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Description
The aim of this project to investigate two geometric structures that played a central role in the description of light rays in the context of solutions to Einstein''s equation. One is the notion of a conformal structure, that is, a means to measuring angles and relative lengths. This is intrinsically connected to the propagation of light along geodesics of `zero length. The other one is the notion of a Cauchy-Riemann (CR) structure, which underlies very special families of light rays. These are said to be non-shearing and feature in many important gravitational objects such as black holes. These two concepts came to shape the development of mathematical relativity in the 1960ies and 1970ies through the work of the Polish relativist Andrzej Trautman and his English collaborator Ivor Robinson. In this project, we shall take their approach further by drawing on contemporary techniques of differential geometry, and apply our results to the study of black holes and related geometries known as horizons. These gravitational objects are significant by virtue of the fact they feature a gravitational singularity. There, the differential geometric fabric of spacetime breaks down and quantum theory, which describes the small-scale structure of the universe, takes over. They thus constitute a fertile playground on which predictions regarding quantum gravity can be put forward. The results expected from this research project will be a more conceptual understanding of the geometric structures behind gravitational phenomena such as found around black holes and their horizons. A thorough analytic understanding of these will allow an invariant description of physical quantities and facilitate their computation. This will in turn provide new solutions to the Einstein field equation in dimension four and higher.
Summary of project results
The aim of this project to investigate two geometric structures that played a central role in the description of light rays in the context of solutions to Einstein''s equation. One was the notion of a conformal structure, that is, a means to measuring angles and relative lengths. This is intrinsically connected to the propagation of light along geodesics of `zero length''.
The other one was the notion of a Cauchy-Riemann (CR) structure, which underlies very special families of light rays. These are said to be non-shearing and feature in many important gravitational objects such as black holes.
These two concepts came to shape the development of mathematical relativity in the 1960ies and 1970ies through the work of the Polish relativist Andrzej Trautman and his English collaborator Ivor Robinson. In this project, we took their approach further by drawing on contemporary techniques of differential geometry, and apply our results to the study of black holes and related geometries known as horizons.
These gravitational objects are significant by virtue of the fact they feature a gravitational singularity. There, the differential geometric fabric of spacetime breaks down and quantum theory, which describes the small-scale structure of the universe, takes over. They thus constitute a fertile playground on which predictions regarding quantum gravity can be put forward. The results expected from this research project were a more conceptual understanding of the geometric structures behind gravitational phenomena such as found around black holes and their horizons. A thorough analytic understanding of these will allow an invariant description of physical quantities and facilitate their computation. This will in turn provide new solutions to the Einstein eld equation in dimension four and higher.
One of the main aims of the project was to bring the study of conformal and CR geometries, and mathematical physics, notably general relativity, closer together, in one form or another. In that respect, the POLS fellowship was very much successful. More particularly, formal mathematical techniques such as the Webster CR calculus,
tractor calculus and Cartan geometry, have penetrated further into the realm of mathematical relativity. This can notably be seen in the way new collaborations have been forged thanks to the POLS fellowship:
- It enabled University of Warsaw, to interact for the first time with the Arctic University of Norway in Tromsø. There has been much mutual interest in what these two institutions can offer scientifically, and in fact, stronger academic ties have since emerged between the two institutions. Such a connection is expected to be long term. We thus see that the fellowship certainly boosted the already growing interest on horizon geometries among pure mathematicians.
- The closing event of the POLS project, which took the form of a conference in July 2023, was a determining factor in the dissemination of the state-of-the-art research in conformal, CR and related geometries, and mathematical physics. The event, which involved experts from various countries including Poland and Norway, was successfully received. It also provided the opportunity to build or
extend collaborations.
Due to the theoretical nature of the project, which is essentially concerned pure mathematics and mathematical relativity, there is no direct socio-economic impact beyond the academic value of the fellowship, such as stronger cooperation between Poland and Norway in the field of geometry and mathematical physics.