Null Raychaudhuri: Canonical Structure and the Dressing Time

TL;DR

The paper develops a non-perturbative, canonical framework for gravity on generic null hypersurfaces, extending the phase space to include spin-0, spin-1, spin-2 and matter data. It shows that Raychaudhuri's equation can be viewed as a Carrollian stress-tensor conservation law and introduces a dynamical dressing time clock that is conjugate to the Raychaudhuri constraint, while dressing all other fields. A Beltrami parameterization and a bi-local propagator enable inversion of the presymplectic form and the full spin-0/spin-2 constraint algebra closes only when both sectors are included, highlighting a non-perturbative mixing between geometry and graviton data. The dressing-time frame yields a positive, monotonic boost charge (area), suggesting a gravitationally emergent observer and providing a potential route toward non-perturbative quantum gravity and generalized entropy concepts on arbitrary null surfaces.

Abstract

We initiate a study of gravity focusing on generic null hypersurfaces, non-perturbatively in the Newton coupling. We present an off-shell account of the extended phase space of the theory, which includes the expected spin-2 data as well as spin-0, spin-1 and arbitrary matter degrees of freedom. We construct the charges and the corresponding kinematic Poisson brackets, employing a Beltrami parameterization of the spin-2 modes. We explicitly show that the constraint algebra closes, the details of which depend on the non-perturbative mixing between spin-0 and spin-2 modes. Finally we show that the spin zero sector encodes a notion of a clock, called dressing time, which is dynamical and conjugate to the constraint. It is well-known that the null Raychaudhuri equation describes how the geometric data of a null hypersurface evolve in null time in response to gravitational radiation and external matter. Our analysis leads to three complementary viewpoints on this equation. First, it can be understood as a Carrollian stress tensor conservation equation. Second, we construct spin-$0$, spin-$2$ and matter stress tensors that act as generators of null time reparametrizations for each sector. This leads to the perspective that the null Raychaudhuri equation can be understood as imposing that the sum of CFT-like stress tensors vanishes. Third, we solve the Raychaudhuri constraint non-perturbatively. The solution relates the dressing time to the spin-$2$ and matter boost charge operators. Finally we establish that the corner charge corresponding to the boost operator in the dressing time frame is monotonic. These results show that the notion of an observer can be thought of as emerging from the gravitational degrees of freedom themselves. We briefly mention that the construction offers new insights into focusing conjectures.