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.
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.
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.
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