Generalized uncertainty principle: implications for black hole complementarity

Pisin Chen,Dong-Han Yeom,Yen Chin Ong

doi:10.1007/jhep12(2014)021

Abstract

At the heart of the black hole information loss paradox and the firewall controversy lies the conflict between quantum mechanics and general relativity. Much has been said about quantum corrections to general relativity, but much less in the opposite direction. It is therefore crucial to examine possible corrections to quantum mechanics due to gravity. Indeed, the Heisenberg Uncertainty Principle is one profound feature of quantum mechanics, which nevertheless may receive correction when gravitational effects become important. Such generalized uncertainty principle [GUP] has been motivated from not only quite general considerations of quantum mechanics and gravity, but also string theoretic arguments. We examine the role of GUP in the context of black hole complementarity. We find that while complementarity can be violated by large N rescaling if one assumes only the Heisenberg’s Uncertainty Principle, the application of GUP may save complementarity, but only if certain N -dependence is also assumed. This raises two important questions beyond the scope of this work, i.e., whether GUP really has the proposed form of N -dependence, and whether black hole complementarity is indeed correct.

Highlights

We will start with a short review of GUP, and investigate its implication for information loss paradox in the context of black hole complementarity principle [9], which proposed that quantum mechanics should only be consistent with causality
If Alice brings in a [localized] quantum state into the black hole and the exterior observer Bob recovers the information in the Hawking radiation, the apparent cloning of quantum information is allowed since these two observers are out of causal contact and cannot compare notes
In order for the complementarity principle to be a correct description, one has to check whether it is possible for the infalling Alice to send her quantum bit to Bob who falls into the black hole at a later time, after he has obtained a copy of the same bit from the Hawking radiation

Summary

The generalized uncertainty principle and black hole physics

One of the most important features of quantum mechanics is the fact that there is a fundamental limit on the precision with which some pairs of observables can be measured This is the Heisenberg’s Uncertainty Principle familiar to physics undergraduates, obeyed by position x and momentum p:. In order for the black hole to remain semi-classi√cal, we require that the black hole should be much larger than the minimum length: rh ≫ N /MP While such an N -dependence is not rigorously proven, as we will later show, GUP provides a good framework to be consistent with black hole complementarity precisely if such N -dependence is allowed, whereas the usual argument that depends only on the Heisenberg’s uncertainty relation fails. This is the optimal limit, that is, given any fixed ∆p, the smallest ∆x is obtained in this limit

Black hole complementarity and generalized uncertainty principle

Discussion