Retardation weakens gravity in an expanding dust ball - retardation explains why the universe is not "frozen"?

We have been struggling to understand what retardation exactly does in a contracting or expanding uniform dust ball which is very large, compared to the "observable universe" of an observer.

In the following, we use a (fuzzy) Minkowski & newtonian gravity model. The speed of light c is the maximum speed of a signal. The gravity force in a static case is as given by Isaac Newton.

Two particles which pull on each other

Let us look at an expansion which is slowing down because of gravity.

          particle 1                           particle 2

      v <---  • --> a                           a <-- • ---> v

                m                   s                     m

  t₀ time                    distance

Let us look the process in global "laboratory" coordinates. We have two dust particles flying away from each other. The global time coordinate is t₀.

Gravity decelerates the particles. Because of retardation, the particles "see" each other farther away than they are "currently" in the global coordinates. The gravity is weaker than in the naive model where we would just look at the current distance of the particles.

Also, the gravity field of each particle is squeezed horizontally in the diagram, because of length contraction (Lorentz correction).

Let us calculate how much retardation and the squeezing affects the force of gravity between the two particles. Let us first assume that v << c. Then Lorentz corrections are negligible.

The particle 1 sees the particle 2 as it was the time

        t  =  s / c

earlier. At that time, the particle 2 was receding faster from the particle 1. The particle 1 extrapolates the location of 2 from the velocity of 2 at that earlier time, and sees the gravity field as if 2 were at that extrapolated location.

But the particle is actually "currently" at a location

       1/2 a t²  =  1/2 a s² / c²

closer. The relative error in the distance is

       1/2 a s / c²,

and the relative error in the gravity force is

       a s / c².

The force is weaker by that factor than the naively calculated on, using "current" distances.

We conclude that the naive gravity force is replaced with one of the form:

      F  =  G m² / s²  -  G m² a / (s c²).

The new force formula affects the expansion of the dust ball in a complicated way?

Effect on an expanding dust ball which is largish but not huge

       

                          largish dust ball A

                                 radius R

           ______

        /             \

       |       •  m   |          ×    center of A

        \_______/

              B 

     environment B                         • m'

     of dust particle m       dust particle in A

                                            s = distance(m, m')

We have a largish expanding dust ball A and an arbitrary dust particle m inside it. Let B be the largest spherical environment of m, which fits inside A.

We assume that the expansion of A is approximately uniform. Because of symmetry, the net force imposed on m by B is zero. Also, because of symmetry, the retardation of the gravity fields of particles within B has no net effect on m.

The gravity of the set difference

       A  -  B 

does pull on m, and the retardation has an effect.

If m is close to the center of A, then all the particles m' in A - B are roughly at the same distance 

       s  ≈  R

from m. The effect of retardation is roughly the same for all m'!

We conclude that the effect of retardation is fairly uniform for any m close to the center.

                           

                       /  A

                     /

                   |O m 

                     \ B              • m'

                       \               Ω

            O = environment B of m

Let then m be relatively close to the edge of A. Particles in A - B pull on m.

Let us look at a narrow solid angle Ω whose tip is at m. Most of the pull on m comes from angles which very roughly point to the right from m. The length of the cone determined by Ω from the edge of B to the edge of A is typically

       2 s  ≈  1.5 R.

The average distance of the particles m' in the cone, weighted by the gravity of m' on m, is

       s  ≈ 0.75 R.

We conclude that the effect of retardation is fairly uniform throughout A, with differences of at most +-15% !

If the dust ball is so huge that the "observable universe" is only a tiny part of it

If the test particle m last saw m' receding at almost the speed of light, then m' had a lot of kinetic energy in the frame of m. Also the field of m' is strongly flattened in a way that the force on m is smaller. Which effect wins?

Let us use the analogy from the electric field. The distance in the frame of m' is

       ~  1 / sqrt(1  -  v² / c²),

and the electric field E strength would be

       ~  1  -  v² / c².

The "gravity charge" is

       ~  1 / sqrt(1  -  v² / c²).

This suggests that we can ignore masses m' which are receding very fast from m?

Retardation interpreted in a naive way would break time symmetry of physics

         o

        /|\                       <---  /\/\/\/\

         /\                             

          A observer               W wave packet

Imagine that a wave packet W approaches an observer A at the speed of light. The observer A is not aware of the gravity field of the packet W because he has not yet received any information about the approaching packet W.

Then the packet W hits the observer A, and is absorbed by his body.

If we reverse time, then in the process, the observer A is aware of the wave packet W that was emitted by his body. Does A feel the gravity field of W now?

We believe that physics is time symmetric. The observer A cannot feel the gravity of W. The gravity field of the wave packet simply cannot extend directly to the front or the back of the packet.

Retardation solves the mystery of why the universe is not frozen, even though it is inside its Schwarzschild radius?

We work in Minkowski space.

Let us make a photon shell to collapse, to form a black hole. The information about the approaching photon shell has not yet reached an observer A inside the shell. Consequently, physics at the location of A will function exactly like before. The proper time of A runs just like the global Minkowski time. There is no "freezing" at A.

The universe looks a lot like this scheme, the time reversed. The matter outside our observable universe is moving away from us very fast.

The universe is not a frozen black hole because no observer has yet received information that the universe is inside its Schwarzschild radius.

If we believe Gauss's law for gravity, then the gravity field of the universe must be immense at its outskirts. The universe would definitely be enclosed inside a black hole. Gauss's law does not hold in this case?

Let us look at a collapse of a very sparse and very large photon cloud. Initially, the cloud is much larger than its Schwarzschild radius. Can any observer know that enough matter at some later time is inside its Schwarzschild radius?

Yes. If the collapse ends into a very compact state, an observer outside the photon cloud will see a very massive compact object. It is a black hole.

However, during the collapse process, no observer inside the photon cloud maybe is aware of this? Then the collapse can proceed without freezing.

Collapse of a photon shell: no gravity felt by photons at all – the end result contains no singularity

We can imagine that the gravity field of a photon is an infinitely thin plane, normal to the velocity vector of the photon. Length contraction has made the field absolutely flat. 

In a collapsing spherical photon shell, no photon feels the gravity field of the other photons. The photons can move as if there would be no gravity at all!

An observer outside the photon shell does feel gravity. He may even see that the shell has collapsed into a black hole.

What is the end result of the collapse? As the photons arrive at the center, thy will collide and produce electron-positron pairs. That is, the photons are converted into massive particles. Since massive particles move slowly, the information about the gravity field of the other nearby particles will reach them. A very strong gravity field slows down everything to an enormous degree. The soup of electrons and positrons will "freeze". No singularity is formed. It is just a very dense, essentially frozen soup of particles.

We are able to avoid the formation of a singularity in this model. The end result is a black hole, which has a frozen soup of particles at the center. The Schwarzschild radius may be large, while the radius of the soup is very small.

If we drop an infinitesimal test mass into this black hole, the test mass will freeze at the horizon? The test mass is aware of the huge gravity field of the soup at the center, or is it?

Let us assume that the photon shell was produced by large lasers in a shell structure far away. As the photons converged into the center, their "flat plane" gravity fields were felt by observers outside the shell? Did the observers have time to know that a photon flew by?

Paradox of the gravity field of a photon

        laser                                   photon

         ===                                       • ---> c

                                                      o

                                                     /|\

                                                      /\

                                               observer

          

There is a paradox: if the observer is directly below the photon in the diagram, he cannot know that the photon exists. He cannot feel the gravity of the photon. But if he is not directly below, the flattening of the gravity field of the photon prevents the observer from feeling the gravity!

Is it so that a photon does not have a gravity field at all? Could this enable a perpetuum mobile?

The solution to the paradox might be this: before some mass m was converted into a photon in the laser, that mass m did possess a gravity field. Maybe this old gravity field gets updated as the photon flies? In that case, the gravity field of a photon is not a flat plane at all, but is mostly the remnant of the old field of m.

Collapse of a photon shell. Let us look again at the collapsing photon shell. If the photons were created from various masses m, then the old fields of the m's get gradually updated in way that the field "knows" the new location of the mass-energy. The fields will eventually know that the mass-energy is now in the electron-positron soup at the center. Thus, an event horizon forms.

Weakening of gravity comes both from deceleration and flattening of the gravity field from velocity

In the case of an expanding universe or a collapsing dust ball, both retardation from the deceleration, and the flattening of the gravity field, reduce the gravity felt by a test mass inside the system.

Friedmann equations and retardation in Minkowski & newtonian gravity

https://en.wikipedia.org/wiki/Friedmann_equations

In the absence of pressure, and a zero cosmological constant Λ, the second Friedmann equation is exactly analogous to a classical newtonian collapse or expansion (classical = the year 1687 version of newtonian mechanics).

Furthermore, we know that the Friedmann equations explain well the observed baryonic acoustic oscillation (sound horizon) phenomena.

Let us interpret this in a collapsing dust ball model. In the classical newtonian version,

1.   we can use Newton's shell theorem at the center of the dust ball, and ignore any dust outside the sphere that we are calculating,

or

2.   we can use the trick in the second section of this blog post: we look at the gravity of the set difference A - B. We can ignore the gravity of B.

The first is a "local" way to calculate, the second a "global" way. These should yield the same results. Do they yield the same results for retardation, too? If yes, then the shell theorem would hold also for retardation.

However, the formula that we derived in the first section says that the relative retardation correction (in a Minkowski & newtonian model) to the gravity force is

       ~  a s / c²

       ~  s².

The relative correction is much smaller in the "local" calculation alternative 1.

We conclude that retardation phenomena gives us information about the entire collapsing dust ball. In cosmology, this means that we get information of the universe outside the observable universe. Dark energy might give us a clue about how large is the entire universe, if the expanding dust ball is defined as "the universe".

Eliminating coupling between particles through retardation: making the particles free

Our example of the collapsing photon shell is an example of a system, where we have been able to eliminate a very strong coupling between particles through retardation.

However, energy has to be conserved. There has to be a mechanism which keeps track of the energy of the system and prevents perpetuum mobiles.

Paradox in the flattening of the gravity field

          •                                    <--- ● 

         m test mass           v ≈ c    M neutron star

We calculated in a preceding section that the gravity field of an object moving almost at the speed of light is very weak to the direction of the movement. In the diagram we would expect the gravity force of M on m to be weak.

But the field of m does pull M with a very strong force, if we look at the field of m. This is a paradox?

We once again encounter the problem of momentum conservation in a force field.

Calculating the effect of retardation and flattening for a huge dust ball

Let us try to calculate the two effects for a dust ball whose edges are expanding almost at the speed of light, and whose density is close to the "critical density".

The role of the gravity potential is unclear. How to handle it? Clocks tick slower close to the center of the dust ball?

Also, how to handle the stretching of the radial spatial metric?

We believe that slow clocks and length-contracted rulers come from a strong gravity field. If an observer does not yet have the information of that he is surrounded by huge masses that are quite close, then we, maybe, can ignore the change in the metric? It would be similar to the case of the collapsing photon shell.

    edge of dust cloud          edge of dust cloud

           • --->          •                              <--- •

    m'  v ≈ c         m test mass           v ≈ c   m''

The expansion of the universe looks uniform and classical newtonian. The particles at the edge of the cloud have gained a very large kinetic energy. The total energy of the cloud has not increased, though. Energy of the gravity field was converted into kinetic energy of particles, especially at the edges of the cloud.

Gravity is pulling a test mass m. Where is the mass-energy of the cloud located?

Hypothesis of mass distribution. The energy of the dust cloud is still uniformly distributed among the dust particles, regardless of the large kinetic energy of the edges of the cloud.

Hypothesis of retardation. If the particles in the cloud are moving at a constant velocity, then the gravity felt by the test mass m pulls it toward the current position of each dust particle m'. "Current" here means a global laboratory time t₀. The field of m' "knows" where m' is located at the current moment.

If the particle m' is in an accelerating motion, then gravity pulls m toward the "calculated" position of m', where the position is calculated based on the last information m can have about the velocity of m'. This is the traditional retardation hypothesis.

                            |   v ≈ c

                            v 

                   ___________

                /                      \

              /                          \

            |              ×                |

              \                          /

                \____________/

                 almost all the mass at the edge

Exponentially dense outer shells. Let us try to describe the dust ball in the frame of the center. If a particle at a distance r is approaching at a velocity 0.9 c, then a particle at a distance approaches at 0.99 c, and so on. Length contraction makes the shells at radii

       n r ... (n + 1) r

exponentially thinner. At the center is a dust ball where the velocities are nonrelativistic. Let its radius be R. The entire huge dust ball has a radius which is only a few times R, say, 3 R.

Almost all the huge mass is close to the radius 3 R, and approaching, say, at a velocity (1 - 0.1¹⁰⁰) c.

We should find a reason why this system roughly satisfies Newton's shell theorem, but not entirely, because of retardation.

      A                r = distance(A, B)                    B

       •                •                •                •               •

  <- 0.99 c    <- 0.9 c                   0.9 c ->   0.99 c ->

The diagram above is drawn in comoving coordinates of dust particles •, but the velocities are in the frame of the mid particle. Is there some reason why we should claim that the gravity force which A exerts on B is

       F  =  G m * m / r²,

where m is the (rest) mass of each particle, and r is their distance in the comoving coordinates?

If the velocities were slow, then it would be the newtonian gravity formula. But now we are dealing with velocities ≈ c, and extreme length contraction in the frame of the mid particle.

Two particles once again

                                    <--- F

                 ============ ruler --> v

                • -> a                  • --> v

               m                        m'

                 ============ ruler

                              r

Let us assume that the acceleration of m is

       a  =  G m' / r²  *  γ,                                       (1)

where

       γ  =  1 / sqrt(1  -  v² / c²).

Let us Lorentz transform a to the comoving frame of m'. Do we obtain consistent results?

https://en.wikipedia.org/wiki/Acceleration_(special_relativity)

In this case, ux = 0, and we have denoted γv by plain γ. We have

       a'  =  G m' / r²  *  1 / γ².                               (2)

In the comoving frame of the m', m is at a (moving) ruler position r γ. The result would be consistent if m would not be moving fast, at the velocity -v in the moving frame. But if v is large, the acceleration is much less. Our assumption was wrong.

Let us assume that the gravity force on m in the moving frame is F'. We calculate the acceleration a' of m in the moving frame. It depends on the inertia of m.

If m would be flying in empty space, the momentum would be

        p  =  v m / sqrt(1  -  v² / c²).

(But m is inside the gravity field of m'. The inertia of m may be different.) Let us calculate the "inertial mass":

       dp / dv  =  m / sqrt(1  -  v² / c²)

                          +  v m  

                              * -1/2  (1  -  v² / c²)^-3/2

                              * -2 v / c²

                      =   m / sqrt(1  -  v² / c²)

                           * (1 + (v² / c²) / (1  -  v² / c²)).

For v² / c² << 1, we can ignore the second summand above, and

       dp / dv  =  γ m.

Then the formula (1) above is consistent with (2). The "inertial mass" is the same as the gravitating mass.

But we are interested in cases where v² / c² ≈ 1. For v² / c² ≈ 1, we have:

       dp / dv  ≈  γ³ m.

The "inertial mass" is larger than the gravitating mass, which is only γ m.

Let us assume that in this case, the "active gravitating mass" (the active mass pulls other masses) for a very fast particle is m' / γ:

        a  =  G m' / r²  *  1 / γ,

        a'  =  G m' / r²  *  1 / γ⁴.

In the moving frame, the distance is γ r, and the "inertial mass" of m is γ² its "passive gravitational mass" (passive mass pulls m toward other masses). We get the same value for a'. This agrees with our calculation in an earlier section where we used the analogy between gravity and an electric field E. However, we have so far ignored the fact that m and m' are immersed in a gravity field. That may change the inertial masses.

***  WORK IN PROGRESS  ***

A large uniform universe necessarily has the spatial metric flat in Minkowski-newtonian model

UPDATE January 12, 2025: The flatness problem really is not about the flat spatial metric in the universe, but the question: why the mass density of the universe is quite close to the "critical density"?

The critical density means that the velocity of a galaxy at a distance R from us is close to the escape velocity from the mass M contained within the distance R from us.

For example, the universe might have a mass density which is only 1/100 of the current one. Then the velocity of a distant galaxy would be 10X the escape velocity.

Our Minkowski & newtonian model does not explain why the velocity is close to the escape velocity.

A possible explanation: a bounce-back model in which an initially almost static cloud of dust collapses, and then bounces back through some unknown physical mechanism. Then the dust cloud has roughly the critical density.

https://en.wikipedia.org/wiki/Flatness_problem

----

In the Friedmann equations, the mass density of the universe has to be set very carefully to the "critical density", in order to ensure that the spatial metric is flat, and will stay flat for a long time. We know that the spatial metric in the observable universe is roughly flat on the large scale.

In our own, Minkowski-newtonian gravity model, the spatial metric is determined by a different inertia of a test mass m to different directions. For example, around a neutron star, the radial metric is stretched because the inertia of a test mass is larger in the radial direction: there is "energy shipping" to the test mass m if it moves radially, which adds extra inertia. The spatial metric bulges at the neutron star, and is not flat.

If the universe is spatially very large (much larger than 15 billion light-years), and almost uniform, then the inertia of a test mass m cannot vary much in any direction, within the observable universe whose radius is only 15 billion light-years. This implies that the spatial metric is almost flat.

We do not need any fine-tuning of the mass density of the universe to a "critical" value. It is enough to demand that the universe is large and almost uniform.

This observation solves the flatness problem of cosmology. We do not need to assume inflation to fine-tune the mass density to the critical one at the beginning.

It may also explain why the expansion looks similar to all directions. If we are dealing with a huge expanding dust ball, the expansion may locally look rather uniform. The uniformity of the CMB would be explained if the dust ball is really large.

Still, we have to assume uniformity of the mass density in a very large dust ball. Why did the uniformity arise?

Also, we do not understand how Minkowski-newtonian handles a dust ball which is much inside its own Schwarzschild radius. Why is it not frozen, like a black hole is?

Retardation in a newtonian explosion of a dust ball

Let us look at a very simple example, to gain understanding of retardation effects in a collapse, or an expansion.

We assume that a uniform ball consisting of dust particles was initially static, and had a very large radius. Then we let it collapse. It is well known that it will stay uniform in newtonian mechanics.

We study the time-reversed process: the dust ball is expanding.

                                                       ^  expansion

                                                       |

       large cloud of dust

             environment of m

                     ______

                   /             \

                 |    • m         |                ×           • m'

                   \_______/                   center

                        R = distance (m, ×)

                       m = test particle

                         r = distance (m', ×)

                                                       |  expansion

                                                       v

     Current laboratory

     time t = 0.

The diagram is an extremely crude description of the process.

In the diagram, the dust ball is large and fills the entire diagram. It is expanding uniformly. Gravity is decelerating the expansion. We have marked a close environment of the test particle m. In that environment m knows the current location of dust particles quite well (current in the laboratory time coordinate).

For the bulk of the dust ball, the gravity field is retarded at m: the field is as if the dust particles would have moved at a constant velocity since m last "saw" them. The field is not aware of the fact that currently (in the laboratory time), the dust particles are located just like in the close environment of m, symmetrically around the center ×.

Let us have a dust shell S whose radius

       r  <  R

at the current laboratory time. Let m' be a dust particle in the shell S.

Let

       s  =  distance(m , m').

The speed of m' at a laboratory time

       t(s)  =  s / c

earlier was larger than now. The deceleration d(t) of the expansion of the dust ball is

       d(t)  =  C M / a(t)²,

where C is a constant, M is the mass of the ball, and a is the scale factor of the ball. We may set a(0) = 1.

On the line going through m and ×, we can calculate an approximate distortion the location of m' between

1.   the true current location in the laboratory time, and

2.   the retarded location of m', which determines the field of m' at m.

The current position of m' differs by 

       ~  -1/2 d(t) r t(s)²

       =  -1/2 d(t) r s² / c²

from the retarded position. There r is negative if m' is between m and ×.

Let us have r fixed. The difference in the position of m' is

       ~  s²,

and its effect on the gravity force of m' on m is at most

       ~  s²  *  2 / s

       ~  s.

                                                 S

                            "cap"                         "cap"

         •   Ω               |      r        ×       r        |

        m                   A           center            B

              R = distance (m, ×)

Let us look at a narrow solid angle Ω drawn from m, such that the angle intersects S. If we draw a straight line through S, then by symmetry, the intersection angle between the line and S is the same at both intersections. The intersection area thus is

       ~  s².

The gravity force of the intersection area is

       ~  s² / s²  =  1.

That is, the intersection "cap" A close to m has the same gravity force as the intersection "cap" B far from m.

Let us then correct the gravities of A and B for retardation. The correction increases the gravity of A, and reduces the gravity of B.

The correction is ~ s, which means that the correction to B wins: the gravity is smaller in the correct, retarded view than in the naive laboratory view.

We can now easily calculate order-of-magnitude estimates for the retardation correction to gravity.

The dependency of retardation correction on R

Does the correction still keep the expansion of the dust ball uniform?

Let us double R, r, and s. Then the acceleration of m doubles. What about the correction?

The correction to the position of m' becomes 8-fold. It must be divided by s to get the relative correction to gravity: the relative correction is 4-fold. We conclude that the relative effect of the correction on gravity is much larger with large R.

That is, m at a large R will feel a surprisingly weak gravity.

Conclusions

We found a very crude formula for calculating the effect of retardation in a collapse, or expansion process where the force field is analogous to the electric field (newtonian gravity with c the maximum speed of a signal). We did not consider "spacetime geometry" effects, or the mass-energy of kinetic energy, at all.

In a collapsing star, or in the universe, spacetime geometry may affect retardation a lot. We need a more detailed analysis.

Suppose that two electric charges are in a free fall, side by side, in a homogeneous gravity field. The electric field certainly is distorted in some way, because of the acceleration, but in which way? An equivalence principle would suggest that in the accelerating frame, the electric field around each charge looks like it would look like in an inertial frame. That is, the equivalence principle holds for the electric field, too.

The gravity field in a spherically symmetric collapse is not homogeneous. The equivalence principle does not hold there.

Retardation spoils the action principle of a global (force) field: loss of information in the sum global field is the problem

The behavior of a global field is often defined through an action integral. We have to find a stationary point of the integral, and that point is then an allowed physical history of the field.

Retardation presents a grave problem to this approach: the behavior of the field at a location A must only depend on what happened inside the light cone of A. If a location B is outside the light cone of A, then nothing that happens at B is allowed to affect A.

But an stationary point of the action is a global minimum, maximum, or an inflection point of the action integral. The value of the global stationary point at A may depend on things which happened outside the light cone of A. That will produce solutions in which faster-than-light communication is possible. And we must not allow faster-than-light communication.

FLRW models

A spherical collapse or expansion history can often be derived as a global stationary point of an integral. The FLRW solutions are (probably) stationary points of the Einstein-Hilbert action.

Does the FLRW solution at a location A depend solely on things that happened inside the light cone of A?

The FLRW universe is perfectly uniform. But suppose that there is a mechanical device which will break the uniformity at a location B at a time t₀. The stationary point of the action integral at a location and time A, t₁ may depend on the events at B, t₀, even if B, t₀ is outside the light cone of A. The time t₀ might even be in the future of A, t₁.

On May 21, 2024 we showed that the Einstein-Hilbert action does not have a solution for any "dynamic" system. That is another problem in the action, but it is different from the retardation problem.

Expansion of a spherical shell of electric charges

On January 3, 2025 we discusses the expansion of a uniform shell of charged particles. The naive solution, which ignores retardation, probably is an stationary point of an action integral. But it is a wrong solution because it allows faster-than-light communication.

A "global field" has to be replaced with "private" interactions between particles?

The way to enforce retardation is to assume that the system consists of particles in Minkowski space, and that the interaction of the particles respects the light speed limit.

We would abandon the concept of a global field.

The self-force of the field of an electron on the electron itself probably cannot be explained with a global field. The concept of a field must be fragmented into the individual fields of each charge carrier. An individual field interacts with another individual particle. There is not much "global" in this.

Since general relativity depends on the existence of a "global" spacetime geometry, it is doomed.

Loss of information in a sum global field

Our arguments above suggest that the global field actually is the collection of the individual fields of charge carriers.

If we try to reduce the global field into a simple sum of individual field strengths, then we lose information about individual fields – and we would need that information in calculating the behavior of the system.

Claim. A "global field", given as the sum of fields of individual charge carriers, is an approximation which simplifies calculations in many cases, but does not handle retardation correctly.

The information loss allows faster-than-light communication

Let us again look at the expanding shell of charges in the January 3, 2025 blog post.

If we want to construct an action integral which prevents faster-than-light signals, we must penalize such signals harshly, so that a history containing such signals cannot be a stationary point of the action integral. Maybe the action integral is not defined at all, if such rogue signals happen. An example is a faster-than-light particle m in a typical action integral. Its contribution would be imaginary:

       m / sqrt(1  -  v² / c²).

But a global field defined as a sum of individual fields loses information. The sum global field can look benign, with no harsh penalty, even though individual fields change faster-than-light!

Thus, a sum global field often allows faster-than-light signals to happen. This is a major shortcoming in the concept of a global field. We must replace it with individual fields of the particles, to avoid loss of information.

A problem with FLRW solutions of general relativity: adjusting the metric using superluminal information

Let us again look at the development of an (approximately) FLRW model at a location A. The solution is not allowed to "know" that the expansion of the universe will slow down uniformly as the time t passes. We have to look at the solution for various possible decelerations of the expansion far away from A. Let a family of possible expansion rates be S(n).

There is a problem in this approach, though. On May 21, 2024 we proved that the Einstein-Hilbert action does not have a stationary point for a "dynamic" system. Thus, there is no solution, unless the expansion rate is the same everywhere! Let us for a while assume that we have been able to correct the action formula, and can find a stationary point.

Setting the metric close to A to some special (different) value for each S(n) may optimize the action, unless the action somehow recognizes that we are using superluminal information, and harshly penalizes such a break of rules. But how could the action recognize that? We are not sending gravitational waves whose energy would be infinite or imaginary. We are simply adjusting the metric in some seemingly innocent way.

An innocent adjustment may amount to a superluminal signal.

Here we again bump into the problem that general relativity does not have canonical coordinates. In Minkowski space, it would be easier to recognize superluminal signals. Though, we still would have to look at the individual field of each particle.

An individual field for each particle is in the spirit of quantum field theory

In quantum field theory, individual particles interact with each other, without any reference to a "global electromagnetic field". It makes sense to introduce an individual force field for each particle.

Conservation of energy and momentum in quantum field theory is implemented through particles exchanging (virtual) quanta. This is a possible solution to the conservation problem in macroscopic fields, though this does not tell us in detail what a macroscopic field does, and how does a macroscopic field implement conservation laws.

Conclusions

We have discovered strong evidence against the traditional global field concept, where the field is understood as the sum of the fields of the individual charges (sum global field). The sum loses information. It cannot recognize and ban faster-than-light signals in some cases.

The self-force of the field on the electron may be hard to describe through a sum global field.

The simple solution to the problem is to split the sum global field into individual fields of each elementary particle. Quantum mechanics likes this solution.

General relativity has major problems, though: there it is not clear what is the field of an individual particle. The mass of the particle acts as a source of spacetime "curvature" at the location of the particle. It affects the curvature also elsewhere, but what is the individual field of a single particle is a fuzzy concept. Nonlinearity of gravity makes this inevitable: how do we assign nonlinear effects to each component field?

Anyway, the individual field of each particle is a useful concept in gravity, too.

The FLRW model is an unusual application of the field concept because the spatial topology is that of a 3-sphere. Can we define the electric field of a single charge in such a topology? Where would the lines of force end? We are not sure if such a topology makes sense at all as a physical model. Is it so that the universe must be a flat Minkowski space?

We will investigate what retardation means in the case of the FLRW model. Does retardation affect the deceleration of the expansion of the universe? Does retardation explain dark energy?

Gauss's law for the electric field fails because of retardation; special relativity conflicts general relativity

UPDATE January 9, 2025: We maybe have to replace the classical law of Gauss with a spacetime version: the average flux of a the electric field E through a closed surface S is a constant times the charge Q enclosed inside S. The instantaneous flux may vary. Conservation of energy requires that the average flu, in some sense, must be constant. Otherwise, we could construct a perpetuum mobile.

Note that in special relativity, time and space cannot be separated. Gauss's law is suspicious because it thinks that space is a separate entity from time. Our retardation argument uncovers the problem.

----

Our analysis on December 30, 2024 proves a surprising result: Gauss's law for the electric field fails!

  E  <------- total electric field 

  E₀ <--  electric field of cap

                                                    ______

                                                 /             \

              •              <--    |                           |

             q               v                 \_______/

      test charge         "cap"     charged shell

Let us have a charged spherical shell which initially is static. Suddenly, the shell starts expanding at a constant speed v. A test charge at some distance learns about the expansion for the nearest part of the shell first. The remaining part of the sphere q still "sees" as static.

The test charge q "sees" the electric field E₀ of the nearest part as it would emanate from a "cap" which is closer to q than the rest of the sphere.

The test charge q sees the total electric field E of the charged sphere now stronger than it was a while ago.

The integral of the electric field E over a spherical surface at the distance of q increased: we broke Gauss's law!

The analysis above uses the retardation law for the electric field: if a charge Q moves at a speed v relative to the laboratory frame, then a test charge which "knows" of the movement of Q, at a laboratory time t, will see Q at the location where Q is at that same laboratory time t. That is, q will see the cap closer.

The cap is moving at a speed v to the left. Let us Lorentz transform its field, in order to make sure that the field E₀ of the cap is now larger than it was when q still saw the cap static.

      electric 

      field

      meter

          O            s          ruler

          -----------------------

           •                <--    |

           q                v     Q cap

                       

     

            -------> x

At a laboratory time t, q's x coordinate is 0 and the x coordinate of the cap is s.

The cap is carrying a ruler which extends over the laboratory distance s to the test charge q. To the ruler there is an electric field meter attached. In the moving frame of the cap, the distance is longer:

       s'  =  s / sqrt(1  -  v² / c²).

The electric field of Q in the moving cap frame is

       E₀'  =  1 / (4 π ε₀)  *  Q / s'²

              =  1 / (4 π ε₀)  *  Q / s²  *  (1  -  v² / c²).

https://www.feynmanlectures.caltech.edu/II_26.html

The Lorentz transformation of E₀' is the identical mapping:

       E₀  =  E₀'.

If the velocity v = c / 10, then q sees the cap 10% closer than the rest of the sphere, and q sees the field E₀ about 20% stronger than when the cap still was static. The correction coefficient in this case is 1 - v² / c² = 0.99, very close to 1. For small v, we can usually ignore Lorentz corrections, since they are ~ v² / c².

The magnetic field of the expanding sphere is zero, because of the symmetry.

What is the effect of the electromagnetic wave which q may "see" from charges in in the sphere suddenly accelerating to the velocity v?

In the analysis above we ignored a possible electromagnetic wave which q may see.

In the laboratory frame, there certainly is no electromagnetic wave from the sudden acceleration. There are no longitudinal waves. Thus, q is not expected to see anything.

But let us analyze in more detail. The test charge q sees that a ring of charges in the shell suddenly accelerates toward q. Does the symmetry eliminate any magnetic field?

       × Be 

       •

      q             

                             v <--  • e

Let e be an elementary charge in the ring. It suddenly accelerates to the left. A magnetic field Be is born at q. In the diagram, the field Be is normal to the screen (hence the symbol ×). When we add the fields Be for each e in the ring, their sum is zero.

It looks like q will not see any electromagnetic wave.

The effect of the extra inertia of q in the field of the sphere

If q and the sphere have a charge of the same sign, then the extra inertia of q is expected to push q to the left. Inertia will not save Gauss's law.

Could there exist a mysterious force which saves Gauss's law? No

The mysterious force should be able to differentiate between the following configurations, drawn in the laboratory frame:

        E 

      <---

                    ______

                  /             \

     <-- |                       |

       v         \_______/

        only the "cap" moves, shell static

                        ^  v

                        |  

                __________

             /                     \

           /                          \

         |                              |

           \                          /

              \___________/

                        |  

                        v   v

        entire shell expands

The mysterious force would compensate the difference of E to what Gauss's law predicts. The force would understand that it must not do anything in the first diagram above, but must do compensation in the second diagram.

For the force to do immediate compensation, it should get information faster than light. Such a force cannot exist if we believe special relativity.

Energy and momentum conservation: an accelerating shell of charges e

We firmly believe that energy and momentum are conserved. The expanding shell creates an electric field E which is larger than what Gauss's law predicts. The impulse to q must somehow eventually have an opposite impulse in the shell.

This is the general problem of conservation of momentum: if particles A and B interact, how does the field store the impulse and eventually deliver it to the other intracting particle?

If we have an accelerating, expanding shell of elementary charges e, then, apparently, any elementary charge e in the shell sees an electric field E which differs from the one given by Gauss's law.

But energy conservation requires that, eventually, the elementary charges e must possess the kinetic energy which we can calculate by integrating E in the Gauss's law solution.

There has to be a compensation mechanism which eventually sets the velocities of elementary charges e to correct ones.

If the acceleration is "too fast" initially, it must be "too slow" later.

How would this compensation mechanism behave? Does it oscillate somehow? In our December 30, 2024 post we speculated that the oscillation between a "too fast" expansion of the universe recently (= dark energy), and "too slow" earlier is due to the compensation mechanism.

Discussion

Gauss's law is one of Maxwell's laws. Let us check the literature, if anyone has realized that retardation spoils Gauss's law.

In our blog we have suspected that Gauss's law fails. We have claimed that the concept of a "field" is too simple to describe complex interactions between particles. The interaction should be calculated individually for each pair of particles. The interaction is "private" between particles. It cannot be simplified into one common "field".

https://physics.stackexchange.com/questions/162744/do-the-integral-forms-of-maxwells-equations-have-limited-applicability-because

In the link people discuss retardation effects in the case where the current through a wire changes. It is much more complicated than our expanding spherical shell.

Is this a miracle that we found a very simple counterexample to Gauss's law? No. Nobody really has claimed that Maxwell's equations are consistent. The self-force of the field on the electron is an open problem. Besides that, there are several well-known, persistent paradoxes.

Birkhoff's theorem fails for gravity: general relativity contradicts special relativity

Weak gravity fields in general relativity should behave much like electric fields. Our argumemt shows that the gravity of a uniform mass shell can vary. The result refutes Birkhoff's theorem which claims that the gravity field of a spherically symmetric object stays constant outside the object.

The refutation uses principles of special relativity. Birkhoff's theorem is derived from the Einstein field equations. We have a proof that the Einstein field equations are incompatible with special relativity.

See our blog post on January 4, 2025 for an analysis why this happens.

Conclusions

We realized on December 30, 2024 that retardation affects collapses and expansions which happen under a force field. A simple consequence is that Gauss's law fails for the electric field.

The speed of light is very large compared to the speed of charge carriers in everyday laboratory experiments. We do not know if we are able to measure the electric field retardation effect in a laboratory.

By far the largest expansion in nature happens in the universe, and the velocities are close to the speed of light. Retardation effects might be important in this huge scale. We will try to calculate an estimate for them in the ΛCDM model. They might explain the peculiarities in the cosmic expansion.

Oppenheimer-Snyder and dark energy in FLRW: finite speed of gravity (retardation) causes dark energy?

On May 26, 2024 we proved that the Oppenheimer-Snyder (1939) solution to a collapsing dust ball is erroneous. It is not an extremal point of the Einstein-Hilbert action. The error is in the choice of the comoving Tolman coordinates: a particle can move backward in the time coordinate. That spoils the derivation of the Einstein equations in variational calculus, since the derivation implicitly assumes a "reasonable" time coordinate where particles cannot move backward in time.

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.

https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric

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.

The Milne model explains the overabundance of hight-redshift galaxies in James Webb photos?

There may be a serious tension between ΛCDM and the observed angular diameter of distant galaxies.

https://arxiv.org/abs/2212.06575

Nikita Lovyagin et al. (2022) have plotted angular diameters of galaxy cores versus the redshift. The red data is from the James Webb space telescope:

There is a serious discrepancy in the observed data from the ΛCDM model. Lovyagin et al. measured the angular diameters from publicly available data. Are their measurements correct?

In the Milne cosmological model, the scale factor a of the universe increases at a constant speed. What is the plot like for the Milne model? Linear?

Density of early galaxies at high redshifts z

The angular diameter of a single galaxy in a photograph is a fuzzy concept. Maybe galaxy cores were much smaller at z = 10 than they are now, and that is why they appear to have a surprisingly small diameter in the photograph?

What about looking at the angular distance of two adjacent galaxies? If galaxies were born all at the same time in the distant past, we can calculate how many of them are at a certain value z in a square degree of the sky, for different cosmological models.

The diagram above suggests that James Webb might see 100X the galaxies at z = 10, compared to what ΛCDM predicts.

https://www.mdpi.com/2218-1997/10/1/48

Stacy S. McGaugh (2024) shows some data about the observed density on massive galaxies at a high redshift z:

On the horizontal axis we have the rest-frame ultraviolet absolute magnitude MU of the galaxy. Blue denotes z = 9, red z = 14. We see that at z = 14, there is 100X the predicted density of galaxies. However, for z = 9, the discrepancy is only 3X.

Dark energy and the expansion rate

https://en.wikipedia.org/wiki/Friedmann_equations

The second Friedmann equation says:

The "dark energy density" is equal to

       -p

where p is the negative pressure. Suppose that the density of matter (excluding dark energy) is m. Then

       ρ  =  m  +  -p / c².

Let us solve for a zero acceleration:

       a''  =  0

<=>

       ρ  +  3 p / c²  =  0

<=>

       m  -  p / c²  +  3 p / c²  =  0

<=>

       -p  =  1/2 m c².

That is, the dark energy density -p is a half of the matter energy density. The "fraction" of dark energy is defined

      fDE  =  -p / (ρ c²).

If the fraction is 1/3, then there is no acceleration in the expansion of the universe – it is like the Milne model.

https://en.wikipedia.org/wiki/Milne_model

The distance versus the redshift z in the Milne model

                 A                                                G

                                    light                  galaxy

                 o  ~~~~~~~~~~~~~~~~  O       

                  |                    

                 /\

          observer                

Let us determine the redshift of light in the Milne model, versus the apparent angular diameter of the galaxy sending that light.

It is like the balloon cosmological model, where the balloon grows at a constant speed.

If the distance between some points A and G grows faster than light, then there is a "horizon" between A and G.

                               A                       G

          ^ x                •                        ●

          |                    <-------  S  ------->

          |                       coordinate

           ---------> y        distance

Metric. Let us model the uniform stretching of space in the x, y plane. Galaxies and observers stay at fixed coordinates. It is the spatial metric which stretches. The flat spatial metric g stretches linearly with the time t:

        g  =  a₀² t² I,

where I is the identity matrix and a₀ is a constant. Note that the stretching factor is the square root of an element in g. The scale factor a of the universe is

       a(t)  =  a₀ t.

A photon moves along a straight line in the x, y plane. We assume that the speed of light, c, is constant (in the metric). The coordinate speed of the photon slows down as time passes.

Angular diameter. Let S be the coordinate distance between the observer A and the galaxy G. The proper diameter d₀ of G stays constant. The coordinate diameter of G at a time t is

       D(t)  =  d₀ / a(t)

                =  d₀ / (a₀ t).

The angular diameter of the galaxy G, as seen by the observer A is

       A(t)  =  D(t₀) / S,

where t₀ is the time at which the light observed by A left the galaxy G. The proper distance s of A and G at that time was

       s  =  S a₀ t₀.

Then

       A(t)  =  d₀ / (a₀ t₀)  *  (a₀ t₀) / s

                =  d₀ / s.

That is, we can determine the angle A(t) either by comparing coordinate distances or proper distances, of course.

Coordinate speed of light. The proper speed of light in the metric is c. The coordinate speed C(t) of light is

       C(t)  =  c / a(t).

Redshift. We assume that the wavelength of a wave packet stretches as the spatial metric stretches. If the redshift is z, then a wavelength λ has stretched to

       λ(t)  =  (1  +  z) λ.

We have

        z  =  a(t)  -  1.

Redshift z as the light reaches the observer. We have to integrate, in order to obtain the time t when the light from an early galaxy G reaches our observer A. The light departs at a time t₀. The coordinate distance traveled by the light is

                  t

       S  =   ∫   c / (a₀ t)  *  dt

              t₀

           =  c / a₀  *  (ln(t)  -  ln(t₀))

<=>

       t  =  exp( a₀ / c  *  S  +  ln(t₀) )

          =  t₀ exp(a₀ / c  *  S).

The age of the universe. The current value of the Hubble "constant" is 73 km/s per megaparsec. A megaparsec is

       3.09 * 10²² meters.

The scale factor of the universe grows by

       73,000 / (3.09 * 10²²)

       =  2.36 * 10⁻¹⁸  *  1 / second

       =  a₀.

The age of the universe is

       1 / a₀  =  4.2 * 10¹⁷ seconds

                  =  13.3 billion years.

Proper distance of the last scattering of CMB. Let us have a photon of the cosmic microwave background which we observe right now. Let us determine the proper distance of the photon to us at the time when the photon was scattered the last time.

The density of the matter was the right one for the last scattering when the scale factor was 1 / 1,100 'th of the current one. Then we have

       t₀  =  3.8 * 10¹⁴ seconds

            =  12 million years.

From

       1,100

       =  t / t₀

       =  exp(a₀ / c  *  S)

we get

       ln(1,100)  =  a₀ / c  *  s / (a₀ t₀)

<=>

       s  =  ln(1,100)  *  c t₀.

           =  7 c t₀

           =  84 million light-years.

The value for ΛCDM is 30 million light-years.

The age of the universe at the last scattering is 12 million years in the Milne model, but only 380,000 years in ΛCDM.

Baryon acoustic oscillations

The CMB angular spectrum matches the baryon acoustic oscillations for a 380,000 year old universe in ΛCDM. Could it be that they also match Milne?

The "sound horizon" determines the size scale of the acoustic oscillations. How far can a plasma sound wave reach in ΛCDM versus Milne?

https://astro.ucla.edu/~wright/BAO-cosmology.html

Edward L. Wright (2014) writes that the most of the mass of the plasma is in photons, which means that the sound waves propagate at an approximately half of the speed of light. The radius of the sound horizon is 200,000 light-years in ΛCDM and 6 million light-years in Milne?

The horizon radius is 10X too large in Milne?

Actually, the formula 

       S  =  =  c / a₀  *  (ln(t)  -  ln(t₀))

tells us that the horizon radius is infinite, if we set t₀ to zero. The sound wave can propagate an infinite distance if the universe starts from a point.

https://en.wikipedia.org/wiki/Big_Bang_nucleosynthesis

What if we claim that the universe starts at, say, t = 10 million years? Then the sound horizon would have an appropriate size – but nucleosynthesis will not work, since the temperature in the synthesis must be around 10⁹ K.

An alternative Milne-like model: a newtonian explosion

Above we defined the Milne model as something which is like FLRW, but has a constant expansion rate. The spatial metric stretches with time.

Another way to make a Milne-like model is to consider a "newtonian" explosion of matter with a positive mass, balanced by strange matter with a negative mass. The explosion cloud keeps growing at a constant rate.

Then the CMB radiation, which we are observing now, has traveled in Minkowski space for ~ 13 billion years. The angular spectrum of the CMB map reveals features of a size ~ 1/60 radians, or 220 million light years. What could these features be? Can they be baryonic oscillations of the plasma?

The last scattering of CMB happened at a time t = 12 million years. The sound horizon cannot have a size of 220 million light years. Is there any way to save this model?

Milne model for times > 380,000 years only?

Most of the mass of the universe in the early stages was radiation, i.e., massless particles. Does the following make sense:

There are particles in dark matter, such that, their interaction, on the average, cancels the effect of the gravity of dark matter on the expansion of the universe.

Those particles might have a negative gravity charge, or an attractive force which produces the negative pressure required to cancel the effect of the gravity of dark matter.

Then the radiation-dominated early stages of the universe would happen just like in ΛCDM, but the dark matter dominated latter stages would happen just like in the Milne model.

https://en.wikipedia.org/wiki/Scale_factor_(cosmology)

But people believe that the radiation domination ended when the universe was just 47,000 years old, or a redshift of 3,600. To reproduce the baryon acoustic oscillations, we need radiation domination to last at least 380,000 years, until the time when the CMB was emitted. Then the universe will develop along ΛCDM long enough.

Let us check how do we know the radiation content of the universe. How much primordial gravitational radiation exists? What is the energy of neutrinos?

The relevant parameter is the "decoupling" time, after which only a small amount of energy flows from the hot plasma to a type of radiation.

As the universe expands, that type of radiation will lose energy and becomes insignificant quite soon.

https://en.wikipedia.org/wiki/Cosmic_neutrino_background

The estimated temperature T of the neutrino background is now 1.95 K and graviton background < 0.9 K. Since the energy density of a type of radiation is ~ T⁴, then most of the radiation energy content is in photons, whose temperature is 2.725 K.

There might be radiation in dark matter, too. Then the radiation-dominated era could last longer than 47,000 years.

Dark matter particles which pull each other create the negative pressure required for dark energy? 

The particles might be massless. Then they would not get caught into gravity wells, and could be distributed quite evenly across the observable universe.

The particles attract each other. They produce enough negative pressure to cancel the effect of dark matter on the expansion of the universe.

Is this plausible?

Let us consider an electron-positron pair. Their common electric field produces a negative pressure. The field acts like the hypothetical "dark energy".

If we imagine that the spatial volume of the universe is finite, then in order to draw the field lines that obey Gauss's law, we must have an equal amount of positive and negative charge.

If the universe is spatially infinite, then there is no need to balance positive and negative charges, since infinity can act as the "other end" for the lines of force.

Would these positive and negative charges clump together and the negative pressure would go away? This has happened with electric charges in our universe.

Dark matter particles with a negative gravity charge?

The energy of the gravity field is negative. Suppose that we have field lines going from a positive gravity charge A to a negative gravity charge B. If we move A and B closer to each other, the field volume shrinks: we conclude that the field energy rises, and there is a repulsive force between A and B.

Negative gravity charges attract each other because the field becomes stronger when we clump them together.

Is there any reason why negative gravity charges should cancel out positive gravity charges during the matter-dominated era of the universe, but not during the radiation-dominated era?

Determining the Hubble constant if the late stages of the universe are Milne

https://physics.stackexchange.com/questions/751762/how-is-hubbles-constant-the-expansion-rate-predicted-from-lcdm-and-the-cmb

The user "gabe" in the link above (2023) gives the formula as:

where DA is the "angular distance" of the light in the CMB. The sound horizon size rs which we see in the CMB map:

A Milne model at a late stage of the universe means that the Hubble "constant" H(z) at earlier stages is surprisingly small. In a pure ΛCDM model, the Hubble constant would grow faster when we go back in time.

How does a late Milne model affect DA? It is larger. Also rs is larger. Who wins?

The Hubble constant in a matter-dominated universe

https://arxiv.org/abs/2309.06795

Marco Raveri (2023) gives us useful formulae for determining the age of the universe in various cases:

where in the present universe, the following must hold:

to make the spatial metric flat currently. The various Ω are the energy densities of radiation, matter, and dark energy, relative to the critical density right now.

The Hubble value H is defined as

       H  =  (da / dt) / a.

Let us calculate the history of the universe, assuming that

       Ωm  =  1

today. Then

        da / dt  =  a H(a)  =  H₀ / sqrt(a).

Let a = 1 now.

Last scattering of CMB. The redshift z is 1,100, which means that a(t) at that time was 1 / 1,100'th of the current a. The speed of the expansion, da / dt, was 33X the current one.

James Webb's largest redshifts z = 15. The size of the currently observable universe was 1 / 15'th of the current one. The speed of the expansion was 4X the current one.

Let us assume that the universe has been almost Milne since the redshift z was something like 100. Then James Webb will see the density of galaxies much larger at z = 15 than in the ΛCDM model.

But the sound horizon size at z = 1,100 must be the same as in ΛCDM, in order to explain the galazy density in the Sloan Digital Sky Survey (SDSS).

To make the universe Milne at z < 100, there has to be some process which slows down the expansion rate surprisingly much around z = 100, and after that keeps the expansion rate unchanged. What could it be?

What about a model in which all matter is counterbalanced by negative mass charges? Then the current spatial metric cannot be flat – but it does look flat.

Let us analyze using the Friedmann equations.

https://en.wikipedia.org/wiki/Friedmann_equations

We assume that k = 0 and the matter density ρ and the cosmological constant Λ cancel each other out.

What kind of a radiation density would produce the age 380,000 years at the last CMB scattering?

The radiation mass density is

       ρr  =  C / a⁴,

where C is a constant. Then

       da / dt  =  sqrt(8/3 π G C) * 1 / a.

Observation. With a suitable pressure p, we can tune the energy density ρ(a) of the universe as a function of a as we like. In that way we can make da / dt anything we like, and a(t) as we like! We can create the pressure by assuming forces between dark matter particles.

Thus, we do not need the cosmological constant, or "dark energy" at all. Dark matter and suitable force fields will do the job.

Conclusions

The universe looks somewhat Milne-like at present. The expansion rate is constant, or may be somewhat accelerating.

Also, the James Webb photographs with "too many" galaxies reveal that either the universe at z = 15 is surprisingly old, or we do not see the angular magnification of the ΛCDM. Both are Milne features.

By assuming dark matter particles with suitable force fields between them, we can tune the pressure of the universe in the way that any scale factor function a(t) is created. We do not need to assume a cosmological constant or "dark energy".

The concept of "dark energy", or a cosmological constant Λ, is very speculative. In everyday physics we do not encounter anything which looks like "dark energy". On the other hand, we can easily imagine particles whose mutual attraction creates a negative pressure.

Certain authors have claimed that a variable dark energy negative pressure breaks the dominant energy condition in the universe. But it is not so: as the universe expands, it does work against the attraction between hypothetical dark matter particles. The energy does not appear from nothing.

The predictive power of the dark matter pressure hypothesis is zero: it can explain everything! In a future blog post we will investigate this.

If we assume that the first 380,000 years happened much like in ΛCDM, then there has to be an extra positive pressure after the last scattering of the CMB. The positive pressure quickly slows down the expansion to the current speed, in the year 2024. After that, an extra negative pressure makes the universe roughly Milne. Is there any reason why this would be a reasonable scenario with hypothetical dark matter particles?

Let us run time backwards. A negative pressure from an unknown attractive force within dark matter initially keeps the collapse of the universe happening in the Milne way. When there are some ~10 million years left until the singularity, a positive pressure (= a repulsive force within dark matter) greatly speeds up the collapse. The extra pressure does not grow appreciably as the universe contracts. The effect of the matter density and the radiation density take over during the last 380,000 years of the collapsing universe.

We need to assume two unexplained pressures. This is very ugly. Maybe we have dark matter particles which like a certain fixed distance between them, and resist both the expansion and the contraction from that equilibrium state? That is very ad hoc.