Extra inertia in an electric field: it cannot be "private"!

Let us calculate the electric potential of an electron next to a kilogram of hydrogen, assuming that the electrons have been removed from hydrogen.

The inertia in an electric field cannot be "private"

A kilogram of hydrogen contains 6 * 10²⁶ atoms, which correspond to a charge of 10⁸ coulombs. If the electron is at a distance of 1 meter, the potential is

       V = 9 * 10⁹ * 1.6 * 10⁻¹⁹ * 10⁸ J

           = 0.14 J.

That is a huge potential compared to the mass-energy of the electron, which is 0.8 * 10⁻¹³ J.

In our March 25, 2023 post we suggested that the inertia of an electron is "private" in the field of each other charge. That is, the amounts of inertia with respect to each other charge have to be summed. Now we see that the hypothesis cannot be true. The inertia of an electron close to any matter would be immense.

We have to return to the hypothesis that the extra inertia has to come from the combined field of charges.

How much energy is shipped in the fields of protons and electrons if we move a test electron?

A ton of matter contains some 10³⁰ electrons or protons. The field energy of these particles around 10⁻¹⁰ m from the particle is

       ~ 10⁻⁵

of the mass-energy of the electron, 511 keV. At longer distances than 10⁻¹⁰ m, the fields of the electrons and the protons cancel each other out almost entirely.

                                                            ^

                                                            |

                     ●                                    •

     proton or electron           test electron

Let us put a ton of matter within a meter from our test electron. The electric field of the test electron close to the other particle is

       ~ 10⁻²⁰

times of the field of the particle. If we move the electron in the diagram one meter vertically, then

       ~ 10⁻²⁰ * 10⁻⁵ * 511 keV

of field energy moves a distance of

       ~ 10⁻¹⁰ m.

That is, the energy shipping in the field of the other particle is 10⁻³⁵ times the shipping in the movement of the test electron.

We conclude that a ton of matter close to the test electron can add a fraction

       ~ 10³⁰ * 10⁻³⁵ = 10⁻⁵

to the inertia of the test electron. Shorter distances than 10⁻¹⁰ m add less.

Let us add R meters of mass around the test electron. The effect for each 1 meter of more mass goes as

       ~ 1 / r,

where r is the distance. The effect of R meters of mass on the inertia of the test electron is

       ~ 10⁻⁵ * ln(R).

It is not negligible for Earth, for which R = 6 * 10⁶ meters.

We see that the energy shipping adds a small, but non-negligible fraction of inertia to the test electron. Polarization of matter adds more inertia. It is probably hard to distinguish these between added inertia from polarization, and added inertia from energy shipping in the electric field.

Conclusions

We must return back to the hypothesis that the extra inertia of a test charge comes from energy flowing in the field. That is case 2 in our March 25, 2023 blog post.

The fact that the Coulomb force can be calculated from the field energy of the combined electric field (if charges move slowly) suggests that the field energy does play some role in the force.

But we face the problem that the field is only updated at the speed of light. We may speculate that the system works as if the field would be updated infinitely fast. Then the Coulomb force does come from the field energy. Maybe the extra inertia, too, is determined by an infinitely fast update of the field?

How does pressure create extra inertia?

UPDATE March 31, 2023: We removed most references to case 1 of the previous blog post.

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The interaction of the gravity field of a test mass, and pressure in nearby matter is a more complex interaction than gravity between two masses. This interaction presumably should add inertia to the test mass. But is the inertia a "package" as in case 1 of our previous blog post, or is it a field effect which is subject to the speed of light?

A test mass within pressurized matter

                            

                  ooooooo

                  ooo • ooo

                  ooooooo

                    •    = test mass m

                 ooo  = matter under pressure

     

The test mass stretches the radial metric around it, adding more space for the pressurized matter to occupy. This is the reason why pressure "generates" gravity. Let us denote by E the energy recovered from the pressure through the stretching of the metric.

                    |

                --- • ---   gravity field lines

                    |

           test mass m

Let us move the test mass. The field moves, and the associated energy E moves with it.

The inertia of E wants to keep the gravity field lines of m static. The field lines bend? Here we (again) encounter the self-force problem. How does the field affect the source of the field? 

Is there any inertia associated with the process where field lines bend?

There may be a significant flow of energy also in the case where the volume does not change but m moves around. The direction of the stretching changes, and the matter orders itself in a new way.

                     •  m

                     |              ooooo

                     |              ooooo   matter

                     |              ooooo

                     |

                     |

                      -----------------> 

Suppose that m moves as shown by the arrow. Energy flows from horizontal stretching to vertical stretching. What route does the energy take? Does it go to m and back to the matter?

We should perform empirical tests. If the field is an electric field and not gravity, experiments might be possible.

The energy associated with pressure probably is not "private" for each test mass

Suppose that our test mass is a symmetric sphere around the matter. The spatial metric inside the matter does not change. It would be strange if pressure would somehow affect a contracting or rotating movement of the sphere.

In this case, the "metric" seems to be different for a pointlike test mass versus a spherical test mass. In our previous blog post we suggested that the gravitational attraction between two particles is their private matter. In the case of pressure, it looks more natural that the interaction is determined by the total field produced by all the particles.

How does general relativity treat pressure?

General relativity has pressure in the diagonal of the stress-energy tensor. If we have a static system, we can calculate a static metric, and pressure affects that metric.

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

The geodesic hypothesis claims that the movement of a test mass obeys that metric. Thus, in general relativity, pressure should cause a "package" of inertia on a test mass.

However, in this blog we have presented several cases where the geodesic hypothesis fails when there are "tidal" effects. Why should the hypothesis work for pressure?

A general observation about the concept of a "field" or a metric: it only works in simple cases

How do we use the concept the "electric field" of a system C?

In certain situations we can pre-calculate the electric field E of a system C, and then we can use that field to predict how a small test charge c moves. However, if the test charge c interacts in a complex way with the system C, or there is a significant backreaction, then c will not obey the precalculated electric field F.

How does general relativity use the "metric"?

We can solve the metric of a system M from the (mathematically) beautiful Einstein equations. Then we use the geodesic hypothesis and claim that a test mass m moves obeying that metric.

Again, if the interaction of m with M is complex, or if m causes a significant backreaction in M, then it would be a miracle if the metric would predict how m moves.

We conclude that the geodesic hypothesis is bound to fail for many complex systems M.

Conclusions

The interaction of a test mass m with pressure in M might be subject to the speed of light, and the extra inertia would not be felt by a test mass m immediately. As m moves, it changes the metric inside M. Pressure inside M causes energy E to flow inside M according to the moves of m. We can even measure where the energy E is located spatially.

It would be ideal if we could empirically measure the extra inertia caused by pressure. However, we do not think it is possible with current methods. We conjecture that the inertia is not a package, and is subject to the speed of light.

Origin and "speed" of extra inertia in an interaction: the field of each particle is "private"

UPDATE May 3, 2023: Our blog post today shows that the pendulum clocks must have the same inertia regardless of the arrangement.

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UPDATE March 31, 2023: For an electric field, case 1 below cannot be true. We have to assume that case 2 is true, or some similar hypothesis.

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UPDATE March 29, 2023: our argument about the atomic clock assumes that the (microwave) oscillator of the clock did not move before we started it. We have to check the technical details to determine if that is the case.

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Our blog claims that an object X which interacts with another object Y, acquires more inertia. We have vaguely claimed that the extra inertia comes from energy moving around in the interaction field. For example, the Poynting vector might (but does not) tell us the energy flux in an electromagnetic field.

If our claim is true, then we would expect a radial pulsing movement of a sphere X to have less inertia than a linear movement of the sphere X. If the sphere just expands and contracts, its field outside the sphere stays constant. There is less field energy moving around => less inertia in the movement.

Also, if we nudge X swiftly, the inertia of X is less because the far field has no time to react.

However, we are not sure if this is the right way to explain extra inertia. Laplace calculated that the Moon would quickly fall to Earth if newtonian gravity would have retardation equivalent to the speed of light. Maybe the inertia arises from a process which is "infinitely fast"?

An atomic clock on Earth; a field must affect an object with no delay

The rate of the clock depends on the elevation of the clock from the sea level. It is the interaction with Earth which slows down the clock. The oscillating process in a cesium atomic clock does some 9 billion cycles per second. Light only travels 3 centimeters during one cycle. Most of the gravity field of Earth only learns about the oscillation milliseconds later.

If the extra inertia would depend on energy oscillating back and forth in the gravity field of Earth, then the clock would run faster for the first milliseconds. There are no reports of such a strange behavior of an atomic clock.

We have two possible explanations:

1. the extra inertia is a "package" carried by the object X itself, or

2. there is a strange mechanism which makes X immediately aware of the inertia properties of the gravity field of Earth, so that X can act as if the field would respond immediately to its movement.

Our problem is equivalent to the corresponding problem of explaining the potential of a field through the concept of field energy. The object X feels the pull of the field instantly. However, if we try to explain the pull by the integral of the combined field energy over the entire space, the integral changes slower: it is limited by the speed of light.

We know that the force of a field affects an object instantly. There is no time for the field energy to be updated in remote locations. If we want to use the field energy explanation, we have to assume a magical, infinitely fast process which updates the field.

This question may be related to the long-standing problem of momentum and energy conservation in an interaction: how do the two objects know to move in a way which conserves momentum and energy for the whole system. Retardation definitely happens in the case of a dynamic field, if not in the case of a static field. How does nature handle retardation?

Does a pulsing sphere really have smaller inertia than the sphere in a linear motion?

If in the preceding section, case 1 is true, then each part of the sphere carries its own package of extra inertia, and the inertia is exactly the same in a pulsing motion as in a linear motion.

But case 2 suggests that the inertia less in a pulsing motion than in a linear motion.

Which alternative is right?

Empirical test for inertia of gravity: circularly arranged pendulums

We wrote about this experiment in our January 28, 2022 blog post.

Let us have several extremely accurate pendulum clocks. Let us put them into a circular arrangement around some point x.

                                |

                ----           x           ----

                                |        pendulum oscillation

Let the pendulums swing synchronously around x. The far field of the system does not change much in the plane of the pendulum movement. If the energy flow in the field affects the inertia (case 2 above), then the inertia should be less than for a standalone pendulum.

Let us calculate how much the Milky Way would affect the inertia. The escape velocity v is 500 km/s, which corresponds to a potential of

       V = 1/2 v² ~ 1.4 * 10⁻⁶ c².

Thus, the effect might be of the order one millionth.

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

It should be easily measurable with the best pendulum clocks.

The Poynting vector is NOT suitable for calculating the energy flow in a static electric field

This is the infamous, unsolved 4/3 problem of physics.

https://physics.stackexchange.com/questions/80856/does-the-frac43-problem-of-classical-electromagnetism-remain-in-quantum-m

            

                  ^    Poynting vector

                 /

                     ●  ----->  charge

                            v

Let us move a spherical charge along a vector v. It acquires a magnetic field which circles around v. The Poynting vector claims that the field energy does not move directly along v, but circles around the charge.

There is more momentum in the flow of the field energy than one would expect from a simple translational motion of the charge.

It looks like the Poynting vector does not calculate the energy flow correctly in this case of a static electric field. Maybe it works for dynamic fields?

The field of each charge is private?

Suppose that we have an electrically charged ring rotating in the plane of the ring

                   O   charged ring

                   ->

             rotation

The electric field of the ring at some distance is constant. Since the charges move, there is also a magnetic field.

If the electric field would be the only important thing, we could imagine that no magnetic field is present since the electric field does not change.

This suggests that we must treat the field of each elementary charge individually. For example, the sum of their fields may be zero, but that is not equivalent to being in empty space with no fields. We wrote about this in our blog on January 28, 2022.

A pulsing charged sphere probably has the same inertia as the sphere in a linear motion

If we treat the field of each elementary charge individually, that field definitely moves when the sphere expands. Thus, we cannot claim that there is no change in the field outside the sphere.

This suggests that the inertia of a pulsing sphere is the same as in a linear motion, after all.

Conclusions

The concept of a private field of each elementary charge might be the key to understanding the inertia in an interaction.

We predict that the experiment above with pendulum clocks will confirm this for gravity. Clocks should run at the same rate regardless of their configuration.

This also means that a collapsing spherical shell has the same inertia as an individual falling particle. Our claims on March 4 and March 19, 2023 about a very fast collapse of a shell were false.

Generally, the idea of a private field de-emphasizes the importance of the field concept. We could say that it is the interaction between particles which matters, and the (sum) field is not fundamental. The 4/3 problem is a manifestation that the field concept is troubled.

We still need to study this more. If the field is created by polarization of a material, maybe in that case the field of a charge is not private?

Also, how does the gravity of a test mass and pressure add to the inertia of the test mass? Is the inertia a "package" on the test mass, or is it subject to the speed of light like in case 2 above?

Reza Mansouri (1977): uniformly collapsing perfect fluid ball with pressure has no solution in general relativity

We wrote a blog entry about this paper on July 13, 2019.

Let us again analyze Reza Mansouri's 1977 proof that a uniformly collapsing perfect fluid ball has no solution in general relativity, if the pressure p is not zero, and p is only dependent on the density ρ of the fluid.

The assumption of a spatially uniform collapse is not realistic. There is no reason why the collapse should be uniform. Imagine, for example, a uniform ball with a uniform pressure. A surface layer will expand outward relative to the interior. If the metric is comoving with matter, the radial metric close to the surface expands, while the tangential metric stays roughly unchanged. It is not uniform.

Matching the internal solution of the ball to the external Schwarzschild metric

https://www.researchgate.net/publication/259392290_On_the_Non-existence_of_Fluid_Sphere_in_General_Relativity_Obeying_an_Equation_of_State

We have not yet checked the correctness of Mansouri's calculations. He uses a formalism developed by McVittie (1967) and Taub (1968).

Mansouri shows in Section 4 that if he tries to match the solution inside the ball to the Schwarzschild metric outside the ball, he cannot get the border conditions satisfied.

He concludes that there is no uniform collapse with the pressure p(ρ) where general relativity would be satisfied.

Question. Can we define ρ and p(ρ) in such a way that the collapse is at least almost uniform for some time, and condition (66) in the picture above fails?

Solving the collapse problem in a computer simulation

Denotations in Mansouri's paper are complicated. Let us attack the problem from a different point of view: using a numerical simulation of the collapse process. We assume that we have stored the initial state into a computer. Let us calculate a short timestep Δt forward, using the metric and the pressure. After that, calculate an approximate new metric and the pressure.

Is there any reason why the numerical simulation would fail? Maybe we are not able to calculate a new metric because there exists no metric which matches the new configuration? Or the calculation may blow up: some parameter runs away to infinity.

These questions are general problems of the existence of a solution. There is no proof of the existence.

The calculation looks innocuous. It would be somewhat surprising if the calculation does not converge.

Conclusions

Reza Mansouri proved that a uniform collapse is not possible in general relativity if there is a pressure p(ρ). It may be that the pressure simply makes a uniform collapse impossible. This does not prove that there is anything wrong with general relativity.

The calculations are complicated and we will not try to develop them further to study nonuniform collapses.

A simple computer simulation argument suggests that general relativity can handle a nonuniform collapse.

We can break Birkhoff's theorem with a sphere which "flexes its muscles" - or break the geodesic hypothesis

In November 2021 we wrote several blog entries about how "tidal" effects can break the metric of general relativity, and Birkhoff's theorem. Here we present yet another example.

Suppose that we have a small test mass m. It possesses the Schwarzschild solution around it. The spatial metric of the solution is not absolutely flat around the mass m.

Let us then imagine a very light sphere, which through some spherically symmetric mechanism can make itself more rigid when it wants. For example, it may contain liquid. When the sphere tightens its surface, it becomes more rigid overall.

              O                     •    test mass m

         sphere

When the sphere is more rigid it pushes m outward with a greater force.

Here we have an example of how gravity together with other force fields can cause unexpected forces. Since the force changes, the "metric" around the sphere changes, which breaks the Jebsen-Birkhoff theorem.

https://en.wikipedia.org/wiki/Birkhoff%27s_theorem_(relativity)

One could claim that we are cheating because the sphere is not totally symmetric; rather it is a little bit distorted by m. There is a "tidal" effect. However, the same objection applies to any application of Birkhoff's theorem with a pointlike test mass. Our November 2021 blog entries point out that using a spherical shell as a test mass around the sphere would save Birkhoff's theorem. There would be no tidal effects.

Let us check various proofs of the Jebsen-Birkhoff theorem. They probably assume that the "coupling" of gravity to other force fields is local and very simple. In our example, the coupling is through the structure of the sphere. The coupling is not local.

The theorem states that a spherically symmetric solution to the Einstein equatioms in a vacuum must be static and asymptotically flat.

The proofs do not assume anything about the couplings. They simply use the Einstein equations. The theorem is true.

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

We could say that our counterexample does not break Birkhoff's theorem. Rather, it breaks the geodesic hypothesis that a point mass obeys the metric derived from the Einstein equations.

The analogy between electromagnetism and general relativity

UPDATE March 25, 2023: Our claim about a very fast collapse of a shell and Cherenkov radiation is probably false. See our new blog post.

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Newtonian gravity looks much like the Coulomb interaction between static electric charges.

Comparison of lagrangian densities

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

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

Their formulae are quite different, though. We should analyze where the analogy is hiding.

In the first formula, the Ricci scalar somehow plays the role of the inner product of the electromagnetic tensor F in the second formula.

Analogous Einstein equations for electromagnetism

The Einstein equations calculate the elliptic orbits of planets correctly. They calculate the newtonian gravity field. They calculate the precession of the perihelion of Mercury.

Let us have a system with no gravity and such that the electric charge of each object is linearly proportional to its mass. The system behaves much like gravity. The precession of the perihelion of "electric Mercury" is caused by the extra inertia which Mercury obtains when it is close to the electric Sun. Mercury moves electric field energy around, and gets extra inertia from that.

If we write the Einstein equations for our electromagnetic system, they describe the system to some degree. However, there is no electric black hole. With extremely strong fields we get a different behavior.

The Einstein equations define a "metric" for our electric system.

We have two descriptions for our system:

1. the classical one with the Coulomb force, magnetic fields, and the Poynting vector which tells how much field energy we move around;

2. the Einstein equations.

These descriptions are (almost) equivalent at least in certain situations.

They are not equivalent for quadrupole radiation. We noted in an earlier blog post that the radiative power of a gravity quadrupole is four times the power of the analogous electric quadrupole. That is a major difference between the electromagnetic field and the gravity field.

A fundamental difference of gravity and electromagnetics is that the field energy of gravity carries a gravity charge, while the field energy of electromagnetism has no electric charge. A graviton possesses a gravity charge while a photon has no electric charge. This fundamental distinction may explain all the differences between gravity and electromagnetism.

Electromagnetism differs prominently from general relativity for a collapsing/expanding symmetric shell; Cherenkov radiation

In this blog we have argued that a collapsing shell in our Minkowski & newtonian gravity does not gain extra inertia. It does not cause field energy to move around.

In the case of electromagnetics, the magnetic field B is zero, and the Poynting vector ~ E × B is zero, which means that no energy flows.

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

Robert Oppenheimer and Hartland Snyder (1939) calculated the collapse of a spherically symmetric dust ball in general relativity. Its surface falls according to the Schwarzschild metric. That is a collapsing shell falls just like a point particle. A point particle a acquires huge extra inertia when it approaches the horizon. We conclude that the shell does acquire extra inertia in general relativity.

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

In our Minkowski & newtonian gravity, a shell can fall faster than what is the speed of an individual photon. If our model is correct, we except to see Cherenkov type radiation from matter falling into a black hole.

Reza Mansouri's proof of the nonexistence for a uniform collapse solution with a pressure p = p(ρ)

https://www.researchgate.net/publication/259392290_On_the_Non-existence_of_Fluid_Sphere_in_General_Relativity_Obeying_an_Equation_of_State

We have already discussed Reza Mansouri's (1977) result in our blog.

Mansouri shows that if the pressure in a uniformly collapsing symmetric ball of perfect fluid only depends on the density ρ of the fluid, then there is no solution in general relativity. The sole case where a solution exists is when p = 0 everywhere, that is, a collapsing dust ball.

Do we have any reason to assume that the collapse should be uniform? That is, the spatial metric at each point (t, x) has contracted as much in the radial direction as in the tangential direction.

Conclusions

Einstein's equations can describe the movement of electric charges fairly well in certain cases. There is a big difference in the power of quadrupole radiation, however.

Another difference is in the collapse of a spherical shell of charge. General relativity claims that it acquires inertia which slows down the collapse.

Reza Mansouri's result about a collapse of a perfect fluid ball raises the question: under what circumstances does a theory of gravity have solutions? An iterative method to obtain a solution may fail if a new iteration of the metric changes the pressure so much that the next iteration of the metric differs a lot from the previous one.

Proving the existence of solutions for nonlinear differential equations is mathematically too challenging. Few results exist. We need to study the problem heuristically. Does our Minkowski & newtonian gravity have solutions for the Mansouri equation of state p = p(ρ)?

Choosing the coordinate system for a fast neutron star is awkward

Let a faraway observer see the speed of light inside the neutron star as v << c, where c is the speed of light in the vacuum.

Let the neutron star move faster than v relative to our observer.

                                         o   observer B

                                         |

                                        /\

                 o                     ● --------> 

                 |           neutron star

                /\

         observer

Let us try to define a coordinate system which is "static" relative to the observer. Let us have another observer B inside the neutron star. Let he be at a position x at a coordinate time t, and at x + dx at t + dt.

Let the metric say that he moved a distance ds and the the proper time interval was dτ.

We have x + dx, t + dt within the light cone from x, t.

The "frame dragging" inside the neutron star is so strong that B has to move. Thus, x, t is not within the light cone from x, t.

We cannot define a sensible metric with these coordinates.

The obvious solution is to let the neutron star to drag along the coordinate system along with it, like the rubber string coordinates of our previous blog post.

But if we try to keep most of the coordinate system static, we end up with a very awkward coordinate system.

The coordinate system for an accelerating neutron star

Suppose that our neutron star is initially static and we have a nice static coordinate system. Then we accelerate the neutron star to a speed larger than v.

Can we still use the the static coordinate system? Apparently, no. We have to switch to a different coordinate system. We see that an accelerating object can "break" a coordinate system.

In a natural, "static" coordinate system the speed of gravity can be superluminal

What is the speed of changes in the gravity field? Usually people assume that it is the local speed of a photon. That is, the local speed of light.

The speed of a photon, as seen by a faraway observer, can be very slow if it travels in a low gravitational potential.

Is it true that changes in the gravity field can only propagate at the local speed of light?

Here we assume a "static", cartesian, initial coordinate system and study how quickly a spherical shell reacts, relative to that coordinate system, to changes in an external gravity field.

If we would define spatial coordinates through using physical, tense rubber strings in space, then the speed of gravity seems to be less than the local speed of light.

A spherical shell under a changing gravity field

Let us have a strong spherical shell which is close to being inside its Schwarzschild radius. Measured by a faraway observer, photons move very slowly close to the sphere.

Let us assume that the speed of gravity is only the local speed of light.

Let us put swiftly a mass M at some moderate distance from the shell. The gravity of M starts pulling on the shell.

               O                          ● <-------

     massive shell        mass M

But the surface of the shell receives the information about the changed gravity field much later. Does the shell stay static for a long time? That would be strange. That would mean that objects behave differently in the field of M - a breach of an equivalence principle.

We conclude that the surface of the shell must react to the changed gravity field faster than what is allowed by the local speed of light. It has to start moving quickly, relative to our coordinate system.

                    |

                    \

                 O  |  --->   shell movement

                    /

                    |

         rubber string

If we would use physical rubber strings to define the spatial coordinates, then the shell would push them in front of it, because the local speed of light is slow close to the shell. The movement of the shell would happen through changes in the metric making the distance between the shell and M shorter. In the rubber coordinate system there probably are no superluminal reactions.

If the shell moves along a complicated path, it may make the rubber strings entangled. The rubber coordinate system is not very practical.

Minkowski & newtonian gravity

On February 25, 2023 we wrote about the speed of gravity in our own theory of gravity, Minkowski & newtonian gravity.

We suggest that the external gravity field of the mass M "grabs" the entire system, the shell and its gravity field. It exerts a force on the entire system. The slow speed of a photon inside the system is a result of interactions within the system. That does not prevent the external field of m

M grabbing the entire system and moving it.

We may interpret the process like this: the combined gravity field of the shell and the mass M tries to get to a lower energy state. The outer field of the shell is "attached" to the field very close to the shell, and ultimately to the matter in the shell.

The field of M exerts a force on the outer field of the shell. The attachment relays the force to the inner field and to the matter in the shell.

The whole system, the inner field & the matter, starts to move under the force. This is essentially newtonian mechanics.

Superluminal communication to an observer on the shell is not possible

Can one send a signal to an observer on the shell superluminally?

Let us use a small, local change in the gravity field as the signal. It propagates at the local speed of light. It is not superluminal.

Suppose that the shell carries an electric charge but the observer is neutral. Let us pull on the shell with an electric field. What does the observer see?

Changes in the electric field lines propagate only at the local speed of light. It takes a long time for the observer to notice anything. The electric field makes the shell to move immediately, but the observer is oblivious of this.

Conclusions

In a "natural" coordinate system we must allow systems to react faster to changes in the gravity field than what is the local speed of light.

The global speed limit has to be the the speed of light in the surrounding asymptotic Minkowski space. Otherwise, we would have the time travel paradoxes.

Our observation has implications for LIGO calculations. It is not clear to us what speed of gravity they use in their computer models. If it is the local speed of gravity, it may be too slow.

The gravity field close to the horizon in a black hole grows exponentially

When we introduced our Minkowski & newtonian gravity model, we explained the huge inertia of a photon with "levels" in the gravity field.

Levels from infinity to close to the horizon

Let us have a straight line which extends from the horizon to infinity. The line is perpendicular to the horizon.

Let rₙ be such radii that the "remaining" energy of a static test mass m at that radius is

       0.9ⁿ m c².

The remaining energy means m c² plus the potential energy.

In an earlier blog post we argued that the radii rₙ converge toward the horizon as n grows to infinity.

Let these rₙ define "layers" of the gravity field of the black hole.

                       •   rₙ

                       •   rₙ₊₁

     

                       ●  center of black hole

Let an observer sit at rₙ and lower a test mass m to rₙ₊₁. If the observer moves the test mass horizontally, he measures that the inertia of the mass is 1.1 -fold compared to if he would move the test mass at his own position.

                   o   observer

                   |\

                   /\

                      |    rope

                      |

                       •    test mass

In earlier blog posts we explained that the inertia is larger because when the observer lowers the test mass with a rope, he gets 10% of the mass-energy of the test mass to the rope system. When he lifts the test mass above at a different location, he must use 10% of the mass-energy. The entire process moved the test mass m, and also moved an additional 10% of mass-energy from one location to another.

We assume that the process moves the test mass from a location X to a location Y, and induces a movement of "field energy" from some other location to some other location.

Let us try to build a model which explains why the inertia grows exponentially at lower levels.

..................................................................................

                                                      <----------- •  field energy   

level n           . ------------>     a little bit of field energy

..................................................................................

                                                      <------------ •  field energy

level n + 1    ● ------------>     test mass m

..................................................................................

                       X              Y             Y'             X' 

Let us look at the level n + 1.

The test mass m is moved from X to Y. The movement induces some field energy on the level n + 1 to move from X' to Y'.

Let us look at the level n.

Let us assume that the field energy on the level n + 1 itself is a source of gravity.

Then the movement of field energy on the level n + 1 from X' to Y' induces a movement of a little bit field energy on the level n, and this time from X to Y.

Also, the movement of the test mass m transports more field energy on the level n from X' to Y'.

When we look successively at n - 1, n - 2, ..., the movement of field energy from X to Y,  and also from X' to Y', grows exponentially.

We have a simple model which might explain why the inertia of a test mass grows exponentially when it comes closer to the horizon. We assume:

1. The mass-energy (inertia) of the test mass alone is m on every level;

2. field energy is a source of gravity;

3. there is a cascading effect on various levels n: on each level, the field energy moved is 10% of the respective mass-energy moved on lower levels in the diagram;

4. a movement of mass-energy on a higher level in the diagram does not affect a lower level.

The field at the horizon and below it

Above we sketched a very crude model which allows us to inspect what happens when the test mass is at the horizon, or inside the horizon.

When we move the test mass horizontally just above the horizon, it causes a cascading effect of field energy movement on upper levels. It is obvious that the moved field energy cannot exceed M c², where M is the mass of the black hole.

Previously in this blog we have simply guessed that the inertia cannot exceed M. Now we have a crude model which supports our guess.

In principle, the movement of the test mass might cause field energy to move in a manner of a "gearbox". Certain field energy could move over large distances when the test mass moves over a short distance. Then the inertia could exceed M.

We conjecture that the inertia cannot exceed M, and is actually much less than M.

The inertia determines what is the speed of a photon as seen by a faraway observer. A local observer will feel gravity if the speed of a photon is less closer to the center of the black hole.

Since the inertia must be less than M, we can only define a finite number of levels n.

Conjecture. The speed of a photon is like in the Schwarzschild solution when we descend from infinity, until we are close to the horizon. At the horizon and inside the horizon, the speed of a photon slows down only moderately when we approach the center of the black hole.

The conjecture means that gravity is relatively weak at the horizon and inside the horizon.

Conclusions

We constructed an extremely crude model which may explain why gravity grows very strong close to the horizon.

The reason for the very large inertia close to the horizon is that a test mass makes field energy to move around, and field energy is a source of gravity, too. It is the "recursive" nature of gravity: any mass-energy carries a gravity charge.

We conjecture that the inertia only grows moderately when we travel from the horizon toward the center of the black hole.

Our model is about a pointlike test mass. We remarked in an earlier blog post that a symmetric spherical shell of mass will not gain more inertia and will collapse to the center very quickly.

Symmetric collapse into a black hole happens very quickly

UPDATE March 31, 2023: The claims might be true, after all. See our new blog post.

----

UPDATE March 25, 2023: The claims in this blog post are probably false. See our new blog post.

----

Our Minkowski & newtonian model of gravity claims that the slow speed of light close to the horizon of a black hole is due to extra inertia that a photon borrows from the large mass of the black hole.

When the photon moves, it transports its own mass-energy and energy in the gravity field. The extra inertia comes from the energy of the field which the photon moves around.

In this analysis it is relevant that it is an individual photon.

We have earlier written about the fact that a radial motion of a symmetric shell of matter does not seem to acquire extra inertia, in contrast to a translational motion of a test mass.

The electromagnetic analogue

                                  |

                                  | 

                                  +

                               _____

                            /            \

       ------        +  |                |  +        --------- field line

                            \______/

                                  +

                                  |

                                  |

Let us have a spherical dust shell with electric charge.  The electric repulsion starts to expand the shell.

Is there extra inertia that the electric field gives to dust particles?

                             |    --------------------------

                             |    --------------------------

                             +

                              ------>      

  zero              wall of           electric field

  field               dust

Probably not. We can interpret the process this way: as the shell grows larger, dust particles harvest energy from the electric field in their immediate neighborhood and convert it to their kinetic energy.

The energy does not need to move over large distances. The field outside the shell remains constant and the field inside the shell is zero.

The process is radially symmetric and essentially one-dimensional.

Let us then have two concentric shells of charged dust. Again, when the shells start to expand, they can collect energy from the field in their immediate neighborhood and convert it to kinetic energy.

A uniform ball of dust is like many concentric shells. We conjecture that there is no extra inertia in the expansion of a ball.

Collapse of a uniform ball of dust into a black hole

This is the Oppenheimer-Snyder collapse.

1. The electromagnetic analogue suggests that there is no extra inertia in the process. 

2. Another argument for no extra inertia: when the entire mass M of the ball moves, then the conceivable extra inertia which it could give to itself might be at most ~ M. Thus, the extra inertia would not be very large.

3. It would be somewhat strange if the mass M would be able to resist a uniform movement of the same mass M by giving itself extra inertia. The mass M cannot give itself extra inertia if we move the mass M translationally through space.

We conjecture that in a collapse of a spherically symmetric mass, there is no extra inertia.

Our conjecture implies that the collapse into a "singularity", or a very dense object, happens almost at the speed of light, as seen by a faraway observer.

Conclusions

Previously, we had thought that the collapse into a very dense object is an extremely slow process in our Minkowski & newtonian gravity. It turns out that the collapse is very fast.

Even though a single photon moves very slowly just above the horizon or inside the horizon, a symmetric large mass can sink very rapidly toward the center.

We need to figure out what happens at the center after the collapse.

Can we recover infinite energy from the gravity between point particles?

Suppose that we have two point particles which attract each other. If we use ropes to lower them very close to each other, can we recover more energy than what was the mass-energy of the two particles?

If that were possible, it would conflict with conservation of energy.

           e- ●   ----->        <-----   ● e+

If the particles are an electron and a positron, then lowering them to a distance ~ 10⁻¹⁵ m would recover more than 1 MeV of energy. Nature has solved the problem by letting the pair to annihilate each other before we can recover too much energy.

If we have a Schwarzschild black hole, and use a rope to lower a test mass m to it, then at the horizon we have recovered the energy m c². The infinite force at the horizon prevents us from lowering the test mass even lower, and recovering too much energy.

However, in our own Minkowski & newtonian gravity, the force at the horizon is not infinite. Does our model allow us to recover too much energy?

No. The gravity charge of the system decreases as we recover more energy. If we could recover the entire mass-energy of the system, then the remaining gravity charge would be zero, and there would be no gravitational attraction left to do work.

This is the familiar rule that one must subtract the binding energy from the mass of a system.

What is the speed of light inside a black hole; the Hawking temperature

In this blog we have argued that time, as seen by a faraway observer, cannot stop completely inside a black hole of a mass M, because the inertia of an arbitrary particle cannot grow larger than M. If we have a photon inside a solar mass black hole, the inertia of the photon can grow at most

       M / m

-fold, where m is the mass-energy of the photon. For a 1 eV photon inside a solar mass black hole, the ratio is ~ 10⁶⁶.

How do we know that M is the maximum possible inertia? We do not know. There might exist a "gear system" which raises the inertia to be even larger than M.

What is the lowest possible inertia? We do not know that either. Let us try to find out.

The increase of the Schwarzschild radius from a falling particle

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

The Schwarzschild radius of a black hole of a mass M is

       R  =  2 G / c²   *   M

            =  1.5 * 10⁻²⁷ m/kg    *    M.

If let a particle of a mass m falls toward a Schwarzschild black hole, we may expect the movement of the particle to differ from the idealized case of a zero mass particle when it is at the distance of

       R'  =  2 G / c²    *    m

from the horizon in the Schwarzschild standard coordinates. This is because the radius of the black hole grows from R to R + R' as the particle is absorbed.

How slow does time progress at that distance, if we use the Schwarzschild metric for the mass M?

The metric is:

We have

       dτ²  =  (1 - R / (R + R'))  dt²

<=>

       dτ  =  sqrt(R' /  R)  dt.

This figure might give us an estimate on how slowly a particle will sink into the black hole. The particle moves essentially at the local speed of light. The speed of the particle as seen by a faraway observer is

       sqrt(m / M) c,

if it moves horizontally. If it moves vertically, then the speed is

       m / M   *   c,

because the radial metric is stretched. Here c is the speed of light in faraway space.

The Schwarzschild metric is probably correct if the particle is "far" from the horizon. Very close to the horizon, and inside the horizon, we expect the particle to move even slower, relative to a faraway observer.

For a solar mass black hole, the vertical speed of a falling proton would be

       10⁻⁵⁷  c,

as seen by a faraway observer. That is

       3 * 10⁻⁴⁹ m/s.

The journey to the center would last 10⁵² s, or 3 * 10⁴⁴ years.

Filling a black hole with black body radiation: the Hawking temperature is much too high

If the speed of light is as slow as calculated above, then the black hole will appear to have an immense volume for a photon inside it. Furthermore, a smaller energy photon will see the volume even larger.

The mass-energy density of black body radiation is

      σ T⁴ / c³,

where σ is the Stefan-Boltzmann constant

       σ  =  6 * 10⁻⁸ W / (m² K⁴).

The Hawking temperature of a solar mass black hole is 60 nanokelvins. Let us calculate what is the mass of the black body radiation if we fill a solar mass black hole with it.

The wavelength at 6000 K is 0.5 micrometers. The wavelength at 60 nK is 0.5 * 10⁵ m or 50 km.

The mass of a photon is

       m = h c / λ   *   1 / c²

            = h / (λ c).

For 60 nK, the mass of a photon is m ~ 10⁻⁴⁷ kg.

The mass of the Sun is M = 2 * 10³⁰ kg.

The mass-energy density of black body radiation at 60 nK is

       ~ 10⁻⁶⁰ kg/m³.

Let us assume that the speed of light inside a solar mass black hole is

       ~ sqrt(m / M)  c

          ~ 10⁻³⁸ c.

The volume of the black hole is then

       V ~ (10³ / 10⁻³⁸)³ m³

           = 10¹²³ m³.

The mass-energy of the black body radiation is

       ~ 10⁶³ kg,

or much larger than the mass of the Sun.

We conclude that the Hawking temperature is much too high, at least by a factor 10¹⁰.

                           |

                           |

                     horizon

                      "hole"

    outside                       black hole

    space                          interior

                           |

                           |

If we lower the temperature by such a huge factor, then the photons will have a wavelength which is too long to let them escape through the "hole" which is the black hole horizon. The diameter of the hole is ~ the Schwarzschild radius.

The time to thermalization, or "scrambling"

https://arxiv.org/abs/0808.2096

Yasuhiro Sekino and Leonard Susskind (2008) claim that a black hole attains a thermodynamic equilibrium phenomenally fast, in a fraction of a second.

Our analysis suggests that it is the exact opposite: it will take an immense time for a black hole to thermalize. This is because the speed of light is so slow inside a black hole, as seen by an external observer. It can take 10⁴⁴ years for infalling particles to collide at the center.

Let us analyze this with an object which is "almost a black hole". Let us have very massive thin spherical shell of matter which is immensely strong so that it can withstand the gravitational pull without breaking. Alternatively, we can assume that the shell is kept from collapsing by filling its interior with very lightweight, incompressible matter.

               _____

            /            \      heavy shell of matter

           |               |

            \______/

We can put the entire shell into a very low potential, without it collapsing to form a black hole.

Now if we pour some gas (or photons) down on the shell, it will take a long time to thermalize because the heat and vibrations propagate at most at the speed of light.

The closer the shell is to be inside its Schwarzschild radius, the slower the thermalization.

If we let the shell to collapse into a black hole, an external observer will see the matter in the shell falling slower and slower toward the horizon. Thermalization in this case probably takes an immense time.

Question. The slow speed of light is due to the photon adopting a lot of inertia from its interaction with the heavy mass in the system. Could it be that this interaction quickly thermalizes the photon?

Let us analyze this for a photon on the surface of Earth. If the photon would lose some of its energy to the 6 * 10²⁴ kg of atomic matter in Earth, we would observe scattering or some other optical phenomenon. We might also see a gravitational lens of a galaxy be somewhat opaque. We conjecture that the extra inertia does not help a photon in thermalization.

Conclusions

An arbitrary physical system, like a neutron star, does not usually have any one temperature. The temperature varies depending on the part of the system, and varies with time. The temperature hypothesis of Stephen Hawking is at odds with this principle of physics. Hawking's hypothesis leads directly to the information paradox of black holes.

The information paradox constitutes strong evidence against the hypothesis of Hawking.

The speed of light inside a black hole seems to be extremely slow, as measured by a faraway observer. If a photon of a certain frequency falls freely into the black hole, its wavelength is extremely short relative to the Schwarzschild radius. Thus, the volume of a black hole is immense from the point of view of an individual photon.

Since speed of light is very slow, it takes a very long time for a black hole to thermalize. Billions of times longer than the age of the universe.

For photons, the black hole appears to have a volume much larger than the visible universe, connected to the outside space through a hole whose size is ~ the Schwarzschild radius.

If we let photons to thermalize in the large volume, their temperature will be 10⁻¹⁰ the Hawking temperature, or less. The photons have such a long wavelength that they cannot escape from the black hole through the hole which is the horizon. Also, the temperature is so low that essentially zero photons will escape. The black hole is truly black.

However, in principle it is possible for a photon to escape. The horizon is not a one-way membrane. We have argued earlier that a physical system cannot have a one-way membrane.