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.

https://blogs.helsinki.fi/kosmokoulu/files/2018/05/Solvalla_2018_detail.pdf

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.

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

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.

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

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.

***  WORK IN PROGRESS  ***

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.

https://en.wikipedia.org/wiki/SN_1987A

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

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.

https://arxiv.org/abs/astro-ph/0612072

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

https://users-phys.au.dk/jcd/explosion/ott09/Ott_Aarhus_September2009.pdf

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.