https://journals.aps.org/pr/abstract/10.1103/PhysRev.56.455
Our result on May 26, 2024 further strengthened our hypothesis that the Einstein-Hilbert action has no "dynamic" solutions at all: we cannot find an extremal point for the action for any system which changes with time.
The problem seems to be that there are no "canonical" coordinates in general relativity, and, consequently, one cannot define the kinetic energy of a particle.
People have recognized that the Oppenheimer-Snyder collapse bears a resemblance to the FLRW solution of the Einstein field equations. The Tolman coordinates comove with matter, just like the standard coordinates do in FLRW.
In this blog we know that the Oppenheimer-Snyder solution is erroneous. Could it be that we must somehow correct the FLRW solution, in order to make it a "reasonable" physical model?
The correction might tell us what is the strange dark energy in the ΛCDM model, and what is the strange process which seems to slow the speed of the expansion after the last CMB scattering.
A "slower" speed of time in the past (coordinatewise)?
If we imagine that the expanding universe is like a collapse of a dust ball run backwards, then the gravitational potential in the past made clocks to tick slower (coordinatewise) relative to the present, and the speed of light was slower in the past.
How would that affect what we see today?
If everything moved slower (relative to the coordinates) then we cannot discern a slower metric of time in the past. We can simply change coordinates in such a way that the metric of time is normalized to, say, 1.
A frozen star is not frozen inside? This explains the expanding universe?
In this blog we have been advocating the frozen star model, where a collapsing dust ball ends in a "frozen state" when its surface approaches a forming event horizon.
We use the Schwarzschild coordinates in the analysis below.
If the interior of the dust ball freezes at the same coordinate time as the surface, how does the inside "know" when to do that?
Let the Schwarzschild radius be R.
If we assume that the "true" metric is Minkowski space, then the center of the ball cannot know about the freezing before an additional coordinate time
R / c
has passes. There, c is the speed of light in Minkowski space. The center may develop much further during that short time.
If the maximum speed of a signal is the local speed of light, then it may take forever for the center to know that it should freeze. Let us try to analyze this case.
forming horizon
• -----> | •
falling photon R particle inside
The falling particle may approach the Schwarzschild radius at the speed of light. It might be a photon. Then the inside of the ball may never know that it was enclosed inside an event horizon!
This may offer us an explanation for why the observable universe was inside its own Schwarzschild radius in the past, but we certainly do not see the universe frozen in the past. We see the CMB which originated when the mass density was 1,100³ times what it is today.
Another question: how does an infalling shell of photons know that they are approaching the Schwarzschild radius? A single photon cannot know that the other photons in the shell continued their journey toward the horizon. This suggests that, after all, a matter shell can fall inside the event horizon it itself creates. But if a horizon was already formed by prior infalling matter, then a new infalling shell must stop at that older horizon, at the latest.
If we have a very heavy neutron star, we can drop an additional shell which forms a horizon around the neutron star. What happens next? Does the shell reach the surface of the neutron star and crush it?
Let us have a particle m inside a forming event horizon. If it is electrically charged, will its field lines become "frozen" at the horizon? The speed of light is extremely slow close to the horizon. How could the field lines move at all? The geometry of the gravity field will probably make the electric field look symmetric to an observer outside the horizon. What happens to field lines inside the horizon?
What about the gravity field lines, if gravity has field lines?
An expanding shell of electrically charged particles: corrections if we do not assume an infinite speed of light
If we run the expansion of the universe backward, it is a collapse in which the edges of the observable universe are approaching us at speeds which may be close to the speed of light.
How does the collapsing matter "know" about other matter approaching it at almost the speed of light at a distance of 10 billion light years?
The "speed" of an interaction is an old problem in physics. Suppose that we have a spherical electrically charged shell expanding under the repulsive force of the electric charges. How does an individual charge "know" that the entire shell is expanding in a symmetric way?
Rule for the field of a moving charge. A test charge q sees the field of Q as if Q would have moved at a constant velocity ever since q received the last information of the location of Q (through some lightspeed mechanism).
● --> a •
Q q
Example. Q and q are initially static. Suddenly, Q is accelerated to the right. The test charge q sees the field of Q constant, until a lightspeed signal tells q that Q has started moving.
Expanding shell of charged particles. Let us have an initially static shell of charged particles. At a time t₀ in the laboratory frame, we let the particles free and the shell starts expanding.
If the speed of light would be infinite, we could calculate the expansion in a simple way, assuming that the electric field is spherically symmetric and adjusted for the current radius
R(t)
of the shell. But the speed of light is finite. A particle does not know that the particles far away started to move in a symmetric fashion.
Initially, the expansion is somewhat faster than in the case of an infinite speed of light. A particle thinks that the particles far away have not moved yet.
The energy and the momentum in the long run must be conserved. There has to be some mechanism which at some point makes the acceleration slower than in the case of an infinite speed of light!
Thus, there is a correction to the simple case of an infinite speed of light.
Assumptions. What did we assume in the analysis of an expanding shell of charges:
1. There is no mechanism which would inform a particle faster than light about the movement of other particles. This is a safe assumption. We would get all the time travel paradoxes if faster-than-light communication would be possible.
2. The energy of the field and the kinetic energies of the particles can be calculated in the standard way if the particles are static, or flying far away from each other at some late time t. These energies must be conserved. This is a safe assumption, knowing the empirical robustness of conservation laws.
Can some of the energy escape as radiation? This is unlikely, because the expanding shell is spherically symmetric. It cannot create transverse waves. Longitudinal waves cannot propagate in electromagnetism.
We conclude that there must be some mechanism through which nature handles the expansion. Ideally, it should be found out empirically how this mechanism behaves. It may be some kind of a self-force which the electric field of a charge q exerts on the charge q itself.
The expanding universe at some times expands faster or slower than derived in general relativity
The universe has a large diameter, and the speeds may be close to the speed of light. The correction described in the previous section may be very large: at some times the universe expands much faster or slower than predicted by general relativity. General relativity assumes an infinite speed of light in updating the forces between distant masses. General relativity must be wrong in this, if our reasoning in the preceding section is correct. Faster-than-light communication would be possible in general relativity.
This observation may explain dark energy, and the overabundance of galaxies in James Webb photographs.
Another example: a shell of particles expanding and an attractive force
Let us assume that a shell of particles initially expands very fast at a constant speed. At a time t₁, the shell "bounces back" and starts contracting at an equally fast speed.
A test particle inside the shell saw the very fast initial expansion of the shell, but because of the finite speed of light, only sees the contraction of the very nearest part of the shell.
shell
______
/ \
"cap" | • test particle
\_______/
The view of the test particle is schematically as in the above diagram. The test particle sees most of the shell expanding far away, but it also sees a "cap" which is approaching the test particle, and is very close.
The cap causes an attractive force on the test particle.
If we look at the configuration in the laboratory frame, the test particle is inside a spherical, contracting shell. In this naive view, there is no force on the test particle.
We proved that a change in the expansion rate of a spherical shell does affect the force felt by a test particle inside.
Conclusions
Our analysis of the expanding shell of electric charges returns us to an old theme which we have touched several times in this blog: how do force fields ensure conservation of energy and momentum? It is an open problem in physics.
The analysis of an expanding shell uncovered something fundamental: there must be corrections to the simplified analysis where changes in the field are assumed to propagate infinitely fast!
Retardation does affect the behavior of spherically symmetric collapses and expansions.
In the case of an expanding universe, these corrections (to gravity) may be very large. They may explain dark energy and the James Webb photographs. We will investigate this further in future blog posts.
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