NOTE December 31, 2020: our criticism of the Edward M. Purcell derivation of the Larmor formula is incorrect. Purcell is right.
The magnetic field makes electric field lines to bend, so that they are continuous.
------
https://en.m.wikipedia.org/wiki/Larmor_formula
The Larmor formula and the accompanying formula for the Abraham-Lorentz force
https://en.wikipedia.org/wiki/Abraham–Lorentz_force
are plagued with paradoxes and inconsistencies. The Wikipedia link above mentions the problem of "signals from the future". In order to calculate the dissipated power now, one must know the motion of the charge in the future.
The Larmor formula and the Abraham-Lorentz force are connected to the famous problem if linearly uniformly accelerated charge radiates electromagnetic wave:
https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field
Separate from the Larmor formula, William Unruh has claimed that a uniformly accelerated charge radiates, or at least sees thermal radiation around it:
https://en.wikipedia.org/wiki/Unruh_effect
We will argue in the following that conservation of momentum and energy imply that a linearly accelerated charge cannot radiate in classical electromagnetism, and that an equivalence principle prohibits the existence of Unruh radiation. Thus, the Larmor formula is erroneous for linearly accelerated charges.
The paradoxes are due to the fact that the Larmor formula and the Abraham-Lorentz force try to calculate the dissipated radiative power from incomplete information - namely the instantaneous acceleration of a single electric charge. What is also needed to do the calculation properly, is the current electromagnetic field in the vicinity of the charge and the motion of other charges which are nearby. It is the interaction between charges that produces the electromagnetic radiation. It makes no sense to talk about the radiation of a single accelerated charge.
Classical radiation from a uniformly accelerated electric charge
Let us first prove that a monotonously linearly accelerated electric charge cannot radiate in classical electromagnetism. We will work in an inertial laboratory frame.
Definition 1. By monotonous linear acceleration we mean a (possibly time-varying) acceleration a(t) to a constant direction, such that a(t) ≥ 0 at all times.
Definition 1. By monotonous linear acceleration we mean a (possibly time-varying) acceleration a(t) to a constant direction, such that a(t) ≥ 0 at all times.
Let us assume that we have a rocket that is accelerated by firing a laser pulse to a mirror at the back of the rocket (or, alternatively, any fast projectiles that will have a perfectly elastic collision with the back of the rocket). The pulse is completely and losslessly reflected back from the mirror. We assume that the speed v of the rocket is much less than the speed of light.
Let us assume that the rocket is carrying an electron that is supported by the static electromagnetic forces of the atoms in a container in the rocket.
Let us assume that the rocket is carrying an electron that is supported by the static electromagnetic forces of the atoms in a container in the rocket.
________________
Rocket ◁ ___ electron e __| <--- laser pulse
Suppose that the electron would emit a photon with energy
E = hc/λ
and momentum
p = h/λ
where h is the Planck constant, c is the speed of light and λ is the wavelength of the photon.
where h is the Planck constant, c is the speed of light and λ is the wavelength of the photon.
The ratio
momentum/energy = p/E = 1/c
for the photon is much less than the ratio
momentum/kinetic energy = mv/(1/2 mv^2) = 2/v
for the rocket flying at speed v. We see that the rocket cannot convert its kinetic energy to a photon that would be radiated by the electron.
Theorem 2. We will work in classical electromagnetism and Newtonian physics. We measure velocities relative to a laboratory frame.
Assume that we have a rocket in vacuum under monotonous linear acceleration, where the acceleration is accomplished through a laser pulse reflected from the back of the rocket (or alternatively, of almost light-speed pellets that bounce perfectly elastically from the back of the rocket).
Assume that the rocket is made of infinitely rigid material which cannot gain thermal energy from the acceleration and the temperature of the rocket is absolute zero. Alternatively, we may assume that the rocket is not infinitely rigid and is under a constant acceleration (which is equivalent to being static under a constant gravitational field) and has radiated away all the thermal energy it had.
Assume further that momentum and energy are conserved in the process and that the laser photons only interact with the rocket system through lossless reflection from the mirror at the back of the rocket.
Then an electric charge, or anything else in the rocket, cannot emit photons.
Theorem 2. We will work in classical electromagnetism and Newtonian physics. We measure velocities relative to a laboratory frame.
Assume that we have a rocket in vacuum under monotonous linear acceleration, where the acceleration is accomplished through a laser pulse reflected from the back of the rocket (or alternatively, of almost light-speed pellets that bounce perfectly elastically from the back of the rocket).
Assume that the rocket is made of infinitely rigid material which cannot gain thermal energy from the acceleration and the temperature of the rocket is absolute zero. Alternatively, we may assume that the rocket is not infinitely rigid and is under a constant acceleration (which is equivalent to being static under a constant gravitational field) and has radiated away all the thermal energy it had.
Assume further that momentum and energy are conserved in the process and that the laser photons only interact with the rocket system through lossless reflection from the mirror at the back of the rocket.
Then an electric charge, or anything else in the rocket, cannot emit photons.
Proof. Suppose that the charge in the rocket emits the first photon at a time t, t_i ≤ t < t_i+1, where t_i is the reflection time (in the rocket local time frame) of the i'th laser photon.
Since the emitted photon has to get its energy from somewhere, and since the rocket is at the absolute zero temperature, the only possible source of energy for the photon is the kinetic energy of the rocket.
The speed of the rocket after the emission is less than it would be without the emission of the photon. To provide the energy for the photon, the rocket must lose its kinetic energy and momentum. But the photon cannot carry away all the momentum. We arrive at a contradiction, which shows that no photon can be emitted. QED.
The above argument is based on the fact that the rocket cannot shed its extra momentum if it emits some of its kinetic energy as a photon. If the rocket would be moving under friction in a medium, say in air, then it would be able to emit thermal radiation because it could pass the extra momentum to the medium.
What about the assumption that the rocket only interacts with laser photons through lossless reflection? In a real-world system, the laser would heat up the mirror and the rocket would emit thermal radiation. But this radiation would be ordinary thermal radiation and there is no reason to call it radiation emitted by an accelerated charge.
Might it be that the electron could emit a disturbance in the electromagnetic field which carries more momentum than a photon of the same energy? That is not possible, because as the disturbance moves far away from the electron, then the disturbance moves under a linear wave equation and we can do a Fourier decomposition of the disturbance. Classical electromagnetism tells us that the momentum/energy for each planar wave in the decomposition is 1/c.
What if the extra momentum is absorbed by hypothetical "vacuum energy"? That would be a radical diversion from the current consensus about quantum fields, but that mechanism would make radiation possible, since then there might exist an "acceleration friction" mechanism of the electron against the vacuum.
Theorem 2 resolves the 70-year-old debate about the existence of classical electromagnetic radiation from a uniformly accelerated charge. The charge itself does not radiate, but the interaction with the system that is accelerating the charge may produce electromagnetic radiation. Our example, which uses a rocket and a laser pulse, in a sense, tries to minimize the interaction between the charge and the acceleration driving system and in that way eliminates radiation.
In a forthcoming blog post we will prove that classical electromagnetism conserves energy and momentum. The key idea in our proof is that electromagnetic waves are actually polarisation of virtual charges in "empty" space. The only real force is the Coulomb electric force that acts between charges, virtual or real. The magnetic force is just a Lorentz transformation of the Coulomb force.
Erroneous "derivations" of the Larmor formula
NOTE: our criticism below is erroneous. See the note at the beginning of this blog post.
---
https://en.m.wikipedia.org/wiki/Larmor_formula
contains two "derivations" of the Larmor formula. Since the Larmor formula is incorrect for linear acceleration, the derivations must be in error, too. The error in the Edward M. Purcell calculation
http://physics.weber.edu/schroeder/mrr/MRRtalk.html
is that it assumes that the electric field lines must be continuous in the laboratory frame at a specific global time t of the laboratory frame.
Let us prove that the Purcell approach is erroneous.
The conventional assumption is that the electric force is transmitted at the speed of light. It is "retarded" in the sense that it does not act at an infinite speed.
Definition 3. If we in a laboratory frame have a moving charge Q and it interacts with a static test charge q then let us define electric field E as
E = F/q,
where E is the electric field strength vector at q at time t, and F is the force vector that would act on q if Q were static in the position where q "sees" it at time t.
The force is in the direction of line r that connects the apparent position of Q and the position of q at time t.
r
Q ----------------- q
In the diagram above, let us move Q briefly up. If we try to draw continuous electric field E lines at a later laboratory frame time t, they will be have this shape:
Q -------------
\
-------------
The curve in the line moves away from Q at the speed of light, c. But now we see that the field line at the curve is not in the direction of r at any time. That is a contradiction. We cannot draw continuous E field lines at a laboratory frame global time t.
The above argument also shows that the Gauss law for a static electric field, that the source of the electric field is just an electric charge q, does not hold for moving charges when E is defined as above.
__________
/ |
/ |
__ / |
|__ y x |
\ |
\ |
\________|
If we have a charge q inside a bottle close to the neck of the bottle at position x, and suddenly move the charge close to the start of the neck to position y, it is easy to see that the sum of electric field E lines passing through the bottle will sum to > q for a brief moment after the move, because the main body of the bottle will see q still at its original position x, while the neck and other areas nearby will see it at its new position y. The rest of the bottle will see q at some point between x and y, and also in the rest of bottle the field lines will pass out of the bottle.
The divergence for E, ∇⋅E, is not always zero in empty space for E defined as above at a global time t.
In a forthcoming blog post, we will try to determine the error in the other derivation in Wikipedia, the Liénard-Wiechert potential method.
Does Unruh radiation exist?
In the classical discussion above, we may dispose of the rocket and let the laser photons directly impinge on the electron. Our calculation would then be about a scattering experiment, and the best way to do that is to use Feynman diagrams.
Laser photons would be scattered by the electron. In double Compton scattering, the electron "splits" the impinging photon into two photons. Could we call this two-photon scattering radiation Unruh
radiation?
Ralf Schützhold and Clovis Maia interpreted double Compton scattering as an Unruh effect:
Quantum radiation by electrons in lasers and the Unruh effect
Ralf Schützhold, Clovis Maia
(Submitted on 14 Apr 2010)
https://arxiv.org/abs/1004.2399
e electron
^
| support
__________
Earth
Let us support an electron on the Earth surface so that it stays static relative to the surface.
We may use a freely falling laboratory where thermal radiation from a black body accelerates the electron upward such that it exactly compensates the acceleration of the laboratory downward. The electron will appear static to a static observer on Earth. We do not use laser light in this thought experiment because a coherent laser beam would make the electron to oscillate strongly in the horizontal direction.
Feynman diagrams show that an observer in the laboratory frame will see a double Compton scattering of a photon from the electron. We let the laboratory to fall freely, because we want this scattering experiment to happen without external potentials.
We will return later to the question if a static observer will see this double Compton scattering.
One might call this double Compton scattering "Unruh radiation", but does that make sense?
Let us then support the electron with the static electromagnetic field of the atoms in Earth's surface. The electron cannot radiate to a static observer, because there is no source of energy for the radiation.
We will return later to the question if the electron will appear to radiate to a freely falling observer.
Equivalence principle 4. If an electron is supported with static electromagnetic forces in an accelerating rocket, and there is an observer who is static in the accelerating frame of the rocket, then the electron will behave in the same way as a similarly supported static electron to a static observer in gravitational field.
The following theorem states that there is no visible Unruh radiation for an observer that is in the accelerating rocket frame. We will treat later the question if an inertial observer might see Unruh radiation.
Theorem 5. An observer in the local frame of an accelerating rocket will not observe any radiation from an electron that is supported with static electromagnetic forces inside the rocket. QED.
Definition 6. Let us assume that an electron in an accelerating rocket is a part of a hydrogen atom or any other system that has more than one bound energy levels for the electron. We say that the electron sees a thermal bath if it will occasionally move to a lower energy level and emit a photon that is seen in the accelerating rocket local frame.
Corollary 7. An electron in an accelerating rocket does not see a thermal bath if conditions of Theorem 5 are met. There is no Unruh radiation in this sense, either. This corollary follows directly from Theorem 4. QED.
Note that since the hypothetical Unruh radiation is a function of the acceleration of the charge, the existence of Unruh radiation would involve the exact same "signals from the future" problem as the Abraham-Lorentz force involves.
In a subsequent blog post we will analyze in what way the usual derivation of Unruh radiation is a result of flawed application of quantum field theory. The basic error is that when moving to an accelerated frame, one has to be very careful when one applies quantum field theory, since quantum field theory is based on inertial frames. Specifically, electromagnetic waves have to be modeled as real-valued waves which always have the same amount of positive and negative frequencies. The approach of Unruh, Hawking and others, where electromagnetic waves are modeled as purely positive frequency probability amplitude waves, is wrong and leads to:
- breach of conservation of momentum and energy;
- nonunitarity of the solution, which in turn produces the information loss paradox of black holes;
- a nonsensical classical limit where part of the energy flux of a laser beam seems to disappear for an observer who is accelerating in the direction of the laser beam;
- breach of an equivalence principle for gravitation.
Definition 6. Let us assume that an electron in an accelerating rocket is a part of a hydrogen atom or any other system that has more than one bound energy levels for the electron. We say that the electron sees a thermal bath if it will occasionally move to a lower energy level and emit a photon that is seen in the accelerating rocket local frame.
Corollary 7. An electron in an accelerating rocket does not see a thermal bath if conditions of Theorem 5 are met. There is no Unruh radiation in this sense, either. This corollary follows directly from Theorem 4. QED.
Note that since the hypothetical Unruh radiation is a function of the acceleration of the charge, the existence of Unruh radiation would involve the exact same "signals from the future" problem as the Abraham-Lorentz force involves.
In a subsequent blog post we will analyze in what way the usual derivation of Unruh radiation is a result of flawed application of quantum field theory. The basic error is that when moving to an accelerated frame, one has to be very careful when one applies quantum field theory, since quantum field theory is based on inertial frames. Specifically, electromagnetic waves have to be modeled as real-valued waves which always have the same amount of positive and negative frequencies. The approach of Unruh, Hawking and others, where electromagnetic waves are modeled as purely positive frequency probability amplitude waves, is wrong and leads to:
- breach of conservation of momentum and energy;
- nonunitarity of the solution, which in turn produces the information loss paradox of black holes;
- a nonsensical classical limit where part of the energy flux of a laser beam seems to disappear for an observer who is accelerating in the direction of the laser beam;
- breach of an equivalence principle for gravitation.
Vladimir Belinski, Detlev Buchholz and Rainer Verch, and some others, have previously studied the existence of Unruh radiation under quantum field theory and have concluded that Unruh radiation does not exist:
On the existence of quantum evaporation of a black hole
V.A.Belinski, Physics Letters A
Volume 209, Issues 1–2, 11 December 1995, Pages 13-20
https://www.sciencedirect.com/science/article/pii/0375960195007857
Macroscopic aspects of the Unruh effect
Detlev Buchholz, Rainer Verch
(Submitted on 18 Dec 2014, last revised 25 Sep 2015)
https://arxiv.org/abs/1412.5892
If Unruh radiation does not exist, that calls into question the existence of Hawking radiation. The problem in Unruh radiation is that there is no place to put the extra momentum if the kinetic energy of an electron is converted to a photon. In the case of Hawking radiation, it might be possible that the gravitational field of the collapsing star in some way could absorb the extra momentum, but the author of this blog has not been able to find a plausible mechanism for that. Thus, it is likely that the existence of Hawking radiation would contradict the basic principles of quantum field theory.
No comments:
Post a Comment