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.

Mach's principle

Mach's principle claims that the frame where large distant masses are not rotating, is the "inertial frame" for rotation. That is, in a frame where the angular momentum of the entire universe is zero, imposes no centripetal "force" on the arms of a static observer.

https://en.wikipedia.org/wiki/Mach%27s_principle

Let us analyze this.

Generalized principle of Mach. Masses in the universe determine the inertial frame in some way, not necessarily like Ernst Mach described it as through the average motion of the masses in the universe.

Newtonian mechanics

Newtonian mechanics satisfies Generalized Mach's principle: an inertial frame is one where noninteracting masses move with a constant velocity vector v.

                      M ● ---> v

                         o

                        /|\   observer

                        /\

                v <--- ● M

But newtonian mechanics does not satisfy the original principle of Ernst Mach. We may well have a two masses moving to opposite directions as in the diagram, and the observer does not feel any centripetal "force".

In this blog (June 20, 2024) we have shown that the Einstein field equations do not have any solution for the two-body problem. In that sense, it is not defined if general relativity satisfies Mach's principle, or any principle.

A "linearized" version of general relativity should give approximately the same prediction as newtonian gravity. Such a model of gravity does not satisfy the original principle of Mach.

The syrup model of gravity: neutron star forces objects on the surface to move with the star – a "linear" principle of Mach

In general relativity, close to a very heavy neutron star, the coordinate speed of light is very slow in the standard Schwarzschild coordinates. It is like syrup: forces everything to move very slowly relative to the neutron star.

              photon

                  • --> v'

                  ● ---> v

                 M

         very heavy

         neutron star

                 

A faraway observer sees that even a photon must move approximately along the the movement of the neutron star. Does this mean that some kind of a principle of Mach is true?

Yes, at least in some sense.

The observable universe is quite close to being a black hole or a "very heavy neutron star"

The amount of visible matter and dark matter in the observable universe is something like 10% of the amount required to place us inside the event horizon of a Schwarzschild black hole.

What does this imply about the inertial frame for rotation?

Could it happen that we could see the entire observable universe to rotate around an inertial frame?

Since the "current" radius of the observable universe is ~ 30 billion light years, any rotation of the universe would have a very small angular velocity. Otherwise, faraway galaxies would move faster than light.

Mach's principle fails for electromagnetism

Let us assume that a spherical shell is electrically charged. The electric field inside the shell is zero. If the shell is rotated, there is no electric or magnetic field inside. Mach's principle inside the shell fails for electromagnetism.

Let us then analyze orbits of test charges outside the rotating shell.

Mach's principle for a rotating, heavy shell of matter

Albert Einstein was excited about the possibility that inside a rotating, heavy shell of matter, the "inertial frame" would rotate along the shell.

Does this make sense?

If the shell is not heavy, then we empirically know that the inertial frame does not rotate appreciably. Could it be that the inertial frame rotates a little bit?

A similar question is what is the orbit of a test particle close to a very heavy rotating neutron star.

Conclusions

Do faraway masses determine what is the non-rotating inertial frame? One form of Mach's principle claims that faraway masses determine that frame.

In newtonian mechanics, the motion of faraway masses has no bearing on inertial frames. Mach's principle is not true.

If general relativity is formulated in a way which matches newtonian mechanics for a few small masses in an otherwise empty Minkowki space, then Mach's principle is not true in general relativity.

For a large moving or rotating mass, the syrup model of general relativity says that all objects are forced to move along. But does this mean that the "inertial frame" moves with the large mass? It is not clear. It is hard to define what Mach's principle precisely means in these cases.

Energy conservation in Tuomo Suntola's Dynamic Universe

We uncovered a major problem in cosmology in our October 10, 2024 blog post. The Einstein-Hilbert action requires the energy of radiation to be conserved. But in ΛCDM, it is not conserved.

We know from measurements that the total energy of the cosmic microwave background (CMB) is only 1 / 1,100'th of its original energy (original = when the radiation was created).

https://www.physicsfoundations.org/dynamic-universe

Tuomo Suntola has developed a cosmological model Dynamic Universe (1990 -), in which total energy is conserved, even though CMB loses energy. Let us check if the Dynamic Universe is consistent with measurement data.

The inflating balloon model

https://www.jpl.nasa.gov/edu/teach/activity/model-the-expanding-universe/

Let us analyze the motion of a freely moving particle m on the surface of a spherical balloon which is being inflated.

The angular momentum of the particle must stay constant. Let v₀ be the velocity when the size of the balloon is

       a  =  1.

The velocity for an arbitrary size a (or scale factor a) is

       v  =  v₀ / a.

Very simple.

A photon, or a wave packet, always moves at the speed of light, c. To conserve the angular momentum, its energy E must go as

       E  =  h f  =  E₀ / a,

where E₀ is the energy of the photon or the wave packet when a = 1.

The balloon model explains well why the cosmic microwavebackground looks like black body radiation. The wavelength changes as

       ~  a,

the density of wave packets goes as

       ~  1 / a³,

and the energy of a wave packet goes as

       ~  1 / a.

The temperature is

        T  ~  1 / a,

and the energy density of radiation is

        ~  1 / a⁴  ~  T⁴,

as it should be. On October 10, 2024 we showed that the Einstein-Hilbert action does not allow the energy of a wave packet to change. General relativity predicts that the energy density is

        ~  T³,

which is clearly wrong.

The Milne model probably gets T⁴ right, but we did not check that yet.

A gravity explosion seems to keep black body radiation black body

Imagine an explosion under newtonian gravity, where baryon matter climbs up from a gravity potential well.

If a photon is moving toward the center of the explosion, it gains energy in a blueshift?

On the other hand, an equivalence principle says that for freely falling matter, things should look symmetric: the orientation of the center of the explosion should not be visible.

Is the CMB spectrum in this case automatically black body?

An observer in the future receives each photon from a lower gravity potential. He should see a redshift regardless of the Doppler effect?

Let us have a clock which tics and produces a laser beam whose cycle time is one tic. When the laser beam was sent, the clock was at a lower gravity potential: it ticked slower. We conclude that the future observer does see a redshift even though there were no Doppler effect included.

The mirror principle of gas. Suppose that the volume of a cloud of uniform gas (or a cloud of photons) somehow changes in a uniform way. Then we can assume that the gas is divided into smaller chambers whose walls reflect the gas atoms.

Proof. Let us divide the volume into smaller "compartments".

           compartment    compartment

         |        • --> v        |       v <-- •       |

               atom                          atom

Let there be a very large number of atoms in each compartment. If an atom moves to the neighboring compartment, then we can imagine that an equivalent atom moves to the opposite direction.

And we can imagine that, actually, both atoms bounce from each other at the location of the compartment wall.

That is, we can think that the gas is under an "adiabatic" process. QED.

Let the compartments (their walls) fall freely under gravity. Then one would expect that the photon gas inside the compartment is compressed (or expanded) adiabtically, and the gas remains black body.

https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/03%3A_Thermodynamics/3.11%3A_Thermodynamics_Applied_to_Light

Literature seems to say that black body radiation remains black body under an adiabatic process. The following conjecture is now plausible:

Conjecture. An explosion where gravity satisfied an equivalence principle, keeps black body radiation black body. It is equivalent to an adiabatic expansion or compression of the radiation.

Energy conservation in the Tuomo Suntola dynamic universe

Let us check how Tuomo Suntola handles energy conservation. In the inflating balloon model, energy conservation follows from newtonian mechanics. Is the Suntola model somewhat similar?

In equation 1.1.2:1, Suntola sets the total energy of the universe to zero. The energy Em above denotes the "kinetic energy" of all matter in the universe. The energy Eg is the negative "gravitational potential" of all matter. If the total energy is defined in this way, it is conserved, as the sum is always zero.

In section 1.1.4, Suntola says that c₀ is the "expansion velocity of space into the direction of the 4-radius". In the balloon analogy, it would be the radial speed of the expanding rubber sheet.

We have to check if energy is conserved also locally.

The energy formula above looks like a newtonian collapse of dust where the particles have a very small rest mass and are initially far away from each other. A half of the negative gravitational potential (the binding energy) is converted into kinetic energy of particles as the dust collapses.

The book by Tuomo Suntola does not seem to contain a clear formula for the energy content of the universe in a mord complex case where there are stars, galaxies, and the cosmic microwave background.

Conclusions

The cosmological model of Tuomo Suntola has energy conservation, in some sense. But the details are missing in his book.

Section 6.2 of the book contains a strange claim that the apparent angular size of galaxies in the sky, for a redshift z > 1 does not match the FLRW model.

https://www.researchgate.net/publication/317074672_Ultra-compact_structure_in_intermediate-luminosity_radio_quasars_Building_a_sample_of_standard_cosmological_rulers_and_improving_the_dark_energy_constraints_up_to_z_3

Modern data from quasars at distances up to z = 2.8 supports FLRW and refutes Tuomo Suntola's model. However, recent data from James Webb fits Suntola's model better. See our December 18, 2024 blog post about James Webb.

Density variations in a young universe break Lambda-CDM?

The cosmic microwave background contain luminosity differences on the order of 1 / 100,000. In the ΛCDM model, these are believed to reflect density variations of the same order in a very early universe. "Very early" here means the end of the inflation phase at ~ 10⁻³² seconds after the birth of the universe.

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

https://www.its.caltech.edu/~esp/ay127/

The book Galaxy Formation and Evolution by Mo, van den Bosch, and White (2010) is available as a pdf file in the link.

The chapters about Cosmological Perturbations handle the fate of density variations in the early universe.

The mass excess at certain locations in the early universe: the Schwarzschild radius

The current density of baryonic matter in the observable universe is

       ~  3 * 10⁻²⁸ kg/m³.

A large galaxy cluster is 10²⁴ meters in diameter, which makes a baryonic mass of

       ~  3 * 10⁴⁴ kg.

Let us assume that the "excess" mass is 1 / 100,000 of that, or

       M  =  3 * 10³⁹ kg.

The Schwarzschild radius of such a mass is

       R  =  2 G M / c²

         

            =  1.5 * 10⁻²⁷  *  M  m/kg

            =  5 * 10¹² m.

We see that if the scale factor of the universe is

        a  =  10⁻¹¹,

then the excess mass is inside its Schwarzschild radius. (The factor a = 1 is the present universe.)

The excess mass should form a black hole in the early universe. Why would a black hole expand like ΛCDM claims?

Playing the Big Bang in reverse: density variations form black holes

https://en.wikipedia.org/wiki/De_Sitter%E2%80%93Schwarzschild_metric

The ΛCDM model claims that the metric of the early universe is very smooth, spatially uniformly expanding metric. If we have two observers at the edge of excess density volume in the early universe, the observers can communicate much like two observers in Minkowski space.

But is that really possible if the dense volume is inside its own Schwarzschild radius, and is a black hole?

If we assume that gravity is time-symmetric, then we can play the Big Bang expansion in reverse. Any (radio) communication between the two observers can happen in reverse, too.

One of the observers is inside the horizon of a black hole, the other is outside it. We see that the observer inside cannot send a signal to the outside.

This is in contradiction with the claim that the observers can communicate without problems.

A small density perturbation in a middle-aged universe amounts to a huge perturbation in the early universe. This is the well-known problem of the instability of an expanding universe.

Note that the Milne model suffers from the instability just like ΛCDM.

A possible solution: the universe does not begin from a singularity, but is relatively "large" at birth. The diameter of the current observable universe might have been much larger than 5 * 10¹² meters at birth. We do not like singularities in this blog. This could be a step forward, actually.

The Nariai metric differentiates between black holes which are smaller than the "cosmological horizon", and larger lumps of matter. Could it be that observers at the edge of a larger lump of matter still could communicate?

Does the literature address the instability problem in the early universe?

Let us look into the the book by Mo, van den Bosch, and White (2010). Their section 4.1 treats "newtonian" perturbations. The section is about how a perturbed gas behaves under gravity and pressure.

In section 4.2 the authors discuss "relativistic" small perturbations. We do not find any analysis of instability.

https://ui.adsabs.harvard.edu/abs/1957RvMP...29..432L/abstract

https://en.wikipedia.org/wiki/Gravitation_(book)

Richard Lindquist and John Wheeler (1957) study a closed universe which consists of many black holes "glued" together. We do not have access to their paper, so that we could check if they handle stability issues. In our blog we have proved that the Einstein-Hilbert action, essentially, does not allow any dynamic solutions. We doubt that Lindquist and Wheeler were able to prove stability.

The density variations in inflation break the Einstein-Hilbert action?

https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action

The basic idea in cosmic inflation is that a scalar field somehow obtains random fluctuations in energy density, and these fluctuations are "blown" spatially very large in the expansion of the universe.

https://en.wikipedia.org/wiki/Lagrangian_(field_theory)

Let us have a history H of the universe. Let us make a variation H' where we keep the metric as is, and thus, the action integral of the Ricci scalar R does not change.

              wave

        -------______--------_____-------   tense string

                     /         /          /

                     \         \          \    springs

The lagrangian describes an oscillating tense string which also is attached to springs, which create the potential V(φ).

In the inflation hypothesis, φ varies from place to place, and the potential energy (or kinetic energy) is greatly increased as the universe expands.

Let us have a "disturbance" W of the field.

Suppose that the disturbance W is not moving substantially. It is the springs in the diagram which make the string to oscillate up and down.

The expansion of the universe adds more length to the oscillating area, and more energy.

Let us have a history H where the oscillating area expands. Energy is produced from "nothing".

Let us make a history H' where we keep the metric the same. The time varies within an interval Δt.

We make a Noether time variation which "measures" the energy of the system at the start and the end of the time interval Δt. The process is like we would have a harmonic oscillator whose mass is zero, but the energy non-zero, created from empty space at some time t during the time interval Δt.

The Noether time variation changes the value of the action close to the end of Δt, because the oscillator there has energy, but does not change the action close to the start of Δt, because the oscillator does not exist there. We conclude that H' has a changed value of the action. H was not a stationary point of the action. H is not an allowed history.

Suppose then that the disturbance W is moving. A Noether time variation can measure the energy of W at the start of Δt, and at the end. We end up in the conclusion that H is not an allowed history.

We showed that a scalar inflaton field is not compatible with the Einstein-Hilbert action.

Note that if there is no disturbance W, and the field φ simply has a constant potential V, then our counterexamples do not work. It may be that dark energy is compatible with the Einstein-Hilbert action.

Conclusions

Density variations in the early universe seem to involve instability: if we play time backwards, we end up with many black holes. It is unclear if such a configuration is consistent with any cosmological model.

Density variations in inflation seem to break the Einstein-Hilbert action.

Our general view of cosmological models:

We looked at some literature about cosmological models. There are many competing hypotheses. There is no proof that the models are stable. Neither is there a proof that the models are consistent with general relativity. Authors typically use newtonian mechanics. Cosmology is a mess.

Lambda-CDM is not a solution of the Einstein-Hilbert action?

The standard ΛCDM model of cosmology explains the black body radiation spectrum of the cosmic microwave background (CMB) by claiming that the radiation pressure does work and CMB loses energy as the FLRW universe expands.

But the Einstein-Hilbert action seems to imply that CMB cannot lose energy because there is no place where that energy could go. Then the history would lose energy, but that is not allowed in a stationary point of the matter lagrangian L in the Einstein-Hilbert action.

https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action

https://en.wikipedia.org/wiki/Lambda-CDM_model

Massive particles in the FLRW universe: they do not lose kinetic energy

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

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

The lagrangian for a single massive particle outside of potential fields is the familiar kinetic energy formula.

In the ΛCDM model, it is familarly assumed that as the time passes, fast moving particles "lose" kinetic energy relative to the spatial coordinates which are fixed to "static" expanding matter content.

                     newtonian explosion 

  "static" particle                        "static" particle

         <- ●                       ●   • ----> v           ● ->

                                            fast particle

This would really be the case if the expansion of the universe would be equivalent to a newtonian explosion where the "static" matter is expanding its volume in Minkowski space. When the fast particle reaches the rightmost "static" particle, the relative speed of the fast particle compared to the "static" particle is less.

Let us have a history H of the FLRW universe, such that fast particles gradually lose speed as the universe expands.

Let m be a fast body of mass such that its density is extremely small. Then the self-potential of m relative to its own gravity field is negligible.

Let v₁ be the initial velocity of m relative to the standard FLRW coordinates, and v₂ its final velocity in the history H.

Let us make the following variation H' of the history H which runs for a coordinate time interval Δt:

1.   we keep the metric H constant;

2.   we let m move at an almost uniform speed v during Δt, except that the speed at the start is v₁ at the start and v₂ at the end.

How much does the Einstein-Hilbert action change? For the (rest) mass m itself, the volume element

can be taken as essentially constant during Δt, because the stretching of the spatial metric does not affect the rest mass. The same holds for the kinetic energy of m.

The changes in the Einstein-Hilbert action:

1.   The kinetic energy action of m in the matter lagrangian L (which is of the form kinetic energy - potential energy) is reduced because the velocity v is almost constant during Δt.

2.   The position of m is displaced relative to its own gravity field: the volume element sqrt(-g) obtains a slightly larger value because m does not reside in its own gravity potential well. However, we can make the effect on the action value as small as we want by reducing the density of the body m. Thus, we can ignore this item.

We conclude that the variation H' reduces the value of the Einstein-Hilbert action relative to H. The history H was not a stationary point of the action. We proved that the speed of m cannot change as the FLRW universe expands.

Massless scalar field waves in the FLRW model

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

The massless scalar field (m = 0) has a very simple lagrangian density formula. Let us have a wave packet. We must determine what history of a wave packet is a stationary point of the action in an expanding FLRW universe.

Let us stretch spatial distances by a factor 2. The naive transformation of keeping the function φ(r) values the same for each coordinate vector r, will make the wave packet volume 8-fold and the squares of the derivatives of φ 1/4-fold. The energy of the wave packet would double. Clearly, we must find a better transformation.

Let us have a history H of the universe for a coordinate time interval Δt. We make the following variation H':

1.   keep the metric g the same;

2.   do a Noether time variation of the wave packet.

By making the energy density of the wave packet very small, we can ignore its potential in its own gravity field, just as we did in the previous section.

The Noether time variation at the start time speeds up the time development of the wave packet, and the variation at the end slows down the time variation of the wave packet.

Is it a problem that the wave packet moves faster than c when we speed it up? Let us for the moment assume that it is not a problem.

For the Noether time variation to keep the value of the action the same, the "energy" measured by the time variation must stay constant. The energy of the wave packet cannot change.

But ΛCDM claims that the energy of the packet changes as the universe expands.

What about doing a Noether time variation of the entire radiation content of the universe?

https://physics.stackexchange.com/questions/130536/when-the-cosmic-microwave-background-radiation-cools-where-does-the-energy-go

"ACuriousMind" on the Physics Stack Exchange claims that since the universe as a whole is not time translation invariant, a Noether time variation will not show that energy is conserved. Does this mean that the history of the universe, H, is not a stationary point of the action? If not, then how do we decide which is a valid possible history?

Our own argument above only does a local Noether time variation. There is nothing which prevents us from doing that. And the laws of nature which govern a local wave packet are time translation invariant.

A precise variation which changes the action value in the massless scalar field wave example

                                        H history

                   |            |

                   |            |

                    •          •  absorptions

                      \      /

                        \  /  decay into two wave packets

                        /

                      /

                     • --> v

                     m massive particle

   ^  t

   |

    ------> x

Let us have a massive particle m which decays into two wave packets. The particle m initially moves at a speed v and then decays into two light-speed wave packets. Both packets are later absorbed by other particles.

We look at the history H during a time period Δt.

We let the FLRW universe expand so much that the energy of the wave packets is negligible at the end of the period Δt. The absorbing particles can be very lightweight, much lighter than m.

Let us vary the history H in such a way that we let the particle m move to the right at a faster speed than v for a short time after the start of Δt. At the end of the Δt, we let the other particles move for a short time to the right, so that they catch the wave packets. Then we let the other particles move to the left, so that the variation H' has the same end state as H.

Since the other particles have negligible masses and the wave packets have negligible energies at the end of Δt, the variation at the end changes the value of the action negligibly.

But the variation at the start of Δt changes the action substantially since it changes the kinetic energy of m.

We showed that an infinitesimal variation H' of the history H changed the action substantially (linearly with the variation). Then the history H is not a stationary point of the Einstein-Hilbert action.

The proof illustrates that if a "packet" of energy is destroyed almost completely in a history H, then it is very easy to construct a variation H' which changes the value of the action.

We did not need to vary the propagation speed of the wave packets. They propagate at the light speed c. There is no problem with faster-than-light signals in the variation H'.

We used the fact that the value of the action is not changed if we translate the middle part of the history H to the positive x direction. That is, we used the spatial translation symmetry.

Why the same argument does not work in a newtonian balloon model of an expanding universe? It has to be because the interaction of the balloon rubber and particles moving on it is more complicated than in general relativity. The spatial translation symmetry does not hold if the particle made a significant "dent" into the rubber. If we keep the history of the rubber as is, but move the path of the particle, then the action of the middle part changes significantly.

Why does the literature claim that energy in radiation is lost when the universe expands?

https://www.scribd.com/document/469722423/Tolman-Relativity-thermodynamics-and-cosmology-pdf

Some papers refer to the Richard Tolman book from the year 1934. Let us find out how Tolman derives the result.

On page 385, Tolman writes:

Tolman claims that a particle moving in an expanding universe will slow down. But in the first section of this blog post we proved from the Einstein-Hilbert action that the particle must keep its velocity. Let us analyze how Tolman arrives at a different result.

                     • ----> v

Tolman derives the path of the particle from the geodesic equation. But if the geodesic equation claims that the particle loses kinetic energy, then it clashes with the Einstein-Hilbert action. How can this be?

A possible reason is that the geodesic equation "measures" the velocity v of the particle relative to the location where it started from. Then there is no loss of kinetic energy.

This leads us to a fundamental problem: if we have a particle which has gone a long way, the expanding universe may make it to recede from its original location faster than light. What is the kinetic energy in that case?

In the Einstein-Hilbert action, is the kinetic energy of the particle reduced or not? The matter lagrangian would require the kinetic energy to be the same, but the Einstein field equations probably require it to be reduced?

All these problems stem from the fact that there are no canonical coordinates in general relativity. The kinetic energy of a particle is a fuzzy concept.

Is the geodesic equation itself ambiguous?

The Einstein-Hilbert action is the fundamental equation. The geodesic equation is more like a guess of how a particle would move under a metric. The guess seems to clash with the action.

Is there any way to make a consistent model where the topology of the spatial metric is isomorphic to a 3-sphere and the geodesic equation holds?

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

In this blog we have been touting the Milne model where space is Minkowski, and we avoid the contradictions produced by the 3-sphere topology.

The balloon analogy

https://www.jpl.nasa.gov/edu/teach/activity/model-the-expanding-universe/

The balloon analogy of an expanding universe has no problem handling radiation pressure and so on, because it is a model in newtonian mechanics. For example, if there is a pressure from "gas" on the surface of the balloon, the pressure does work as the balloon expands. The gas cools, and the energy is given to the rubber in the balloon, stretching the rubber and giving it kinetic energy outward. The action of this system conserves energy.

https://www.physicsfoundations.org/dynamic-universe

Tuomo Suntola's Dynamic Universe has the 3-sphere expanding to a fourth spatial dimension, and the kinetic energy of a particle for the velocity in the fourth dimension is included. It is a step toward the newtonian balloon model. We have to check how Suntola's model handles energy conservation.

A locally Minkowski space?

What about using "canonical" coordinates which are locally Minkowski at each small "patch" of the the universe? We will worry later about gluing these coordinate systems together.

The kinetic energy of a particle would be defined relative to these canonical coordinates. Energy would be conserved.

The problems start when a particle moves from one patch to another. Suddenly, the kinetic energy of the particle changes.

A reverse Oppenheimer-Snyder collapse: make that the new ΛCDM model?

https://journals.aps.org/pr/abstract/10.1103/PhysRev.56.455

In the Oppenheimer-Snyder collapse, uniform dust collapses to form a black hole. If we run the collapse in reverse, it is much like an expanding FLRW universe. However  there are natural canonical coordinates (the Schwarzschild coordinates), and we avoid the various problem with the topology of the 3-sphere in the FLRW model.

A radiation pressure really does work in that model, accelerating the explosion of the matter outward. Energy is conserved.

Maybe we should abandon the 3-sphere, and claim that the natural expanding universe model in general relativity is the reverse Oppenheimer-Snyder collapse? The spacetime is asymptotically Minkowski. There may be several Big Bangs at various locations of the asymptotic Minkowski space.

In our blog we have been claiming that the Oppenheimer-Snyder collapse "freezes" when a horizon forms. No singularity can form. But we see that our own universe has had a much larger matter density in the past, and we would have been inside the horizon. There was no freezing. The universe kept expanding. Currently, the observable universe would be outside the horizon, if there is not much matter which is farther than the observable volume.

We have supported the Milne model in this blog, to avoid the freezing. If there is an equal amount of negative gravity charge (in dark matter) as there is positive charge, then no horizon forms and the Milne model is the one.

Another option is to design a model where somehow the Oppenheimer-Snyder collapse can go past the forming horizon. We could claim that the Big Bang is the only case where such a behavior is possible, and it cannot happen in an ordinary black hole. As if an infinite time could pass in this special, Big Bang process, and take matter past the horizon. That is very ad hoc.

Conclusions

Let us close this blog post. In subsequent posts, we will further analyze the problem of missing canonical coordinates in the ΛCDM model.

Currently, our best bet is to replace the 3-sphere spatial topology of ΛCDM with a reverse Oppenheimer-Snyder collapse. The Schwarzschild coordinates would act as canonical coordinates against which we measure the kinetic energy of a particle.

If the ADM formalism is correct, then energy is conserved in a reverse Oppenheimer-Snyder collapse. We avoid the problems of energy non-conservation in the 3-sphere spatial topology.

In our blog we have been claiming that the gravity field merely simulates the "metric" of spacetime. The metric is not fundamental, and the true underlying spacetime metric and topology is the flat Minkowski space. We have said that the 3-sphere spatial topology of ΛCDM cannot exist. In this blog post we showed that the 3-sphere spatial topology causes serious, maybe insurmountable, problems for ΛCDM.

On the Internet there is a lot of discussion about energy non-conservation in expanding models of the universe. However, no one seems to have analyzed what the Einstein-Hilbert action says about energy non-conservation: it prohibits that. We are probably the first to recognize the serious problem in the ΛCDM model, and in the 3-sphere spatial topology.

We have noted in this blog that while there is ample empirical evidence that gravity can simulate "curved" spacetime, there is no empirical evidence that gravity could alter the topology of spacetime, like it does in the ΛCDM model. The hypothesis that gravity can alter the topology is a bold one. Maybe the hypothesis is incorrect.

It may be that an action in physics can truly work only if energy is conserved. Energy non-conservation in ΛCDM may be a fatal error in the model.