Wednesday, May 13, 2026

Dark energy: a balloon model

In December 2025 we had problems finding a reasonable mathematical model for retarded gravitation in an asymptotic Minkowski space. Let us try it again, this time for a spherical balloon model. The spacetime is then topologically different from a plane.














Picture publicdomainpictures.net



Can the equation of spacetime be local for a balloon model?


Let us think of the Schwarzschild solution. The solution extends through the entire asymptotically Minkowski space. The central mass M affects locations very far away.

For a spherical rubber balloon, the equation governing its stretching is local, if we assume that the pressure inside is uniform.

Friedmann equations in general relativity look local, but that is only because we assume a uniform mass distribution. Any retardation effects are masked.


A rubber string and a weight


                 rubber string
           ●----------------------------●
      point                            point


Let us draw points uniformly around a sphere, and connect them with tense rubber strings which run on the surface of the sphere. The rubber strings simulate the gravity between the mass points. If the balloon is static, or inflates or deflates at a constant rate, this model looks like an ordinary balloon: the tension in the strings is uniform along the string.


A rubber string and a weight which the string slows down


Let us consider the simplest possible model of a tense rubber string plus weight:
          

                                              force
                                         F  <------ 
         ==| ~~~~~~~~~~~~~~ ● ---> v
    wall         rubber string            M weight


We are interested in what kind of longitudinal waves form in the string when the weight moves to the right while the tension in the string pulls the weight to the left.

Free waves in the string are sine waves. Since the speed of the waves is finite, there is "retardation".

As the weight slows down, it creates new waves. The weight perturbs the string. Calculating the precise form of the wave probably requires a computer. But we are only interested in very crude estimates.


Do binary pulsars prove that there are no longitudinal gravitational waves? Are longitudinal waves confined between masses?


Binary pulsars have confirmed that transverse waves and linearized Einstein field equations explain the energy loss of a binary pulsar up to a precision of 0.1%.

In an earlier blog post we suggested that longitudinal waves must be "absorbed into" matter quickly, because they cannot propagate in empty space. This hypothesis would explain why binary pulsars match the transverse wave model so well.

Another hypothesis: longitudinal wave effects can only exist in the field between two masses M₁ and M₂ and can never escape to empty space.


Longitudinal electromagnetic waves exist in plasma, but they do not exist in empty space. Could this be analogous to our model of retarded gravity? Langmuir waves exist in a plasma which is, on the average, neutral. Since gravity charges always are positive and gravity always pulls, Langmuir waves cannot exist in a gravitating system.


A model with solid rubber inside


Let us have two masses M₁ and M₂ on the surface of the balloon. Visualizing a rubber string between the masses, and running on the surface of the balloon, is somewhat hard. Let us imagine that the rubber string runs inside the balloon.

The simplest case is when M₁ and M₂ are on the opposite sides of the balloon. A rubber string runs through the center of the balloon and connects them.

The balloon expands. The rubber string is not immediately aware of the expansion slowing down. This will cause oscillation in the expansion rate.










Photo publicdomainpictures.net

Let us have a solid rubber ball. The rubber will resist the expansion in most cases. But if there is a shock wave moving from the center toward the surface, the expansion of the rubber ball may actually accelerate at a certain time. "Dark energy" in this case is the energy of the shock wave.

This model predicts that the expansion rate of the universe will oscillate, while it, on the average, is quite similar to an FLRW model.

But we should find a mathematical formula, so that we could compare it to the observed dark energy.


The strength of dark energy seems to be weakening right now




The DESI project measures the 3-dimensional mass density variations of the universe right now, and compares it to the baryon acoustic oscillations after the Big Bang. The results of DESI, when combined to other observations, suggest that dark energy is weakening right now. Wikipedia says that dark energy density right now might be 10% less than 4.5 billion years ago.

This is consistent with our own rubber ball model: on the average, the expansion should be like an FLRW model, but there should be "bounces" from the retardation of gravity.


In the link we have the definition for the CPL parametrization of the dark energy equation of state, w(a), where a is the universe scale factor. The current value of a is set to 1.


Abdul-Karim et al. (2025) write that the energy density of dark energy may have increased in the past. This is called a "phantom crossing". In our rubber ball model, a phantom crossing is expected: at some points the bouncing slows down the expansion, at other points speeds up the expansion.


The Hubble tension


Let us measure the baryon sound horizon size in the cosmic microwave background, and match it to the measured density fluctuations in the current universe. Assuming, for example, the standard ΛCDM model, we end up with a Hubble "constant" current value of

       67 km/s / Mpc.

But a direct measurement, based on Cepheids and type 1a supernovae yields a value

       73 km/s / Mpc.

The values differ too much to be just a statistical fluctuation.


Marco Raveri (2023) tries to use dark energy to make these values compatible. Let us check what he suggests. His conclusion is that dark energy modifications at "late" times, 0.01 < z < 3 cannot solve the Hubble tension.


Early dark energy (EDE)



Xuejian Shen et al. (2024) write about an early dark energy (EDE) model, where an unknown field boosts the expansion of the universe just before the recombination (last scattering) which happens when the universe is 380,000 years old in a standard ΛCDM model.

Shen et al. say that this model can solve the Hubble tension. Is it so?

Let us start from the observed uneven matter distribution in the recent universe, and match it to the the uneven matter distribution in the cosmic microwave background (CMB). The baryon acoustic oscillations (BAO) make bumps into the uneven distribution. Let us match the bumps in recent matter and in the CMB.

Let us calculate time backward from the present time, and model matter distribution at earlier times. We use a standard ΛCMB model. When the universe has shrunk to one 1,100th of its size, the CMB has a temperature of 3,000 K, and the last scattering occurs.

The Hubble tension is that the BAO bumps in this calculation have a larger angular scale than in the measured and mapped CMB in the sky. That is, the "features" in the actual CMB map are smaller than our calculation backward in time predicts.

The early dark energy (EDE) hypothesis talks about what happened before the last scattering. The hypothesis does not help in any way in solving the Hubble tension. We have to check the literature. Can the EDE hypothesis be so much wrong?


Karsten Jedamzik et al. (2020) write that EDE cannot resolve the Hubble tension.





 
Above, θ* is the angular size of the sound horizon, that is, the angular size of "features" in the CMB. We use coordinates comoving with matter. Then r* is the spatial size of the sound horizon (in those coordinates), and D(z*) is the spatial distance to the last scattering.

Jedamzik et al. say that only the quantity

       Ωm h²

affects the spatial distance to the last scattering, in a flat ΛCDM model. There, Ωm is the matter fraction of the critical density, and h is the value of the Hubble constant now.

The authors state that the Dark Energy Survey and the Kilo-Degree Survey have set strict limits on Ωm h², so that any attempt to solve the Hubble tension through manipulating Ωm h² will fail.

Let us check what they say about manipulating r*. Suppose that we calculate the development of a ΛCDM model backward in time. The matter density variations of galaxy clusters eventually turn into variations of the CMB as the temperature rises.


Solving the Hubble tension with a very slow expansion during the first billion years of the universe: also the James Webb paradox of too mature galaxies is solved


The James Webb telescope has observed galaxies which in a standard ΛCDM model are only 500 million years old, but look much more mature, as if they were 1.5 billion years old.

Let us assume that the expansion during the first billion years of the universe was much slower than the ΛCDM says. We add an "extra" 1 billion years to the age of the universe, to the early phases of the universe. Adding that extra means that the universe expands much slower than in the ΛCDM model, during the first 1 billion years or so.

Then galaxies have much more time to mature. This solves the James Webb paradox.

During that slow expansion during the first 1 billion years, photons from the last scattering had time to move a "longer" path than in the ΛCDM model. Let us explain what "longer" here means. If we model the expanding universe with coordinates comoving with matter, then the ruler of the spatial metric (say, a ruler 1 meter long) shrinks as time progresses. The "length" of the path of the photons is measured in these comoving coordinates.

Since the photons moved a longer path before coming to our eyes, the features in CMB will have smaller angular diameters in our eyes. This resolves the Hubble tension.

Let us make a quantitative calculation. We assume a matter-dominated universe with a zero pressure. Then we can use newtonian mechanics to calculate the development of the universe backwards from the present time.


              "feature" size
               <----->
              ●        ●        ●       ●

              ●        ●        ●       ●
                             o ~~~~~~  last scattering
                             |   c = 1 / a
                            /\
                      observer


We set a = 1 as the current scale factor of the universe. In comoving coordinates, the speed of light

       c  ~  1 / a.

Let us look at ten "epochs" where

        a ≈ 1 / 2ⁿ,

0 ≤ n ≤ 10. The epoch 10 corresponds to the time of the last scattering, that is z ≈ 1,100.

In ΛCDM, the kinetic energy of a particle m in static (not co-moving coordinates) is very roughly

       1/2 m v²  ~  1 / a  =  2ⁿ.

That is, v ~ sqrt(2ⁿ).

This implies that the time the model spends in an epoch n is

       t  ~  (1 / 2ⁿ)  /  sqrt(2ⁿ).

The speed of light c ~ 1 / a = 2ⁿ. Light in that time propagates a distance in comoving coordinates:

       Dₙ  ~  2ⁿ  *  (1 / 2ⁿ)  *  (1 / sqrt(2ⁿ))

              = 1 / sqrt(2ⁿ).

We are interested in the distance

                10
       D  =  ∑ Dₙ
                n = 0

in comoving coordinates, which light travels from the last scattering to the eye of the observer. The value of s forms a decreasing geometric series for n. If the contribution of epoch D₀ is 1, then the sum of the series is

       D  ~  3.4.

In particular, the contribution of epoch D₁₀ is only 0.03. We see that we have to modify the physics drastically during epoch 10, if we want to increase D by 8%.

The first one billion years of the universe corresponds to a redshift z ~ 10. That is, the epochs 4, ..., 10 are relevant. The contribution D₄ = 0.25, and the the sum of D₄, ..., D₁₀ is 0.85. We see that if we modify physics substantially, but not drastically, during epochs 4, ..., 10, we can increase D by 8%.
















***  WORK IN PROGRESS  ***
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Thursday, March 26, 2026

de Broglie and Bohm: photon creation from a radio transmitter – the problem of new particles

Let us consider a simple semiclassical process which creates photons. Let us have two charged particles (charge Q) whose mass M is macroscopic. We let the particles pass close to each other, so that they send a radio wave with many photons.


                                                                    screen
                                                                          |
                                                                          |
                                                                          |
                              )   )   )   )   )  radio wave
           ● ---->              
          M, Q                 <---- ●
                                         M, Q


The radio wave will hit a screen far away, and we can determine the positions of individual photons.

We can calculate the shape of the radio wave classically, but we cannot calculate where the individual photons will hit the screen.

The process creates new particles. Is there a way to extend the de Broglie-Bohm model to this case?

Could this work: the photons "pre-exist" in the electric fields of the charges Q, and are "freed" in the scattering process? Then all the particles could have a hidden variable already at the outset of the experiment, and would sail in a deterministic way to the screen – just as in the double-slit experiment.

The outgoing photons have an uncountable number of possible momenta p. Even if we would have pre-existing photons in the field of Q, how would these obtain the required p?


Time asymmetry in the process


The initial configuration of two charges Q flying at each other is quite different from the final configuration where we have the wave function for the two charges Q plus many alternative configurations of photons. The entropy of the system grows, and there are more degrees of freedom in the final configuration.

What if there would be no growth of entropy? Then the initial and the final configurations should look similar. Maybe the number of the degrees of freedom does not grow any more?

What about inverting the direction of time? Then the degrees of freedom will shrink as time progresses.


Absorption of a photon by a hydrogen atom


An excited hydrogen atom can decay and send an individual photon. The atom can absorb the required momentum from the reaction as it launches the photon.

What about the reverse process? A wave function of the photon contains many cases in which the photon does not hit the atom at all. The photon passes the atom.

Can we somehow add more "worlds" in such a way that the the graph of possible worlds would look more time-symmetric?

In the case of a hydrogen atom absorbing a single photon, the degrees of freedom only shrink in the case of the absorption. In other cases, there is no change. A "wave function collapse" only happens in one branch of many-worlds.


Ignore photons and only consider (charged) particles which existed at the outset?


The de Broglie-Bohm approach might work if we only had undestructible particles. Let us have N such particles. Let us have a foliation of spacetime such that spacelike folios define "simultaneousness". Then we could replace the N particles with one particle moving in a space with 3 N spatial dimensions.

But we have a problem already with electrons: pair production and pair annihilation. There are no undestructible particles.


Give up determinism?


Albert Einstein said that God does not play dice. That is, the "next", measured, state of a physical system should follow deterministically from the "previous" state.

Since simultaneousness is not well defined in relativity, we maybe should say this: what happens at some point in time and space is determined by the light cone which precedes that point.

Why not require determinism also in the "initial" states of the particles? Why can the initial states be random? Does God play dice, after all?

Why is the time dimension different from spatial dimensions?


Defining a "branch" in many-worlds


Let us have a human living in spacetime. Most of the massive elementary particles in his brain survive for a long time. They are not annihilated. However, quite a few particles are replaced with other similar particles over his lifetime. How could we define the "branch" in many-worlds, in which he is living?

The brain consists mainly of electrons and of nuclei. These particles are, in most cases, conserved over 100 years.

Then we can apply the de Broglie-Bohm model to these particles, and obtain well defined "branches" of their configurations.

We do not try to define a branch in a more general sense: we do not define, e.g., which photons on Mars belong to his branch.

What if two photons collide and form an electron and a positron, and the electron subsequently is used as a new building block in his brain?

Could we simply accept a model in which the branch of this brain chooses "by random" the initial state of the electron, as it is integrated into the brain?


A new problem: which paths of separate particles are in the "same" branch of many-worlds?


Suppose that two electrons are created in pair productions, and these two electrons are integrated to the brain of the observer.

We can use the de Broglie-Bohm model to describe the paths of the two electrons. But what paths are in the "same" branch of many-worlds?

This might be the solution: we simply choose one de Broglie-Bohm path for each electron. Then the combination of those paths is taken to be in the "same" branch as the brain of the observer.

When a new particle is included into the brain, the brain performs a "quantum jump" to include a certain path of that particle.


Do we need precisely defined branches at all?


Definition of a branch does not change anything in the wave function. In the de Broglie-Bohm model, the path of a particle does not affect the wave function in any way.

However, picking a certain path may define the "real world".

Let us have an observer. Maybe he lives in the "real world", and the copies of him in other branches are just "empty" wave functions?

This does not sound natural. Why should a conscious observer always exist in the "real world" and not in the "empty" branches?

In the de Broglie-Bohm model, the empty branches do physically exist, too. They are just different branches of the wave function.

Maybe it is necessary to drop the concept of the "real world" and treat all the branches equal. This means that Schrödinger's cat is both dead and alive at the same time.


Do we have a reasonable definition of a "branch" now? A robot


Instead of an observer we may consider a computer or a robot which possesses a camera and other sensors. It also has arms for manipulating external objects.

Above we suggested that we can use the de Broglie-Bohm model to define a "branch" of many-worlds, such that an instance of the robot lives in that branch.

The robot will encounter various random (or pseudorandom) quantum phenomena, and adjust the contents of its memory accordingly. Its camera may register random photons hitting its CCD sensor. 

The robot is an almost classical machine. It probably has relatively exactly definable states, and it in most cases functions deterministically.

Is the de Broglie-Bohm model an awkward way to define a branch in this case? Why not use a traditional Copenhagen model where the next state of the machine depends probabilistically on the previous state?

Assigning hidden variables is awkward?


The observing subject


Suppose that we have a reasonable way to define a branch for the robot. The computer in the robot, in that branch, will have a similar world-view as a human observer. It will remember its history. In almost every branch, the robot notices that physical processes in its history behaved according to the probabilistic rules of quantum mechanics.

According to Rene Descartes, cogito ergo sum. That is, I as an observing subject do exist.

My subject might be attached to one branch of a robot, my physical body, or it might "jump" from branch to branch. I would not notice anything if my subject jumps around the branches of many worlds.

It sounds strange if I alternate between different branches. I very much feel that I am in just one branch. This paradox is similar to another paradox about time. In what sense is now different from moments in the past or in the future?


Conclusions


It would be nice if we could define a single "branch" in the many-worlds interpretation. Then we could imagine that a human observer lives in that branch.

The de Broglie-Bohm model gives us a way to define a branch for a robot. Simply plot the paths of the nuclei and the electrons in the mechanism of the robot.

However, that model is not very natural. Why should the branch exactly follow the strange paths of de Broglie-Bohm model?

Maybe a more natural way to define a branch is more fuzzy: it is the collection of those states of the robot which are "similar". For example, we may require that the bits in the memory of the robot have identical values in the collection of the states. This would be a "fuzzy branch".

We still have the question if an "observing subject" is somehow attached to a certain branch. This is a metaphysical problem which is hard to clarify and solve. What makes a certain branch "real" for the subject? How would the subject know if it jumps from branch to branch?

A similar metaphysical problem is in what sense is the world "now" more real than the past or the future. We have no answer.
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Thursday, March 19, 2026

de Broglie-Bohm model and the many-worlds interpretation of quantum mechanics

In the Copenhagen interpretation, the wave function collapses. That destroys information and spoils the time symmetry of the laws of physics.

The many-worlds interpretation would be better. If my "subject", that is, I as an observer, is located in a certain "branch" of possible world histories, then no collapse is required. It just happens that I as a subject am in a certain branch. Then all the other laws of physics can be time-symmetric. If we further assume that the path of the subject through the tree is determined by a time-symmetric method, then we have a model which is completely time-symmetric, and does not destroy information. This would be desirable.


A freely readable .pdf file:


The de Broglie-Bohm model might provide us with a deterministic path for the observing subject, if we can extend it to cover the creation of new particles. The classic Bohm model is limited in that that it only covers a fixed number of particles, and uses the non-relativistic Schrödinger equation.


What constitutes a branch in the many-worlds interpretation?




The definition of a branch is a question which comes up regularly on the Physics Stack Exchange.

Suppose that a single photon hits a charge-coupled device (CCD), and starts a cascade which registers the location which the photon hit. A quantum mechanical object photon "decoheres" into a macroscopic object or into a macroscopic process.

Many people seem to believe that this explains what a branch means in many-worlds. It is a local process of decoherence. However, this definition of a branch is not exact, nor mathematical. If the photon happens to hit the CCD at a slightly (1 micrometer) different location, is that another branch? Do branches actually form a continuum?

The de Broglie-Bohm model might give us a mathematical definition of a branch: it simply is the time development for fixed initial hidden variable values.


We might define a branch with certain initial hidden variable values the "true state of the material world". Albert Einstein wrote that "God does not play dice". The branch would be deterministic, and Einstein would be happy. Does this approach lead to any contradictions?


de Broglie-Bohm as a flow of probability?




        |      |                        |
        |      |                                )        )
        |      |                        |
        |      |                                )        )
        |      |                        |

           --->
      incoming             double   diffracted wave
         wave                    slit       and interference


Let us think about the double-slit experiment. The incoming wave can be understood as a probability distribution, where the probability density is the squarw of the wave function value.

We may imagine a flow of probability, a flux, which takes a "probability parcel" forward from the double slit. That would give us a deterministic "path" for an incoming particle? As is well known, a de Broglie-Bohm particle will move along a strange, curvy path.

In a Feynman path integral, also fully unrealistic paths of a particle are calculated and added. If a de Broglie-Bohm path can be designated as the "true world", what about an unrealistic Feynman path?


Werner Heisenberg writes in his 1958 book Physics and philosophy that the de Broglie-Bohm model has problems handling stationary states. The flow of probability is zero in a typical stationary state, say, for a particle in a box. That suggests that a particle would remain in a fixed place.

A particle in a box, intuitively, has two flows of probability, to opposite directions. The sum of these flows is zero.

    
             wave packet
     <--    ~~~~~~~~~~
                       ~~~~~~~~~~    -->
                       wave packet


Let us have a very long wave packet which describes a single particle. We use a beam splitter to divide the packet into two wave packets and let the packets collide so that they travel to opposite directions.

The flow of probability in the overlapping part of the two packets is zero. But the natural many-worlds interpretation for the process is that a single particle is moving either to the left or to the right in the diagram? 

No. In the many-worlds interpretation, "branching" means a process where the waves for various alternatives will not interfere much. The branching has not happened yet when the wave packets overlap.


Do other branches "exist"?


If we designate one de Broglie-Bohm branch as the "true world", the other branches still exist, in the sense that they interfere with the true world, and affect the probabilities how the true world develops in time. This is quite different from, say, newtonian mechanics.

Certain people have criticized the fact that all branches exist, also those that are not the "true world". But we cannot help it: if we prune some branches, we lose information, and the model no longer corresponds to quantum mechanics.


Generalizing to relativistic wave functions and creation and destruction of particles: quantum field theory


The de Broglie-Bohm model is based on the Schrödinger equation, which is nonrelativistic. A generalization to relativistic particle physics requires that we develop a de Broglie-Bohm model for quantum field theory.

Quantum field theory, in its present form, contains dubious methods like regularization and renormalization. It is unlikely that we can develop a satisfactory deterministic model for quantum field theory unless we can remove regularization and renormalization. In fact, a de Broglie-Bohm model might help us to solve the problems.


Hrvoje Nikolić (2009) describes a model where particles can be created and destroyed.














Nikolić assumes that a particle can have a finite lifetime. In the diagram, x⁰ is the time coordinate. "Virtual particles" pop up and disappear in a very short time. They are denoted with dots.

For the model to be deterministic, the position of those dots, (x⁰, x¹) should depend on the initial conditions (data) at some earlier time. Does it?

We can, of course, formally define that the locations of the dots are decided in the initial data, but that looks ugly.


Flow of probability when new particles are created


                        ~~~~~~~~~~~  real photon
                      /
       e- ----------------------------------
                           |
                           | virtual photon
       e- ----------------------------------


Let us consider a collision of two electrons. Photons and pairs can be created. We can draw Feynman diagrams and calculate the probability of each outcome. Above is a diagram for bremsstrahlung.

In the de Broglie-Bohm model of the double-slit experiment, there is a beautiful way to define the "path" of the particle, along the flow of probability in the Schrödinger equation. The path is meandering, but it is defined in a beautiful way.

But for a Feynman diagram, there is no such beautiful way. The real photon which appears above "breaks" the system into more degrees of freedom. If we have a probability flow in 6 spatial dimensions, there is no obvious way to make it to branch into a flow in 9 dimensions. What would decide the path for an arbitrary initial configuration of two electrons? How does the path suddenly branch to more dimensions?

In our blog we have several times written about the problem of "breaking" into more degrees of freedom. This seems to be associated with regularization and renormalization problems for Feynman diagrams.

Let us assign an arbitrary (irrational) real number r for the initial two electrons. We could design an ad hoc way to reproduce the correct Feynman probability distribution from the infinite number of decimals in r, and obtain a deterministic model. But that would be a very ugly solution.


Schrödinger's cat


The de Broglie-Bohm model for the Schrödinger equation solves a paradox associated with Schrödinger's cat, for the treatment with the Schrödinger equation.

The cat always is either dead or alive, in the "true world" determined by the hidden variables. Opening the box does not change anything. The cat does not suddenly "collapse" into either state.

Let us assume that we find the cat alive.

What remains of the paradox is the "empty" branch in which the cat is dead. The branch does exist in the de Broglie-Bohm model, though that branch is not "occupied" by the values of the hidden variables.

Schrödinger's cat involves a radioactive decay process which decides if the poison is released inside the box. Such a decay must be treated in quantum field theory, since new particles are created. And the de Broglie-Bohm model has no beautiful extension there. Thus, the paradox remains.


Conclusions


The de Broglie-Bohm model is a beautiful way to introduce determinism into the Schrödinger equation. We can define in a beautiful way what is a branch in the many-worlds interpretation. We avoid the "measurement problem" and many paradoxes in quantum mechanics.

But extending the de Broglie-Bohm model into quantum field theory depends on if we can find a beautiful model quantum field theory. Presumably, divergences, regularization, and renormalization should be removed. The problem of breaking into more degrees freedom may be a mathematical problem. Can we solve it?
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Tuesday, December 30, 2025

Dark energy from retardation of gravity?

In our previous blog post we were able to derive an estimate for retarded gravity when a supernova core collapses to form a neutron star. We introduced a natural hypothesis which yields a numerical value.


Energy of the retarded gravity field in a collapse of a mass shell: the energy is significant


In the case of a collapsing sphere, the retarded field steals energy from the kinetic energy of the collapse.

Suppose that retardation keeps time running at a fraction h / 2 faster at the center of the collapsing shell than it would run if the signal speed were infinite. Then most of the retardation energy is stored in the interaction of the mass near the center, M, and the retarded field. The energy is

       ~ h / 2  *  M c²,

that is, linear in h.

If there is no mass near the center, the energy of the retarded field might be similar to the analogous electric field. Another way to estimate the energy of the gravity field is to calculate the pseudotensor. Generally, the energy density is quadratic in h.


Mark Hindmarsh (2018) gives the following approximate formula for the energy density of gravitational waves:








The dot means the derivative over time. The brackets ⟨ ... ⟩ denote an average over an entire wave. The components hij are differences from the flat Minkowski metric.

Let us calculate a very rough estimate for the collapse into a neutron star. We assume that the mass 3 * 10³⁰ kg forms a thin shell. What is the energy of the retarded gravity field inside the shell?

In the previous blog post we calculated that h₀₀ ≈ 0.13 at the center, that is, time runs 6.4% faster at the center than for a flat metric.

Let us assume that the metric becomes flat when the signal to the center reaches from the radius r = 10 km of the neutron star which just formed. Then

       dh₀₀ / dt  =  h₀₀ c / r.

The energy density of the retarded gravity field is something like

       egw  =  c² / (32 π G)  *  h₀₀² c² / r²

               =  c⁴ / (32 π G)  *  h₀₀² / r²,

and the total energy

       E  =  1/24  *  c⁴ / G  *  h₀₀²  *  r

            = 1/24  *  8 * 10³³  *  1.5 * 10¹⁰

                * 0.13²  *  10⁴ J
     
            = 8 * 10⁴⁴ J.

This is interesting. The energy is of the same order of magnitude as we calculated in the previous blog post (13 versus 8) from the interaction of the mass near the center and the retarded gravity field (gravity potential).

Maybe the radial metric g₁₁ is stretched, too? In that case the energy density ggw must be doubled.

The energy of the pure retarded gravity field, without any mass in it, is significant!

What is the energy of the retarded field if we calculate it from the analogous electric field?

In the force formula,

       G  ~  1 / (4 π ε₀).

The energy density of a gravity field is then

       e  =   1 / (8 π G)  *  Eg²,

where Eg is the strength of the gravity field (force / mass).

If the potential is h₀₀ / 2 * c² over a distance of r = 10 km, the field strength is

       Eg  =  h₀₀ / 2  *  c² / r.

The energy density of the retarded gravity field is then

       e  =  c⁴ / (32 π G)  *  h₀₀² / r²

           = egw.

We see that the energy density e agrees with the one which we calculated from the formula for gravitational waves!


Does the energy of the retarded gravity field approach the total released energy in a collapse into a black hole?


The gravitational energy released in a collapse into a neutron star of a mass 3 * 10³⁰ kg is roughly 300 * 10⁴⁴ J. The energy in the retarded gravity field is something like 13 * 10⁴⁴ J – not a very large portion of the total energy.

What if the collapse would go further, so that the system becomes a black hole? Could the energy of the retarded field approach the total energy released in the collapse?

As the shell approaches the Schwarzschild radius r = 4.5 km, its speed approaches c.


               signal
               ----------------------------------------------->

               --> 0.4 c               --> c
              |                           |                          ×
       r = 10 km              r = 4.5 km           center
  
       potential              potential
       -0.2 c²                    -c²


Let the shell at r = 10 km send a signal to the center. When the signal arrives, the shell is already close to the Schwarzschild radius r = 4.5 km. But the observer at the center thinks that the radius is 6 km.


The potential difference at the center versus close to the shell is then ~ 0.5 c². Then 

       h₀₀ / 2  *  c²  =  0.5 c²,

or h₀₀ = 1. The energy of the retarded gravity field is

       E  =  1/24 * c⁴ / G * h₀₀² * r.

The energy is (1 / 0.13)² * 0.45 = 30 times that which we calculated in the previous section, or 240 * 10⁴⁴ J.

The potential energy released is 

      M c²  =  3 * 10³⁰  *  9 * 10¹⁶ J

                =  2700 * 10⁴⁴ J.

Our calculation is extremely crude. It shows that the energy of the retarded gravity field approaches a significant portion of the released potential energy when a shell collapses into a black hole.

This suggests that in the reverse process, an explosion, the retarded gravity field might boost the expansion speed greatly, which might explain "dark energy" in the universe.


An explosion of a mass shell


Can a retarded gravity field release energy enough, so that it could accelerate the expansion?

In a collapse, some of the energy goes to the retarded field, so that kinetic energy grows slower. But kinetic energy does not decrease at any point.

Let us then imagine an explosion of a shell. As the shell expands, gravity slows down its expansion. A clock at the center "thinks" that the shell has expanded more than it actually has. The clock ticks faster than it would if the speed of signals were infinite.

Let us assume that the speed of the expansion approaches some constant value v > 0 as the radius r becomes very large.


                                              signal
                                             -------------------->

 0.4 c <--                        <-- c
              |                            |                       ×
           r = 10 km           rs = 4.5 km       center

           potential           potential
           -0.2 c²                -c²


Above we have the start of an expansion from an almost-black-hole. When the center receives the signal, a clock there "deduces" that the shell is at r = 9 km. But the velocity of the shell has already fallen substantially. Let us guess that the shell is at r = 8 km.

In the Schwarzschild metric, the potential at a radius r is

       (-1  +  sqrt(1  -  rs / r))  *  c².

The potential at 9 km is -0.29 c², and at 8 km it is -0.34 c². Then h₀₀ = 0.1.

Let us calculate the energy of the retarded gravity field:

       E = 1/24 * c⁴ / G * h₀₀² * r

           = 1/24  *  81 * 10³²  *  1.5 * 10¹⁰

              * 1² / 10²  *  8 * 10³

           = 16 * 10⁴⁴ J.

The negative potential is -900 * 10⁴⁴ J at r = 8 km. The energy of the retarded field is not very large compared to this.

Is there any reason why we should assume that the potential at the center is very different from -0.29 c²? Maybe it should be zero?


A cosmological model is uniform: the retarded gravity field is visible only in "static" coordinates?


In the collapse to a neutron star, we claimed that the retarded gravity field holds a substantial energy. Our claim is based on the potential difference between the mass shell center and close to the shell.

But in the universe, no such potential differences seem to be present. However, the equipotential state happens in the comoving coordinates. In "static" coordinates, there is a potential difference. That difference slows down the expansion of the universe.

If dark energy is in the retarded gravity field, then that energy is visible in static coordinates.

Let us assume that the universe is a large uniform expanding ball, embedded in an asymptotically Minkowski space.

Let us assume that gravity behaves, for some reason, "uniformly" in comoving coordinates, so that the ball expands uniformly, just like in cosmological models.

We probably can study the center of the ball in such a way that we ignore other parts. We only look at the "observable universe" whose radius is something like 30 billion light-years.

Then our considerations above about an expanding mass shell are relevant. The main difference is that the shell expands at almost the speed of light, and its radius is huge.

       E = 1/24 * c⁴ / G * h₀₀² * r.

The energy of the retarded field is linear in r for an almost-black-hole. The mass of a black hole is linear in r.

The mass inside a radius r is proportional to r³ in a cosmological model.


Can the retarded gravity field energy be released in a cascade?


Is there a reason why almost all the energy in the retarded gravity field should be released just now when the mass density of the universe is 0.3 times the "critical density"?

Yes: if the expansion no longer slows down, then all the energy in the retarded field must be released. Recall that the energy is there only because the expansion was slowing down. If the expansion slows down enough, it may start a cascade in which all the energy is released from the retarded field.

This would mean that the expansion of the universe may accelerate a little now, and then start slowing down again, as predicted by cosmological models. In the long run, the expansion, on the average, obeys cosmological models.

Let us try to estimate if the cascade happens in the expansion.


                           ^  bulge
                           |
      ____                                ___
              \         _____         /
                 \ • /            \ • /
            v <--                    --> v
       weight                     weight


Let us first consider a rubber membrane model. A ring of weights expands, and the membrane between them bulges upward as the velocity of the weights, v, slows down.

Could it be that no cascade happens?

If the energy stored in the bulge is just a few percents of the negative potential energy, then probably no cascade occurs.

We need better models and ideas to explain dark energy.


The length contraction of the gravity field of a mass moving at almost c


            \     /
              \ /
         m  • -->  ≈ c
              / \
            /     \
       "squeezed"
        gravity field


Suppose that a mass shell expands almost at the speed c. If the gravity field behaves like the electric Coulomb field, then the gravity field is very much "squeezed" in the direction of the movement of m. The field of m is not much felt inside the shell. This would mean that the a clock at the center of the shell would not be much slowed down. 

Does this make sense? If the shell is initially static, and the shell consists of photons, what happens?

The potential initially corresponds to a static shell. Then there is a pressure shock, as the shell is shot outward at the speed of light.

The center "knows" that the shell has moved far away, and can "calculate" the potential and adjust the clock rate accordingly.

If we shoot a photon shell at the speed of light toward the center, then the center does not "know" that it is at a lower potential as the shell approaches. In this case, a clock at the center should run at almost the rate of outer space. In this case, the gravity field behaves as it would be "squeezed", and not affect the center.

In the contracting case, the retarded gravity field can hold a lot of energy.

If the shell contracts at a velocity slightly less than c, an observer will receive a warning of the approaching shell wall early on.

Could it be that the speed of light inside the shell remains faster than at the shell? In that case, an observer at the center might see the contraction slowing down as the shell wall approaches?

We deduce the expansion speed of the universe from the redshift. In the case of a contracting universe, we look at the blueshift. The blueshift can, indeed, decrease if the clock rate at the shell wall slows down.


The redshift inside an expanding mass shell


                            ^ bulge
                            |
                            o  observer
      ____                                 ___
              \          _____          /
                 \ • /              \ • /
            v <--                    --> v
            weight           weight


Above is the rubber membrane model. Let us have an observer at the center, on the "bulge".

He sees a redshift in the receding shell. The redshift depends both on the velocity v and the rate of the clock on the bulge. If the bulge grows taller, then he sees a larger redshift.


The energy of the distorted radial metric in an expansion of a mass shell: it is 6 times the energy of the metric if time?


Thus far, we in this blog have for the retarded gravity field calculated the presumed energy of the distorted metric of time inside the sphere.

However, most of the energy in a gravitational wave seems to be in the distorted spatial metric. Let us try to estimate that energy heuristically.


      shell wall
         <-- |                    ×                      | -->
                                   clock rate
                                   d faster


Suppose that time flows a fraction d faster at the center of an expanding shell than close to the shell.

If we imitate the Schwarzschild metric, then radial distances are stretched by that same fraction 

       d  = h₀₀ / 2.

Let us put an extremely rigid material inside the shell. When the retarded gravity field tries to stretch the material, a negative radial pressure forms.

The negative pressure tries to flatten the spatial metric.


The formula of the Komar mass gives us a guess of how much pressure is needed to flatten the metric. In the Komar formula, the  pressure in each spatial direction x, y, z is counted as an "energy density".

A positive mass-energy M c² stretches the radial metric. It can be "compensated" with an "equal" negative pressure.

In our example in a preceding section, h₀₀ = 0.1 and d = 0.05.

How much positive mass density do we need to create the d = 5% potential difference? Let ρ be the hypothetical energy density inside the shell. The potential difference between the center and the shell is then

       d c²  =  1/2 G M / r

                = 1/2 G * 4/3 π * 1 / c² * ρ r³ / r

                = 2/3 π G / c²  *  ρ r²
  =>
       ρ  =  3/2 * 1 / π * c⁴ / G  * 1 / r²  *  d.

The energy which we could hypothetically extract from a pressure ρ and a stretched metric by d is

       W  =  1/2 ρ d * 4/3 π r³

             = c⁴ / G  *  d² r

             = 1/4 c⁴ / G  *  h₀₀²  *  r.

The energy is 6X the energy which we got for the metric of time, g₀₀. In the expansion case of a preceding section, the energy thus is

       96 * 10⁴⁴ J.

The negative potential energy is -900 * 10⁴⁴ J.

The energy of the retarded field is probably too small to start a cascade which would accelerate the expansion.

However, for the collapse into a neutron star, this is big news. Above calculated that the energy of the retarded g₀₀ is something like 8 * 10⁴⁴ J, which is a bit low to explain the supernova explosion. A 7-fold energy 56 * 10⁴⁴ J explains it better.


But is the radial metric stretched for a retarded gravity field? Probably not


Retardation causes an observer at the center to miscalculate the radius of the mass shell. That should his clock to tick at a "wrong" rate.

But the spatial metric does not change if we slowly lower the shell down. Thus, retardation should have no effect on the spatial metric.

It seems likely that retardation does not affect the spatial metric.


The energy of the retarded gravity field in a Big Bounce


The observable universe came out from an explosion or a Big Bounce. Could it be that a lot of energy at that time was uploaded to the gravity field, and that energy only is released now when the mass density falls below the critical density?


                        pit
       ____                          ____  membrane
               • ------___------ •
         <-- a                     a -->
               mass       mass


In the first phase of the expansion, the expansion is accelerated. The retarded gravity field lags behind. In the rubber membrane model of an expansion of a mass shell, there is a "pit".

As gravity slows down the expansion, the pit becomes a bulge. What happens to the energy of the pit in the transition? If it becomes kinetic energy of the masses, then they might be accelerated, for a little while.


Dark energy is the expected result from any large explosion?


Let us assume that an unknown law of nature produces a large explosion. Retardation is present. Then it is expected that the gravity field of the system will contain a lot of energy which comes from retardation. The gravity field does not have time to settle down to its "lowest energy state". We can claim that "dark energy" is the expected result of such an explosion.

Since the energy in the retarded gravity field cannot escape as longitudinal gravitational waves from a spherically symmetric explosion, the energy has to be absorbed by the matter. The absorption process must be spherically symmetric, and must increase the energy of the matter. The natural way is to increase the radial velocity of the matter.

When the mass density of the universe falls a lot below the "critical density", we can regard that as the end stage of the explosion. Then the energy in the retarded gravity field must be absorbed by the matter. This explains why the expansion of the universe is accelerating right now.

When the energy has been absorbed, we can expect the universe to expand at an almost fixed rate ever after. Gravity will slow down the expansion only a little.

Our hypothesis has a serious weakness, though: it does not give any numerical prediction of how much dark energy we should observe. The hypothesis is purely qualitative.

The energy in the retarded gravity field gravitates, too. How does that affect the expansion of the universe? The mass density of the universe is significantly larger than the amount of matter and dark matter.


Dark energy implies that the metric of time is "upwards convex" close to Earth


By "close" we mean distances less than 5 billion light-years. Dark energy means that "close" galaxies are accelerating away from us – even though gravity should decelerate them.

Matter tends to be accelerated toward a place where the gravity potential is lower, or where the metric of time, |g₀₀|, is smaller in absolute terms.

If the potential is "sphere-like", so that its curvature is the same to all directions, we will observe all nearby objects accelerating uniformly away from Earth.


                  bulge
       -------                 -------  rubber membrane
               \•__----__•/

        v <--              --> v
       weight      weight


Let us consider a rubber membrane model of gravity. Let the weights recede from the center. The weights slow down, because of gravity. Then the membrane at the center of the ring of weights will form a "bulge" upward. The potential will be convex upward. But why would the potential be spherical?

Another problem: if we have a uniform, exploding dust ball, why can we assume that only its edges have a lot of mass, so that it is like the expanding ring of weights in the rubber model? The edge moves at almost the speed of light and has a lot of kinetic energy. This suggests that almost all the mass would be at the edges?

If the interior of the dust ball has only a little mass, then the upward bulge in the rubber membrane model can form. The interior will feel "dark energy".

Another problem: if the edge of the ball moves at almost the speed of light, then gravity will not decelerate it a lot, since it keeps moving at almost the speed of light. Why would a bulge upward form in the rubber model in that case?


The matter density of the universe has recently grown "thin" relative to the "critical" density – squeezed gravity fields


One of the implications is that we can use almost newtonian gravity to describe the gravity field locally. Maybe the effects of gravity are linear and can be summed for each mass element dm?


                                   squeezed
                                 gravity field

                                       \   |   /
                                         \ | /   ≈ c
                    ×                     • --->  
                                         / | \
             observer                   mass element dm


Let us consider a shell of mass expanding rapidly, at almost the speed of light.

The gravity field of each element dm is "squeezed" in the direction of the movement of dm. Does this mean that a clock at the center of the shell will tick faster than a clock close to the shell? If yes, then an observer inside the shell will feel a force which pulls him toward the shell. This could be the "dark energy"?

If we compare this to an expanding shell of electric charge, the squeezing of the electric field does not matter because the field is zero inside the shell, and we do not have a clock which would be affected by the electric potential.

If we have an expanding dust ball, the kinetic energy of its edges causes most of the gravitating mass-energy to be at the edges. Observers at a "typical" position in the dust ball will see an "upward convex" gravity potential.

But why would dark energy only show up after the universe is 8 billion years old and its density has dropped significantly below the critical density?


Conclusions


Let us close this blog post. We were not able to find a mathematical model for the "dark energy" caused by retardation, yet. We will return to this later.
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Friday, December 5, 2025

Type II supernovae explode because general relativity is wrong?

There is a major open problem in how type II supernovae are able to explode when their core collapses.






















In the photo (middle right), we have the bright type II supernova SN 1987A in the Large Magellanic Cloud. The photo was taken with the ESO Schmidt Telescope.



The leading hypothesis of a type II supernova is that the collapse creates a huge number of high-energy neutrinos, and an unknown mechanism makes the neutrinos to interact with the outer layers of the star.

The interaction then blows the outer layers to space.

Another hypothesis is that some kind of a shock blows the outer layers away.


H.-Th. Janka et al. (2006) discuss various models.


Is there a connection to dark energy?


In this blog we have remarked that the only large collapse/explosion which we can monitor in detail is the expansion of the universe, and it does not follow the FLRW model derived from the Einstein field equations.

Dark energy spoils things. The expansion seems to be speeding up, though it should slow down.

Similarly, the other example that we have of a large collapse/expansion, the type II supernova, fails to follow the path predicted by general relativity and particle physics.

In our blog we in May 2024 tentatively proved that the Einstein equations do not have a solution for a collapse or expansion. This opens the possibility that the hypothetical correct theory of gravity could explain dark energy. It might also explain the type II supernova explosion.


A longitudinal gravitational wave?


Let us model gravity with the traditional tense rubber membrane, on which we have weights resting.

If a set of weights "collapses" toward each other, and they collide, a longitudinal, circular wave will form, and carry some energy out.

If longitudinal waves are banned, like in electromagnetism, the no wave can form in a spherically symmetric collapse.


          --------             --------    rubber sheet
                     \         /
            v <--   • --- •   --> v

                     ring of 
                     weights


Let us assume that the initial state is an expanding ring of weights sliding on the rubber sheet.


                     bulge
                      ____
         ------ • --       -- • ------   rubber sheet
        v <---                 ---> v


In this blog we suggested on May 22, 2025 that retardation causes a "bulge" which can explain dark energy. Could it be that this phenomenon necessarily creates longitudinal gravitational waves?

Above we have a model of the expanding universe. The rubber membrane has kinetic energy upward. This kinetic energy should go to acceleration of the ring of weights outward. The kinetic energy of the membrane explains why the expansion speeds up.

If the bulge rises above the rubber sheet (membrane) level, we are in trouble. That would enable superluminal communication. We cannot allow that. Could it be that the expansion of the universe must be fine-tuned in such a way that all the kinetic energy of the membrane is spent to accelerate the masses outward?

If we run the process backward, the bulge never rises above the membrane level, since it starts to from the membrane level and starts descending. In the reversed process, there is no superluminal problem.


         -----                ----- tense rubber membrane
                 \          /
                   ••••••  weights
           --->   stop  <---


But what happens if the reverse process (collapse) suddenly stops and a neutron star is formed? Where does the kinetic energy of the membrane moving downward go?

If we would allow longitudinal waves, it would go to them.

What about a "transactional" model? Retardation must eventually be "settled" to ensure energy conservation. The energy could go to the kinetic energy of the weights. But what law would decide who gains the energy and how?

When matter is around, gravitons couple to it, and probably gain a mass. An analogue for electromagnetism is plasma. In plasma, longitudinal waves do happen. This implies that longitudinal gravitational waves probably are present inside a collapsing mass, and probably also close to it.

The plasma analogue may help us. There are no longitudinal electromagnetic waves in empty space. If we have an exploding ball of electrons, then longitudinal waves can happen inside the ball, but they cannot escape outside the ball.

Could it be that the outer layers of a star must absorb the outgoing longitudinal gravitational waves, and that causes the explosion outward?


             )       )       )      •   •   •   •  

      longitudinal         masses
      wave
      (stretched x metric)

      ---------------> x


Let us analyze a longitudinal wave which would periodically stretch distances in, say, the x direction. Suppose that there is some density of mass in space.

Is there a mechanism which would allow the masses to absorb and re-emit the longitudinal wave? A plasma, of course, temporarily absorbs some of the energy of an electromagnetic wave. But if the masses are sitting still, they will not gain any kinetic energy in the longitudinal wave. This means that masses will not allow longitudinal gravitational waves to propagate.

A plasma contains both positive and negative electric charges. But a mass density only contains positive gravity charges. These configurations are very different.

We do not see a mechanism which would allow longitudinal waves in gravity.


The energy of the curved metric inside a collapsing mass shell


Let us have a massive mass shell which collapses. Because of retardation, clocks inside the shell will tick "surprisingly fast". This because they do not yet know that the gravity potential is low.

That is, the metric inside the sphere will not be flat. People often assume that it would be flat, even if the collapse is a dynamic process, and clocks inside cannot "know" what the gravity potential is supposed to be right now.

Mass tends to move to the direction where clocks tick slower. In this case, mass would move toward the shell. Since the metric inside the shell can do work, it must "contain" energy.

Note that this breaks Gauss's law. Gauss's law would imply that there is no force inside the shell. Also, this breaks the Einstein field equations. Einstein says that the metric inside the shell should be flat. But it is not flat because a clock at the center ticks faster than clocks close to the shell.

What happens if the mass shell suddenly stops contracting? The metric inside the shell contains energy. Where does that energy go? It cannot escape as longitudinal gravitational radiation. Because of spherical symmetry, it cannot escape as transverse radiation either.

Since energy goes to the gravity field inside the shell, the shell will contract somewhat slower than in Einstein gravity.

Let us analyze a simpler configuration.


                 ● ---> a                a <--- ●
                M                                   M


Two masses M are accelerated toward each other, and quickly stopped. What does the retarded gravity field between them do to the masses?

The stopping force F gains energy from the system. If the stopping is done slowly, then the deformation energy of the gravity fields should make F to gain more energy.

Another way to interpret this is to consider the gravity field inside the sphere as a "spring" which is deformed by the accelerating collapse of the sphere. In the spring interpretation, the energy of the spring will push the sphere outward, when the collapse stops. In this model, there will be a shock which will make the sphere to expand. If the shock outward is passed to the outer layers of the collapsed star, that can explain the supernova explosion.

If the core collapse ends up as a neutron star, then there necessarily is some kind of a shock as the pressure outward wins the gravity. There is some kind of a "bounce back".

Our hypothetical "gravity shock" adds to this. But is this enough to explain the explosion?

The entropy of the deformed gravity field is low. The energy in it does not easily end up as heat. The collapse to a neutron star generates a lot of heat to the neutrons.


Energies involved in a supernova explosion



Christian David Ott (2009) writes about the "supernova problem". The gravitational energy released in a neutron star collapse is

       ~ 300 Bethe = 300 * 10⁴⁴ J.

The initial bounce back of the neutron star only has ~ 1.2 Bethe of kinetic energy. Almost all energy escapes in neutrinos as the neutron star cools. The collision of the matter, as it forms the neutron star, is almost perfectly inelastic.

But the energy required to throw the outer layers into space is ~ 12 Bethe. The energy of the bounce, 1.2 Bethe, is too small.

We have to calculate what is the energy of the retarded gravity field in the collapse.

The collapsing body is a sphere, not a shell. We have to figure out what happens in the case of a sphere.

How much is the energy of the retarded gravity field?

Hypothesis. The energy stored in the retarded gravity field may be approximately the "extra energy" of the mass inside the forming neutron star, because the mass "does not know yet" that its gravity potential has fallen.


How much does retardation distort the potential at the center?

The radius of a neutron star ~ 10 km.

The velocity of the surface at the bounce back, which happens at r = 10 km, is ~ 0.4 c.

We picked the value 0.4 c from some literature.

We assume a standard retardation rule: an observer "sees" the field as if the source of the field would have moved at a constant velocity. The surface of the collapsing neutron star is accelerated. Thus, the retarded view sees the "current" radius of the star larger than it actually is.

We assume that the density of collapsing matter is constant, and it is a free fall.


            --------------------------------------------->
                                           signal
     surface
           | ---> 0.32 c     |                            × center

           r = 16 km         r = 10 km


Suppose that the radius r was 16 km when a signal to the center left. The signal travels at the speed c. The speed of the surface is 0.32 c, and it will increase to 0.4 at r = 10 km.

The average speed of the surface between r = 16 km and r = 10 km is 0.36 c.

When the signal arrives at the center, the speed has grown to 0.4 c, and r = 10 km. The retarded view sees the radius too large by an amount

       (16 km / c)  *  (0.36  -  0.32) c

       = 640 meters.

The gravity potential at r = 10 km for a neutron star of 1.5 solar masses is

       -G M / r  = -6.7 * 10⁻¹¹  *  3 * 10³⁰ / 10⁴

                      = -2 * 10¹⁶

                      = -0.2 c².

The retarded view at the center sees the gravity potential

       640 m / 10 km  =  6.4%

too high.

The energy in the retarded gravity field would then be something like

       10³⁰ kg  *  0.2 c²  *  6.4%

        = 13 * 10⁴⁴ J

        = 13 Bethe.

We have assumed above that 0.5 solar masses (10³⁰ kg) is located in the volume which sees the potential too high.

We see that the energy of the retarded gravity field might cause a bounce back strong enough, so that the outgoing shock wave can explain how the outer layers of the star are blown to space.

But is there some reason why neutrino radiation does not carry away this 13 Bethe? The bounce, presumably, does not affect elementary particle reactions in the star. Then neutrinos should not carry away the energy.


Gravity almost certainly is retarded, and that breaks the Einstein field equations


        ● ---> a                      O 
       M                               clock of an observer
    static                            static


Suppose that we initially have a static large mass M and a static clock some distance away.

Let us start accelerating the mass M toward the clock. It would be very surprising if the rate of the clock would slow down before the clock "knows" that we started accelerating M.

Suppose that M is very far away from the clock. Suppose that the rate of the clock would change instantaneously in the static frame when we move M. We could observe the change with another clock at some distance from the first clock. We could send signals superluminally. This leads to all the paradoxes in time travel. Obviously, the gravity field must be retarded, in its effect to clock rates.

Since gravity must be retarded for a single mass M, it probably is retarded also for a collapsing shell of mass. If the collapse is accelerated, a clock at the center will tick faster than a clock close to the collapsing shell.

That implies that the metric inside the shell is not flat. But since the inside is empty of matter, the Einstein field equations imply that the metric must be flat. We broke the Einstein field equations.

If the metric were flat inside the shell, could we implement superluminal communication? Probably not. The configuration inside the shell would then be like a movie projector whose frame rate we can change.

What about a nonuniform spherical shell?

Suppose that we can represent the shell as a perfectly uniform shell S plus local deviations S'.

Then the gravity field of S would not be retarded, while that of S' would be retarded.

Let us assume that initially the shell is static and the metric is g inside the shell. Since the inside of the shell is empty, the Ricci tensor R is zero there.

Let the shell then start to collapse. If we assume that the metric associated with S is instantaneously updated inside the shell, then the metric of time, g₀₀, due to S, gradually slows down as the shell contracts.

We probably can adjust the metric which is due to S', so that the Ricci tensor stays zero inside the shell. Tie the metric due to S' to the (proper) time determined by S. Basically, slow down the frame rate of the movie projector.

But the definition of S is ambiguous. Suppose that the shell is thick and dynamic, so that mass moves around in the shell. How should we define S?

Suppose that the shell develops in such a way that all the mass gets concentrated to one side. Why should we assume that a part of the metric gets instantaneously propagated to a certain spherical volume, while another part is retarded?

We showed that defining a non-retarded gravity field is fraught with problems. Our conclusion is that the gravity field must be retarded, and the Einstein field equations are broken inside a collapsing shell.

We have not seen any literature which would analyze the metric inside a collapsing shell. Authors simply assume that the metric stays flat, and the metric of time instantaneously propagates to the entire volume.

In this blog we showed in May 2024 that the Einstein field equations probably have no solution for any dynamical system. Our new result says that the Einstein field equations break reasonable retardation rules in the very simple case of a collapsing shell.


The energy stored in the retarded gravity field


Above we used an ad hoc assumption that the retarded gravity field stores the same energy as how much the "retarded potential" of the mass differs from the final, static potential.

     ____                               ____  tense rubber
             •  -->              <-- •          membrane
                 _____•_____
          

            • = m mass element


This assumption may be the right one for a rubber membrane model of gravity. The mass m at the center in the diagram is too high because it does not "know" yet that the masses on the circumference have fallen lower as they are accelerated.

But what if there is no mass in the middle?


The energy of the dynamic gravity field inside a collapsing shell


Since the metric is distorted in the middle of a collapsing sphere, it certainly can do some work. How much work? Most of the energy in a gravitational wave is in the stretching of the spatial metric.

But is there any stretching? In the electromagnetic analogue, the field inside stays zero.


         shell
             • ----> a               o  observer
           dm


If we assume that the metric can be derived by linearly summing the metric perturbations caused by small elementary masses dm, what is the result?

In this blog we believe that gravity can only stretch distances, not contract them. In a rubber membrane model, spatial distancds are stretched because the membrane is stretched. In the rubber model, radial distances are stretched inside the sphere. How much energy can we extract? If we have a rigid structure, the stretching distance causes a negative pressure, which acts like a "negative mass".

A naive way to calculate the energy of a distorted metric is to use the formula for an analogous electric field.

Let us assume that time runs by a fraction h faster at the center of the shell than close to the shell. Then the gravity potential at the center is

       h c²

higher. If the shell radius is R, then the gravity field strength is

       ~  h / R,

and the energy density of the field is

       ~  h² / R².

The formula is very different from our estimate above where the energy of retardation was h M c², where M is the mass close to the center.

This suggests that the retardation energy is not energy of the gravity field, but an interaction energy of the field with the mass which resides in the volume in question.

The retardation process is not long-range. It is not like ordinary (transverse) gravitational waves which propagate in empty space. There is no surprise in the fact that the energy is an interaction between matter and the field. This is like plasma and the electromagnetic field. Also there it is an interaction between matter and a field.


The connection to dark energy


In the case of a collapse to a neutron star, the retardation energy must return to the kinetic energy of matter when there is a bounce-back, and the neutron star stabilizes. There is no retardation then, and the energy has to assume an ordinary form.

In our blog we believe that the Big Bang actually is an ordinary explosion, embedded in an asymptotically flat Minkowski space.

In the case of an explosion, the retardation energy must return to an ordinary form when the explosion "ends" in the sense that retardation no longer can store a lot of energy. 

The mass-energy density of the universe is right now rapidly falling relative to the "critical density". Could it be that retardation no longer can store much energy, and the energy must be released as kinetic energy? This would explain the accelerated expansion of the universe. It would also mean that we are now in a "late stage" of the Big Bang.


Conclusions


For a collapsing shell of mass, the Einstein field equations require the metric of time to propagate infinitely fast inside the shell. This breaks a natural retardation rule which says that no influence in nature can propagate faster than light.

In May 2024 we tentatively proved that the Einstein field equations have no solution for any "dynamic" system. Our new analysis reveals yet another weakness in the equations.

We introduced a simple hypothesis: in a collapse into a neutron star, the energy contained in a retarded gravity field plus the mass in that field, is released as kinetic energy when the energy no longer can hide in retardation. The kinetic energy is released as a shock wave which pushes matter outward from the center.

In a tense rubber membrane model of gravity, the energy would be released as a longitudinal wave in the rubber. But there are no longitudinal waves in gravity. The energy has to find another way to escape.

The hypothesis implies a shock wave whose energy is on the order of 13 * 10⁴⁴ J, and might explain how the outer layers of the star are blown out in a core-collapse (type II) supernova. This would solve an open problem in astronomy.

Our hypothesis might explain "dark energy" in cosmology. We will inspect this in detail in the next blog post.
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