He also believed that a black hole can evaporate through radiation, and that the radiation does not contain the information that was dumped into the black hole.
This is the famous black hole information paradox: can a black hole destroy information by first devouring it, and then losing the total energy of the black hole in radiation which does not contain that information?
We in this blog have not been able to derive the existence of Hawking radiation. If the radiation would destroy information, that would be strong evidence against its existence.
The information paradox of singularities
However, we have another information paradox in a black hole. If the information which it devours, is squeezed into a zero volume, we can say that it destroyed information. A singularity, whether a point or a ring, is a bad thing. A rotating black hole in the Kerr metric is assumed to have a ring singularity.
Note that an idealized collapse of uniform dust to a point does not destroy much information, since uniform dust contained just a few bits of information. A realistic collapse of some 10⁶⁰ elementary particles into a point or a ring singularity would destroy lots of information.
Quantum mechanics bans singularities
Quantum mechanics comes to our rescue and saves the information. Let us assume that there exists a global frame of reference where we can measure things.
We believe that the true metric of spacetime is the Minkowski metric, and gravity only creates an illusion of curved geometry. Thus, we can take an inertial frame in the Minkowski space, and work in that inertial frame.
Assume that elementary particles, like photons, continue their existence. The particles are very tightly bound together through gravity. They collectively form an "atom" with ~ 10⁶⁰ particles in a solar mass black hole.
Just like in the case of the electron in the hydrogen atom, we cannot know the position of an individual particle too precisely, because then the particle would have more energy than is available.
We cannot know the relative positions of particles precisely, either. There would be infinite energy in the interaction if we did.
When a particle interacts with a large mass, the wavelength of the particle becomes lower. An example is a photon on the surface of a neutron star. A global observer, and also a local observer, see the wavelength shorter. The global observer interprets that the photon has acquired more inertial mass in the interaction, and the speed of light is consequently lower. The local observer thinks that the photon was blueshifted when it came down.
In principle, the global observer might see that a photon inside the black hole has gained the entire mass of the black hole as the inertial mass of the photon. In our syrup model of gravity, that would mean that the black hole is made of perfect syrup.
If that is the case, the wavelength of a photon inside a solar mass black hole is at least
λ = h c / (M c²)
= h / (M c)
= 10⁻⁷² m,
where M is the mass of the Sun. The wavelengh is tiny, but it is not zero.
Thus, a hypothetical singularity cannot be a singularity at all but is a soup of waves (or particles).
Assumptions. In our argument we had to assume:
1. there exists a global frame of reference;
2. energy is conserved in the global frame: particles cannot gain infinite energy;
3. particles do not merge in a way which destroys information;
4. uncertainty relations prevail over classical general relativity.
Are our assumptions self-evident? Assumption 1 is controversial. Assumption 2, conservation of energy, is a rock-solid principle of physics. Assumption 3 is implied by unitarity. Assumption 4 is true in everyday physics - it would be surprising if it does not hold in general relativity.
We are suggesting that uncertainty relations prevent singularities from forming. In a uniform dust collapse, no classical force can resist the contraction, dictated by Birkhoff's theorem. However, quantum mechanics works with waves. Waves cannot be squeezed into a point.
Does a realistic classical collapse produce a singularity?
Uniform dust is not a realistic configuration. If we let a real star to collapse, the spacetime geometry will initially be very complicated behind the event horizon.
Let us assume that the angular momentum of the star is zero.
Particles will initially zigzag behind the horizon. Some particles are still close to the horizon and try to move at the light speed outward. They can stay close to the horizon for a long time.
However, we expect most particles to move downward, because the light cones point down.
Intuitively, we should have a cloud of particles, where the upper layers of the cloud become thinner as time passes. More and more particles come close to the center.
Is there information loss when black holes merge and lose a lot of mass-energy in gravitational radiation?
If we have one million black holes of equal mass M, we can, in principle, release about 99.9% of their mass-energy in gravitational radiation by letting them merge together in pairs of equal mass.
99.9% is a lot, but it does not break the Bekenstein entropy formula, because the area of the horizon is a million times larger in a black hole of a mass 1,000 M than in a black hole of a mass M.
Gravitational radiation is coherent, and does not carry much entropy. The bulk of the entropy must stay in the black hole.
We can probably store a million times more information if we make the number of particles 1,000-fold. There is presumably no information loss.
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