In the Hamiltonian thinking, the Schwinger process happens because it lowers the energy of fields and frees heat. Heat has a large entropy. Therefore, the process happens to one direction.
Problem 1. What about a strong gravitational field? Can we find a process which frees energy as heat?
A Feynman diagram point of view
If a virtual electron has zero or negative energy, it can gain energy by moving in the right direction in an electric field and become a real electron. In a gravitational field, a virtual particle cannot do the same, because as soon as its mass-energy becomes zero, it will no longer gain energy by moving in a gravitational field.
In Feynman diagrams we require that our process must eventually produce only real particles.
Thought experiment 2. What about the metaphorical Hawking process where a negative energy particle A enters a black hole and a positive energy particle B escapes?
The positive energy particle must have momentum upward to be able to escape. Is there any way that we could get rid of the negative energy particle and its downward momentum? If we would have its antiparticle A' coming up with the right energy and momentum, then we could let them annihilate and we would be left with zero energy and momentum close to the forming horizon. But we do not have such upward-traveling particles at the horizon. Furthermore, such annihilation would lower the entropy, which means the process is extremely unlikely to happen. Also, the process is better described as the upward shooting particle A' escaping on its own. There is no need to invoke A and B.
The metaphorical Hawking process is banned by Feynman diagram type of thinking, or has an infinitesimal probability to happen.
Thought experiment 3. What about a process where the positive energy particle drops to the horizon and the negative energy particle escapes? Again, the negative energy particle cannot gain enough energy from the gravitational push and cannot become a real particle.
A Hamiltonian point of view
Let us then forget about Feynman diagrams and think in terms of fields and their energy, momentum, and all conserved quantities. That is, use the Hamiltonian way of thinking.
Thought experiment 4. If we are collapsing a photon sphere to form a black hole, then we can satisfy conservation laws by letting two photons to reflect upward on the opposite sides of the forming black hole. But this is better understood as reflection rather than Hawking radiation or the Schwinger process. Our optical gravity predicts such process.
Let a dark matter shell collapse to form a black hole. We can satisfy conservation laws by letting two photons to materialize and escape from the opposite sides of the forming black hole, and by deducting the energy for photons from the kinetic energy of dark matter particles.
Is this process possible? Suppose that we have two dark matter particles approaching each other in empty space. We could satisfy conservation laws by deducting their kinetic energy and letting that energy to escape as two photons. Is such a process possible?
Let us think about the Hamiltonian. We have some fields and a function which assigns an energy to a field configuration. The fields themselves have energy and the fields have interactions which add to the energy.
If there is no direct or indirect interaction between two fields, then it is obvious that the fields cannot exchange energy.
The gravitational field does have an interaction with all fields. Can it transfer energy from any field to another field?
If we have two field configurations with the same energy, momentum, electric charge, etc., is it always possible that one configuration evolves to the other? Or are there other conserved quantities that ban such an evolution? Intuitively, most such evolutions are banned or have just an infinitesimal probability of happening. For example, the electron-positron annihilation process is extremely unlikely to happen in the exact same way in the opposite direction, though the annihilation happens very easily. The opposite process would require fine-tuned photons meeting just in the right way.
If we have a time-varying electromagnetic field, it will always produce a time-varying gravitational field. Actually, such electromagnetic field is always accompanied with a non-zero gravitational field, since mass-energy always produces a gravitational field.
What about a time-varying gravitational field? If the electromagnetic field has zero energy, can a time-varying gravitational field transfer energy to it?
If we have a totally smooth surface of water and a breeze of wind, ripples will appear to the water. The ripples get their energy from the breeze. The breeze amplifies tiny fluctuations in the surface of water.
An electromagnetic ripple can gain substantial energy from a gravitational field by moving closer to a black hole horizon. But from where does the ripple get its original energy? If it borrows it from a virtual photon, how do we get rid of the virtual photon? If the ripple originally has some energy of its own, we can describe the process by the blue shift of a photon moving down a gravitational field. Furthermore, if we measure the energy of the photon from the point of view of a global Schwarzschild observer, the photon really does not gain any more energy because it is offset by a lower gravitational potential. The ripple changes the gravitational field very little as observed from far away.
The above is in contrast with the Schwinger effect. A strong electric field around a big charge can tear electric charge from "empty" space and the charge will change the electric field in a significant way.
What if the electromagnetic ripple gets energy from a gravitational "elevator" process where it first goes down a gravitational potential, say, close to a neutron star and then the neutron star is suddenly taken apart so that the ripple moves to a higher gravitational potential without using its own energy? If we would have such elevators in nature, they could greatly amplify small electromagnetic ripples. Maybe they could create electromagnetic ripples from space also in the case where the electromagnetic field originally has zero energy? But again, how do we get rid of the negative energy ripple from which our ripple has to borrow energy?
Classically, a zero energy electromagnetic field cannot gain any energy from a time-varying gravitational field because there is no "handle" in it to which the gravitational field could get a grip.
In quantum mechanics, we can never know that a field has exactly zero energy. We must assume that there are small disturbances around. But the author of this blog has not been able to find a way to transfer much energy to those disturbances from a time-varying gravitational field.
Conjecture 5. A time-varying gravitational field cannot in natural circumstances transfer much energy to a zero or almost zero energy electromagnetic field. There is no Schwinger process for a gravitational field.
Collision of gravitons
Let us return to Feynman diagrams and think about a scattering experiment.
A collision of electrons and positrons always produces photons. A collision of high energy photons can create electron-positron pairs:
https://en.m.wikipedia.org/wiki/Two-photon_physics
A collision of electrons and positrons always produces photons. A collision of high energy photons can create electron-positron pairs:
https://en.m.wikipedia.org/wiki/Two-photon_physics
What about a collision of gravitons? Can it produce photons or electron-positron pairs?
Since gravitons themselves carry mass-energy, the collision of gravitons will produce more gravitons.
Pair production from photons looks like this:
/ e-
^
photon / ~~~~~~ photons
~~~~~~
|
^
|
~~~~~~
photon \ ~~~~~~ photons
^
\ e+
Time flows to the right. The wave equation is in a way symmetric with respect to time and position. We can interpret the scattering in various ways. One such interpretation is that a positron travels backward in time and meets a transient electric potential. The positron cannot enter the zone with a potential but scatters to the opposite direction in time. The time reflection reverses the sign of the frequency of the wave function of the positron: it becomes an electron.
If we replace the photons in the diagram with gravitons, there is a prominent difference from the original diagram: the gravitational charge is positive for both the electron and the positron.
If we would replace the e+ in the diagram with e-, that would be a better analogue for the graviton scattering. But we cannot do such replacement because that would break conservation of charge.
If it were possible to create an electron-positron pair + some other particles from a collision of just gravitons, then by time symmetry, it would be possible to run the reaction backwards. The reaction run backwards would mean that we could with some probability > 0 convert all electromagnetic energy that we feed to the process to gravitons. Is such a process possible?
Conjecture 6. In a Feynman diagram, if a photon, electron, or positron enters the diagram, then the scattering amplitude for a process where a sizeable portion of its energy is converted to gravitons is zero or infinitesimal.
As a simple example, let us think about a diagram where just two photons enter and they are totally converted to two graviton.
photons electron-positron
loop gravitons
~~~~~~~~ O --------------------
~~~~~~~~ --------------------
We need 2 particles to make the total spin 0 in the diagram.
A brief look at literature reveals that calculating scattering amplitudes in a simplified quantum gravity is difficult and at two loops the formulas diverge. There is not much hope of proving our conjecture anytime soon.
Since gravitons themselves carry mass-energy, the collision of gravitons will produce more gravitons.
Pair production from photons looks like this:
/ e-
^
photon / ~~~~~~ photons
~~~~~~
|
^
|
~~~~~~
photon \ ~~~~~~ photons
^
\ e+
Time flows to the right. The wave equation is in a way symmetric with respect to time and position. We can interpret the scattering in various ways. One such interpretation is that a positron travels backward in time and meets a transient electric potential. The positron cannot enter the zone with a potential but scatters to the opposite direction in time. The time reflection reverses the sign of the frequency of the wave function of the positron: it becomes an electron.
If we replace the photons in the diagram with gravitons, there is a prominent difference from the original diagram: the gravitational charge is positive for both the electron and the positron.
If we would replace the e+ in the diagram with e-, that would be a better analogue for the graviton scattering. But we cannot do such replacement because that would break conservation of charge.
If it were possible to create an electron-positron pair + some other particles from a collision of just gravitons, then by time symmetry, it would be possible to run the reaction backwards. The reaction run backwards would mean that we could with some probability > 0 convert all electromagnetic energy that we feed to the process to gravitons. Is such a process possible?
Conjecture 6. In a Feynman diagram, if a photon, electron, or positron enters the diagram, then the scattering amplitude for a process where a sizeable portion of its energy is converted to gravitons is zero or infinitesimal.
As a simple example, let us think about a diagram where just two photons enter and they are totally converted to two graviton.
photons electron-positron
loop gravitons
~~~~~~~~ O --------------------
~~~~~~~~ --------------------
We need 2 particles to make the total spin 0 in the diagram.
A brief look at literature reveals that calculating scattering amplitudes in a simplified quantum gravity is difficult and at two loops the formulas diverge. There is not much hope of proving our conjecture anytime soon.
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