We had problems determining the behavior in a case where the velocities are close to c, and if the ball is heavy, maybe inside its Schwarzschild radius.
Let us study the hypothesis which we raised at the end of the previous blog post: in comoving coordinates, the dust cloud behaves in a "newtonian" way.
If the observable universe is a part of a giant expanding dust ball, then we have empirical evidence that the newtonian hypothesis is correct, to an extent. The expansion of the universe has very crudely followed a newtonian expansion. If we assume a "shell theorem" which says that the masses from a distance > R from us do not affect the behavior inside the radius R, then masses inside R behave in a newtonian way. Though, recall that dark energy breaks this model recently.
We can quite easily calculate the effect of the retardation for a newtonian model. We simply assume that gravity cannot "predict" the acceleration of a collapse, or the deceleration of an expansion. This prediction error makes gravity to oscillate between the naive model (= with a perfect predictive capability) and the retarded model. We believe that the oscillation is able to explain dark energy and other anomalies in the expansion of the universe. Retardation is expected to cause anomalies in the expansion.
Why does gravity have to oscillate? Because conservation of energy forces gravity to have, on the average, the naive strength.
Electric lines of force in a collapsing dust ball: the field E looks just like in newtonian physics
Let us first try to draw electric lines of force for a charge q falling along the dust in a collapsing ball.
^
E | m m'
• • dust particles
| /
v v
● q electric charge
In the comoving frame of the charge q, the lines of force form the usual radial pattern. The electric field E is parallel with the velocity vector of an approaching dust particle m.
Let us then switch to the comoving frame of m. What does the electric field E look like there?
At m, the field E has the same value in the comoving frames of q and m.
If the frame is not accelerated, then Gauss's law for the electric field E holds there.
• m'''
r = distance(q, m''')
^ E
|
• m''
\ v
v
• m
• q
Each dust particle, in its own comoving frame, sees the electric field E pointing radially toward the charge. Let us take a dust particle m'' very close to m, so that its relative velocity v to m is very slow. Then the Lorentz transformation of E at m'' to the comoving frame of m keeps the direction of E essentially unchanged. The configuration looks locally very simple.
We conclude that we can draw the lines of force of E close to m, radially relative to q, in the comoving frame of m. Let m''' be yet another dust particle on the line from q to m, but a little bit farther from q. The strength of the field E at m''' obeys the usual
~ 1 / r²
formula, where r is the distance measured in comoving coordinates of dust particles.
In summary, the field E obeys Gauss's law in the comoving coordinates, even though the coordinates are quite exotic compared to standard Minkowski coordinates.
The formula for the electric field is then
E = 1 / (4 π ε₀) * q / r²,
just as in newtonian physics and coordinates.
The problem with using the proper time of each dust particle as the time coordinate
On May 26, 2024 we observed that using the proper time of each dust particle as the time coordinate leads to very unnatural coordinates: one is able travel to an earlier time coordinate if one approaches the center of a collapsing star! How does that affect our analysis above?
Why clocks do not freeze in a collapsing dust ball? A retardation hypothesis
We can argue that gravity behaves much in the same way as an electric field in a collapsing dust ball.
But how do we explain why the nonlinear effects in gravity are absent in the dust ball? We believe that extremely strong nonlinear effects exist near a neutron star and close to the horizon of a black hole.
Could it be that a dust particle never gets a message that it is inside an extremely strong, forming gravity field? Particles at the edges of the dust ball are falling at almost the speed of light – which reminds us of the collapsing photon shell in the previous blog post.
Hypothesis of the retardation of slowing clocks. A dust particle in the collapsing cloud is aware of the "newtonian" gravity field at all times. It knows that its motion must accelerate. But the dust particle is not immediately aware that it is in a low gravity potential. The information about a low potential must come from the surrounding, asymptotically Minkowski space, and that may take a very long time, possibly an infinite time if the speed of light drops to zero at a horizon.
The slowing of clocks may be somewhat analogous to freezing: the frozen portion begins from the surrounding space and spreads to deeper gravity potentials.
Various authors have remarked that we can only know the location of a horizon afterwards. The redshift of a photon sent from a coordinate location x at a coordinate time t cannot be calculated unless we know also future events. This shows that the rate of a clock at a forming horizon can be quite fast.
On and inside a static neutron star, the gravity potential is known at every location. Clocks know how slowly they should tick. Also, close to the horizon of a static black hole, clocks know that they should tick very slowly.
Retardation as a general natural law: its mechanisms are unknown
~• --> c c <-- •~
photon photon
Imagine a collision of two photons. Neither one can know that it is inside the gravity field of the other one. A scientist measuring their progress in a laboratory cannot know that either. The photons move as "free particles". There is no interaction.
When the photons collide, massive particles may form and take all the energy of the photons.
These massive particles are inside their common gravity potential. Locally, they have to have more energy than what was in the arriving photons. That is because the massive particles are in a low gravity potential. Energy conservation requires that suddenly, some extra energy appears. How do the new particles know how much extra energy they must possess?
Is it so that the flattened gravity fields of the photons somehow "combine" and release the extra energy? In what time does this happen?
To eliminate outgoing gravitational waves, we may let a spherical photon shell collapse. When and how does the gravity field settle into the Schwarzshild metric?
We realize that retardation in many cases profoundly affects processes in nature. There must exist unknown mechanisms which restore conservation laws. Also, if the end state is static, there must be mechanisms to create the eventual static fields from earlier, dynamic ones.
We do not currently know anything about these mechanisms. Field theory is generally developed for very slow processes, where retardation has a negligible effect.
In general relativity, Birkhoff's theorem (that is, the Einstein field equations) assumes that the field can magically, instantaneously, know that a system is spherically symmetric. Gauss's law for the electric field contains the same assumption, which is very unlikely to be true. See our January 3, 2025 post.
We have uncovered a whole new field of physics: physics of retardation.
Retardation comes from the finite speed of light. Field theory has, so far, only studied few phenomena which arise from the finiteness of the speed.
The universe, as well as a collapse of a star into a black hole, are processes where retardation must have significant effects.
Retardation might have major effects in collisions of elementary particles. We have to think about that.
What is inside a black hole: the observable universe shows us what it is like
The observable universe most probably is inside its Schwarzschild radius. Thus, in the observable universe we probably see what it is like to be inside a collapsing star. Nothing very special. Gravity seems to obey the newtonian law. But what happens when collapsing dust particles collide into each other?
There will be a lot of pressure. That pressure may speed up the collapse. On the other hand, if the particles come to a standstill in collisions, they may have time to receive the information that they are now in a very low gravity potential, and clocks will essentially stop. The end result is a very dense, "frozen" object.
Retardation in the rotation of galaxies: could it explain some of the dark matter?
One rotation of the Milky Way takes 300 million years, and its diameter is 100,000 light-years. Are there retardation effects in the rotation? How does the gravity field know that stars at the other end of the galaxy stay in circular orbits, so that the gravity field does not change with time?
Retardation effects may be significant, but we do not think that they can explain all the dark matter, which makes up 90% of the mass of the Milky Way.
We have to figure out what type of retardation effects does a centrifugal acceleration create. So far, we have only considered an accelerated expansion of a shell of charges.
If we could trust Gauss's law, we could model the lines of force of the gravity field of an individual star with radial "steel wires" or "rubber strings" which start from the star. The centrifugal acceleration would bend the steel wires. The field would probably be "stationary", since the motion of the star is periodic and uniform.
Let us assume a very simple form of retardation: the gravity field at a test mass m behaves as if the stars of the galaxy would have obeyed straight line orbits at a constant speed, since m last "saw" each star.
Let m be in the plane of the galaxy, at some distance.
Then the gravity field is as if the stars at the far end of the galaxy would be very slightly farther than they are. Gravity at m is reduced slightly, maybe one part in a billion.
Retardation cannot explain dark matter.
Conclusions
We now have some ideas about how gravity might work inside a collapsing dust ball. We have to do a lot more analysis.
Today, we realized that our January 3, 2025 proof of the failure of Gauss's law is erroneous. If we assume as an axiom that the field of each elementary charge at each moment satisfies Gauss'd law in the laboratory frame, then, of course, the entire shell of charges must obey it, too. However, we do not see how a changing magnetic field could create a radial electric field which restores Gauss's law for an expanding shell of charges. We will look at this problem again.
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