of Coulomb's force.
We do not have any empirical measurements of gravitomagnetism. We do not know if it can attach the lines of force of the newtonian gravity force.
Thus, it may be that Gauss's law does not hold in the collapse of a dust ball, or in the expansion of the observable universe. This opens the possibility that dark energy is a manifestation of Gauss's law failing.
Retardation effects may at some phases of the collapse make gravity surprisingly weak. Conservation of energy requires that gravity, on the average, is as strong as Gauss's law requires. Then the acceleration of the collapse can oscillate in a surprising way.
Can we deduce something about this process? Can we prove that Gauss's law cannot hold?
Pressure breaks Gauss's law for gravity
In the fall of 2023 we realized that Tolman's paradox breaks Birkhoff's theorem, and consequently, also Gauss's law.
Pressure arises, for instance, from moving gas molecules. In the collapse of a dust ball, dust particles acquire radial speeds.
If we move a test mass m closer to a collapsing dust ball, the metric stretches in the radial direction from m. The volume of the space increases, and, effecticely, the fall of dust particles slows down in certain directions. Some kinetic energy is released, and that energy pulls m closer to the dust ball?
Looking from far away, the collapsing dust ball is much like a gas cloud with a pressure. The velocities of the dust particles are nicely ordered, though. That is the difference from an ideal gas where particles move randomly.
It seems likely that the pressure does create a gravitational pull outside the collapsing dust ball.
In the expanding universe, the speeds of galaxies are close to the speed of light. The effects of the pressure may be very large. We have to investigate how this breach of Gauss's law affects the expansion.
The Oppenheimer-Snyder paper (1939) uses the comoving coordinates of Tolman. They can ignore the effects of the pressure which arises from the movement of dust particles. But on May 26, 2024 we showed that the Tolman coordinates will produce flawed results. Maybe it is better to use "static" coordinates, and take into account the pressure.
Does pressure cause oscillation in the speed of the collapse?
*** WORK IN PROGRESS ***
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