James Clerk Maxwell in his 1865 paper writes that energy is "essentially" positive. He cannot make sense of the gravity field, whose energy density seems to be negative, and he does not try to develop further an electromagnetic theory of gravity.
We recognize at least the following problems which concern the energy density of a force field:
1. The electric field of a pointlike electron contains infinite energy, if we assume that the energy density is 1/2 ε₀ E².
2. If we try to renormalize away the infinite energy by making the mass-energy of the electron particle minus infinite, then the combination the electron & its inner field has negative energy. Negative energy behaves in a strange way in newtonian mechanics.
3. Electromagnetic waves and gravitational waves contain positive energy. How are they born if a static field contains zero energy?
4. Since gravity is attractive, in a naive "electromagnetic" model, the gravity field has negative energy.
5. A point particle in gravity is a singularity and a black hole.
6. How do we implement energy and momentum conservation when particles interact?
Retardation and the Wheeler-Feynman absorber theory
If we have a proton and an electron, we can assume that the proton has an electric potential, and the electron moves in the potential with no retardation of the interaction. Then there is no need to assume the existence of any "electric field" of the proton, where the field would contain energy. It is enough to assume that there exists an abstract potential.
In quantum electrodynamics (QED), the interaction in the static field is carried by a virtual (off-shell) photon. A virtual photon may have zero energy though it carries momentum.
Conservation of momentum is a problem in the QED model. When do the proton and the electron feel the kick from the exchanged momentum? If we assume a finite speed of of a signal, where is the momentum stored when the signal travels?
The exchange of momentum is like a transaction which the proton and the electron decide together. If the proton emits a virtual photon, then the transaction guarantees that the electron will receive it. It is all or nothing. The virtual photon cannot be left dangling in space.
If two electromagnetic systems at a large distance from each other exchange energy and momentum, then we assume that they exchange a real photon. In this case it is hard to believe that the exchange is a transaction. The real photon can be left propagating in space on its own.
The Wheeler-Feynman hypothesis is that even real photons are inside a transaction. Every photon will be absorbed in the future. Somehow the universe decides which photon will be absorbed by which system. A "free electromagnetic field" does not exist. There are no independent photons flying around. All photons are virtual, or off-shell.
We in this blog believe that the Wheeler-Feynman hypothesis is too radical. We certainly observe electrons losing their kinetic energy in a radio transmitter. Why would the photon lack independent existence, while the electron can exist on its own?
However, we do not know how to decide the demarcation line between virtual photons, which exist inside a transaction, and real photons, which have independent existence.
How does retardation create waves in the field?
We have been claiming that when an electron is accelerated, its "static electric field" lags behind and extracts energy from the movement of the electron. That energy is the energy of the electromagnetic wave.
Edward M. Purcell derived the Larmor formula of the extracted energy from:
1. the retardation of the electric field;
2. the fact that the electric lines of force cannot break;
3. the energy density of the dynamic electric field is 1/2 ε₀ E².
In the link Daniel V. Schroeder presents his derivation.
In retardation the crucial fact is that when the electron is moving at a constant velocity, its electric field is centered at the current position of the electron, not its position where the line of force "departed" the electron. There is no "retardation" at all if the electron moves at a constant speed. That is, the potential of the electron's electric field is fully up-to-date.
Connecting the lines of force in the diagram above requires very substantial deflection of the lines of force at the circle which marks the sudden acceleration of the electron.
A magnetic field is required to deflect the electric lines of force substantially. The wave thus contains both an electric field and a magnetic field.
The deflection of the lines of force makes them more dense at the circle, and increases the calculated energy density of the field, compared to a static field. The dense part of the field moves away at the speed of light, carrying the energy of the electromagnetic wave.
In the derivation we do not need to assume that a static electric field has any energy at all. It is enough to define that the extra density of the field lines in the circle carries energy.
To extract energy from the movement of the electron, the far field has to possess "inertia". We have suggested that the inertia is not from mass-energy of the far field, but a result of retardation: the lines of force of the far field cannot keep up, and simulate "inertia". Thus, retardation can replace inertia in the extraction of energy from a periodic movement of a charge.
Why the static electric field contains no energy, but the dynamic field does carry energy? The Fourier transform of the electric field
If the static electric field of an electron is just an abstract potential which does not contain energy, why does the retarded dynamic field then carry energy? Why cannot it be another abstract potential?
Let us determine the Fourier decomposition of the electric potential of the electron. Let us assume that the electron is eternally moving at a constant velocity.
If the Fourier transform is taken in an inertial frame comoving with the electron, then the decomposition only contains waves whose 4-momentum has zero energy. The waves are time-independent.
What if we take the Fourier transform in an inertial frame which is not comoving with the electron?
We have to Lorentz transform the 4-momenta of the time-independent waves to a moving frame. They are then time-dependent. Some waves acquire positive energy and some negative energy. One could claim that the total sum still has zero energy.
If we have an electromagnetic wave, then the Fourier decomposition of the electric field contains positive energy in the 4-momenta of the waves.
We gave a heuristic explanation why a static field contains no energy, but a dynamic field carries energy.
From where does the energy in the Coulomb interaction come? What about gravity?
Coulomb attractive force
● <---------------------------------------> ●
positron electron
Let us have a positron and an electron. If we move them closer to each other, we can harvest energy from the Coulomb attractive force. From where does this energy come?
One way to explain the harvested energy is to say that the total energy of the electric field
∫ 1/2 ε₀ E² dV
all space
grew smaller. The combined field grew weaker when we moved the particles closer to each other.
Another way is to say that we harvested energy from the Coulomb force between the particles.
If we claim that the energy of the static electric field is zero, then we have to explain the process with the Coulomb force. We cannot refer to the energy of the static field.
newtonian attractive force
● <---------------------------------------> ●
test mass test mass
In the case of gravity, we apparently have to claim that the harvested energy comes from the newtonian attractive force between two test masses if we move the masses closer to each other.
In the case of gravity, the combined field of the masses grows stronger. If we define that the static field has zero energy, then the field is irrelevant for the process.
Note that in the rubber membrane model of our previous blog post, the energy of the gravity field is the deformation energy of rubber. It is positive.
In general relativity, there is no gravity force. From where does the harvested energy come? The ADM mass of the system does not change if we store the harvested energy close to the test masses. Apparently, general relativity has "implicit" potential energy associated with the system. In the "implicit" description, there exists a force of gravity.
Another interpretation is that the mass-energy of the test masses drops to a lower potential. A part of the released energy can be harvested, and the rest goes to the "deformation energy" of spacetime. This model would be analogous to the rubber membrane model.
There is positive energy in gravitational waves, but why it is 16-fold?
We can argue just like in the case of the electric field that a static gravity field contains no energy.
A dynamic gravitational wave does contain energy. Robert C. Hilborn calculated that the energy is 16-fold compared to a naive electromagnetic model of gravity.
We will try to find out the reason why it is 16-fold. We could take that as an axiom, but that would not be nice.
The simple way to harvest energy from a gravitational wave is to have a ring of test masses, and put rods as "shock absorbers" between them. The shock absorber harvests energy from (apparent) changes in the distance of the test masses.
shock absorber
● =================== ●
test mass test mass
The creation of a gravitational wave is the opposite process: the shock absorber stretches and contracts, moving the test masses.
We have to analyze the process in detail. It may explain the energy content of waves.
How much energy we can extract from a wave by destroying it?
If we have a spatial volume V with an electric field E, we can cancel the field E by putting suitable positive and negative charges in the volume. For example, consider a flat volume between capacitor plates.
When we cancel the field E, we extract
∫ 1/2 ε₀ E² dV
V
of energy. If we cancel an electromagnetic wave, we actually get double the energy above because a half of the energy is in the magnetic field.
Canceling a wave with an equal amount of positive and negative test charges does not affect the creation of the wave since far away the field of the test charges is almost zero.
How would we cancel a gravitational wave? We cannot use positive and negative charges because negative mass-energy does not exist. What is the correct analogy in electromagnetism? Consider canceling a wave created by a negative electric charge with a large number of positive test charges. The test charges create an electric field which may affect the creation of the wave.
We need to analyze this. It may be that a wave of a purely attractive force must hold 16-fold energy compared to an electromagnetic wave.
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