The black hole horizon is a perfect mirror also in classical general relativity?

Our hypothesis of optical gravity suggests that the forming horizon of a black hole will act as a perfect mirror for all incoming waves in any field.

The paper of Jahed Abedi, Hannah Dykaar, and Niayesh Afshordi, in turn, suggests that gravitational waves do indeed reflect back from the horizon:

Echoes from the Abyss: Tentative evidence for Planck-scale structure at black hole horizons

https://arxiv.org/abs/1612.00266

But is it really so that in classical general relativity, arbitrary waves should travel down the geometry of a black hole without reflecting back?

Let us study the Schwarzschild solution of the geometry.

Close to the horizon, the radial speed of light as measured in the global coordinates of the Schwarzschild solution goes as ~ d where d is the Schwarzschild global distance from the horizon and the tangential speed goes as ~ √d.

Definition 1. Let us have a ray of light whose wavelength far away in the Minkowski space is λ. Let the ray of light enter a static gravitational field. We can use the wavelength of the light as a measuring stick for distances. By optical coordinates we mean coordinates where spatial distances are measured in wavelengths λ, that is, we measure spatial distances by the optical path length of optics.


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

We can choose the wavelength λ of a radially ingoing wave such that, close to the horizon, each one wavelength step even closer will make the radial speed of light to halve, when measured in the global Schwarzschild coordinates. We call such a step a λ-step.

The λ-step will make the tangential speed of light to go to 1/√2 of the previous step. The wavelength λ far away from the black hole is of the order of the Schwarzschild radius.

Let us try to visualize the geometry of the Schwarzschild solution. We measure the radial and tangential distances in terms of the local length of the wavelength λ. By local we mean what is the wavelength of the incoming wave at that location, measured by a static observer. For tangential distances, we imagine that a mirror at a 45 degree angle is used to turn the radial ray of light into tangential.

Let z = 0 in the Schwarzschild solution. We try to visualize the x,y-plane. If we try to keep λ constant in the visualization, we get a kind of surface.

Our surface looks like a funnel with a wide upper end, the flat Minkowski space, and an infinitely deep exponentially widening pipe that descends to the horizon:

____            ____

        \        /

          \    /

           |   |

          /    \
       /          \

     .               .
 .                      .

The pipe is infinitely deep before we reach the horizon. That is, the ingoing wave will have an infinite number of cycles λ before it reaches the horizon.

The width of the pipe grows exponentially when we move downward because the tangential λ decreases exponentially. We cannot really embed the pipe into a 3-dimensional Euclidean space because the circumference grows exponentially at each step λ closer to the horizon. The figure above is only to help imagination, not to be taken literally.

The funnel neck is at its narrowest at 3/2 Schwarzschild radii in the global Schwarzschild coordinates.

https://en.m.wikipedia.org/wiki/Fermat%27s_principle

Fermat's principle states that a ray of light can travel a path which is a stationary point of the optical length of the path, with respect to small variations in the path. A ray of light can have a circular orbit around the black hole at that radius.

Abedi et al. assume in their paper that part of outgoing waves are reflected back roughly at the narrowest neck. Abedi et al. consider the LIGO black holes which are rapidly spinning and have the Kerr solution while we have been looking at non-spinning Schwarzschild black holes.

Open problem 2. Where does the backreflection of outgoing waves actually occur? The geometry is not flat anywhere close to the horizon. The narrow neck in the figure is not special in that respect. In optical gravity, the optical density grows smoothly as we approach the horizon. Why would the narrow neck somehow produce more backreflection than other points around it? What does optics tell us? How does the spinning affect the backreflection?


Open problem 2 can probably be solved with a numerical simulation.

As an aside, note that if we measure the depth of the pipe with a ruler (= proper length) then the pipe is not infinitely deep. If an observer is at distance d from the horizon in global Schwarzschild coordinates, then the proper distance from the horizon is 2 * √(d * r_s), where r_s is the Schwarzschild radius. An ingoing wave will be blueshifted close to the horizon. That is why the pipe is infinitely deep if we measure in wavelengths λ.

If we have an incoming planar wave, how much of its energy can travel down the pipe? The Huygens principle tells us that we can calculate a new place for a wavefront by assuming that each point in the old wavefronts is a new source of oscillation.

Close to the horizon, the space is very much curved. If we try to draw a square where each side is λ and the upper line is horizontal, then the sides differ very much from the radial direction. The lower line would be √2 λ if we put the sides radially.

          λ
   _________
   \             /    A "square" close to the horizon
 λ  \_____/  λ

          λ

According to the Huygens principle, there will always be waves reflected back if there is not a total destructive interference of the reflected waves. If a planar wave proceeds in the flat Minkowski space, the reflected waves are completely canceled out by destructive interference.

Intuitively, it is likely that the destructive interference cannot cancel out reflected waves in the very much curved geometry close to the horizon. A numerical simulation may confirm this.

The reflection would be very strong for waves of length λ, that is, waves whose length far away from the black hole is of the order of the Schwarzschild radius. The reflection might be up to 10 % (?).

For shorter waves the reflection is less, but since the distance to the horizon is infinite in terms of λ, even a small ratio of reflection will, after an infinite number of iterations, reflect everything back.

There is some uncertainty, how exactly we should simulate waves in a curved spacetime geometry. Our optical gravity hypothesis is one possibility.

Could it be that all energy of the downgoing waves will eventually be reflected back?

Open problem 3. If planar waves hit a black hole, will all wave energy be reflected back before the downgoing waves reach the horizon?


If the wavelength is of the order of the Schwarzschild radius, then Problem 3 can be solved with numerical methods. An analytic solution is unlikely.

For short waves, the Bogoliubov transformation might offer a way to estimate the reflection. We can use geometric optics to trace the wave a certain distance closer to the horizon. Then do the Fourier decomposition of the wave in a freely falling reference frame. The wave will appear as a chirp in such a frame. Its decomposition contains negative frequencies. The negative frequency waves can be interpreted as reflected waves that are traveling upwards.

Open problem 4. Is there a depth where a sizeable portion of downgoing waves has been reflected back? Is the Planck length related to this depth in some way?


Open problem 5. If we have a static observer very close to the horizon, then any falling particle will have an almost infinite mass-energy as observed by the observer. How do these almost infinite masses affect the local geometry?

Critique of resonant cavity of Abedi et al.

Abedi et al. calculated that the echo repeat time should be of the order 0.1 seconds in the LIGO data. That is, if we assume reflection points at about 1.5 Schwarzschild radii and at 1 Planck length proper distance from the horizon.

In our model, the backreflection of waves may happen smoothly and not be concentrated at the narrow neck at 1.5 Schwarzschild radii.

Abedi et al calculated that in the LIGO data, the Schwarzschild distance from the horizon is of the order 10^-74 meters when the proper distance from the horizon is the Planck length 1.6 * 10^-35 meters. They expect the reflection to happen around that position.

Gravitational waves that the LIGO can observe have a long wavelength, that is, of the order of our λ above. Each one wavelength λ step towards the horizon halves the Schwarzschild distance from the horizon. If we start at a Schwarzschild distance of, say 10 kilometers from the horizon, it will require roughly 250  λ-steps to get to 10^-74 meters from the horizon in Schwarzschild coordinates.

If we assume a 10 % reflection at each λ-step, then the horizon has reflected most of the wave back already at 10  λ-steps, not 250, as Abedi et al. assume.

The author of this blog will next study the Kerr solution and the backreflection at 1.5 radii.

How do black holes orbit and merge in optical gravity?

https://en.wikipedia.org/wiki/No-hair_theorem

The name of the black hole no-hair theorem is misleading. Actually, it is a conjecture. It claims that after a black hole is formed, it can be described, for all practical purposes, with the mass, the angular momentum, and electric charge: that is, with just three real numbers.

Optical gravity, on the other hand, implies that the inside of the horizon is essentially frozen, but the Newtonian gravitational pull of the matter inside horizon can still be felt by an outside observer. Thus, the entropy of a forming black hole is large in optical gravity. We cannot describe a black hole with just three real numbers.

A problem in the optical gravity hypothesis is how do we model the merger of a binary black hole? LIGO has provided us with some empirical data of mergers.

In an optical black hole, the local speed of light inside the forming event horizon is zero. If we bring two black holes together, how do their forms adjust in the merger?

To explain the orbiting of two black holes, we have to add the following hypothesis to optical gravity:

Hypothesis 1. The local speed of light for signals between two spacetime points inside or close to the forming black hole horizon is essentially zero, but the forming black hole can move collectively and rotate collectively.


Hypothesis 1 is required by Lorentz invariance. If a horizon would stop and stick at "one position" in space, a black hole would have an infinite inertial mass.

Open problem 2. In optical gravity, what type of global transformations are allowed to deform the mass inside or near a forming horizon? Inside or very close to the horizon, the effective Newton or Coulomb force between masses that are close together, is essentially zero. All strictly local change has stopped from the point of view of a global observer. But there could be global deformations to the wave function of the system that still could happen and shape the insides of a forming horizon.


Open problem 2 exists also in traditional general relativity. We cannot map the proper time of an observer inside of the forming horizon to the proper time of an outside observer, because no signal can reach the outside.

If we have a cigar-shaped star and make a black hole out of it by collapsing a dust shell on it, does the cigar-shaped gravitational field persist or is it deformed to be spherical? If it becomes spherical, then an outside observer would see the mass inside the horizon to "move", but how can that happen if no signal can come out from inside the horizon?

Echoes of gravitational waves are evidence for optical gravity?

http://backreaction.blogspot.fi/2018/04/a-black-hole-merger-merger-merger.html

Sabine Hossenfelder's blog post mentions an interesting result from LIGO:

https://arxiv.org/abs/1612.00266

Gravitational waves seem to bounce off from the horizon of a black hole.

Our blog post on April 20, 2018 introduced the optical gravity hypothesis:

http://meta-phys-thoughts.blogspot.fi/2018/04/optical-theory-of-gravity.html

The hypothesis implies that the forming horizon is a perfect mirror for all fields - it has an infinite optical density for all fields. That could explain why gravitational waves bounce back.

But is the reflection of gravitational waves from the horizon actually predicted by standard general relativity?

UPDATE: Yes, it is! See the post:

http://meta-phys-thoughts.blogspot.fi/2018/04/the-horizon-is-perfect-mirror-also-in.html?m=1

Actually, optical gravity is not a new theory of gravitation - it is a new interpretation of classical general relativity. Therefore, all consequences of optical gravity can also be derived in classical general relativity.

Optical gravity does clarify some vague aspects of general relativity: optical gravity claims that the information in the Newton force propagates at the global speed of light while the Einstein equation leaves the speed vague. Also, optical gravity describes the structure of a forming black hole, while there are various views of what Einstein equation says.

Optical gravity may be amenable to quantization while the Einstein equation has presented insurmountable problems.

Pauli exclusion principle comes from the repulsion between electrons?

According to the spin statistics theorem, particles with a half-integer spin are fermions, that is, they obey the Pauli exclusion principle.

But that does not explain why electrons are fermions in the first place.

An electron has a relatively strong magnetic moment. If we have two electrons whose spins are in opposite directions, the electric repulsion between them is partly canceled.

If the distance is just 10^-15 m, then the magnetic attraction between two electrons is still just 10^-20 of the electric repulsion.

The magnetic attraction may be the underlying reason why every quantum state in a stationary atom can accommodate both an electron with spin +1/2 and spin -1/2. The magnetic attraction only works well in a system of two electrons, not three. That is why a state can hold exactly 2 electrons.

The force of magnetic attraction is too weak. Why can two electrons then fit on each quantum state?

But why the Pauli exclusion principle? Consider the particles in a box model of quantum mechanics. If 3 electrons would be in the ground state, then we would have an electron cloud that is denser in the middle in the box, and furthermore, the magnetic attraction does not help in keeping all 3 electrons close to each other.

To achieve a smoother distribution of the electron cloud in the box, one has to populate energy levels above the ground state.

Todo: prove that the minimum energy is obtained with 2 electrons on each level.

If we would have an atomic nucleus that is orbited by negatively charged bosons (hypothetical boson electrons), then all those bosons would fall into the lowest possible orbit? Classically, if we have a positive charge in the middle and compensating negative charges, those negative charges come as close to the positive charge as they can get.

Conclusion: we cannot explain the Pauli exclusion principle in an atom by the electric repulsion alone.

The Bell inequality does not require faster-than-light communication

Suppose that we have a source that produces pairs of correlated photons, such that the polarization of the photons is opposite to each other.


Polarization                            Polarization
filter   <-----photon 1   photon 2----> filter

We measure the polarization of each photon relative to a polarizarion filter. The output is either +1 or -1. The filters can be turned so that they are at an angle relative to each other.

https://en.m.wikipedia.org/wiki/Bell%27s_theorem

The famous theorem by John Bell states that the distribution of measurement results versus the angle cannot be reproduced if we assume that each photon already had its polarization determined before the measurements.

Some researcher claim that this shows there must be some kind of faster-than-light communication between the measuring devices.

Their claim is based on the following:

Assumption 1. The measuring devices make the wave function of the pair of photons to "collapse" at each end of the measurement apparatus. The measurement result can then be compressed to a single binary number +1 or -1 and sent to the scientist to tabulate and analyze.


Assumption 1 already highlights what is the error in this thinking: we make the wave function of the apparatus to collapse before the data is in the head of the observer, the scientist.

If we cut off part of the wave function of a system before the observer interacts with the system, then, of course, that will lead to strange behavior, like the need for faster-than-light communication or loss of unitarity.

The correct way to treat this experiment is to let the wave function propagate to the measurement devices and then to the head of the scientist. The interference pattern, that is, the table of measurement results, is formed in his head. There is no need for faster-than-light communication.

It is like a complex double-slit experiment where the screen is the head of the scientist. It is the screen where we finally let the wave function to collapse, if we use the Copenhagen interpretation.

If we use the Bohm model, then the hidden markers of the photons will sail on the wave function all the way to the head of the scientist where the markers finally "hit the shore" and determine in which of the many alternative worlds the observing subject will live in after the experiment. In the Bohm thinking, the wave function does not collapse at all. All the alternative worlds continue to exist. The markers just choose in which world our observing subject will be afterwards.

Open problem 2. The observing scientist receives the measurement outcomes from the different ends A and B of the apparatus at different times. For the interference pattern to form, the wave function must not collapse before the scientist has the outcome from both A and B in his head. How do we formalize this? Can the Bohm model help?


https://en.m.wikipedia.org/wiki/Entropy_of_entanglement

Entanglement is traditionally seen as a purely quantum phenomenon, but we can also interpret it classically. Entanglement just means that a classical wave was born in a single place in spacetime in a single process. When the classical wave propagates, there is correlation between its values in different points of spacetime. This correlation is called entanglement.

Entanglement entropy in quantum mechanics has the following meaning: if we have a combined system C & D in a pure state, and we measure C as accurately as we can, what is the von Neumann entropy of the mixed state in D after that?

Open problem 3. Can we really measure half of the system first and make its wave function to collapse, and later compare the result to the measured values in the rest of the system? That would require faster-than-light communication in the Bell inequality experiment. Does entanglement entropy have any physical meaning?

The vacuum is a semiconductor for electrons

Conjecture 1. The vacuum is filled with virtual electron-positron pairs. In a creation of a real, "free" electron-positron pair, the 1.022 MeV of energy that is given to the pair raises them to the "conduction band". This is analogous to a semiconductor. A free electron in space is actually a denser zone, or a cloud, of electrons where the charge of positrons does not cancel their negative charge; vice versa for positrons. We cannot pinpoint which electron in this zone is the real free electron. It is a collective phenomenon.

Conjecture 1 explains why an electron in a particle accelerator appears as a point particle whereas its classical radius is quite large. The higher the energy of the electron, the freer it appears to be. In a typical semiconductor, the gap between the valence band and the conduction band is just 5 eV. An electron in the conduction band is a collective phenomenon in a semiconductor, and a hole is another collective phenomenon which behaves like a positron.

We do not know what the energy of a true free electron would be if we could lift it out of the vacuum.

The existence of the antiparticle for electron is required by our model: it is the counterpart of a denser electron zone.

Conjecture 2. The angular momentum and the magnetic moment of the electron are produced by the rotation of the cloud of virtual electrons and positrons.

The angular momentum (spin) of the electron is 1/2 h/2π. It is 1/2 of the minimum nonzero value one can attain through rotation of ordinary matter in an orbit. Maybe the relevant metric around the electron squeezes everything in the tangential direction so that the circumference of a circle around the electron is effectively 4πr for the rotating cloud?

Why does the cloud always rotate, and always has the same angular momentum? The following may explain that: an electron-positron pair is born as an "atom" of positronium. Tidal forces make the clouds of the electron and the positron not to rotate relative to each other but they do rotate relative to an inertial observer. When the clouds are freed, they keep rotating.

Why does the Dirac equation describe the cloud? What happens in the cloud in the zitterbewegung?

The orbital angular momentum of an electron on the 2p orbital in hydrogen is 2X the spin angular momentum of the electron. Why are these numbers related?

                                          ↑
         e-  electron           2e+ virtual positron

Maybe the electron is surrounded by a rotating cloud that is equivalent to a virtual positron with roughly 2X the normal charge of a positron? The 2X is approximately the gyromagnetic ratio of the electron.

The orbit of the virtual positron would be responsible for the angular momentum and the magnetic moment of the electron. The values of those are far too large to be explained by a spherical electron whose size is the classical electron radius. But they could be explained as the motion of virtual electrons and positrons in a cloud.

In a sense, the virtual particles circling the electron are permanent virtual particles, in contrast to the short-lived virtual particles in a Feynman diagram.

How does Lorentz invariance require the existence of the electron spin and the antiparticle in the Dirac equation? Our positronium model above does not have anything to do with Lorentz invariance, and it still predicts the existence of an antiparticle and an electron spin.

Conjecture 3. The annihilation of an electron-positron pair just means that the density variations of electrons and positrons are evened out. The energy escapes as electromagnetic waves. Thus, the annihilation is the LED light of empty space.


In annihilation, no electron or positron is really destroyed. They are returned to the pool of virtual particles. If the Dirac equation would describe just a single free electron in otherwise empty space, then it would be mysterious how the electron wave and the positron wave can exactly cancel each other out so that only electromagnetic waves remain. Classically, if we have coupled fields, it never happens that the coupling completely erases the wave in one of the fields. That would be a miraculous coincidence, if a wave would completely disappear. Similarly, we never observe the annihilation of two photons, even though the photon is formally its own antiparticle.

Conjecture 4. Photons are temporary polarization of the virtual electron-positron pairs. There is really no such thing as a wave of the electromagnetic field. It is always just a polarization wave of virtual pairs. The Coulomb force makes the virtual electrons and positrons move.

A photon has spin 1, which means that it carries a large angular momentum. If we have a system where a negative charge is orbiting a positive charge, their motion will produce circularly polarized waves that carry away the angular momentum of the system. We can envisage virtual electron-positron pairs in space starting to mimic the rotation of the system. The angular momentum can be carried arbitrarily far by the rotation of the virtual pairs. Eventually, the angular momentum can be absorbed far away by a copy of our system.

We may also envisage that our system sends virtual copies of itself to space. The collective motion of virtual electron-positron pairs adds up to a single virtual orbiting electron-positron pair where the distance of the virtual electron and positron is large. The virtual copy can be "absorbed" by another system far away.

In annihilation, as the electron and positron clouds even each other out, the result is a disturbance of the virtual electron-positron pairs. That disturbance is what we call electromagnetic waves.

Conjecture 5. The only electromagnetic force is the Coulomb force. Its Lorentz transformation produces the illusion of a separate magnetic force. The Coulomb force always propagates at the speed of light of the global Minkowski space. Locally, polarization of a refractive medium, e.g., glass slows down the propagation of polarization waves, but the information in the Coulomb force always travels at the global speed of light.


Conjecture 5 reflects the analogy of our optical theory of gravity and this blog post.

Open problem 6. Why is the propagation speed of the Coulomb force the same as the speed of polarization waves in the vacuum, that is, the speed of light? Why is light not slower than the Coulomb force? Or maybe it is?


Conjecture 7. Particles that move in the vacuum, in a sense, themselves build the virtual electron-positron pairs around themselves. That is how we achieve Lorentz invariance. An observer himself supplies the energy to build the vacuum polarization around him. He cannot measure his absolute speed relative to virtual pairs in the vacuum. The energy content of the vacuum is exactly zero if there are no field excitations present.


Our conjectures resurrect the aether theories of the 19th century. An observer drags the aether along with him and therefore cannot measure his speed relative to the aether.

We could say that an electron is a slower-than-light disturbance in the sea of virtual electron-positron pairs, while a photon is a light-speed disturbance. A free electron-positron pair can be seen as a frozen photon. We can convert the pair back into live photons through annihilation.

Optical theory of gravity

UPDATE Sept 14, 2018: the optical theory apparently gives wrong predictions for the phase of the wave function of a particle. See blog post Sept 12, 2018.

...

Isaac Newton had thoughts about unifying optics and gravity:

Eric Baird: Relativity in Curved Spacetime: Life Without Special Relativity

Since gravity bends light and a refractive material also bends light, there is an analogy between them.

Suppose that we have an observer in a gravitational potential well, say, on the surface of Earth. Suppose he is holding two electric charges in his hands and he has to use energy to pull them one meter apart. If another observer far away in space sends him this energy as light, the light will be blueshifted and the observer on Earth thinks he receives more energy than was sent by the space observer.

From the point of view of the space observer, the Coulomb force on Earth appears to be weaker than in space. The same holds for every force, also the Newton force of gravity itself.

The weakening of forces is the reason why clocks tick slower and light travels slower in a gravitational potential well.

Conjecture 1. The real geometry of spacetime is Minkowski everywhere, also in a gravitational potential well. The apparent curving of the spacetime geometry is due to the weakening of effective forces in the gravitational potential well. Weakening of the Coulomb force causes the speed of electromagnetic waves to slow down.

We call the above conjecture the optical theory of gravity.

Conjecture 2. Gravitation is just the Newton force of gravity, which propagates at the global speed of light in the Minkowski space. The global speed of light is not affected by gravitational potential wells but is the same everywhere in the Minkowski space. The Newton force is completely analogous to the Coulomb force.

Conjecture 2 resolves a problem of general relativity: how fast does the information of a changed position of a black hole travel? It takes an infinite time for light to travel from the horizon to the environment. How do the black holes in a binary black hole know the position of each other? Our solution is that the local speed of light does not affect the speed of the information that is carried in the Newton force.

The apparent curvature of spacetime is the result of differences in the optical density, or refractive index, of empty space. The density is not the same to every direction. In the Schwarzschild global coordinates, the speed of light is less in the radial direction than in the tangential direction. In optics, this phenomenon is called birefringenge. Crystals have refractive indexes that depend on the direction of light relative to the crystal lattice.

Conjecture 3. The Newton force causes polarization of virtual particles of positive and negative mass-energy (gravity charge) in empty space, just as the Coulomb force causes polarization of virtual electron-positron pairs in empty space. This polarization makes all forces to be weaker in the radial direction than in the tangential direction in the Schwarzschild geometry. The space around a gravitating mass is like a crystal whose lattice has the radial direction as a special direction.

Polarization causes a round hydrogen atom to squeeze in the radial direction in the Schwarzschild solution. All measuring rods become shorter in the radial direction but not in the tangential direction.

Open problem 4. Is quantum gravity easy to formulate in optical gravity? Is there a renormalization problem? How does the weakening of forces in a gravitational potential well affect quantum field theory?

Open problem 5. How do we model mathematically the polarization of virtual gravity charges in empty space?


Open problem 6. The horizon of a forming black hole would be optically infinitely dense, that is, all forces have zero strength and the speed of light is zero. Particles with non-zero rest mass will reflect from the horizon and repeatedly bounce from it. If the particles are electrically charged, they will radiate their energy away in photons? What is the end result of this process? Can this explain the relativistic jets that shoot from astronomical black holes?


Would the reflection of waves and particles from a forming event horizon break the Strong equivalence principle 4 of our blog post below?

http://meta-phys-thoughts.blogspot.fi/2018/04/does-hawking-radiation-exist.html

A freely falling laboratory would observe mysterious reflection of photons and other particles when it is close to the horizon. Eventually, the particles that form the laboratory would themselves be reflected. The laboratory would be destroyed before it reaches the forming horizon.

UPDATE: this may be the first experimental evidence that supports optical gravity:

http://meta-phys-thoughts.blogspot.fi/2018/04/echoes-of-gravitational-waves-are.html

Does Hawking radiation exist?

Stephen Hawking has claimed that a time-varying gravitational field, or a time-varying geometry of timespace, creates electromagnetic radiation. The idea is originally due to Leonard Parker:

Leonard Parker, The creation of particles in an expanding universe, Ph.D. thesis, Harvard University (1966). Publication Number 7331244.

Our previous blog posts suggest that Unruh radiation does not exist as a phenomenon independent from the acceleration mechanism of the detector. There is no place to put the extra momentum if we try to convert kinetic energy of the detector to photons.

In the case of Hawking radiation, there is gravitating mass available nearby. Maybe we could put the extra momentum to this mass or its gravitational field?

Hawking's 1975 derivation

Hawking radiation is not specific to black holes. It is produced by any time-varying gravitational field that makes a gravitational potential well to deepen. Then a light signal that goes through the well will have a longer delay.

S. W. Hawking
Comm. Math. Phys. Volume 43, number 3 (1975), 199-210
Particle creation by black holes
https://projecteuclid.org/euclid.cmp/1103899181

Hawking used a canonical transformation, similar to Definition 6 in our April 10, 2018 blog post, to deduce that a wave packet that is tracked back in time through a collapsing star will deform to a chirp.

Hawking then used the Bogoliubov type reasoning that the number operator for the original wave packet must be > 0 because the negative frequencies in the chirp "come for free".

That means that an inertial radiation absorption detector far away of the collapsing mass will "click" as the wave packet lifts it to an excited state. These clicks of the detector are called Hawking radiation. The derivation of Hawking is called a "semiclassical" derivation.

The energy to excite the detector seems to appear from nothing. That is the origin of the black hole information paradox.

If we assume conservation of energy, the energy has to come from the kinetic energy of the collapsing mass, or from its gravitational field energy, or some even more exotic process.

Our Claim 7 from April 10, 2018 says that an electromagnetic wave packet should be described as a real-valued wave packet relative to an accelerated observer. Let us extend our claim to a varying gravitational field:

Claim 1. An electromagnetic wave packet should be described as a real-valued classical wave packet also under a time-varying gravitational field, that is, under a time-varying spacetime geometry.


Our claim is that, for example, a laser beam through a time-varying gravitational field will behave as a classical real-valued electromagnetic wave.

The derivation of Hawking, on the other hand, describes an electromagnetic wave packet as a complex-valued probability amplitude wave packet that is built from purely positive frequencies. If our Claim 1 is correct, then the derivation of Hawking is incorrect.

As an aside, note the very close analogy between optically dense material and a gravitational field potential. Both affect the apparent speed of light and bend light rays accordingly. Gradient-index optics studies refractive material whose refractive index varies from place to place:

https://en.wikipedia.org/wiki/Gradient-index_optics

The analogy of a time-varying gravitational field is material whose refractive index changes with time. We could extend our Claim 1 to gradient-index optics: an electromagnetic wave is a real-valued classical wave also under those conditions.

If the semiclassical derivation of Hawking radiation is wrong, there could still be another mechanism by which kinetic/gravitational/mass energy under a varying gravitational field is converted to electromagnetic radiation. Let us explore the possibilities.

Gravitational symmetry of positive and negative electrical charges

If we have interacting electric charges, for example, in bremsstrahlung, they produce electromagnetic radiation. The electric field E(r, t) changes with time and place in the process. If we model the process with classical electrodynamics, the electric field E(r, t) can be calculated deterministically. We can calculate deterministically the phase of the electromagnetic wave w(r, t) that is born in the process.

On the other hand, if we have electrically neutral particles, for example, particles of hypothetical dark matter, that interact gravitationally, these generate a time-varying gravitational field g(r, t). But there is no reason to associate a specific electric field E(r, t) with g(r, t). If we claim that there should be an electric field E at a point in timespace, why not -E? If positive and negative charges are symmetric under gravitational interaction, there is no reason why the electric field vector should point to one direction and not the other.

If a time-varying gravitational field would produce an electromagnetic wave w, why not a wave whose phase is shifted by π?

Theorem 2. If positive and negative electrical charges behave in a symmetric way under gravitation, then a time-varying gravitational field cannot produce electromagnetic radiation in a deterministic way. The proof is by symmetry. QED.



Hawking radiation is traditionally thought to be indeterministic, black body radiation. Is there any process in quantum field theory that produces truly indeterministic radiation, such that also the phase of a wave w of the radiation is indeterministic?

If we let a large number of electrically charged particles collide in a scattering experiment, there is a very large number of ways in which the process can emit an electromagnetic wave w(r, t). The radiation from the collision will appear as almost indeterministic and its spectrum will be close to black body radiation. We can use a Feynman diagram to calculate various scattering probabilities.

The scattering process is, however, unitary, or deterministic, in the sense that if we assume a certain wave function for the system at the start, we can calculate the end wave function deterministically.

For most initial wave functions of our scattering experiment, the end wave function is not symmetric with respect to electric charges. For example, the probability P that the electric field E(r, t) is equal to vector E (which is not zero) in a point (r, t) in timespace after the collision, P is typically not the same as the probability P' to have an electric field -E there.

The discussion above leaves open the possibility that a time-varying gravitational field would produce an electric field in a nondeterministic way. For instance, there would be a 50 % chance of field E at certain point of timespace, and 50 % chance of field -E at the same point. Spontaneous symmetry breaking in nature is a process where a system in a more or less random way "crystallizes" and finds a preferred direction in space.

Let us compare gravitation to the Higgs field. The Higgs field forms a condensate that is electrically neutral. The Higgs boson has a weak hypercharge.

The Higgs boson can decay in many ways, even to 2 photons with a 0.2 % probability. Maybe its weak hypercharge causes asymmetry in other fields which can produce ripples to those other fields?

Equivalence principle suggests that gravitation is symmetric with respect to all non-gravitational charges. Does that mean that a time-varying gravitational field cannot produce any other particles than gravitons?

Are all non-gravitational charges symmetric such that if charge +a exists, so does -a? If yes and if gravitation is symmetric, why would a time-varying gravitational field produce a field vector A instead of -A?

Conjecture 3. Gravitation is symmetric with respect to all non-gravitational charges. A time-varying gravitational field cannot produce any other particles than gravitons. A collision of two gravitons only produces gravitons.


If Conjecture 3 is true, then no Hawking radiation can exist in the form of photons.

Since excitations of other fields can emit gravitons, and quantum field theory is time-symmetric, does that mean that gravitons alone can produce excitations of other fields? Assume that two photons annihilate each other and all that is left is ripples in the gravitational field, that is, gravitons. But that process would break conservation of energy since the energy in the photons is lost. Also, classically it would be miraculous if the waves in the two photons could exactly cancel each other out.

What about tunneling? Assume that we have a very strong gravitational field and a virtual particle which has a positive mass-energy. If the particle moves in the direction of the gravitational field, can it gain enough energy to become real? That would break the equivalence principle, because a scientist in a freely falling laboratory would observe particles to pop up from empty space. We can use this same argument against particle creation by a time-varying gravitational field.

Strong equivalence principle 4. A freely falling "small" laboratory in a (time-varying) gravitational field will not observe excitations of other fields to pop up from nothing - the behavior is the same as in a laboratory that is inertial in empty Minkowski space.


Principle 4 is strong in the sense that "tidal" effects of an inhomogeneous gravitational field might still produce photons that appear into the laboratory.

UPDATE: We do not expect Principle 4 to be true for waves whose wavelength is of the order Schwarzschild radius. Waves of that size cannot be studied in a small laboratory that is freely falling. Their behavior is affected by the global geometry around the black hole. Principle 4 does not say anything about the existence of Hawking radiation far away from the horizon of a black hole.

What about Hawking radiation very close to the horizon? It is blueshifted by enormous factors. But its wavelength is still very big relative to the proper distance to the horizon. Thus, "tidal" effects could produce Hawking radiation regardless of Principle 4.

Freely falling observer and a static charge - Unruh radiation does not exist

Let us treat a problem that we left open in the blog post about the Larmor formula and Unruh radiation. If we have an electron that is statically supported on Earth using static electromagnetic forces, does it appear to radiate to a freely falling observer?

It does not radiate to another static observer, because there is no source of energy for the radiation.

From the point of view of a freely falling observer, the static electron is being accelerated by a huge spaceship, namely Earth.

Suppose that the freely falling observer would see radiation, that is, a wave packet. According to Claim 7 of our previous blog post, then a static observer would see a chirp.

Since the static observer does not see chirps or any other type of radiation, the freely falling observer cannot see either.

Let us conjecture an equivalence principle:

Equivalence principle 1. If we have an electron that is supported with static electromagnetic forces in an accelerating rocket and an inertial observer, then the electron will behave in the same way as a similarly supported static electron in a gravitational field to a freely falling observer.


Equivalence principle 1 implies:

Theorem 2. An inertial observer will not see any Unruh radiation from an electron that is supported with static electromagnetic forces in an accelerating rocket. QED.


Why did we resort to equivalence principles and did not study directly the radiation from an accelerating rocket in space? That is because the author of this blog is not aware of generally accepted methods of studying complex systems like a rocket in quantum field theory. If we have an electron supported in the rocket, what kind of quantum field interactions might happen between the electron, the frame of the rocket, and the propulsion system? There will be vibrations, phonons, in the frame. It looks like the vibrations cannot produce Unruh-like radiation, but it is hard to prove that.

Summary

In our blog posts April 5, 2018 and April 15, 2018 we have shown that Unruh radiation does not exist if two equivalence principles hold and a claim about photons in an accelerating frame holds.

On the other hand, if an electron is accelerated with laser or impinging other electrons, there will be radiation. Some of that radiation could be interpreted as Unruh radiation.

Our blog posts about the Larmor formula and Unruh radiation have a common message: radiation is a result of dynamical interaction between charges or photons. It is not the acceleration itself that produces any radiation, as can be seen from our treatment of electrons that are supported with static electromagnetic forces.

We have left open the question if a varying gravitational field might produce electromagnetic radiation. Does Hawking radiation exist? The next blog post will be dedicated to studying that question.

Unruh radiation and the error of Unruh and Hawking

Since quantum field theory in accelerating frames is not well understood, the most trustworthy way to study the existence of Unruh radiation is to work in an inertial frame and analyze what each observer, accelerating or not, will see.

William Unruh and several other authors have tried to apply quantum field theory to accelerating frames, but they have not done a careful analysis of conservation momentum and energy, the existence of a sensible classical limit, and some other cornerstones of traditional quantum field theory.

Actually, the term quantum field "theory" is misleading, because many central problems of the framework, like vacuum stability, remain open. It is not a theory in the sense of mathematics, but rather a toolpack of heuristic algorithms.

In the previous blog post we considered two thought experiments, one of which was:

Thought experiment 1. An electron statically supported in gravitational field by impinging photons.


      e   electron
     ↑   support

____________
   Earth

We left open the question what a static observer will see.

Double Compton scattering

The central pillar of quantum field theory is the Feynman diagram method of calculating scattering experiments. When the electron is supported by a flux of photons impinging on the electron from downward, that amounts to a scattering experiment.

In Thought experiment 1 above, let us perform the scattering experiment in a freely falling frame, such that it is an inertial frame.

Definition 2. In a Feynman diagram, a virtual particle is any particle whose energy and momentum do not match any free particle in vacuum. A real particle is such that its energy and momentum in the diagram could match a free particle. Real particles are said to be on-shell and virtual particles are said to be off-shell.


A virtual particle can return to the "shell", for instance, by emitting as a photon the the extra energy it has.

photon A     e    photon B
  》                 /   《
   《              /   》
       》        /《
        《     /~~~~~~~~
          《 /                      | virtual photon
           》 \                     | momentum p
          《     \  ~~~~~~
           》       \
          《           \
           》            \
photon A     e electron

Diagram 3.

In the Feynman diagram above, time flows upward.

An electron emits a virtual photon with momentum p, which takes the electron off-shell, that is, the electron is also virtual after the emission, its kinetic energy and momentum do not match. The virtual electron collides with photon A.

After the collision, the electron absorbs back the virtual photon. The electron is still off-shell because it has too much kinetic energy after the collision. To get back as a real electron, the electron has to emit the extra energy it has as photon B.

The above process is called the double Compton scattering. Ordinary Compton scattering does not produce photon B. The virtual photon in the diagram is sometimes called a "self-energy" photon, and the diagram above calculates a "self-energy" correction to probability amplitudes.

Classically, an electron that interacts with a wave in the electromagnetic field, makes the combined system electron & field nonlinear. The wave that scatters from the electron will have a very complex form.

Conjecture 4. We conjecture that at low energies, double Compton scattering and other similar quantum electrodynamical processes can be treated as fully classical phenomena. In the classical analysis, the electromagnetic wave replaces the wave function Ψ of the photon. The square of the electromagnetic field replaces the Born probability |Ψ|^2. At high energies, the electron has to be modeled with the Dirac relativistic wave equation.


The classical analysis of the scattering does not determine where the quantums of energy, that is, the photons, will be observed. We only obtain a probability distribution, or an interference pattern of electromagnetic waves. To make the process deterministic, one might use a de Broglie - Bohm type hidden variable interpretation, where markers that designate actual particles "sail" on the waves without affecting the dynamical behavior of the waves. In a future blog post we will elaborate on this idea.

If Conjecture 4 is true, that explains why optical phenomena, like the double slit experiment, can be handled fully classically besides the usual quantum mechanical wave function treatment.

The scattering matrix S for double Compton scattering has been calculated in several publications, e.g.:

Radiative Corrections to Compton Scattering
L. M. Brown and R. P. Feynman
Phys. Rev. 85, 231 – Published 15 January 1952
https://journals.aps.org/pr/abstract/10.1103/PhysRev.85.231

Let us now return to Thought experiment 1. An observer in the laboratory will see photons B as well as photons A scattering from the statically supported electron. What does a static observer on Earth see? Recall that the laboratory is falling freely, which means that the Earth static observer appears to accelerate upwards in the laboratory frame. Could it be that a static observer cannot detect photons A or B?

The question is how an accelerating observer will interpret the photons that an inertial observer sees to come out of the scattering process.

Photon in an accelerating frame

In Thought experiment 1, photons are emitted in the inertial laboratory frame. A photon can be modeled as a wave packet.

When the wave packets go far from the electron, they move under an essentially linear wave equation, and can be Fourier-decomposed into plane waves.

How does an accelerated observer see an electromagnetic wave packet?

Definition 5. An absorption detector of electromagnetic waves is a system where an electron is in a bound state and it has several metastable energy levels besides the ground state. A hydrogen atom, for example, is an absorption detector. A field detector is a free electric charge whose movement we can measure.


Classical electromagnetic waves are real-valued and contain the same amount of "positive" and "negative" frequencies. That is, at each moment t, the waves are of the form:

E(r, t) = ∫ f(⍵) * e^(i(k ⋅ r + ⍵ t)) + conjugate(f(⍵)) * e^(-i(k ⋅ r + ⍵ t)) d⍵,

where ⍵ is the angular velocity of the oscillation, k is the wave number vector, r is a vector in 3D space, t is time and the function conjugate(x) returns the complex conjugate of x. The sum above is real-valued because the sum is of two complex conjugate numbers. The function f is the Fourier decomposition of the wave into plane waves of different frequencies.

Definition 6. If we have an accelerating source of radiation, then a wave packet w(r, t) sent by this source will be distorted in an inertial frame.

We assume that the waveform as a function of the proper time of the source t, w(R, t) measured at a point R close and static relative to the source, will stay the same for an inertial source and an accelerating source. That is, the acceleration does not affect the output of the source.

Let w'(r, t) be the waveform in the inertial frame. We say that the distorted form w' is a result of the canonical transformation of w for this acceleration.

Conversely, if we have an accelerating observer, and a wave packet w(r, t) in an inertial frame, let w'(r, t) be the waveform that would have resulted if the source at the time of emission would have accelerated like the observer now, but in the opposite direction. Also in this case we say that w' is the result of the canonical transformation of w for this acceleration.


In some cases, Definition 6 is equivalent to the geometric optics approximation used in electrodynamics.

Definition 6 is very complicated, but still vague. It is very hard to do physics in an accelerated frame. The safe and most reliable way is to work in an inertial frame. In the case of wave packets, the Huygens principle makes their propagation complex. It is very hard to calculate the canonical transformation exactly. A rough approximation, however, is easy to calculate, if one just maps the proper time of the source to the proper time of the observer using a light speed signal.

Claim 7. An accelerated observer "sees" an electromagnetic wave packet as a classical real-valued electromagnetic wave. We can do a canonical transformation to the wave packet and use the Fourier transformation to decompose the real-valued wave packet into a spectrum of real-valued plane waves. An absorption detector carried by the observer will approximately detect this spectrum.


Note that Claim 7 again resurrects the "signals from the future" problem of Larmor radiation, because in order to do the Fourier transformation, we have to know the form of the whole wave packet, and that will we know only after we know the acceleration of the observer also in the future. The word approximately contains many sources of errors.

William Unruh, Stephen Hawking, and several others have claimed that photons should be modeled as purely positive frequency probability amplitude wave packets and that the packets should be manipulated with the Bogoliubov transformation when we switch to an accelerating frame:

https://en.m.wikipedia.org/wiki/Bogoliubov_transformation

A problem in the Unruh et al. approach is that when we do the canonical transformation, positive frequency wave packets become a sum of positive and negative frequencies. Negative frequencies come by because an accelerating observer sees a standard wave packet as a "chirp" due to the Doppler effect. A chirp in radar technology means an oscillation whose frequency goes up or down as time passes. One cannot build a chirp from purely positive frequencies.

In classical waves, there is no problem with the Doppler effect: real-valued waves will stay real-valued in an accelerating frame. On the other hand, if we model photons as purely positive frequency waves, we have no sensible representation for a chirp, that is, no classical limit that would be a chirp.

Based on Claim 7, we can now answer the question how a static observer sees photons A and B in Thought experiment 1: he will see them as real-valued wave packets that are chirps. He can calculate the spectrum with the Fourier transformation.

Particle in an accelerating frame

In a very simple case, we can describe a classical system completely at a time t with two real-valued parameters: the position of a particle on the x axis and its velocity v:

       particle   • --> v
0 ------------------------------> x axis

We can then attach a complex-valued probability amplitude to each classical configuration. The probability amplitude is a complex number of a fixed absolute value, and the value will rotate around the origin of the complex plane as time passes. This is the usual Feynman path integral way of thinking. The angular velocity of rotation depends on the total energy of the configuration. Since the energy is, by definition, always positive, the rotation will happen to the counterclockwise direction in the complex plane - in this sense, the "probability amplitude wave" is always positive frequency.

The probability amplitude wave lives in the abstract configuration space, with time t as an additional coordinate. The probability amplitude wave does not live in the simple spacetime coordinate space (x, t).

In the simple case above, we can usually work in quantum mechanics with a wave function which has just x and t as its parameters:

Ψ(x, t).

The wave function appears to live in the simple spacetime coordinate space (x, t).

Suppose that we prepare the system such that our particle is described by a wave packet. It is then a sum of infinitely many positive frequency plane waves.

If we try to switch to an accelerating frame and transform Ψ(x, t) with the canonical transformation, we end up with a chirp, which contains also negative frequency plane waves, not just positive. Does that mean that an accelerated observer may measure the particle to have a negative kinetic energy? That is nonsensical, which shows that we cannot do meaningful quantum mechanics in an accelerating frame by transforming a wave function with the canonical transformation.

We can do meaningful classical mechanics in an accelerating frame with the canonical transformation. If we have a classical electromagnetic wave packet, we can do the canonical transformation to it and we obtain its frequency spectrum as seen by an accelerated observer.

The error of Unruh, Hawking, and others, is that they have mixed the abstract configuration space & time t of quantum mechanics with the simple timespace (x, t) of classical mechanics and have applied the canonical transformation to a wave packet of quantum mechanics. The Bogoliubov method then shows that one can get negative frequency components of the transformed wave packet "for free", that is, the expectation value for the original wave packet is > 0.

That error is the origin of the strange claims that the vacuum would appear to contain particles to an accelerating observer, or that a black hole could evaporate in a non-unitary way.

Since the configuration space & time is an abstract framework, it actually makes no sense to talk about accelerating the whole framework. Accelerating in which coordinate, and what is accelerated?

As a parallel, we may think of a movie in a movie theater. It makes sense to say that a car in the movie is accelerating on a road, but what would it mean to accelerate the whole movie to some spatial direction? The movie is the framework, the car is an object in that framework for which "acceleration" is defined within that framework.

What if we accelerate an electron with electrons?

We can model this as a scattering experiment.

              e     e1         e
                 \    |         /
                   \  |       /
                     \|     /
                     /|   /
                   /  | /
                 /    |\
               /      |  \
             /        |    \
           /          |      \
         /            |        \
      e             e1         e

In the diagram above, time flows upward. Electron e1 is being accelerated by bombarding it with other electrons.

When an electron collides with another electron, photons may be emitted. Those photons can be interpreted as electron-electron bremsstrahlung:

https://en.m.wikipedia.org/wiki/Bremsstrahlung

Bremsstrahlung has a spectrum that is quite similar to a thermal spectrum. One could associate a temperature with bremsstrahlung. An electron in bremsstrahlung is able to convert some of it kinetic energy to a photon because another electron will absorb the extra momentum of the electron. Recall that a photon cannot carry away all the extra momentum.

The error in the Larmor formula for an accelerating charge and Unruh radiation

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.

------

J.J. Larmor in 1897 published a formula for calculating the power of electromagnetic waves emitted by an accelerating electric charge:

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.

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.

                    ________________

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.

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.

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.

---

The Wikipedia article

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?

Theorem 2 above is a classical result. Hypothetical Unruh radiation is claimed to be a quantum phenomenon.

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.

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.