Does the merge of black holes lead to a loss of information?
In 1976, following his work on black holes, Stephen Hawking raises a paradox: according to general relativity, the information absorbed by a black hole is lost when it evaporates. However, the laws of quantum mechanics impose a conservation of information. In the same way, when two black holes merge, they lose part of their total mass. Does this phenomenon also lead to a loss of information?
During the ten black hole mergers detected by the LIGO and Virgo interferometers in the last two years, each of the black holes involved lost a fraction of total mass during the process, around 5% on average. If the information is encoded in the mass of black holes, then it should be lost.
In any case, this is what general relativity says. When a particle falls into a black hole, all its properties - baryonic number, leptonic number, isospin, etc. - no longer play any role in the physics of the black hole. Information related to these properties is believed to be lost. In other words, according to Einstein's theory, the entropy of a black hole is zero.
Black hole and entropy: information stored on the event horizon
However, this consideration contrasts with the laws of thermodynamics and quantum mechanics. Any object with a defined temperature, energy and physical properties has a non-zero entropy, which can never decrease. If the material from which the black hole originates has a non-zero entropy, then throwing material into it would only increase its entropy. The black hole must therefore have a finite, positive and non-zero entropy.
According to these rules, all the properties of a particle (spin, charge, mass, polarization, etc.) falling into a black hole constitute information which must therefore be stored somewhere. If it’s not the singularity, then it’s in another place. And it was physicist John Wheeler who was the first to suggest that this information could be stored on the event horizon.
According to the formula of the Schwarzschild radius Rs = 2GM / c², it is the mass of a black hole which determines the size of its event horizon. It is therefore natural to think that the entropy can actually be located on the surface of this horizon.
As the mass of a black hole increases, its event horizon expands, storing the entropy / additional information absorbed. According to the work of Jakob Bekenstein and Stephen Hawking, this information would be encoded as qubits in Planck areas.
The fate of information when two black holes merge
During the fusion of two black holes, the mass of the resulting black hole is equivalent to the sum of the mass of the two black holes, reduced by 10% (5% of mass lost for each of the objects). Thus, if each black hole has a mass of 1 M, the final black hole will have a mass of 1.9 M. This means that, simultaneously, gravitational waves are emitted and transport an energy of 0.1 Mc².
From this observation, three scenarios are possible:
- the information of the two initial black holes remains entirely encoded on the event horizon of the final black hole, the gravitational waves do not therefore carry it
- the majority of the information is found encoded in the gravitational waves, the final black hole keeping only a very small amount
- information is shared more or less equally between gravitational waves and the final black hole
The entropy of a black hole is proportional to the area of its event horizon, the latter itself being proportional to mass squared. This means that if two initial black holes have an entropy of S, then a final black hole of 1.9 times the mass of the two black holes has an entropy of 3.6 S, which is clearly enough to store the information of the initial black holes. This is the premise of the Bekenstein-Hawking entropy.
However, gravitational waves must carry some of this information. Indeed, they are generated by the changes imprinted in the geometry of space-time during fusion, and their energy comes from the change of distribution of matter-energy of space-time. However, without the theory of effective quantum gravity, it is impossible to determine how much information is retained by the final black hole and how much is transferred to gravitational waves.
In any case, when two black holes are merged, there is no loss of information, since the entropy of the final state is higher than that of the initial state. But there is currently no way to extract the amount of entropy or information from gravitational waves or the event horizon of a black hole. Only theory here is capable of providing a few pieces of information.
