Coupling gravity to electromagnetism helps to position energy and momentum?

In the previous blog post we got the insight that gravity is a way to "grab" moving energy in an electromagnetic field.

If the electric field is a complex "mechanical machine", then the packet of energy in E E' could move without inertia

                           E + E'

                           |  |  |

           --------------------------------------  - cylinder

               - rod   ---------  ---> v

           --------------------------------------  - cylinder

                           |  |  |

The cylinder is uniformly electrically charged and its field is E. The rod is uniformly charged with the same sign (-), and its field is E'.

The field E + E' contains an extra energy density from the overlap:

       ε₀ E E'.

How do we know where is that extra energy located?

                |

            ---------  -

            ---------  +

                |

We can use capacitor plates to harvest the energy in the field very quickly, by moving the plates apart. 

How quickly? The plates could, in principle, move at a relativistic speed. It is a reasonable assumption that the energy of the field E is stored locally. We know that an electromagnetic wave stores its energy locally – that is further evidence.

It could still be that the extra energy E E' does not add to the inertia of the rod. We can imagine that the field is a complicated mechanical machine which takes "instructions" from the rod, and moves E E' around. The rod does not need to give momentum to E E'. The machine could be lossless and would not lose energy in friction.

We cannot "grab" the packet E E' directly because energy is not coupled to electromagnetism.

Gravity comes to the rescue: it allows grabbing an energy packet

The energy in the packet E E' probably does gravitate. That offers us a method to "grab" the momentum in E E'.

Let us assume that mass-energy (energy divided by c²) in E E' is much more than in the rod and its own field E'.

                     ● M

                   E + E'

                   |  |  |   ---> v

We can use gravity to make the energy E E' to "bump into" a mass M. The energy in E E' gives up some of its momentum and kinetic energy to M.

Could the system still conspire in the way that M cannot receive a substantial amount of momentum? It is possible. Then, the mass M has to take part in the conspiracy. That is unlikely.

Conclusions

It is likely that the extra energy in the combined field E + E' does add to the inertia of the charge which produces E'.

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

This solves the Klein paradox, as we wrote on October 31, 2018.

However, we have to solve the mystery why the electric potential does not affect the spectrum of an atom. Also, we have to analyze how an electron or a positron behaves with a potential energy less than -511 keV.

Kinetic energy of overlapping electric fields: electromagnetic action ignores it?

UPDATE September 10, 2024: Our blog post today suggests that the extra field pressure of the field E + E' contains "moving" pressure whose speed is v / 2. The corresponding Poincaré stresses contain the canceling negative pressure which moves at the speed v / 2. 

----

UPDATE September 9, 2024: The post contained an error. We cannot make horizontal forces negligible by making the cylinder very narrow and reducing the charges. This is because in a narrow cylinder, the lines between two random charge elements become more horizontal. In the Maxwell stress tensor for the cylinder, the horizontal positive pressure of the field close to the cylinder has the same formula

       1/2 ε₀ E²,

as the energy of the field.

----

Our August 24, 2024 post uncovered a possible shortcoming in the electromagnetic action, or Maxwell's equations. The equations may be aware of the kinetic energy of a static electric field. But are they aware of the kinetic energy of the energy in overlapping two fields?

Kinetic energy of the electric field between capacitor plates, sideways movement relative to E

             |

        ----------  -    capacitor

         |  |  |  E

        ----------  +

             |

              • ---> v

       observer

Let us calculate the kinetic energy of the field in the case of the above diagram. The length contraction is by the factor

       1 / γ  =  sqrt(1  -  v² / c²)  =  1  -  1/2 v² / c².

The energy density ρ, i.e., the value of 

       ρ  =  1/2 ε₀ E²

in the moving frame of the observer is increased by the factor

       γ²

in the moving frame of the observer. The volume of the electric field is reduced by the factor

       1 / γ.

The total energy is increased by the factor γ. This makes sense. In the moving frame, there is also a magnetic field

       B  =  1 / c²  *  v E.

Its energy density ρ' is

       ρ'  =  1/2 * 1 / μ₀  *  B²

             =  1/2 * 1 / μ₀  *  1 / c⁴  *  v² E²

             =  1/2 * ε₀ / c²  *  v² E².

The density ρ' is insignificant for v << c.

Capacitor plates, movement parallel to the field E: problematic

                   F

              <------>

               

              +   E    -

         ----| === |---    capacitor

                   • ---> v

           observer

  ^ y

  |

   -----> x

What if the observer moves to the direction of the electric field lines? There has to be a force F which keeps the plates of the capacitor from touching each other.

In the moving frame of the observer, the electric field

       E

stays the same. The plates are closer to each other, due to length contraction. The electric field has lost energy.

Do the effects of F and E cancel each other, so that there is no kinetic energy in the moving field?

Or is this another case where electromagnetism does not understand the kinetic energy of two overlapping fields?

Special relativity comes to the rescue: the problem solved

https://en.wikipedia.org/wiki/Electromagnetic_stress%E2%80%93energy_tensor

The electric field between the plates contains both energy and a pressures. What do the pressures contribute?

In the field between the capacitor plates, there is a negative pressure in the x direction and positive pressures in the y and z directions.

The component T⁰⁰ increases T⁰⁰' in the Lorentz transformed T':

       t'  =  γ (t  -  v t).

The 00 component of T Lorentz transforms with the factor

       γ².

In the Lorentz transformation,

       x' = γ (x - v t).

The negative xx pressure component T¹¹ reduces T⁰⁰' in the Lorentz transformed T'. 

The contribution to T⁰⁰' from T⁰⁰ is an increase of

       (γ²  -  1)  *  1/2 ε₀ E²

       ≈  v² / c²  * 1/2 ε₀ E².

The contribution from the negative pressure T¹¹ to T⁰⁰' is a decrease of approximately

       v² / c²  *  -1/2 ε₀ E².

The contributions cancel each other. The electric field itself in the moving frame has the same energy density as in the static frame.

The extra kinetic energy in the moving frame is contributed by the positive pressure exerted by the force F. It is the "Poincaré stress" in this case. The force F increases the overall energy density by a factor

       (1  +  v² / c²)  *  1/2 ε₀ E².

A half of the increase v² / c² is canceled by length contraction. Thus the kinetic energy of the moving field energy W is

       (1  +  1/2 v² / c²) W.

This is correct and sensible. The kinetic energy of the system is hidden in the field of the opposing force F.

This may explain our August 2023 observation that the field E is not smart enough to understand the inertia of a test charge q moving in the field E. The inertia may be hidden in the field of the opposing force F.

The overlapping electric fields

Let us check if the stress-energy tensor and hidden "Poincaré stresses" solve the paradox of August 24, 2024.

Let us first check if it is possible in special relativity to move a packet of energy by giving it a momentum for a while, but no kinetic energy.

                      |       •    •    •    • 

                   wall       packet

The packet is a system of particles. If we let particles of packet bounce from the wall, then they gain momentum to the right, but the wall does not give them any energy. Thus, it is possible to give momentum to the system without giving new energy.

                                                    F

                            E + E'              |Poincaré stress

                                                    v 

           E   |     |  |  |  |  |     |            

            -  ----------------------------------  cylinder

                      -  -----------  rod             

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

           E   |     |  |  |  |  |     |   

                                                    ^                                                        E + E'              |Poincaré stress

                                                    F

        F' --->                                 <---  F'

Let us have a static uniformly charged cylinder and a uniformly charged rod inside it. The potential energy of the rod is expressed by the dense E lines of force emanating from the cylinder at the rod. Poincaré stresses F, F', etc., keep the cylinder from exploding from the repulsion of the charges, and keep the charge attached to the rod.

The pressures of the electromagnetic field and the Poincaré stresses must cancel each other if the system does not explode. Thus, the contribution of the pressures to the momentum or the energy of the system should be zero in most cases.

                                E                  E + E'

                o          ------------------------------------

                |\ ------------  F'' --->   ----------  rod    --> v

               /\           ------------------------------------

  observer   stick    force

                                           <---->

                                               s

The cylinder is attached to a laboratory table. An observer uses a stick to push the rod to the right a distance s. The observer does the work

       W  =  F'' s

in the pushing. Where does the energy go?

The potential energy of the rod in the field E of the cylinder is stored in the stronger field

      E  +  E'

around the cylinder at the position of the rod. There, E' is the field of the rod. We assume

       E'  <<  E.

The energy density is

       1/2 ε₀ (E  +  E')²

       =  1/2 ε₀ (E²  +  2 E E'  +  E'²).

The potential energy density there is

       ε₀ E E'.

Let us assume that the rod moves at a speed v. The Poynting vector S gives us the energy flow density of the field:

        S  =  ε₀ c²  *  (E  +  E')  *  1 / c²  *  v  E'

            =  v ε₀ E E'  +  v ε₀ E'².

                        |   | E'

                     -----------    rod   --> v

                        |   |

              -->  Poincaré  <-- 

                     pressure

The Poynting vector for solely the field E' is double of what we would expect from the energy density 1/2 ε₀ E'². But the negative horizontal Poincaré pressure inside the rod cancels one half of the energy flow. That is, a double energy flow moves to the right in the field of the rod, but a half of it returns back to the left inside the rod because of the negative Poincaré pressure inside the rod.

The Poynting vector for the combined field term E E' seems to reflect the movement of field energy to the right.

Is there a "moving positive field pressure" in the horizontal direction in the combined field E E' when the rod moves to the right? 

What about the Poincaré negative pressures inside cylinder and the rod, which cancel the positive pressure from E E'?

We can argue like the following: we move the rod forward in steps. Before a step, we take the rod far from the cylinder in the vertical direction, and store the harvested energy. Then we let the rod and the harvested energy take a step to the right. Then we push back the rod inside the cylinder. The pressures did not "move": they were created anew in the pushback. What moved was the harvested energy.

Thus, the Poynting vector for E E' does reflect correctly the energy flow, or the momentum, of the potential energy in the field.

If the rod moves faster, it is length contracted. However, the integral of the term E E' does not change, as there is a fixed number of field lines from the rod.

Electromagnetism does not seem to understand that the potential energy in E E' can gain kinetic energy. If this is the case, then it may be an error in the electromagnetic action. When the observer pushes the rod, only a part of the energy can go to the kinetic energy of the rod and its own field. The term E E' should get its share of kinetic energy, but electromagnetism is not aware of that. Or is it?

Analysis of the problem in special relativity, using a stress-energy tensor

The term E E' comes from interactions between the parts of the system. In the stress-energy tensor it just shows up as an energy density.

The pressures from E E' show up like any pressure in the tensor. How does special relativity know what pressure "moves" with an object and what does not?

If a pressure is derived from forces between particles, then it is clear that if both particles move, then the pressure "moves". If only one of two particles moves, then the pressure does not "move"?

     

          1     cylinder charges    2

          ●                                       ●

               •   • --> v'      -v' <-- •   •    pebbles

            -v' <--  •   •      •   •  --> v'     pebbles   

                               ● -> v

                       rod charge

Let us analyze this using the pebble model from our previous blog post. In this case, the observer is static relative to the cylinder. The pressure does not accelerate the rod charge. The pebbles thrown from it must have the same velocity backward and forward in the laboratory frame.

The pebbles thrown by the rod charge represent the momentum flows that it creates.

The charge 2 absorbs the momentum in the incoming flow, and sends it immediately back.

The number of pebbles flying to the right is as large as the number flying to the left. The sum momentum is zero. Apparently, the "interaction pressure" does not contribute to the momentum of the system.

Special relativity probably thinks that when the interaction energy E E' moves, it gains momentum. In this, special relativity agrees with the Poynting vector.

Does the interaction energy E E' gain kinetic energy as it moves? An indication of this would be that we can harvest from E E' more energy if it is moving. Can we do that?

           E + E'

          |  |  |

         -----------------------    cylinder

           --------  v -->    ● obstacle

         -----------------------

            rod

Let the moving charged rod bump into an obstacle. The rod gives the obstacle some momentum and kinetic energy. Let the rod stop then.

We can harvest the interaction energy E E' that the static rod had. We can also harvest the kinetic energy of the obstacle. The kinetic energy is determined by the momentum that the system the rod & the interaction energy had. Now it is clear that the interaction energy E E' must possess kinetic energy if E E' moves!

It looks like that the electromagnetic action contains an error. It does not understand complicated forms of energy.

What about special relativity itself? Where is it specified which energy "moves" and which not?

                         W extra energy

                       ####  --> v       ●  obstacle

    ##########################  energy

The hump W might acquire its kinetic energy from its own energy W. Then it would possess "no kinetic energy". It could also acquire its kinetic energy from the energy pool at the bottom of the diagram.

Bump and harvest. Let W bump into an obstacle. The obstacle acquires kinetic energy. The system W & energy then has less energy than it had earlier. The bump and harvest test is a practical way to determine the existence of kinetic energy.

There is no momentum in E E' as it moves? Yes there is

If E E' is large, then the extra pressure of the electric field is large, and the balancing Poincaré pressure is large.

In this case, Poincaré stresses are negative pressure inside the cylinder and inside the rod. Lorentz transforming a negative pressure creates negative momentum and negative energy.

Could it be that the positive momentum in the interaction E E' is canceled by a negative momentum in Poincaré stresses? Then E E' would not give any momentum to the obstacle in the collision.

The Poincaré stress inside the rod is small, since moving the charges in the rod apart does not reduce the energy E E'. The interesting thing is the Poincaré stress in the cylinder.

      -------------------------------------- cylinder charge

                          ×/\/\/\/\ ------- rod charge

      -------------------------------------- cylinder charge

                            spring

To prevent the rod from accelerating under the weak horizontal electric field, we must have a spring which cancels the horizontal field force on the rod. The spring is attached to the center of the cylinder.

If the rod charge is static, then the electric repulsion pushes the rod weakly to the right, and the pressures in the cylinder and the spring must pull the rod weakly to the left.

The analysis based on the center of mass

                         E + E'

                        |  |  |

      o /----------------------------------------  cylinder

      |\ F  <---> s   --------  rod

     / \   ----------------------------------------  cylinder

                        |  |  |

Let us assume that the cylinder and the rod are floating freely in space. A man hanging from the cylinder pushes with his hand the rod a distance s to the right, and then stops the rod. The center of the mass of the system must not change.

Suppose that the interaction energy E E' would not contribute momentum when the man pushes the rod the distance s to the right. The man feels a surprisingly weak force F resisting the movement of his hand.

The force -F pulls the cylinder to the left. For simplicity, assume that F = 0. Then the man is able to move a substantial mass-energy E E' to the right, but the cylinder remains at the original position? That would move the center of mass, which is impossible.

Could it be that the energy E E' pushes the cylinder to the left as it moves to the right?

The charge in the rod moves and produces a magnetic field B. But the cylinder does not move, and B does not exert any force on it.

The electric field of the rod does exert horizontal forces on the cylinder. However, we can lower these forces to a negligible level if we make the cylinder narrower and reduce the charges, keeping the energy E E' constant.

We conclude that the energy E E' must possess momentum if it moves.

How can the spectra of atoms stay the same under an electric potential?

We proved that an electric potential E adds an "inertia" of ~ E E' to the charged rod. If we put a hydrogen atom close to large negative charges (i.e., to a high potential energy), then the effective mass of the electron should be larger, which should dramatically alter the spectrum. But no such effect has been observed. Possible explanations:

1.    The potential energy E E' is stored so far that it does have time to react as the elecron makes one loop around the proton in 1.6 * 10⁻¹⁶ s, in which time light propagates 5 * 10⁻⁸ m.

2.   Other charges around the electron "shield" the energy E E' from moving. The proton has the opposite field.

3.   Quantum effects prevent any "visible" movement of energy outside the atom. The hydrogen atom appears electrically neutral from the outside.

4.   If we could observe the energy E E' moving outside the atom, we could probably extract energy from the movement. Then the atom would fall to a lower energy than the ground state: impossible.

How to repair the electromagnetic action?

The kinetic energy of the energy in overlapping fields is negligible in most cases. The action is approximately correct. If we add more terms to the action, in order to account for the missing energies, the action can become very complicated.

Can we observe the increased momentum (inertia) of an electron empirically?

It does not show up in the spectrum of an atom. What about observing the paths of individual electrons under a magnetic field, varying the electric potential at the same time?

On November 16, 2018 we wrote about an experiment. We have to do a net search again.

The apparatus has to be analyzed carefully. If an electron moves, it makes other electrons to move in metal parts of the apparatus. That can easily mask any effect of an electric potential.

If the electric potential does not affect the inertia of the electron at all, that is surprising. It might even break momentum conservation in nature.

Conclusions

Our reasoning above suggests that the electromagnetic action, or Maxwell's equations, is only approximate. It does not take into account complicated interaction energies.

In this blog we have been claiming for six years that the electric potential affects the inertia of a charge. We now understand the effect better, analyzing the Poynting vector and the stress-energy tensor.

Note that the electric field could, in principle, be a complex mechanical machine which moves the interaction energy E E' around "on the instructions" of the charge in the rod. Then the rod would not get extra inertia from E E'. We have not proved that E E' increases the inertia of the rod. Empirical experiments are required to establish the truth on this matter.

If we could "grab" the moving energy, as E E' moves, then we could prove that it must contain momentum. We can "grab" it through gravity! That may resolve the question theoretically.

We will also tackle the question of what happens if we put a positron near large negative charges, so that the potential is less than -511 keV. The positron has a "negative" total energy and a "negative" inertia? How does this show up in its behavior?

4/3 and Poincaré stresses

In our blog post on August 25, 2024 we realized the stress-energy tensor of the field of a static electric charge Q contains both an energy density and pressures. When we Lorentz transform to a moving frame, the pressures contribute to the momentum of the system, which is unintuitive. The Poynting vector in the moving frame contains strange values, because of the pressures.

The momentum of the field is

       4/3 m v,

if m is the mass-energy of the field, and v is its linear velocity.

The stress-energy tensor of the field of a static charge Q

https://en.wikipedia.org/wiki/Electromagnetic_stress%E2%80%93energy_tensor

                      r

            ●                  • ----> Ex 

            Q          observer

    

    ^ y

    |

     ------> x

Do the pressures in T make the electric field "stable", in the sense that if it would be a solid matter object, it would not collapse or explode?

                Q              __-----

                 o     r     |          |   1

                                ----___|   ring section,

                                      1       points distance 1

                                               out from screen

                 cylinder "o" points

                 out of the screen

Let us look at a very long uniformly charged cylinder, and the electric field pressure forces on a section of a ring drawn around it.

Let the radial electric field at a distance r be

       E  =  1 / r.

Let the negative radial pressure be

       pr  =  -1 / r²,

and the tangential positive pressure be

       pt  =  1 / r².

The radial component of the force from the positive pressure is

       Fr  =  2  *  1/2 / r  * 1 / r²

             =  1 / r³.

The negative pressure difference between the left and the right side of the section is 2 / r³. That adds a force 

       Fr'  =  -2 / r³.

The area of the right side is larger by a factor 1 + 1 / r. That adds a force

       Fr''  =  1 / r³.

We conclude that the pressures balance each other and make the electric field stable, if it were a solid matter object.

Poincaré stresses

https://www.feynmanlectures.caltech.edu/II_28.html

We have not yet found a precise formulation of the Poincaré stresses (1906). Henri Poincaré proposed that non-electromagnetic forces must hold the "charge elements" of the electron together, to make the electron stable. In literature, it is claimed that the momentum in these force fields resets the factor 4/3 back to 1.

              •    •     •

                \  |  /

           • -----×----- • dQ  string keeps dQ in place

                /  |  \

              •    •    •

                  Q

We can keep the "charge elements" dQ of Q stable by adding strings which attach them to the center of Q.

The 4/3 problem for a solid matter sphere

Above we showed that we can imitate the stress-energy tensor of the static electric field with a solid matter object where the radial negative pressure is balanced by a positive tangential pressure.

But the negative pressure tends to infinity at r = 0. We cannot realize this in a solid object?

If we move such an object, does it exhibit the 4/3 problem of the electric field?

The repulsion on the charge elements dQ comes from the fact that the energy in the integral of T⁰⁰ over space is reduced if they move farther from each other.

In the case of a solid object, there is no such repulsion.

If we take an almost cubical section of a spherical shell, then there is a negative pressure -p in the radial direction, and a positive pressure p in the tangential directions. If we integrate over the entire sphere, then positive pressures win?

If we Lorentz transform for a velocity v to the x direction, that "mixes" the time coordinate t and the spatial coordinate x. The pressure T¹¹ in the original T will contribute momentum to T⁰¹' in the Lorentz transformed stress-energy tensor T'.

Maybe we forgot shear stresses inside the sphere?

                                    -----> push shear

                                  -------

      Q ●     pull <--   |        |   cube

                                  -------

                                    <-- push shear

If we look at a perfect cube, the radial negative pressure would pull it toward the charge Q. But shear forces on its sides must balance the pull from the negative pressure.

The shear forces arise from the components -σxy, etc., in the stress-energy tensor T. They are not zero on the sides of the cube.

The integral of T¹¹ over the electric field of a charge Q

Let us look at a spherical shell of a radius r around a charge Q. Then

                       •

                     /

                   /    r

                 /  θ

           Q ● -------------->  x

       T¹¹  =  ε₀ Ex²  -  1/2 ε₀ E²

               =  ε₀ E² (cos²(θ)  -  1/2),

where θ is the angle from the positive x axis. On a shell, |E| is constant.

The integral of cos²(θ) - 1/2 over a half shell is 

              π / 2

               ∫      cos²(θ) * 2 π sin(θ)  

              0

                        -  1/2 * 2 π sin(θ)   dθ

       =  2 π (1/3  - 1/2)

       =  -1/3 π.

We see that the integral of T¹¹ is not zero over the whole sphere. The positive pressure wins.

We calculated in a frame where Q is static. Let us Lorentz transform to a frame which moves to the x direction at a speed v. Then both T⁰⁰ and T¹¹ contribute to T⁰¹' in the new stress-energy tensor T'. No other non-zero elements of T contribute.

Lorentz transforming a pressure to a moving frame

                         pebbles

                         •    •    • -->

       -a <--  ●                           ● --> a

                 M    <-- •    •    •     M

         • ---> v

   observer

         -------> x

We implement a positive pressure between the two masses M in the way that both throw pebbles to each other.

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

Let us have an observer moving at a speed v to the right. What does he see at his own time t' = 0? In his coordinates, the time t' = 0 if

       t'  =  0 

           =  1 / sqrt(1 - v² / c²)  *  (t  -  v x / c²).

The observer sees the rightmost M at a later t time than the leftmost M.

The observer would think that momentum conservation is broken if he only looks at the masses M. There is too much momentum to the right.

          ^ t          pebbles flying to the right

          |           •     •     •

          |         /     /     /

          |       /     /     /   

          |      _____--------  t' = 0

           -----------------------------------> x,  t = 0

Since the line t' = 0 is skewed up, the density of pebbles flying to the right is reduced on the line t' = 0, and there are more pebbles flying to the left. The observer sees that the "pebble field" holds some momentum to the left. This is from length contraction.

We conclude that the "pressure field" of pebbles does hold momentum, if viewed in the moving frame of the observer. Special relativity is right in this.

The solution to the paradox of the solid sphere: the negative pressure singularity at the center

Henri Poincaré found the right explanation for the apparent 4/3 paradox in 1906.

In the case of a solid spherical object, the positive pressure does win the negative pressure in the outer layers of the sphere. But at the center we must have a core where there is just a negative pressure. Otherwise, we would have infinite pressures at the center.

If we think of the electron as a pointlike object, then we may imagine that the core of a negative pressure is a limit, an infinitesimal volume with an infinite negative pressure.

Discussion of the "paradox"

Is there any paradox in the pressurized solid sphere?

Let us have a moving sphere. Then we add there a negative radial pressure and the positive tangential pressures to the outer layers of the sphere. After that, the outer layers possess more momentum, and the core less momentum.

We can add the pressures extremely quickly, essentially instantaneously. The momentum moves in an instant into a new location in the sphere. Is this paradoxical?

It could not happen if the momentum was bound to moving mass-energy. But pressure can, in principle, contain zero energy. 

One cannot "grab" the momentum contained in a pressure field, like one can grab moving particles.

The apparent "paradox" is in the nonintuitive behavior of special relativity. The momentum contained in pressure does not behave in the same way as momentum in moving particles.

In the case of the electron, the problem or the paradox is in the renormalization, or regularization, of the infinite energy of its field. The 4/3 problem is subject to that greater problem.

Conclusions

We conclude that Henri Poincaré solved the 4/3 problem already in 1906. The problem is equivalent to bookkeeping of the momentum in a moving solid sphere where there is a negative radial pressure and positive tangential pressures. In special relativity, pressure can carry momentum, and that momentum can behave in a surprising way.

The 4/3 problem shows that a significant portion of the momentum in a moving electric field comes from the positive pressure in the field. But in many cases we can ignore this pressure momentum because there always exists a negative pressure which counteracts it. If there is no negative pressure, then the system of charges is "exploding". Then the configuration is like in the Lorentz pebble example above. The role of the negative pressure is replaced by the change of simultaneousness in a moving frame.

The 4/3 problem in the case of an electron, or a pointlike charge, is a renormalization or regularization problem of the infinite energy in its electric field. It is not clear if 4/3 makes the renormalization problem any harder.

Now that we understand how the 4/3 problem is solved, let us look at the kinetic energy of the field paradox that we formulated in our August 24, 2024 blog post.

The 4/3 problem of electromagnetism

UPDATE August 28, 2024: Henri Poincaré found the correct solution to the 4/3 problem in 1906.

----

UPDATE August 26, 2024: Our solution below for the 4/3 problem forgets that we must have a complete action which treats also the mass-energy and the momentum of the body M which carries the electric charge q. Then we can calculate how M behaves inside an electromagnetic field.

The "Poincaré stresses" in M are intended to solve this problem. If for any pressure in the static electric field of M, we have a "counter-pressure", then the contribution of pressures to the momentum is zero if we Lorentz transform to a frame where M is not static.

In that case, there would be no 4/3 problem at all. It is like moving a pressurized vessel. The negative pressure in its wall cancels the momentum effect of the positive pressure inside the vessel.

But is it plausible that the body M has such counter-pressures? We have to check what Henri Poincaré suggested in 1906.

The electric field spans to infinity. Is a counterpressure necessary at all? If the universe is the surface of a sphere, we can make a mechanical system of wires or rods which only contains negative pressure, or only positive pressure. There is no need for a counterpressure.

----

In our August 20, 2024 blog post we tentatively solved the 140-year-old 4/3 problem. Let us analyze the 4/3 problem further.

The Enrico Fermi 1923 resolution

https://www34.homepage.villanova.edu/robert.jantzen/temp/fermi-43/index.htm

Enrico Fermi, in a paper in Nuovo Cimento 25, pp. 159 - 170 (1923), suggests that one must calculate the momentum of an electromagnetic field in a frame (or, frames) where the charge Q creating the field does not move. Only after that, one is allowed to Lorentz transform the momentum to a moving frame. This, of course, would resolve the 4/3 problem. The Poynting vector ~ E × B is zero in the frame where Q is static. Fermi simply works around the 4/3 problem by banning most frames. It is an unsatisfactory resolution of the 4/3 problem.

The Noether x, y, z coordinate variations

We want to show that the electromagnetic action conserves momentum, and that the Poynting vector has the role of revealing the momentum stored in the electromagnetic field "as a whole".

https://homepage.univie.ac.at/thomas.mieling/pdf/electrodynamics-noether.pdf

Thomas Mieling (2017) writes about Noether variations for classical electromagnetism. He is able to derive the momentum conservation formula:

Let us write the last formula for the x momentum (t is the coordinate 0, and x is 1):

       dT⁰¹ / dt  =  -f¹.

At a location, the momentum density of the field to the x direction grows with time, if the field there is pushing a current j to the negative x direction with a force f¹.

Is this consistent with the fact that the Poynting vector exaggerates the momentum of the field of a spherical charge distribution by a factor of 4/3?

The energy and momentum flow are in the realm of "gauge freedom"

                Q ● -                   +|      |-           

                                    capacitor plates

Let us have a charge Q sitting still. We use the capacitor trick of our August 20, 2024 post to cancel its electric field and "extract energy" from the field of Q.

A. Calculate in a static frame and Lorentz transform to a moving frame. Let us then switch to a moving frame. Since electromagnetism is Lorentz covariant, everything happens in the moving frame in an essentially "same" way, save the Lorentz transformations.

B. Switch to a moving frame and calculate there. Electromagnetism, Lorentz transformed to the moving frame, contains the Poynting vector ~ E × B, which, if interpreted as a flow of light-speed particles, behaves in a really strange way.

However, electromagnetism in the moving frame predicts the exact same measurable behavior for the capacitor plates, as in the procedure A above.

The 4/3 problem does not affect anything which we measure. In this sense, the problem does not exist.

The Poynting vector is a tool in calculation. We can use the method A above, and avoid using the Poynting vector.

In a sense, we have "gauge freedom" here. One is allowed to calculate with any method, as long as it gives the same predictions for measurements.

The energy density of the field is another concept which enjoys "gauge freedom". As long as the theory predicts measurements correctly, we are free to place the energy anywhere we want.

Since energy and energy flow do not interact with electric charges, it does not matter where and how our model stores these.

Gravity casts light on the 4/3 problem: it is not a problem at all

For gravity, we would like to know the location of the energy in a field, and also the movements of the energy. Is there any way to couple gravity to electromagnetism without specifying explicitly where the field energy lies, and how does it move?

https://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric

The Reissner-Nordström metric tells us what is the static solution of the Einstein field equations for the stress-energy tensor of a static electric field. The stress-energy is not just the energy density of the electric field,

        ε₀ / 2  *  E².

There is also a pressure, since the field lines of the electric field repel each other.

https://en.wikipedia.org/wiki/Electromagnetic_stress%E2%80%93energy_tensor

Assuming that both electromagnetism and the Reissner-Nordström metric transform correctly under a Lorentz transformation, the electromagnetic stress-energy tensor contains Poynting vectors in a moving frame, and the Lorentz transformation of the metric is the solution for the Einstein field equations in the moving frame.

This casts light on what the 4/3 problem really is about. The static electric field contains pressure. Because of this, its Lorentz transformation to a moving frame is not a simple mass density flow moving linearly. The correct Lorentz transformation does have the momentum strangely 4/3 of the value that pressureless matter would have! The excess 1/3 comes from pressure in the stress-energy tensor of a static field.

The physics in the moving frame works just as it would in the static frame. The 4/3 problem is not a problem at all. There is no experiment which would reveal that the momentum has a strange numerical value.

Thus, the origin of the 4/3 problem is a misunderstanding of what is the Lorentz transformation of stress-energy, specifically, pressure. The transformation does not mean that one can "grab" the energy flow, as one would grab a flow of particles.

Conclusions

We probably solved the 140-year-old 4/3 problem. The solution is that it is not a problem at all.

The confusion comes from the wrong intuition that one can "grab" the energy flow in the Poynting vector, as if the flow would consist of particles.

The strange value 4/3 comes from the Lorentz transformation of the pressure in the stress-energy tensor of a static electric field. The extra 1/3 is not anything which one could "grab". One can only "grab" the part which comes from moving energy density.

We will next analyze the energy flow problem of the two static electric fields that we presented on August 24, 2024. Can we find a satisfactory resolution for it, too?

The electromagnetic action fails for the sum of two static fields?

UPDATE August 25, 2024: Where is the kinetic energy of the "extra energy" W stored? It may be stored by the field of q being squeezed horizontally by length contraction. More field lines of q become vertical, and strengthen the vertical field of the cylinder.

But what if q moves close to the speed of light? Then almost all field lines of q are vertical. How can we store ever more energy as q approaches the speed of light?

            field lines

               |  |  |

           q  --------   ---> v

               |  |  |

Also: we can make the charge q rod-shaped. Then its field points vertically, and length contraction has no effect on W.

----

Maxwell's equations and the corresponding electromagnetic action were not written with field energy and the kinetic energy of a moving field in mind. Therefore, it would be surprising if they could handle the kinetic energy correctly.

The kinetic energy of moving field energy

                                      \       |      /

                                         \    |    /              -

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

     P,  v' <--      Q ●       m, q • ---> v, p

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

                       M              /    |    \             -

                                       /      |      \  --> pW

                       "extra" field energy W

                        moves with the charge q

Let us have a long uniform cylinder with a uniform negative charge. We put a small negative charge q inside the cylinder. We assume that we have to do a lot of work when we put q inside. Let the work be

       W  =  N m c²,

where m is the mass of q, and N could be 10. Let the mass-energy of the cylinder plus Q be

       M  =  N² / 2  *  m.

If we move the charge q inside the tube, we expect most of the inertia of q to come from moving the extra field energy W in the combined field of q and the tube.

We attach a small negative charge Q inside the cylinder, close to q. The charge Q pushes q to the right. We ignore all other forces, except the repulsion between Q and q.

The charge q is accelerated to the right. Let it move relative to the laboratory at a speed v. Let the cylinder move at a speed v'.

The Poynting vector

      S  =  1 / μ₀  *  E   ×  B

is aware that the extra field energy W moves along with the electron to the right.

The electromagnetic action knows about the momentum of W, but is not aware of a possible kinetic energy of W. If the charge q moves a distance s to the right, all the released field energy must go to the kinetic energy of q and its own field. None to W. The released energy is

       p² / (2 m)  =  1/2 m v².

The total momentum to the right is

       (N + 1) m v.

Momentum conservation requires that the momentum of the cylinder to the left is

       P  =  -(N + 1) m v.

The kinetic energy of the cylinder is then

       P² / (N² m)  ≈  m v².

This is nonsensical. The cylinder is much heavier than the charge q. The repulsion between Q and q did much less work to move the cylinder to the left. But the kinetic energy in the formula is twice the kinetic energy of q.

We assume that the cylinder and its field behaves approximately like any ordinary object of a mass M. This is plausible because q does not alter the field of the cylinder that much.

The paradox of a positron in a very low potential

In laboratory, it is possible to produce voltage differences of up to 32 MV.

Let us assume that we put a positron inside a sphere which has a very large negative electric charge. Then the positron may be in a potential much less than the 511 keV mass of the positron.

Does the positron then have a negative energy? Does it have a negative inertia?

A possible solution: even though the positron formally would have a negative inertia, the electromagnetic action may still make it to move in external fields in a reasonable way, as it would possess a positive inertia. We have to study this question.

Conclusions

No one in the literature claims that Maxwell's equations, or the associated action, would be perfect. There are several long-standing open problems in electromagnetism. The best known one is the renormalization or regulation of the electron field energy. The 4/3 problem is another one.

The question where is the energy of a field located, and how does the energy move, is open for most field theories. But we would need that knowledge, in order to determine the gravity field of another field (or the gravity of the gravity field itself).

Does our thought experiment above prove that Maxwell's equations or the action are flawed? We do not know. Maybe we overlooked something, which would save energy conservation and momentum conservation. We will analyze this further. In this blog we have been very interested in the inertia of energy flowing in a field. The example above suggests that we have to include the inertia of the extra field energy W to the inertia of the charge q. But what happens if W is negative?

Electromagnetic waves as stationary points of the action; the 4/3 problem

UPDATE August 28, 2024: Henri Poincaré's 1906 solution to the 4/3 problem is correct.

----

Let us check if the electromagnetic lagrangian correctly predicts electromagnetic waves.

https://en.wikipedia.org/wiki/Lagrangian_(field_theory)

If the waves propagate in empty space, then the electric currents j are zero, and the lagrangian density is the very simple formula:

The values of E and B must somehow depend on each other. Otherwise, we can vary just one of them and change the value of the action.

https://en.wikipedia.org/wiki/Maxwell%27s_equations

We have to include at least some of Maxwell's equations, in addition to the lagrangian.

Since the fields in an electromagnetic wave vary with time, a history must match the last two equations.

Deriving the speed of light

https://www.wikihow.com/Derive-the-Speed-of-Light-from-Maxwell%27s-Equations

We eliminate B from the two last equations of Maxwell, and obtain a wave equation for E.

Can we define B solely from E for a wave packet, using the last equation of Maxwell?

Probably yes. For each sine wave solution of E, we easily find the associated B. Then combine a wave packet from sine waves. Then the value of

is zero. What is the role of the lagrangian density if we know that each valid history has it zero? Should we write the lagrangian solely in terms of E, and then drop the requirement that Maxwell's equations must be satisfied by a history?

Conservation of energy

https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node10.html

(Richard Fitzpatrick, 2014)

Deriving the Poynting energy conservation formula from Maxwell's equations is simple. What does it prove about energy conservation?

Let us have electric charges Q under non-electromagnetic forces under various orbits. What is the shape of the field of Q, and what kind of a self-force does its own field exert on Q?

Proving conservation of energy with the Noether time variation

Using the Noether time variation to a history of an electromagnetic wave packet makes the waves to move faster than light in the "speed up time" phase. Is that nonsensical?

Let us assume that E is the fundamental field, and B is just derived from dE / dt. Let us have a sine wave of E.

  ^  E                       

  |                                c

  |                               ---->

   -------> x       ___/\/\/\/\/\___

                          wave packet

   

Let us apply the Noether time variation to a history H of a wave packet. We "speed up time" at the start, and slow down time at the end.

The contribution of E² to the action stays the same after the variation. Speeding up time increases |B|, because B is derived from dE / dt. Slowing down time reduces |B|. The changes to B² cancel each other. We see that the history H is a stationary point of the action.

We were allowed to make the wave packet to propagate faster than light in the variation of the history H.

The Noether time variation shows that 

is conserved. That is a sensible result.

The electromagnetic action makes a faster-than-light signal "singular"

Let us start from a wave packet moving at the speed of light, c. What about a history H, where time is sped up, and the packet moved at a speed > c?

Let us keep E as is, but speed up time. Then the value of dE / dt is larger, and

is larger. The value of the lagrangian density

is negative. We can change the value of the lagrangian density by reducing the amplitude of the wave, E. If we set E to zero in the intermediate stages of the history H, then we find a stationary point. But then the initial state suddenly "collapses" into a zero wave, and the end state "jumps" from zero to the final state. The history H becomes nonsensical, containing "singularities".

We conclude that the electromagnetic action correctly predicts that an electromagnetic wave packet moves at the speed c.

The problem of the self-force

Let us have an electric charge Q. In the lagrangian density above, the charge Q is represented by a "current" j⁰ which flows to the time direction.

Let us try to find a stationary point of the action for Q and the electric field, such that the system is static. We can ignore B and kinetic terms in the lagrangian.

The action says that the charge Q creates an electric field E around itself, such that Q falls into a lower electric potential φ. The price we pay is that the energy of the electric field, ~ E² increases.

         ----        -----  drum skin

               \●/ 

                M weight

It is like a weight M which is put on a drum skin. The weight M creates a depression into the skin, so that M can go to a lower gravity potential.

A side note: the field E of a positive charge Q then looks like the field of a negative charge. The "true" sign of an electric charge is the opposite of what its electric field suggests!

Since the electron and the proton have very large electric charge densities, they are very deep in the pit (or hill) which they created into the electric potential. If we move the elementary charge, its electric field follows it faithfully.

The problem of the self-force is the interaction between Q and the electric field which it created. But is the problem solved by the action itself? The action does tell us the interaction between a charge and its own field.

The open 4/3 problem in electromagnetism

There is at least one open problem in the electromagnetic action.

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

https://www.feynmanlectures.caltech.edu/II_28.html

The 4/3 problem of electromagnetism: what is the momentum p of a static electric field moving at a velocity v?

If we calculate the field momentum using the Poynting vector, that is, integrate 

       1 / μ₀  *  E × B,

we obtain a value which is 4/3 times the value which we would get by multiplying the integral of 

       ε₀ / 2  *  E²

by the velocity v of the field E. 

                Q ●  --> v

                                ^   1 / μ₀ * E × B energy flow

          \__________/

The Poynting vector claims that the energy in a field makes a detour and does not travel linearly along v. How could such a detour be possible?

                        F force           gears

                          --->                  ● ●   

                        ========== ● ● ● ●  ^

                         shaft                ● ●    /  gears turn

                                               M mass

Suppose that we have a mechanical device, into which we have attached a shaft. If we accelerate the device using a force on the shaft, that will inevitably make the gears to rotate. As if the device would possess more momentum than M v, where M is the mass of the device. Could this explain the detour in the Poynting vector?

On January 14, 2024 we analyzed a pure magnetic field B, and concluded that the Poynting vector does not understand the associated momenta. Maybe the solution to the 4/3 problem simply is that the Poynting vector is wrong?

When an energy distribution changes spatially, there are many ways to explain it with energy flows. Why would the Poynting vector ~ E × B be the correct description? We can easily imagine a physical model where the formula

is true, but the actual energy flow in the field is very different from ~ E × B.

Harvesting energy from the field of a charge Q, using capacitor plates

                         -        + 

                           \   V   \   --> move the + plate

                             \         \

       ●                      capacitor plates

       Q +

If we have suitably charged capacitor plates, we can reset the static electric field E of Q between the plates, and harvest the energy ~ E² there.

       ● ---> v

       Q +

What does this look like if Q is moving to the right at a velocity v? 

The Poynting vector claims that the field energy is then flowing to the right and down at the plates. Should the plates start to move downward? That would break Lorentz covariance, since the plates do not move that way if Q is static.

Is there some force which cancels the downward momentum which the plates gain?

The plates make an empty volume V to the energy flow of the field. Since the energy still has to flow, the routes of the flow must take a detour around V.

It might be that there is no 4/3 paradox at all! The force on the + capacitor plate does not have any downward directed component which would come from the Poynting vector ~ E × B. Consequently, the plate does not harvest any extra momentum downward. Where does the momentum of the "missing" field in the empty volume V go then? It may be retained in the remaining field!

It is like flowing water. If we have a balloon filled with air there, the flow might have less pressure, and that makes the balloon larger. But that does not necessarily mean that the balloon absorbs the momentum of the "missing" water.

We must check what the Noether x coordinate variation says about conservation of momentum. The integral of ~ E × B probably is included there.

Does the Poynting vector break the speed of light? The solution to the 4/3 problem

Question. If we have a charge Q moving at almost the speed of light, does the Poynting vector ~ E × B claim that the energy flow moves faster than light around Q? The energy flow makes a detour. Does it necessarily break the speed of light?

                Q ●  --> v

                                ^   1 / μ₀ * E × B energy flow

          \__________/

Breaking the speed of light is no problem if one cannot send a signal through the Poynting energy flow. How could one send such a signal? Let us try to block the energy flow with capacitor plates. It is not possible to block the flow, because the flow will always find a route around the block!

What is the role of the Poynting energy flow then?

The role seems to be as a formal tool which is used in calculations of momentum conservation in the system. The energy flow is not like a flow of particles which we can grab, and absorb the energy and the momentum in the particles. One cannot "grab" the Poynting vector.

This solves the mystery of the 4/3 problem. The problem arises from a misunderstanding that the Poynting vector would describe a flow of energy in the particle form, so that one could grab those particles. If the flow would be particles, then we would have a paradox. But the Poynting vector is a formal tool, it is not particles.

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

The 4/3 problem has been open since 1884, when John Henry Poynting discovered his vector. We may have solved the problem.

The momentum stored in a magnetic field

On January 14, 2024 we wrote that the Poynting vector cannot explain the (large) momentum which is stored in a magnetic field relative to the coil.

This would not be surprising. The Poynting vector is a tool for the calculation of momentum conservation in the entire system. There is no reason why it should be able to explain phenomena associated with "stored momentum" of A relative to B.

                    spring

           A ● /\/\/\/\/\ ● B

In the diagram, if A and B are moving toward each other, then the spring can "store" their relative momentum. Momentum conservation of the entire system A & spring & B is a different thing.

Conclusions

Let us close this blog post. We will analyze electromagnetic waves further in future posts. So far, the electromagnetic action seems to fare well. It may be able to explain also the self-force of the field on an electric charge Q.

We may have solved the 140-year-old 4/3 problem of electromagnetism. The solution is in the insight that the Poynting vector is a formal tool which does not describe the flow of physical particles. One cannot "grab" the energy and the momentum in the Poynting vector. We will think about this more. Did we overlook something?

One problem remains unsolved. To determine the gravity exerted by energy in the electromagnetic field, we probably need to know the "true" flow of energy in the field. What is the true flow if the Poynting vector does not tell it?