DYNAMICS OF NULL HYPERSURFACES IN GENERAL RELATIVITY AND APPLICATIONS TO GRAVITATIONAL RADIATION, CONSERVED CHARGES AND QUANTUM GRAVITY

Abstract

Noether’s theorem is one of the most beautiful pillars of classical mechanics and field theory. It unravelled a relation between symmetries and conservation laws, and found applications in all domains of physics. Among its applications, the case of general relativity is probably one of the most subtle ones. The only symmetry of general relativity is the invariance under coordinate transformations, or diffeomorphisms. But this is more like a local gauge symmetry, and like for local gauge symmetries, a direct application of the theorem says that there are no non-trivial conserved charges. A more careful analysis shows that if one correctly takes into account the boundary conditions, there are non-trivial charges, but these are not integrals over a Cauchy hypersurface, like in applications to field theories on flat spacetime, but rather surface integrals over the two-dimensional boundaries of a Cauchy hypersurface. Such surface charges have played a key role in the understanding of general relativity since the ADM (Arnowitt-Deser-Misner) Hamiltonian analysis and later on the seminal paper by Regge and Teitelboim. In current research, these surface charges play an important role in phenomenological applications: for instance the quantities measured by LIGO and Virgo, like the mass and angular momentum of coalescing black holes carried away by gravitational waves, are understood as surface charges. They also play a role in theoretical developments: they describe the first law of black hole mechanics, enter the description of black hole entropy and have been used to explore resolutions of the black hole information paradox. There is currently an active area of interest around the charges, and various open questions are on the table, from the inclusion of gravitational multipoles to understanding their correct quantization. There is a second subtle aspect of general relativity that I addressed in this thesis. A consequence of the diffeomorphism invariance of the theory is the presence of first class constraints, like the Gauss law in gauge theories. This first class constraints limit the choice of admissible initial conditions for the Cauchy problem in a non-linear, highly non-trivial way. This is a problem that shows up very clearly in numerical relativity, where one has to carefully implement the constraints and make sure that the approximations used by the numerical grid don’t introduce too strong violations. Good initial conditions are known for very simple solutions, and a general solutions of the constraints is unknown. This fact has an important consequence also for approaches to quantum gravity. In loop quantum gravity for example, there of space, and geometric operators with discrete spectra and non-commutativity properties. This picture holds at the kinematical level, namely prior to the imposition of the quantum version of the Hamiltonian diffeorphism constraints, and it is not proved that the same quantum geometry would also describe the physical Hilbert space of the theory, defined on-shell on the constraints. A different perspective to the problem can be gained if one switches attention from a Cauchy initial value problem on a space-like hypersurface to a characteristic initial value problem on a null hypersurface. In this case, it is known since the work of Sachs in the sixties that one can identify constraint-free data, in the form of the shear of the null geodesics congruence of the hypersurface. The question is whether these constraint-free data can be given an interpretation in terms of connection variables, and then the loop quantum gravity techniques be applied.