Soft Heisenberg hair on horizons

Black holes have no hair is a slogan that captures the apparent simplicity of black holes. Unlike other stars, like our Sun, it is believed that black holes are specified uniquely by providing a handful of numbers, like their mass, angular momentum and charge. If true, black holes would be the simplest macroscopic objects in our Universe, characterized by the same numbers as elementary particles. Quantum mechanically, however, black holes are very complicated objects. In fact, there is evidence that they might be the most complicated objects that can possibly exist, in the sense that they maximize the so-called entropy, a measure for information and disorder. This apparent tension between maximal simplicity at the classical level and maximal complexity at the quantum mechanical level has been appreciated by researchers for a couple of decades and is at the core of numerous unresolved conceptual issues in black hole physics, like the information loss problem. In 2016 Hawking, Perry and Strominger introduced the notion of soft hair, which denotes structure on a black hole that does not change its energy (hence the name soft). According to this proposal, black holes may have hair if it is soft enough, so that in principle microscopically distinct black holes with the same mass can be distinguished according to their soft hair charges. They also expressed the hope that soft hair on black hole horizons may help to address the information loss problem. A few months later we provided a concrete realization of soft hair for specific black holes in a three- dimensional toy model of gravity. We called this realization soft Heisenberg hair, since we discovered, quite unexpectedly, that the soft hair for these black holes falls into representations of the Heisenberg algebra, which is familiar from quantum mechanics but usually does not appear as symmetry algebra in gravitational theories. One of the main goals of the project Soft Heisenberg hair on horizons is to construct soft hair for black holes of astrophysical interest and to study to what extent this permits to address and resolve the information loss problem. Numerous generalizations and theoretical checks are envisaged to verify (or falsify) the generality of the Heisenberg algebra as the basic algebraic ingredient governing the symmetries near a black hole or cosmological horizon.

Stephen Hawking died soon after the start of this project. His last cycle of papers (with Malcolm Perry and Andy Strominger) promoted the idea of soft hair, referring to zero-energy excitations of black holes. The main goal of the project "Soft Heisenberg hair on horizons" was to construct soft hair for black holes of astrophysical interest. We succeeded in doing so. The slogan "Black holes have no hair" captures the apparent simplicity of black holes. Unlike other stars, like our Sun, black holes are specified uniquely by providing a few numbers: mass, angular momentum, and charge. If true, black holes would be the simplest macroscopic objects in our Universe, characterized by the same numbers as elementary particles. Quantum mechanically, however, black holes are very sophisticated objects. There is evidence that they might be the most complicated objects that can exist in the sense that they maximize the so-called entropy, a measure for information and disorder. This apparent tension between maximal simplicity classically and maximal complexity quantum mechanically is at the core of numerous unresolved conceptual issues in black hole physics, like the information loss problem. In 2016 Hawking, Perry and Strominger introduced the notion of soft hair, which denotes the structure of a black hole that does not change its energy. According to this proposal, black holes may have hair if it is soft enough. Thus, black holes with the same mass can be distinguished microscopically by their soft hair charges. One of the key results of this project was the analysis of near horizon symmetries and soft hair excitations for astrophysical black holes, published in collaboration with a postdoc at TU Wien, Celine Zwikel, and collaborators from Chile and Iran, Phys.Rev.Lett. 124 (2020) 041601. We found a non-linear extension of the Heisenberg algebra as relevant near horizon symmetries. Different boundary conditions are specified by a functional of the dynamical variables, describing inequivalent interactions at the horizon with a thermal bath. Our construction allows equipping not only black holes but also cosmological horizons with soft hair. We also addressed the implications of soft hair for black hole thermodynamics and entropy. In nine additional soft hair publications, we generalized these findings, e.g., to describe black holes interacting with gravitational waves. Our soft hair results entered a textbook on black holes that we wrote for Springer. Since several junior team members received competitive grants, additional funds were available to host more postdocs and Ph.D. students and to organize the prestigious conference Strings 2022 in Vienna. This extension of the project scope led to 13 additional scientific publications, partly on related topics and partly on surprising new insights into gravity and the holographic principle.