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