Black Holes, Flat Space Holography and the Infrared Triangle

Symmetries play an important role in the formulation of physical theories. They help physicists in selecting from all mathematically possible theories those that describe our universe. On the other hand, the finding that some symmetry is realized in nature has very often profound consequences for our understanding of the universe. Einstein`s theory of general relativity describes gravity as curvature of space and time. A spacetime can therefore be seen as a particular geometry caused by the presence of mass. A specific geometry is very often associated with a set of symmetries that leave it invariant, e.g., a sphere can be rotated arbitrarily without changing its appearance. In the same way one finds that a spacetime can have symmetries. In the seventies it was shown that the spacetime, i.e., the gravitational field of an isolated object ( a star, a galaxy or a black hole) has infinitely many symmetries far away from the source. These symmetries are called BMS symmetries. Although they have been known for quite some time, it was found only recently that BMS symmetries can be seen as the cause of other long-known phenomena in gravity, such as the so- called gravitational memory effect or the soft-graviton theorem. This newly uncovered underlying structure was called the infrared triangle. This finding becomes even more interesting when it was realized that it could help in understanding the so-called black hole information problem. The information problem, which was posed by Hawking forty years ago, is one of the most important unsolved problems in theoretical physics. He showed that black holes can destroy information although the laws of quantum mechanics do not allow these processes. The above mentioned new insights concerning BMS symmetries indicate that some of the basic assumptions in the derivation of the black hole information problem are incorrect. One should therefore reconsider the problem in the light of the new findings. However, although some arguments point in that direction there are no concrete calculations available, since they are far too difficult and probably not feasible without further insights into the problem. In similar cases physicists like to employ so-called toy models, that allow the study of conceptual problems by reducing technical difficulties. When studying quantum-mechanical aspects of gravity one of the most widely used toy models is three-dimensional gravity. Although our universe is four-dimensional (three space dimensions and one time dimension) it turns out that by dropping one space dimension, Einstein`s theory of general relativity becomes much simpler. This project is concerned with the study of the above mentioned new aspects of BMS symmetry in three-dimensional gravity. The insights gained from these toy model calculations will then be put to use to understand aspects of the black hole information problem. This project will be carried out at Harvard university, Cambridge, U.S.A, in the group of Andrew Strominger who is one of the leading figures in the renaissance of BMS symmetries.

In the course of this project, it was shown that the so-called self-dual sector of general relativity not only has an infinite number of symmetries, but that these symmetries are not destroyed by quantum corrections and are therefore exact symmetries of a theory of quantum gravity of this sector. The search for a theory of quantum gravity has occupied theoretical physicists for more than a hundred years. While quantum physics describes the world on a small scale, i.e. molecules, atoms and elementary particles, general relativity (GR) shows that gravity, which controls our world on the largest length scales, is nothing but the curvature of space and time. At very high energies, gravity and quantum theory must unite to form this new, unknown theory of quantum gravity. One of the central objects of quantum theory is the so-called scattering amplitude, which allows to calculate how elementary particles are deflected by their mutual interactions. A theory of quantum gravity should allow to calculate this scattering amplitude for gravitons, i.e., elementary particles of gravity. In so-called flat spacetimes, i.e. spacetimes with vanishing cosmological constant, GR makes interesting predictions for processes that involve very low energies or very large distances: an infinite number of symmetries arises, the so-called Bondi-van der Burg-Metzner-Sachs (BMS) symmetries, and, related to this, an infinite number of vacua, i.e. states of lowest energy. Although some of these results have been known for several decades, only recent research has shown that these infinitely many symmetries must also have consequences for the scattering amplitudes of a theory of quantum gravity. However, up to this point it is not yet clear whether all of these infinitely many symmetries can be realized in a quantum theory of gravitation or whether they will be changed or even destroyed by quantum corrections. As mentioned above, it was shown that these symmetries are not destroyed when restricting GR to its self-dual sector. Even more, one finds infinitely more additional exact symmetries of the corresponding quantum theory. Furthermore, it was shown that also the above mentioned infinitely many vacua have interesting consequences for scattering amplitudes of gravitons. A calculation of these scattering amplitudes typically leads to so-called divergences, i.e., results that are either zero or infinity, from which the physically meaningful answer must be extracted. As shown in this project, the infinite number of vacua is responsible for these divergences. In particular, it is possible to construct a so-called effective theory that computes the form of these divergences, facilitating the extraction of the physically meaningful answer. This project was carried out as part of an Erwin Schrodinger-fellowship at Harvard University in the group of Prof. Strominger.