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.
----
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
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?
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