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.

A critique of Adrian Kent's "One world versus many: the inadequacy of Everettian..."

A funny coincidence: many-worlds interpretations are the cover story of New Scientist this week
http://www.newscientist.com/article/dn26267-four-ways-you-can-see-the-multiverse.html#.VCVPb3YcR2E

Adrian Kent tries to refute many-worlds interpretations of quantum mechanics:
http://arxiv.org/pdf/0905.0624v3.pdf
In his paper "One world versus many: the inadequacy of Everettian accounts of evolution,
probability, and scientific confirmation", Adrian Kent aims to show that the many-worlds interpretation of quantum mechanics, first proposed by Hugh Everett, leads to absurdities when one tries to define a "rational" betting strategy for an inhabitant of the multiverse.

Toy multiverses and betting strategies of an inhabitant

In section III of the paper, Kent defines some toy multiverses and discusses possible betting strategies for an inhabitant.

In my opinion, the inhabitants of a multiverse are "robots" created by evolution. They do not have "free will". We have to study them like laboratory mice in a labyrinth. It does not make sense to think about their betting strategies from the viewpoint of an outside observer who sees the whole branching universe of the many-worlds interpretation.

One specific question is if it makes sense for an inhabitant to take a maximum risk in betting, because then he maximizes the win in ONE branch of the multiverse. Again, I do not see this as a relevant question at all. Rather, we should study what kind of players thrive in a "typical" branch of the multiverse. Obviously, in most branches, evolution favors players who take only moderate risks. An animal species which would always take huge risks to maximize the return in ONE branch of the universe, would quickly become extinct in most branches of the multiverse.

Probabilities in tossing of a coin

In section IV A, Kent worries about a coin toss experiment in a toy multiverse. Kent describes an experiment where a quantum process prints either 0 or 1 to a tape. The inhabitants of the toy multiverse should determine the relative frequency of 0s and 1s experimentally. Kent is worried that in some branches of the multiverse, the inhabitants will see a wildly biased distribution of the binary numbers.

I do not see any problem in this. Even in the one-world case, we can run a similar experiment where we have a big number of participants tossing a coin. If we have 1 000 000 participants in the experiment, then one will probably see a string of 20 tosses that are all heads. What is the problem then? Most of the participants will get roughly 50 % heads and 50 % tails. The ones who get a very biased distribution may deduce from the physics of the coin toss experiment that they have been incredibly "lucky" to get a very biased distribution. There is no philosophical problem in this, whether the experiment happens in one world or in a multiverse.

Multiplication of species in a multiverse

In Appendix C, Kent describes a thought experiment. There are two species, A and B. B takes huge risks and tries to increase its population wildly, while running also a big risk of extinction. Then, there is one branch in the multiverse where B's population grows exponentially. And the sum of populations of B in all the branches quickly dominates the sum of A populations when time increases. If we live in a multiverse, does it mean that it pays to take huge risks for huge returns? If we get a huge population of our species in one branch of the multiverse, do we need to care if our species becomes extinct in almost all other branches?

The flaw in the thought experiment is that the resources of the visible universe are not infinite. The population of species B cannot grow exponentially for very long. In the long run, the risk-avoiding species A will dominate the total population of A + B in the multiverse because B becomes extinct in almost all branches.

De Broglie - Bohm interpretation is very close to many-worlds interpretations

Adrian Kent seems to sympathize with de Broglie - Bohm interpretation of quantum mechanics because that interpretation is "one-world".

http://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory#Occam.27s_razor_criticism
But David Deutsch has said: "pilot-wave theories are parallel-universe theories in a state of chronic denial.^"

Since de Broglie - Bohm interpretation does not involve a collapse of the wave function, and since it evolves the wave function using the exact same equations as many-worlds interpretations, that means that de Broglie - Bohm interpretation is almost equivalent to many-worlds interpretations. The difference is that hidden variables in de Broglie - Bohm pick one branch of the multiverse as "more real" than other branches. De Broglie and Bohm believe that "conscious" observers only exist in this one branch.

But since the other branches may still interfere with the "real" branch of the universe in de Broglie - Bohm interpretation, we may conclude that the other branches also "exist" in some sense. Furthermore, physical phenomena in the other branches are similar to the "real" branch. There are humans living in the other branches, doing physics, and thinking about quantum mechanics. That raises the question: in what sense is the "real" branch more real than the other branches? This question makes de Broglie - Bohm interpretation look awkward.

In many-worlds interpretations, we might define that the branch where I am currently living as an observing "subject", is more "real" than other branches. But it is not mandatory to claim it is more real. Rather, we may say that I as a subject have landed through some random process in one branch, but that one branch is not more "real" than other branches.

I as a living subject was born in Finland. Does that mean that the UK is less "real" than Finland? I do not think so. Similarly, I think all the branches of the multiverse may be as real as the one where I am living.

The problem in one-world interpretations

We saw in the preceding section that de Broglie - Bohm interpretation is essentially a many-worlds interpretation of quantum mechanics though it pretends to be a one-world interpretation.

What is the problem in genuine one-world interpretations, like the Copenhagen interpretation where there is a "collapse" of the wave function when a quantum system is measured?

The problem is that there is no natural way to decide what kind of interactions with a quantum system constitute a measurement that collapses the wave function. In the traditional Copenhagen interpretation, a "conscious being" does the measurement and causes the wave function to collapse.

The paradox of the Schrödinger cat raises the question whether a cat is a "conscious observer", or if it is just humans that are "conscious observers". This all looks very ugly, and in recent years experiments in quantum mechanics have shown that almost macroscopic lumps of matter can be in a superposition of quantum states, that is, such a lump of matter cannot be a "conscious observer". Maybe in the future we are able to show that even humans can be in a superposition of quantum states.

True one-world interpretations need some way to prune the branches of the multiverse in a many-worlds interpretation. Such pruning involves a "collapse" of the wave function of a kind. I do not believe such collapse exists, and that is why I have long been a supporter of many-worlds interpretations.

Many-worlds interpretations can explain some miracles in the origin of life

How the first living cell was formed is a mystery. No pathway from simple molecules to a complex cell that contains, say, a billion atoms is known. Many-worlds interpretations can explain such miraculous happenings. However improbable it is that simple organic molecules spontaneously react and form a living cell of one billion atoms, that DOES happen in some branches of a multiverse.

Using a similar "anthropic" reasoning we could also explain the origin of intelligent life, if the evolutionary pathway to intelligent beings is extremely improbable.

But evolution in general cannot be explained using an anthropic argument in a many-worlds interpretation, because evolution has created many wonderful creatures that are not necessary in the pathway to intelligent life.

Black hole firewalls - real or not?

I have been combing through the web about arguments for and against so-called black hole firewalls. I will present some of my thoughts.

http://en.wikipedia.org/wiki/Firewall_(physics)
Joe Polchinski et al. caused commotion in the theoretical physics world in 2012 when they published their paper that seems to imply there is an infinitely hot "firewall" just behind the event horizon of an old black hole.

That is, assuming that black holes do not lose information, and that they radiate their information away in Hawking radiation, and assuming that the no-cloning theorem of quantum mechanics is true, then there MUST be an infinitely hot firewall just under the event horizon. The firewall prevents any observer from entering the black hole alive.

Their observation is in a stark contrast to Einstein's equivalence principle which states that a freely falling observer should not observe anything special as he falls through the event horizon (this is called the "no drama" hypothesis).

The Black Hole Information Loss Problem

http://en.wikipedia.org/wiki/Thorne%E2%80%93Hawking%E2%80%93Preskill_bet
Stephen Hawking initially claimed that black holes DO destroy information. The information would be forever lost in the singularity of the black hole. The Hawking radiation would be totally random black body radiation and would not carry away the lost information. In 2004, Hawking famously reversed his opinion and conceded that black holes do not destroy information, after all.

Information loss in a black hole is mathematically an ugly phenomenon, as it breaks the "unitarity" of quantum mechanics. That is, in quantum mechanics, time is reversible in the CPT symmetry, and the earlier state of a system is uniquely determined by a later state of the wave function. But if a black hole would destroy information, then there would be no way to deduce the early state - before the formation of the black hole - from the later state - where the black hole has already formed.

2004: Hawking Claims Black Holes Preserve Information, After All

http://arxiv.org/pdf/hepth/0507171.pdf
In 2004, Hawking presented his argument that black holes do preserve information, after all. His idea is to consider the formation and the radiating away of a black hole as a scattering experiment from the point of view of a very faraway observer. Particles are sent into the system. If the particles at some point in time become dense enough, a black hole forms. But the black hole is later radiated away in Hawking radiation.

Hawking seemed to claim that since there is a possible history where the black hole does not form at all, that is, the particles involved never condense enough to collapse into a black hole, then the information of the initial state is preserved in the wave function of that history.

Hawking's argument seemed flawed to me, and it looks like no eminent physicist has endorsed Hawking's idea. In quantum mechanics, we have to consider ALL possible histories. We cannot discard any. We cannot omit the case where a black hole forms and radiates away. The probability of black hole formation might be, say, 99.999 %. It does not help if the information is preserved in the remaining 0.001 % of cases. Unitarity must hold for all histories. It is not enough that it holds for 0.001 % of histories.

Quantum Mechanics Seems to Imply a Firewall in a Black Hole

http://arxiv.org/abs/1207.3123
In their paper, Polchinski et al. show quite convincingly that we have to give up one of the basic principles of quantum mechanics, the no-cloning theorem, or we must give up Einstein's equivalence principle that states there is "no drama" for an observer that falls freely inside a black hole.

I personally now favor the quantum mechanical point of view. Unitarity is a beautiful mathematical property. Giving up unitarity would be a much greater loss than giving up Einstein's equivalence principle.

Einstein's Solution is Mathematically Ugly

The black hole model of general relativity contains a singularity at the center of the black hole. Needless to say that a singularity is an ugly phenomenon from the mathematical point of view. As is laws of nature would break down at a point of space.

To do away with the singularity, people have suggested that some theory of quantum gravity prevents the singlularity from forming. Let us assume that there are some unknown laws of nature that stop the collapse of a black hole into a singularity. If there are some unknown laws that cause nature from differ from general relativity, it might be natural that the differing happens already at the event horizon of a black hole?

Decay of Protons in a Black hole

http://en.wikipedia.org/wiki/Virtual_black_hole
http://en.wikipedia.org/wiki/Proton_decay
As a sidenote, let us bring up the question about the decay of protons. If Hawking radiation really makes a black hole to evaporate, and the radiation is mostly photons, then we have a mechanism to turn a hydrogen atom, a proton plus an electron, into photons. That would imply proton decay. Proton decay is predicted also by some grand unified theories in physics.

Does a Black Hole Have an Interior at All?

http://en.wikipedia.org/wiki/Holographic_principle
If there is searing hot firewall just behind the horizon of a black hole, we can ask if the black hole has an interior at all. According to Bekenstein and Hawking, a black hole is the structure which has the maximal possible entropy in a given area of space. The entropy is NOT proportional to the volume of the enclosed space, but proportional to the area of the black hole. This has been taken as evidence for the holographic principle that the universe really has only two large spatial dimensions and a 3D universe is an illusion.

Since the black hole, in a way, reveals the holographic nature of the universe, it does not sound too far-fetched to speculate that a black hole does not have an interior at all in the sense of our 3D universe. Maybe material falling in a black hole gets distributed in some exotic form on the horizon of the black hole, and the "inside" of the black hole does not exist at all.

Is a Black Hole Analogous to an Atom for Quantum Mechanics?

The hydrogen atom is an example of an object which can superficially be viewed as a classical ball from a long distance. For example, if hydrogen atoms form ideal gas, the atoms in the gas seem to behave like in classical mechanics. But when we go close to the atom and start to study its internal structure, we notice that classical mechanics does not work at all. We have to study the structure of a hydrogen atom in quantum mechanics to get right results.

Maybe a black hole is an "atom" of universe in the way that its internal structure can only be understood through quantum mechanics? That is, we have to forget about the theory of general relativity when we study the internal structure of a black hole. In the previous section, we argued that since black holes reveal the holographic nature of our universe, it might be that they have to be treated in a quantum mechanical way and we must forget about general relativity in their internal structure.

Attempts to Reconcile Quantum Mechanics and the Equivalence Principle

After Polchinski et al. published their seminal paper about firewalls, there has been a flurry of papers where different authors try to get rid of firewalls by reconciling quantum mechanics and Einstein's equivalence principle in some way.

http://arxiv.org/abs/1306.0533
Juan Maldacena and Leonard Susskind suggest that entangled systems in quantum mechanics are not separate entities after all, but are connected through a wormhole in general relativity. Then the interior of a black hole would NOT be separate from the exterior, and the no-cloning theorem for an infalling observer is not violated. That is, we do not need any firewall behind the event horizon. The observer falls with no drama inside the black hole. The authors coin a slogan "ER = EPR".

http://arxiv.org/abs/1402.5674
In another paper, Leonard Susskind suggests that the no-cloning principle is preserved in the way that it is exceedingly hard to compute the state of the interior of a black hole from the Hawking radiation. That is, though the no-cloning principle is violated, it is impossible for anyone to see the violation because the calculation to show the violation would require too big computing resources.

To my mind, the two above suggestions to do away with firewalls sound as complicated and far-fetched as hidden variable theories that try to revert quantum mechanics back to classical physics.

http://arxiv.org/abs/1401.5761
Trying to solve the firewall problem, Stephen Hawking himself suggests that a black hole does not contain an event horizon at all, but just an "apparent horizon". Then there would be no need for a firewall, as there does not exist a true black hole in the classical sense. If Hawking is able to form a mathematical theory from his ideas, it would be a nice solution to the problem. If I understand correctly, the "firewall" in Hawking's solution is the matter that is densely packed very close to the apparent horizon of his quasi black hole.

Fuzzballs

http://en.wikipedia.org/wiki/Fuzzball_(string_theory)
A fuzzball is an attempt to explain a black hole as a string theoretic object. A fuzzball does away with the singularity in the black hole, and it predicts that information is not lost in a black hole. Maybe a fuzzball is the right description of a black hole, if we abolish Einstein's equivalence principle and use quantum mechanics as our guide?

Conclusion

I have been browsing the World Wide Web for a few days now, and it is obvious that there is no consensus among physicists whether black hole firewalls exist or not. A large number of papers have been published in the past two years, and many of them try to do away with firewalls. But none of these papers seems convincing enough. My guess at the moment is that we really have to give up the equivalence principle and treat black holes as purely quantum mechanical objects. Maybe that involves a firewall or maybe we have to assume that a black hole has no interior at all.

Faster-than-light travel

http://en.wikipedia.org/wiki/Alcubierre_drive
NASA is studying the Alcubierre drive as a possible mechanism of faster-than-light travel. Lubos Motl remarked that the drive would break the special theory of relativity:
http://motls.blogspot.fi/2013/07/relativity-bans-faster-than-light-warp.html,
though the drive is consistent with the equations of the general theory of relativity. Let us discuss if faster-than-light travel makes sense, and whether it might be possible.

Faster-than-light travel enables a time machine

The above Wikipedia article states that a faster-than-light rocket would make closed timelike curves possible, that is, the rocket could travel backwards in time. Suppose that we have a rocket that can travel at a "moderate" speed of  2 times the speed of light, relative to the frame of the Earth.

Using the formulas of time dilation and length contraction:
http://en.wikipedia.org/wiki/Time_dilation,
http://en.wikipedia.org/wiki/Length_contraction,
we can calculate how we can travel back in time with our rocket.

Suppose that stars A and B are traveling at the speed 0.99c, where c is the speed of light, relative to the Earth. They move to the direction of the vector (A, B):

A ------------ 1 light year ---------- B ------> speed 0.99c

                           O <--- the Earth

The distance of stars A and B is 1 light year, measured from the reference frame of the Earth.

The time dilation coefficient for A and B is:

             1
______________ = 1 / sqr(1 - 0.99^2) = 7.0888.
sqr(1 - v^2 / c^2)

That is, a time interval t measured at star A appears for an observer in the frame of the Earth to last 7.0888 t.

The length contraction coefficient for A and B is the same as the time dilation coefficient. That is, the distance of stars A and B in the frame of A and B is 7.0888 light years (recall it is only 1 light year in the frame of the Earth).

Suppose that we have a rocket that can move at speed 2c relative to the Earth. Let us make the rocket to fly from star A to star B. As the rocket starts from A, we also send a ray of light from A towards B.

The rocket arrives at B after 1 year of the Earth's time. But the ray of light only moves at a relative speed of only 0.01c relative to B (the speed of light observed from the Earth is always c, regardless where the light originated from). Thus, the ray of light arrives at B after 100 years of the Earth's time.

The time interval (rocket arrives at B, light arrives at B) is 99 years, relative to the frame of the Earth.

Since the time dilation coefficient is 7.0888, the time interval of (rocket arrives at B, light arrives at B) is 99 years / 7.0888 = 13.966 years in the reference frame of A and B.

But remember that the distance of A and B in their own reference frame was just 7.0888 light years. Thus, in the frame of A and B, the light arrives at B 7.0888 years later than it left A. Since the rocket arrives at B 13.966 years earlier (in the frame of A and B), we see that the rocket flew 6.877 years backwards in time!

In the calculation above, we assumed that our rocket can fly at speed 2c relative to the frame of the Earth. Since in special theory of relativity, all inertial frames are equivalent, we can let the rocket fly at speed 2c also relative to the frame of A and B. Let us make our rocket fly back from B to A at speed 2c, relative to the frame of A and B. The flight back takes 3.544 years. The rocket moved back in time 6.877 years when it flew from A to B. And it moves forward in time 3.544 years when it flies back to A. That is, the rocket arrives at A 3.433 years BEFORE it left A! Our rocket has acted as a time machine and sent us backwards in time 3.433 years on star A.

Time travel makes faster-than-light travel impossible?

If we allow faster-than-light travel only relative to the frame of the Earth or the Milky Way, then there is no time travel with respect to our frame. Ww just travel immensely fast within our galaxy.

But if we allow time travel relative to any frame, like in our calculation above, then time travel is possible, and we have to face the numerous paradoxes involved with time travel. Suppose that I use my fast rocket to travel one week back in time, and then blow up the rocket before it started its journey. How can I (or my copy) start my journey one week later? The rocket was blown up, and does not exist any more!

We instinctively believe in the principle that the past cannot be changed. If the past can be changed, then there essentially does not exist the past, because the past could be changed arbitrarily in the future. Any kind of flexible time travel, where we can transport intelligent robots to carry out missions in the past, breaks the principle that the past cannot be changed.

To avoid the paradoxed of time travel, we could use the many worlds interpretation from the philosophy of quantum mechanics. In that interpretation, the universe is constanly branching according to the results of "experiments" we perform on microscopic systems. For example, in the double-slit experiment, the branching happens according to the spot where a photon hits the screen behind the double slit.

If our time travel machine takes us to another branch of the branching universe, then no paradoxes arise. In that brach I can blow up the copy of my time travel rocket, and that does not spoil my departure with the rocket which I did from another branch of the universe. But we do have the restriction that our time travel rocket cannot take us to the past of our current branch.

A generalized travel machine: travel between universes

The previous line of thought takes us to the concept of  generalized travel machine. We already have machines that can can take us to a different position in space. We can also imagine time machines that take us to the future or the past (the past of another branch of the universe). A generalized travel machine would take us to any universe, to any place in it, and to any time in that universe. Again, our generalized travel machine cannot take us to the past of our branch of the universe, though.