One can escape from a black hole in our Minkowski & newtonian model; the black hole information paradox is solved?

People believe that in general relativity, the event horizon is a one-way membrane. Particles can go down but cannot come up. We write "people believe" because there apparently exists no description of the exact spacetime geometry when, for example, an electron, falls to an event horizon.

In our Minkowski & newtonian model the apparent "spacetime geometry" is an illusion created by the newtonian gravity and increased inertia. To make a one-way membrane, one would need an infinite force, and such a force is not available. Thus, the event horizon cannot be one-way. Escaping from a black hole is very hard, but in principle, it is possible.

In our October 20, 2021 blog post we calculated that a 1 MeV photon passes through the horizon of a solar mass black hole in < 1.38 millisecond of the global Minkowski time. An "infinitely strong" event horizon would keep the test mass hovering above the horizon for an infinite time. Our result suggests that the real event horizon is not infinitely strong. If it is not infinitely strong, maybe it does allow photons to pass upward?

Conservation of energy dictates that the electron and the positron must annihilate; no need in gravity

An electromagnetic analogue of a black hole is a point charge, for example, an electron. We will discuss this ignoring quantum mechanics.

Let us imagine a device which lowers a positron toward an electron. The device can harvest ever more energy from the process and send the energy away. If the particles would not annihilate, then we could harvest infinite energy from the process.

Either there is no conservation of energy, or we end up with an object which possesses negative energy. Both alternatives are bad. Nature has saved us from this paradox by making the electron and the positron to annihilate.

                gravity                  gravity

     mass  --->                               <---  mass

          ● |-------------------____________| ●

                      energy harvesting

                                device

In gravity, the problem of infinite energy does not exist.

Let us assume that the original mass-energy of the system in the diagram, as seen by an outside observer, is E.

Suppose that the device harvests some amount of energy. The device sends the energy away to an outside observer. Let an outside observer receive W of energy. Now the gravity charge E of the system has become W less. If the outside observer receives the entire E of energy, then the gravity charge of the system is zero, and there no longer is any pull of gravity. Thus, the outside observer cannot receive more than E.

For a local observer in the system, it may appear that energies grow toward infinity, but that is not relevant for the physics of the system. The physics obey the gravity charges as measured by an outside observer.

The difference to the analogue of the electron and the positron is that the gravity system loses charge as it outputs energy.

Canceling gravity with an electric charge

In a blog post on January 12, 2022 we argued that one can cancel all effects of gravity with electric repulsion. If we have an electrically charged black hole, then a suitable electrically charged object can move down through the event horizon as if it were moving in empty space. It can as easily come up through the event horizon. The horizon is not a one-way membrane for such an object.

Having a one-way membrane breaks classical thermodynamics; Hawking radiation

Jacob Bekenstein and Stephen Hawking argued that a black hole contains a lot of entropy, and in classical thermodynamics, an object with lots of entropy should radiate energy, when it is seeking a lower entropy state.

A perfect crystal does not radiate because it is in the lowest entropy state. A block of glass radiates because atoms there slowly order themselves into crystals, releasing heat.

The black hole of general relativity is problematic because it contains a lot of entropy, but cannot radiate at all. It seems to break classical thermodynamics.

Stephen Hawking invented hypothetical Hawking radiation to remedy the problem. A black hole does radiate, after all. But the famous information paradox arose: how is the information in the entropy carried away in Hawking radiation?

In our Minkowski & newtonian model, photons probably can go up through the event horizon. They lose almost all of their energy in the redshift. That is why the black hole appears almost black.

The black hole information paradox

The temperature of a black hole appears almost zero for a faraway observer, whereas a local observer will feel extremely high temperatures.

To obey classical thermodynamics, there must be very many degrees of freedom in a black hole, so that the thermal energy in it has a very low temperature, as seen by a faraway observer.

It may be that our Minkowski & newtonian model solves the information paradox.

A black hole radiates away its entropy as thermal radiation. Since there are many degrees of freedom, the temperature of a solar mass black hole is extremely low, and the radiation is very weak. The information in the entropy is carried away in the thermal radiation. There is no information paradox.

A similar effect happens with a neutron star, too. A faraway observer sees it having a lower temperature than a local observer.

Could it be that particles inside a black hole collide and produce gravitons? Gravitons collide, too, and produce smaller gravitons. The kinetic energy and heat escape quickly into a huge space of gravitons. There are so many degrees of freedom in the space that the temperature becomes extremely low.

A similar process cannot happen in electromagnetism because the photon has no electric charge.

Maybe the thermal radiation from a black hole is very low-energy gravitons?

The final fate of a black hole

A black hole keeps radiating as long as its entropy is above the minimum possible level. It is like a block of glass which is cooling down in space.

Conservation laws suggest that the black hole will keep baryons and other matter inside. If matter remains inside the black hole, the final state is some kind of a "crystal" where matter has reached its minimum energy state. The final fate of a black hole is just like the final fate of a block of glass.

https://www.nature.com/articles/248030a0

If all matter somehow decays into massless particles, then the final fate probably is a "black hole explosion", about which Stephen Hawking speculated in his 1974 note in the journal Nature.

Conclusions

General relativity has yet another problem: the one-way membrane in a black hole breaks rules of classical thermodynamics.

Stephen Hawking tried to remove the problem with Hawking radiation, but could not solve the information paradox.

We will investigate if our model solves both problems. Since our model is classical physics, thermodynamics should work in the right way.

The analogous problem with neutron stars is a good starting point.

Electric repulsion can cancel ALL effects of gravity? The origin of inertia and the flatness problem

Our previous blog post brought up the question what happens if the Coulomb force cancels the attraction of gravity between masses m and M. Let M be a large mass and m a test mass.

In the 18th century newtonian mechanics, if the force between m and M is canceled, then gravity has no effect whatsoever.

General relativity claims that the large mass M determines the "geometry of spacetime", and that electric repulsion cannot cancel its effects.

For example, the speed of light close to M is slower, as seen by a faraway observer. The mass m cannot move faster than a photon, claims general relativity.

Shipping of energy between different force fields

                     flow of energy

          M  -------------------------------> m

                                                  s

           ●                                <---   •

          Q  <-------------------------------  q

                     flow of energy

We have two test bodies: a large mass/charge M/Q, and a test mass/charge m/q.

In the diagram, there is attraction of gravity between M and m, and repulsion of the Coulomb force between Q and q. The forces cancel each other.

If we move m a short distance s closer then energy seems to flow to both directions. Is the total energy shipping distance zero? That is, does the energy from the large body ● take a shortcut and go directly back to ●, without passing through the test body • ?

             • e- electron 

            ● proton                      • q test charge

In the case of positive and negative electric charges, the answer is clear: energy flows do take any shortcuts available.

Earth contains some 2 * 10⁵¹ protons whose combined charge is Q = 3 * 10³² coulombs. The combined Coulomb force on a single electron on the surface of Earth is

       F = k Q e / r²

          = 1.4 * 10¹⁰ N.

If we move the electron one meter closer to Earth, we ship

       1.4 * 10¹⁰ J

of energy over 6,000 kilometers. That is equivalent to shipping 10¹⁷ J, or one kilogram of matter, over 1 meter! The inertial mass of the electron would be two kilograms, instead of 10⁻³⁰ kg.

The 10¹⁷ joules of energy cannot travel through our test electron. The energy must go directly from the fields of protons to the fields of electrons inside Earth.

Hypothesis. Shipping of energy always takes the shortest route between fields, even if the fields are different force fields, like gravity and the Coulomb field. Also, if an object appears electrically neutral (because of quantum mechanics, for example) then the shipping distance of electric energy is zero. A neutral object does not contribute to the inertial mass of a test charge.

Our hypothesis does not differentiate between different force fields. If we have a hamiltonian where different force fields have complex interactions, we cannot in a reasonable way tell what part of an effect is due to which field. Our hypothesis is compatible with complex hamiltonians.

Shipping of energy: the concrete view

We might be able to define the energy shipping distance in a very concrete way. It simply is distance over which we ship recoverable energy.

                 W energy               W

              pulley                     pulley

                  O                             O

                 |                               |

                 |                               |

                 |                               |

                 ¿                              ¿

                 • e-   ----------------->

                                 ●   Q  positive charge

A positive charge pulls on an electron. We lower the electron with a pulley and harvest the energy W. Then we move the electron horizontally and lift it with another pulley. The second pulley uses its quota of energy, W.

The process shipped the electron to the right and the energy W to the left. The electron possessed this energy W as extra inertia when we were moving the electron to the right.

Now imagine that we cancel the charge Q by putting an opposite charge -Q close to it. We can no longer ship the energy W between the pulleys. There is no extra inertia on the electron.

If we cancel the pull of gravity with electric repulsion, we cannot concretely move energy, like in the pulley example. This supports our Hypothesis that energy does not differentiate between force fields.

The gravity of galaxies: how much energy we ship when we move a test mass? Mach's principle

Could it be that the entire inertia of a kilogram of mass comes from its attraction with masses far away in the universe?

Let us do a quick calculation. The mass density of the close visible universe is assumed to be

       D = 10⁻²⁶ kg/m³.

The "current" radius of the visible universe is 60 billion light-years, or

       R = 6 * 10²⁶ m.

The total mass is

       M = 2 * 10⁵⁴ kg.

Let us calculate the inertia of a test mass m in a horizontal movement at a distance R from a mass M.

The Schwarzschild radius of the mass M is

       r_s = 2 G M / c²

              = 2.6 * 10²⁷ m

              = 4.3 R.

At the distance of the Schwarzschild radius, the potential of gravity is -m c². The extra inertia from the force field is then m.

Since R < 4.3 R, our mass m has 4.3 m as the extra inertia?

The whole universe seems to be inside the Schwarzschild radius of its own mass, but quite narrowly.

It looks like that the entire inertial mass of our test mass m might come from its gravity  with the rest of the universe. This would satisfy Mach's principle.

Also, the inertial coordinate system would be determined by other masses in the universe. The other masses would do frame dragging which totally determines what coordinates are inertial. This is Mach's principle.

Close to the horizon of a black hole Mach's principle holds in the sense that the black hole almost totally determines what is the inertial coordinate system. An observer is like in "syrup" and follows movements of the black hole, even if we accelerate the black hole with some mechanism.

Let us look at the Big Bang. Suppose that the gravity charge of elementary particles stays constant. In the early universe, the inertial mass of a particle could have been much larger than now. What would that imply?

If inertia is born from gravity, that would explain why the inertial mass and the gravity charge have a constant ratio for all known objects.

There may be a vicious circle in our definition of inertia. If we move a test mass m, then energy is shipped in the global gravity field. But the shipped energy itself holds mass-energy, and causes more energy to be shipped in the global gravity field. How do we stop the vicious circle?

The flatness problem of cosmology

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

Our Hypothesis solves the flatness problem of cosmology.

Let us assume that the energy of mass is

       E = m c²,

like it is in special relativity. In special relativity m is the inertial mass, since there is no gravity in special relativity.

According to our Hypothesis, the inertial mass of a body is its negative potential in the gravity field of the universe.

https://en.wikipedia.org/wiki/Zero-energy_universe

However, we have to differentiate between the negative potential energy of a test mass, and the average binding energy of a test mass. The negative potential usually has a larger absolute value.

The extra inertia for a test mass inside a neutron star is the negative potential. It is not the average binding energy of the neutron star. The potential usually is double the binding energy.

Suppose that the universe is empty, except for the neutron star. The energy in the inertial mass inside the star will appear to be double the binding energy. That is close to what we are currently observing in the universe.

Does this bring problems in the early universe? Or do laws of nature then, at the small scale, appear to be identical to the ones now?

Electromagnetism contains all the phenomena of general relativity: implications for quantum gravity

Gravity is not a special force in our Minkowski & newtonian model. It turns out that electromagnetism contains analogues for almost all the peculiar phenomena of general relativity.

Slowing down of clocks

Let us make a mechanical clock whose parts possess an electric charge. If we put the clock close to a large electric charge Q, then the parts have more inertia => the clock ticks slower.

All charged objects have more inertia close to Q. Everything with charged objects happens slower. As if time itself would have slowed down.

Since all material objects possess a gravity charge, in general relativity one is led to think that time, as an abstract entity, really has slowed down in a gravity field. In this blog we do not believe that time itself slows down. We want to use the straight Minkowski time.

Can we cancel the extra inertia of a gravity field with electric repulsion?

Question. A test mass m in the gravity field of a mass M acquires as the extra inertia the work W which the gravity of M did when we lowered m close to M. Suppose that we cancel the attraction of gravity by putting electric charges of the same sign to m and M. Does this cancel the extra inertia in m?

If the answer to the Question is yes, that is strong evidence for our claim that time itself does not slow down in a gravity field. A mechanical clock slows down for two reasons:

1. its parts have greater inertia in a gravity field;

2. the energy to drive the clock did work when we lowered the energy down in the gravity field: there is less energy to drive the clock.

We might be able to cancel 1 with electric repulsion. Can we do something to cancel 2? Ship the energy to drive the clock as an electrically charged particle? But we cannot annihilate a charged particle without having the opposite charge.

Stretching of the radial metric

If a charge q moves radially relative to Q, then energy is shipped to q over the distance between q and Q. The inertia of q is larger in a radial movement than in a horizontal (tangential) movement. As a result of this, charges tend to move slower in the radial direction. It is as if the radial metric would have stretched. The charge Q makes radial distances to appear longer.

Frame dragging

If a test charge q goes closer to Q, and Q is moving, then q is dragged to move along Q, because q gains more inertia in the field of Q.

To prevent q from following Q, we have to apply a force. The "inertial frame" is dragged to move along Q.

Precession of the perihelion of Mercury

Let us have a test charge q in an eccentric orbit around a charge Q. When q comes close to Q, it acquires more inertia. It takes q a surprisingly long time to pass Q, and Q has more time to give an impulse to q. This larger impulse makes q to turn more. The elliptic orbit of q precesses around Q to the same direction as q orbits Q.

Bending of light close to the Sun

Since the photon possesses a charge in gravity, its path is bent when it passes close to the Sun.

The analogue is a particle q which has an electric charge and moves almost at the speed of light. When q passes close to Q, the path of q is bent because of the electric Coulomb force. Also, q acquires extra inertia in the field of Q, and moves slower, just like a photon moves slower close to the Sun. The slowdown bends the path of q toward Q.

Gravitational waves and the undulating metric

If we move a test charge inside an electromagnetic wave, the test charge has some extra inertia since energy is shipped around. This creates similar effects to the ones in a gravitational wave.

Gravity of pressure

Let us have a test charge q bouncing back and forth in a long box. The box has "pressure" in it. We lower the box toward a large charge Q, in such a way that q bounces radially relative to Q.

                        box

                         ----

                        |   |      ^

                        |   |      |

                        | • |     q   bounces

                        |   |      |

                        |   |      v

                         ----

                          |

                          |

                          v

    

                          ●  Q

There is some frame dragging. The "natural inertial frame" for q is somewhat "attached" to Q. The test charge q sees that the bottom of the box is moving downward in this natural inertial frame. When q bounces from the bottom, q loses some momentum in the natural inertial frame. It is like pressure doing work in a box whose volume is growing. Where does this work go? It might go to pulling Q closer to q. That is, "pressure" in the box would attract Q.

We have to think about this in more detail. Frame dragging and the natural inertial frame need clarification.

Conclusions

Electromagnetism contains analogues for all peculiar phenomena of general relativity. This is strong evidence for our claim that gravity is an ordinary, newtonian force, and that gravity does not affect the underlying metric of spacetime in any way. The underlying metric is the Minkowski metric.

A quantum theory of gravity should be quite similar to the quantum theory of electromagnetism. In electromagnetism, the phenomena with the "metric" are side effects of the electromagnetic field. The same holds for gravity: the apparent metric of spacetime is just a side effect of a very ordinary newtonian force field.

The role of negative pressure in general relativity versus our Minkowski & newtonian model

We have been struggling to find substantial differences between general relativity and our own Minkowski & newtonian model. For example, both predict the same Schwarzschild metric around a spherically symmetric mass.

We mentioned in our previous blog post about the Alcubierre drive that negative mass-energy behaves very differently in general relativity versus our own model.

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

The Einstein field equations are linear in the Ricci curvature. If we flip the sign of the stress energy tensor T on the right side, then the sign of the Ricci curvature is flipped.

Thus, if positive mass causes positive curvature, then general relativity predicts that negative mass flips the sign of the Ricci curvature of the metric. Time close to negative mass would run faster than in the surrounding asymptotic Minkowski space. Radial distances around negative mass would be contracted. This would open doors for faster-than-light travel. By faster than light we mean that one could move from one location in the asymptotic Minkowski space to another location faster than would be allowed by the Minkowski metric, if the Minkowski metric would hold everywhere in the space.

                             light

                 x • ---------------- O --------------- • y

                                   donut of            

                                negative mass

An example: we want to send a signal from x to y faster than light in the Minkowski space. The solution is to put a donut of negative mass between x and y. A Minkowski observer sees light to move surprisingly fast through the donut. It is like a gravitational lens, but instead of slowing down light, it makes light to go faster.

In our own Minkowski & newtonian model an interaction can only increase the inertia of a test mass, which means that clocks will run slower, and optionally, some distances become longer. Thus, in our model the donut has the opposite effect: it slows down light.

As far as we know, there does not exist negative energy. However, there does exist negative pressure.

For a massive rod, negative pressure acts like negative mass

Suppose that we have a very rigid rod. A gravitational wave arrives, normal to the rod, and causes spatial distances along the rod to get longer for a while.

                                    s

                            m  • -->

            =====================  rod

                       ---------------------- 

                                                         ^   gravitational 

                                                         |  waves

                       ----------------------

We assume that the wave was born from a movement of positive masses, for example, from a binary black hole.

We assume that the gravitational attraction of the rod is stronger than repusion generated by negative pressure in the rod. This is the weak energy condition.

                                   s

                             m • -->

        =======                      =======   rod

             M1                                M2      positive mass

         - - - - - -                         - - - - - -  negative pressure

In the diagram we have removed the central part of the rod to concentrate our attention on the ends of the rod.

If there is no negative pressure, then the masses M1 and M2 cause extra inertia to the test mass m, because if we move m to the right, then energy flows from the field of M2 to the field of M1.

However, if there is a little of negative pressure, then energy flows also to the opposite direction. We claim that the opposite energy flow decreases the extra inertia of the test mass m if we move it a short distance s.

What does the rod do to the wave?

The stretching of the metric on the upper side of the rod is decreased by the negative pressure. That means that the wave has problems going "through" the rod. Some of the wave is reflected back.

The quadrupole moment of the rod, relative to the center of the rod, is, in a sense, reduced by negative pressure. The negative pressure in the rod acts as a source of a new gravitational wave and causes some destructive interference to the passing gravitational wave.

The behavior for a massive rod is similar in general relativity and our model.

What if we would have a gravitational wave generated by a movement of negative mass? The negative pressure could strengthen the wave? That cannot happen. We have to think about this.

Gravitational waves generated by pressure: does pressure possess a charge?

In our Minkowski & newtonian model gravity is an ordinary force whose charge is mass-energy. Accelerating mass-energy generates waves just like an accelerating electric charge.

But in general relativity, changing pressure generates gravitational waves, too. How do we explain those waves in our model? Does pressure possess a charge?

In our model, the attractive or repulsive force on a test mass is an indirect consequence of the gravity of the test mass. The Schwarzschild metric around the test mass interacts with the force that causes the pressure. It is a different interaction than direct gravitational attraction, or is it?

Should we assign a charge to pressure?

Another option is to claim that any long-distance interaction creates "waves" if an object of that interaction is in an accelerated motion or periodic motion.

Then we do not need to specify if pressure has a charge. It is enough to claim that "waves" will carry that interaction over large distances.

In this blog we have tossed the idea that a wave carries a "copy" of the transmitting system over a large distance, and puts it close to the receiving system. The copy is distorted, though, because it does not contain longitudinal fields or static fields, and the copy is "weaker" than the original transmitting system.

Thus, it may be that we do not need to specify if pressure has some kind of a charge.

Another option is the following: the interaction with pressure is relayed from the test mass through gravity, just like gravitational attraction. We could claim that any interaction which is relayed through gravity can cause gravitational waves. That is, the force on the test mass is defined as the gravity field. We assume that the test mass does not possess any other charges but mass.

What about more complicated interactions? Suppose that the test mass also possesses an electric charge, and the hamiltonian of our system contains some complex formula of gravity and the electric field. If we move the system, does it generate some kind of "hybrid" waves?

Maybe the "copy" model is the way to define the waves generated by very complicated interactions?

We need to think about this in detail. Can we always treat pressure as if it would generate a gravity field? That would be simple, but do any problems arise?

Negative pressure in general relativity generates waves which have the opposite phase to our Minkowski & newtonian model?

The weak energy condition prevents us from creating a static negative gravity field in general relativity. But it does not prevent us from creating a gravitational wave which does have a negative gravity field. For that, we need negative pressure.

                                      rod

    pull  <----   ==================   ----> pull

                                  mass M

We periodically pull the ends of the rod, which generates periodic negative pressure inside the rod. The rod also contains the mass M, which ensures that the sum of gravity forces is always attractive, despite the negative pressure.

However, the gravitational wave, which is created, ignores the static field of the mass M. The wave is born from the negative gravity of the negative pressure.

We assume approximate linearity: static fields do not produce a wave. The wave is born from the dynamic field.

Let us ignore the mass M and concentrate on the negative pressure. In the diagram we have cut the rod to two pieces, to emphasize the ends of the rod.

                                   s

                                 • -->

                                m test mass

          =======                 =======    rod

          - - - - - - -                  - - - - - - -    negative pressure

In the Minkowski & newtonian model, the negative pressure produces repulsive fields. If we move the test mass horizontally a short distance s, that ships energy between the fields. The inertia is higher, and horizontal distances are longer.

In general relativity, negative pressure generates a wave which makes horizontal distances shorter? We have to check this. Probably no one has calculated what kind of waves pressure generates in general relativity.

Conclusions

Lots of open problems remain.

If we have a gravitational wave generated by negative pressure, how does the wave react to a rigid wall?

Can we consistently say that an indirect interaction of the field of a test mass and pressure causes "gravity"? Does that "gravity" generate gravitational waves just like masses do?

Pressure generates "gravity". What is the analogous effect in electromagnetics. Can pressure generate an electric field?

The Alcubierre drive is forbidden by the Minkowski & newtonian model: even negative mass makes trips longer, not shorter

Miguel Alcubierre in 1994 proposed a faster-than-light travel method in general relativity.

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

If the drive were possible, then we would have all the paradoxes of traveling back in time.

The drive works by contracting the spatial metric in front of the drive, using negative energy.

What does our Minkowski & newtonian model say about the drive?

In our model, all interactions, even interactions with "negative energy", slow down time, and can also make spatial distances longer, never shorter. This is because any interaction increases the inertia of a test mass. An example is the Schwarzschild metric: time slows down and radial distances become longer.

Negative energy would create a repulsive force of "gravity". In our model, that is analogous to the repulsive force between electric charges of the same sign. If we move a test charge, then we also move some energy in the field => the inertia is larger.

We conclude that the Alcubierre drive is not possible in our model, even with hypothetical "negative energy" matter.

In the Conclusions of the previous blog post we discussed the possibility that gravitational waves in general relativity may allow faster-than-light travel. Then a kind of an Alcubierre drive could be possible in general relativity, using gravitational waves. There would be no need for negative energy matter.

We repeat our opinion that faster-than-light travel has to be forbidden in an acceptable theory of physics. We cannot allow the time travel paradoxes.

Quadrupole wave: a better model

The quadrupole wave model which we introduced in the previous blog post does not reproduce the propagation of energy correctly. The area of the XXXX zone only grows as 1 / R. Since the electric field in the zone is 1 / R, the transmitted total power goes as 1 / R, which cannot be correct.

Let us introduce a model which does not lose energy.

Let us have a binary black hole, or two equal electric charges orbiting each other. The fundamental reason why destructive interference does not cancel the entire wave is that the dipole wave produced by a single orbiting charge is asymmetric: the wave is stronger on the outer side than on the side of the center of the orbit. The weak wave is also in other ways deformed relative to the strong wave.

                                              ^

                                              |

  weak wave           •            ●         strong wave

                             center      Q

                

Let us then have two symmetric orbiting charges.

                                               ^

                     Q                       |

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

                     |           r           Q

                     v

The quadrupole wave is only zero on the axis which is normal to the screen and goes through the center of the system. On the right side of the system it is the wave produced by the right Q which dominates.

Let us try to draw the asymmetric wave of Q.

                      

                               E1             XXXX

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

                                                           | X

                                                           | X   E1

                                                           v

                                                           

                                                           

                          •         ●                    

                                     Q1                

                                                          

                                                           ^

                                                           |  X E1

                                                           |  X

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

                                E1            XXXX

We assume that Q1 is moving toward the screen. We assume that the wave is hugely asymmetric: there is no field at all on the far left side.

If we would draw the wave of the other charge Q2, it would cancel the wave of Q1, except in the zones marked with XXXX.

Now we see that the volume of the XXXX zones actually grows as R². Energy is no longer lost.

The asymmetry of the dipole wave

Let us construct the upper part of the quadrupole wave using polarization of a hypothetical material.

E1 field

strength

    ^

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

    |                  /                          \

    |             /                                 \

    |        /                                        \

    |      /                                            \

    |                        •         

                        center

     ---------------------|----------------------|-----> x

                              0                          R

The diagram shows the strength of an asymmetric field E1. The strength of E2 is the mirror image around the center.

If we implement the fields using polarization, what is the energy shipping distance S?

It might be

               R

       S = ∫  |E1|  -  |E2| dx     /     max(|E1|).

           0

Since E2(x) = E1(-x), the integral is a measure of the asymmetry of E1(x) with respect to the plane x = 0.

S is the "average translation" we have to do to convert the graph of |E1| to the graph of |E2|.

If Q1 and Q2 were static, then |E1| would be symmetric around the location of Q1, and S would be the separation r of Q1 and Q2. In a dynamic wave, S is not necessarily r. Could it be as large as 4 r?

How large is the asymmetry of a dipole wave?

                  reflection          P           symmetry

                                            |

                                            |

                                            |

                                       a

                                    <--- ● Q1

           -------------|--------------|--------> x

                           0               r / 2

Suppose that Q1 is orbiting around the center in the x-y plane. It is located at r / 2 on the x axis.

It creates a dipole wave whose absolute field strength |E1| is almost reflection symmetric relative to the plane P which goes through Q1 and is parallel to the y-z plane.

Assume then that Q2 is orbiting at x = -r / 2.

Assume first that |E1| would be perfectly symmetric relative to P. Then we could translate |E1| to |E2| by moving Q1 left the  distance r. That is, S = r.

However, if |E1| is not completely symmetric, but stronger on the right side of P, then the average translation S is larger than r.

Let us try to calculate what S might be.

Assume that the static electric field rotates with Q1 around the center. We imagine that Q1 and its electric field are "glued" to a rod which is attached to the center and rotates around the center.

                                                             rotation

                                                           ^

                                                           |

              • ---------- ● ----------------------- rod

  center     r / 2     Q1

Since the electric field cannot move faster than light, the dipole wave is "detached" from the static field at the distance of 1 radian, or λ / (2 π).

The detachment distance on the left is the distance r farther from Q1 than on the right. The field of the wave on that side might be

        2 r / [λ / (2 π)]

weaker because the static field is weaker at a larger distance, and the angular velocity of Q1 is smaller there. If r is 1% of 1 radian, then the static field is 2% weaker and the angular velocity is 1% less. Since the point of detachment is 1 % farther from Q1, and the wave field goes as 1 / R for the distance, we conclude that the dipole field is 2% + 1% - 1% weaker on the left side than on the right side, at the same distance R from Q1.

How much does the asymmetry contribute to S?

Let us calculate the contribution to S at the distance where the wave is "detached" from the static field at 1 radian.

Let us assume that 1 radian is one length unit, r < 1, and max(|E1|) = 1. Then the integral for a symmetric graph is

        1

       ∫  |E1|  -  |E2| dx

      0

       ≅ r.

If the field |E2| is only 1 - 2 r times the field |E1|, then asymmetry adds roughly 2 r to the above integral.

We conclude that the average translation S might be roughly 3 r. That is the distance over which energy is shipped if we move a test charge q at a distance 1 from the orbiting charges.

Our calculation is extremely crude. Mainly, it shows that the energy shipping distance S might be several times r.

The asymmetry of a dipole wave diminishes at large distances R

                             r <----

The diagram of the previous blog post, by Daniel V. Schroeder (1999), helps us to analyze the asymmetry of a dipole wave.

A charge is suddenly moved to the left a distance r, and then stopped. We have to connect the electric field lines in the circular region. The density of the lines in the region becomes large, which means a lot of energy concentrated there. The circular region is the electromagnetic wave which propagates at the speed of light. The radial lines are static electric fields.

Now we see that when the radius of the circle, R, grows, the wave becomes less asymmetric, relative to the size of the circle. The distance r is a kind of a measure of asymmetry, and r stays constant.

We did not find in literature calculations of the asymmetry of a dipole wave. The symmetry becomes (relatively) almost perfect at large distances R. That may be the reason why authors have ignored this question.

If the asymmetry would stay as is for large R, then our calculation of the integral |E1| - |E2| from 0 to R, would show that S grows without bounds, which makes no sense. If the asymmetry is ~ 1 / R, then S stays constant.

Conclusions

We used a model where an electromagnetic wave is created by polarization of a material. The energy flow is

       the electric field <-> "elastic energy" of the material. 

We do not yet know if we can generalize this to waves in a vacuum, where the energy flow is

       the electric field <-> the magnetic field.

Calculating the energy shipping distance S in a quadrupole wave is hard, except if one uses a numerical computer calculation (?).

We argued that the energy shipping distance S in a quadrupole wave is several times r, but does not grow as the distance R from the source increases.

We argued that the relative asymmetry of a dipole wave grows smaller as R increases, probably by a formula 1 / R.

We were able to explain why a quadrupole wave transmits a constant power at large distances R, even though the energy shipping distance S stays constant.

We did not need to assume that the waves of Q1 and Q2 "exist" separately. We can calculate the effect from the sum of these waves.

Our arguments show that the energy shipping distance S might be 2 r or 4 r, which is required in our model to explain the metric perturbation of a gravitational wave. In this blog we claim that stretching of the spatial metric is caused by inertia which is larger in one direction than the other.

We need to analyze how a mechanical clock ticks inside a gravitational wave. That will tell us what is the perturbation of the metric of time.

In our blog we claim that any interaction increases the inertia of a test mass. This implies that a wave can only slow down time or make spatial distances larger. No wave can enable faster-than-light communication between points inside or outside the wave.

General relativity may allow faster-than-light communication inside a gravitational wave. If that is the case, we think that would be a fatal blow for general relativity, since that would bring all the paradoxes of time travel.

If general relativity allows clocks to run faster inside a gravitational wave than in the asymptotic Minkowski space, that is a breach of the weak energy condition. We need to check literature if anyone has studied this.

What is the distance over which a dipole or a quadrupole wave ships energy?

If a static electric charge pulls on a test charge over a distance R, and we harvest energy from the process by moving the test charge, then energy flows from distant places to the test charge. The energy flow shows up as extra inertia in the test charge. Shipping an amount of energy E over a distance s is like shipping a mass E / c² over s.

What is inertia? Inertia means that we must commit some amount of energy as the kinetic energy of an object A when we move the object. If the object A is attached with a rope to another object B, then we must commit more kinetic energy. The object B does not need to consist of atoms. B can also be energy which flows around in a field, or between atoms.

Energy that is harvested from an electromagnetic wave

              |

              |

              |  r

              |                            • q

              |                            | s

              |                            v

             ● Q        

              |               

              v

Let us assume that a charge Q is moving up and down at a relativistic speed over a distance r. The charge Q produces a dipole wave.

The wavelength of the dipole wave might be

       λ = π r

if Q moves very fast. The wave is "detached" from the local field at a distance ~ 1 radian or r / 2. Let us put the test charge q at the distance r / 2. Let us assume that in the diagram, the charges Q and q are opposite. Because of retardation, q "sees" Q quite far down, and the wave pulls q down.

If we move q a short distance s down, from where is the energy shipped to q? We can harvest the energy

       F s,

where F is the force which the wave exerts on the test charge. From which location does that energy come from?

An equivalent dipole wave, but a different quadrupole moment?

Let us look at the electric field strength E which the oscillating charge produces. We can produce the same |E| and the wavelength λ either with:

1. a large charge Q which oscillates over a short distance r, or

2. a smaller charge Q which oscillates over a large distance r, and may move at a relativistic speed.

The oscillating dipole moment

        Q r

is the same in both cases, but the oscillating quadrupole moment

       Q r²

is larger in case 2.

In gravity, the stretching of the spatial metric in a gravitational wave is proportional to the quadrupole moment - it is not proportional to the dipole moment.

This follows from the definition of a metric. Let the metric be 1 to the direction of the x axis, and 1 + d to the direction of the y axis, where d > 1 is small. If we rotate the coordinate axes through an angle α, then the metric on the new x axis is

       1 + sin²(α).

A small α is linearly proportional to r. Thus, r² is the relevant figure.

Linearized Einstein equations are wave equations for each component. The behavior of the metric close to the source must determine the amplitude of the wave.

We see that if we have almost planar dipole waves which have (almost) equal field strength, their effect on the metric depends on r, or on the quadrupole moment of the dipole.

We have been claiming that the stretching of the spatial metric is due to energy being shipped over a large distance. We have to show that this distance is linearly proportional to r. It does not depend on the wavelength λ.

In other words, the mechanism must be able to distinguish between almost planar waves which have almost the same field strength, but a different r or a quadrupole moment. How is this possible?

The distance over which energy is shipped: the Edward M. Purcell diagram

https://physics.weber.edu/schroeder/mrr/MRRtalk.html

Edward M. Purcell derived the Larmor formula using the diagram below. The talk in the link and the diagram are due to Daniel V. Schroeder (1999).

A charge is suddenly moved a distance r to the left, and then stopped. The diagram shows the electric field lines after some time. The field lines far away are radial from the original position of the charge. Close to the charge, the field lines are radial from the new location. There are two zones, which are separated to a circular transition zone.

Imagine a very lightweight opposite test charge in the outer zone. It will be pulled to the left when the transition zone passes it. The following claim looks natural, looking at the diagram:

The energy to pull the test charge to the left comes from the field lines of the inner zone, from a distance r to the left from the test charge.

The quadrupole wave as two dipole waves separated by a distance r in a polarizable material

Suppose that we have two equal oscillating charges Q separated by a distance r. The test charge q has the opposite sign.

  

       <--------------------- ~ R -------------------->    

          E'

        ----->

       XXXX  --------------------------------------> E2

 E1 <---------------------------------------  XXXX  

                                   • --> s

                                  q

                                  ^

                                  |

                                  |

                                  R

                                  |

                                  |

                                  v

                       ● <--- r ---> ●

                       Q1             Q2

The charges Q oscillate in the horizontal direction over the distance r. They could also orbit each other, and the plane of the orbit would be normal to the screen. The test charge q oscillates over a short distance s at the same frequency as charges Q.

We have schematically drawn the electric fields E1 and E2 of the waves produced by the charges Q1 and Q2 at a distance R.

Note that our approximation of the form of E1 and E2 is extremely crude.

The fields do not overlap completely. There is a displacement of r in the fields. We have marked with XXXX the zones where there is no overlap.

The displacement obviously creates the quadrupole wave. There is no complete destructive interference of the fields in the XXXX zones.

How to model the electric fields E1 and E2? The fields are only λ / 2 thick but extend over a large distance R. We cannot create such narrow fields easily with static charges in empty space.

Let us instead assume that the electromagnetic wave propagates in a polarizable medium.

Also, let us change to a frame which moves at a relativistic speed up in the diagram. The wave then is redshifted and its wavelength can be many times R.

                                  |   q

                                  v

      C1     C2

       -         +                                 +         -

       -         +                                 +         -

       -         +                                 +         -

        XXXX ------------------------------------> E2

    E1 <---------------------------------- XXXX

        <----->                                   <----->

            r                                            r

In the new frame, the test charge q moves at a relativistic speed downward.

The polarization of the medium carries both dipole waves. We have marked in the diagram the extra charge which the polarization for each field E1 and E2 have concentrated in certain areas. It is like capacitor plates.

Let us imagine that we move q a short distance s to the right and q is a negative charge. Then the charge of C2 pulls q and C1 repels q. Energy is shipped from C1 to C2. The energy moves over a distance r.

We did not need to assume that the fields of Q1 and Q2 "exist" separately. The sum field E1 + E2 was enough to explain the shipping of energy.

In a polarizable material, the energy of the wave flows between the electric field and the elastic energy of the "attachments" of the electrons to the atoms or molecules. In an electromagnetic wave in empty space, the energy flow is between the electric field and the magnetic field. Can we somehow extend the argument above to empty space?

Conclusions

We argued that inside an electromagnetic wave, moving a test charge q ships energy over a distance r, where r is the amplitude of the movement of the charges which generate the wave.

We have claimed that for a static field of a charge Q, the energy is shipped over the distance r of the test charge q from Q. In the last section above, we used our claim to derive the shipping distance in a wave in a polarizable material.

In empty space, the electric field is generated by a changing magnetic field. How could we show that the shipping distance is r? Maybe we have to take that as an axiom?

In a previous blog post we derived the gravitational wave metric by assuming that energy is shipped over 2 r (or actually, 4 r, because in the earlier blog post we counted the two dipoles separately). Why is it not over r? We have to find out what is the reason for the discrepancy.