Black holes, information, and the universal coefficient theorem

Abstract

General relativity is based on the diffeomorphism covariant formulation of the laws of physics while quantum mechanics is based on the principle of unitary evolution. In this article I provide a possible answer to the black hole information paradox by means of homological algebra and pairings generated by the universal coefficient theorem. The unitarity of processes involving black holes is restored by demanding invariance of the laws of physics to the change of coefficient structures in cohomology.

The Universal Coefficient Theorem and Black Holes

“…If however the integration is being done through surfaces described by cohomology groups with cyclic coefficients, the non-trivial topologies disappear and the emerging integral corrections amount to the same terms required to provide the proper Page behaviour of the entanglement entropy. I showed this extensively in [13] and partially in [14]. The link between twisted coefficients and the relativity of the circles I also showed in [19].…”

“…I showed in ref. [13] following [14] that the universal coefficient theorems can express the (co)homology groups of a space with a certain coefficient group in terms of (co)homology groups of the same space with a different coefficient group. I also showed in [13] following [14] that some information visible when a certain group is used becomes invisible when another group is used.…”

“…If one accepts the idea of invariance to topology changes as mentioned in ref. [13] one has to reformulate the theory in a topologically invariant way, and from a universal coefficient theorem point of view this would amount to describing the theory starting with the general UCT exact sequence for general types of coefficients. This is of course quite complicated as the number of different types of coefficients is technically unlimited.…”

“…The details starting from basic principles have been shown in ref. [13]. Calculation of the path integral required for the estimation of the results of the replica trick must involve summation over all topologies connecting the various replicas, involving therefore spacetime wormholes.…”

“…Hence, the evolution of the Page curve follows a transition from a dominant disconnected topology to one that is fully connected. This requires for a topology independent description which has been performed in [13] with the observation that the correction terms arising in the universal coefficient theorem must be taken into account in order to correct the replica trick path integral. Therefore let us apply the universal coefficient theorem to the replica manifold transition from a disconnected topology to a connected topology in the context of the Page curve.…”