When the electron and the positron are very close, the immense acceleration converts ALL the energy in them into electromagnetic waves.
The annihilation process ensures conservation of energy:
If we could through some mechanism lower the electron and the positron closer than the classical electron radius 3 * 10^-15 m, we could extract more energy than the mass of the pair.
An analogous question about gravity:
Does a particle falling into a black hole radiate away ALL its energy in gravitational waves?
If yes, then we have a solution for the evaporation of black holes, as well as for the black hole information paradox: the evaporation happens as gravitational waves, and the information is preserved in the outgoing waves!
Why a black hole does not evaporate immediately then? Because it takes a huge amount of time (measured by an external observer) for light to climb up if the light is born close to the horizon.
Imagine a static observer close to the horizon. He sees an infalling particle accelerate at an enormous rate. How large gravitational waves does the particle produce?
What about an infalling photon?
Takashi Nakamura (2006) has calculated the gravitational wave of an infalling particle. Some 3% of its mass energy is radiated away in gravitational waves.
Did Nakamura take into account that gravitons which the horizon devours are particles themselves, and also they will radiate gravitational waves?
For an infalling electron, how much energy is radiated away in electromagnetic waves?
Can a black hole devour a photon at all?
Richard Hanni and Remo Ruffini calculated that the horizon of a black hole behaves like a metal sphere for electric field. Maybe even like superconducting metal?
Suppose that we have an electric charge q outside a black hole, and we suddenly move the charge some distance L.
We can use Edward M. Purcell's reasoning to calculate the energy of the outgoing electromagnetic pulse. The lines of force of the electric field will move suddenly, and their "bending" carries away the energy.
metal
charge sphere
q ----------------O---------------
field line field line
If there is a superconducting metal sphere nearby, there will be induced charges on its surface, and the electric field lines will be perpendicular to the surface. When we suddenly move the charge q, the field lines will move accordingly, and carry away the energy also along the field lines starting from the metal sphere.
Can the metal sphere absorb any of the energy? Yes, if there is resistance in it - moving charges on its surface will heat up the surface.
What about induced electric currents and the energy of their magnetic field?
If the black hole horizon behaves like a superconducting metal sphere, can the bends in the outgoing field lines become less sharp than in the incoming field lines? If yes, then the black hole could devour some of the energy in the sudden change in the electric field of the charge q.
Solved! Yes, a black hole can devour energy from electromagnetic waves. Imagine a neutron star where the gravitational potential is very low, and the speed of light very slow (as measured by an external observer).
Then a bend in a field line can spend a very long time inside a neutron star. When the bend finally comes out, it is still very sharp and carries the same energy which went in. The neutron star can keep the energy "captive" for a long time. A black hole is the limiting case where the energy stays captive forever.
This may break the "no-hair" conjecture of black holes. Does the black hole electric field store information about the earlier position of the charge q?
Freeman Dyson's argument that the QED perturbation series cannot converge, and "electric black holes"
Dyson argues that if the perturbation series converges, then it will also converge for a pathological theory where charges of the same sign attract each other.
But in the pathological theory we can create macroscopic "electric black holes" of charges, and extract more energy than the rest mass of the charges. That would break energy conservation. Dyson claimed that the perturbation series of this pathological theory cannot converge => the QED perturbation series cannot converge.
Can we somehow prevent electric black holes from forming in the pathological theory? No, because the photons which carry the energy away do not reduce the amount of charge.
In gravity, we have a better chance. Gravitons do carry away the gravitational charge (= mass-energy).
The annihilation of an electron and a positron is an empirical quantum phenomenon, and does not follow from classical electromagnetism.
There might exist a similar mechanism for the annihilation of a black hole. The annihilation just lasts an enormous time because gravitons climb up very slowly (as measured by an external observer).
What about a black hole connected to a white hole?
If a forming black hole would eventually evaporate into gravitational waves or whatever, could the same happen to anything which drops into a black hole that is connected to a white hole?
We like to think that a spaceship can pass unharmed through such a wormhole. If it would evaporate, it would be harmed, or there would be a magical process which both allows the spaceship to pass and makes it evaporate.
A magical process does not seem plausible. This is an argument against evaporation through gravitational waves, or through any mechanism.
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