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.

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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.

Frame dragging is a universal effect in interactions

In this blog we have studied the inertia of an electric test charge close to a large charge. We observed that the inertia is larger, because if we move the test charge, we have to move some energy in the electric field, too.

The same holds for a test mass close to a large mass. The effect is dramatic near the horizon of a black hole, and adds some 10% to the inertia of a test mass on the surface of a neutron star.

The speed of light is slower on a neutron star than in faraway space.

Slow speed of light is "frame-dragging"

The slow speed of light can be regarded as a kind of "frame-dragging".

The neutron star restricts the speed at which a test mass can move relative to it. Near the horizon of a black hole, the speed of light is very slow (relative to a faraway observer). The frame-dragging is extreme.

What is usually called frame-dragging is the effect that a rotating mass forces a test mass to orbit along with the surface of the mass.

The rotation effect is a special case of frame-dragging.

The syrup model of a neutron star or a black hole

Syrup is a substance where there are strong interactions between molecules. Frame-dragging is strong in syrup for small test objects, like an ant crawling in the syrup. This is the underlying reason why the syrup model is nice for neutron stars and black holes.

The speed of gravity is the speed of the photon?

In this blog we have our own Minkowski & newtonian model of gravity which claims that gravity is an ordinary force and does not affect the "true" metric of spacetime.

We have not been certain of what is the speed of gravity in the sense that how fast can a the static gravity field of a mass spread if we move the mass. Is the speed the light speed of the underlying Minkowski space, or is it the speed of the photon? Photons move slower in a low gravitational potential.

Let us analyze an electromagnetic analogue

A charge inside a polarizable ball

Let us put an electric charge at the center of a ball made of an electrically polarizable material.

                                                

                                                   field lines

                    _____                          \         /

                  /           \                   

                |      -        |              <-------   +

                  \______/

                                                       /         \

                charge in            opposite charge

       polarizable material

The polarization takes some of the internal central charge to the surface of the ball.

Let us move the external charge very quickly closer to the ball.

                    _____ 

                  /           \                   

                |       -       | -               +

                  \______/

                            induced

                             charge

As the external positive charge moves closer, it induces more polarization at the surface of the ball. It draws more negative charge at that edge of the ball which is closest to it. The induced polarization "shields" the charge at the center from seeing that the external charge has moved closer.

How quickly is the charge inside the ball aware that the external charge moved?

When electric field lines are suddenly bent, there is an associated magnetic field. The bent field line is like a half of a photon which propagates. What is the speed of the propagation?

In our example, bent field lines become denser and turn to be closer to the normal of the line between the external charge and the ball. They will induce more opposite charge to the edge of the ball. That may be enough to shield the central charge from seeing that the external charge moved closer.

Electromagnetism in a medium is determined by the permittivity ε and permeability μ of the material. It seems to be an empirical fact that field lines and their movement inside a medium obey these values.

Then it is not possible to change the field felt by the central charge faster than the speed of light in the polarizable material.

When the external charge moves close to the ball, it immediately starts pulling on the charge at the surface which the central charge induced.

But the central charge itself will not feel a stronger pull until the the field lines change, and changes in them only propagate at the speed of light in the material. We may assume that the speed of sound in the material does not exceed the speed of light. Then there is no mechanical pull either, until the field lines change.

              ------->

             |

             |            -   central charge

             |

      moving

      surface

As the surface of the ball moves, then the center of the ball would eventually bump into the surface. Then the center, at the latest, feels that something has changed and should start moving.

The speed of "local" gravity is the same as the speed of the photon

Our analogue suggests that in gravity, changes in the gravity field lines only can propagate at the speed of the photon.

If we move a mass quickly close to a neutron star, the surface of the neutron star will feel the pull earlier than the center.

A black hole is an extreme case. If we use the time of a faraway observer, does the horizon of the black hole ever feel the pull of the external mass?

But the black hole itself starts to move. Can its horizon stay at the "same" location indefinitely?

The mystery of the static location of the event horizon: frame dragging lets the horizon to move

We have our own Minkowski & newtonian model of gravity. It is counter-intuitive if the horizon stays static while the black hole seems to move. How can we solve this paradox?

The solution might be the following: the slow speed of light is caused by the collective action of all the mass-energy in the black hole. The slow speed is not relative to a fixed Minkowski coordinate system. When the external mass pulls on the outer parts of the black hole gravity field, the whole system starts to move. A kind of "frame-dragging" moves the center of the system, even though the center does not yet know that anything has changed.

The slow speed of light close to the horizon is a result of a photon "borrowing" inertia from the whole system. As most of the system starts to move, then the photon - and any field lines - must move along the whole system.

The maximum speed of a low-energy signal. Frame dragging seems to be a bad way to send data to an observer. An observer does not notice anything if his whole frame is dragged.

Local changes in the gravity field can carry information and their speed is restricted by the speed of the photon. We conclude that the speed of the photon is the maximum speed of a weak signal, a signal whose mass-energy is much smaller than the gravitating system.

On the other hand, if we use a mass similar to the gravitating system as a "signal", and let it collide with the system, the signal will drag its own frame, and may move faster than a weak signal.

Mach's principle

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

Albert Einstein wished to show that general relativity satisfies Mach's principle: inertial frames are determined by the collective movement of faraway large masses.

The frame dragging, which we described, is reminiscent of Mach's principle. Close to the horizon of a black hole, the inertial frame is almost entirely determined by the movement of the black hole.

Question. Minkowski space is empty of matter. Could we somehow simulate Minkowski space by putting a massive shell of mass far away from us?

In this blog we have argued that nearby masses increase the inertia in a linear motion of a test mass, while a radial motion of a test mass shell has no extra inertia. Is it really true, in the light of the analysis above?

In our electromagnetic analogue, the slow speed of light resists moving an electric test charge linearly. If we have a test shell of charge expanding, the electric field lines do not change outside of the sphere. The inertia should be less?

Empirically, the inertia of a test mass in a linear motion is the same as in a radial motion. That suggests that there cannot exist any large faraway large masses which would dictate the inertial frames in our universe. The answer to the Question above is negative.

Conclusions

"Local" propagation of changes in the field lines of the gravitational field happens at the speed of the photon. That is what our reasoning above strongly suggests.

But the speed caused by "frame-dragging" may exceed the local speed of the photon. If we start to move the outer layers of a massive spherical system, then the center will follow in the movement. This is because the slow local speed of the photon is a result of the collective action of the masses in the system. If we move the masses, the photon moves with them. The local speed of light does not restrict the movement.

The universal speed limit is the speed of light in the underlying Minkowski space.

We did not find a semiclassical model for the electron - a Feynman path integral is the way?

For the past six months we worked very hard to construct a semiclassical model for the electron. The goal was to build an intuitive model which would explain the magnetic moment and the gyromagnetic ratio of the electron.

We failed.

The magnetic moment and the spin of the electron

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

The magnetic moment follows quite simply from the nonrelativistic approximation of the Dirac equation. The ultimate reason for it is that the momentum operators

       p  =  i d / dx

act on the vector potential of the magnetic field, too, when we construct a solution for the Dirac equation in the nonrelativistic case under a magnetic field B.

If we take the axioms:

1. the Dirac equation describes the electron wave, and

2. the "correct" way to add a magnetic field B is to use the minimal coupling to the vector potential,

then we get the correct magnetic moment. Why is the minimal coupling the way to add a magnetic field? We do not know.

The spin of the electron follows from the Dirac equation if one guesses that the sum of the angular momentum J and the spin S should be conserved.

Using a Feynman path integral as a "particle model" of the electron

In our blog we hold the view that empty space is strictly empty of fields, except of the Higgs field. This solves the infinite energy problem of empty space (except for the Higgs field).

We would like the electron to be a particle. Then empty space is an intuitive concept: it contains no particles.

In a Feynman path integral, the electron is, in a sense, a particle. A single path can be viewed as a path of a particle.

In a path integral it is important that alternative paths of the particle must not interact. The only "interaction" allowed is the linear superposition of the end results. The probability amplitude of a final result is obtained by summing the amplitudes for each path. The weight of each path is a fuzzy concept, though.

The propagator, or the lagrangian action over a path is the probability amplitude of that path.

Conjecture. A wave equation must be linear for the Feynman path integral approach to work. We assume that the action of a path is calculated using the propagator of the wave equation.

Conclusions

We wanted to construct a semiclassical particle model for the electron, but failed.

We should determine if we can build a satisfactory particle model for the electron using the Dirac equation and a Feynman path integral.

For example, in high-energy collision experiments the electron behaves quite like a classical point particle with an electric charge. Can we explain this in an intuitive way using a Feynman path integral?