Noncommutative Heisenberg algebra in the neighbourhood of a generic null surface

TL;DR

This paper demonstrates that near a generic null surface, the preserving diffeomorphisms form a noncommutative Heisenberg algebra, leading to a derivation of null surface entropy via the Cardy formula, supporting the emergent gravity paradigm.

Contribution

It generalizes previous work by showing the noncommutative algebra applies to any null surface, not just Rindler horizons, and derives entropy without using equations of motion.

Findings

Null surface diffeomorphisms form a noncommutative Heisenberg algebra.

The algebra is applicable to generic null surfaces in any spacetime.

Null surface entropy is derived using the Cardy formula in an off-shell framework.

Abstract

We show that the diffeomorphisms, which preserve the null nature for a generic null metric very near to the null surface, provide {\it noncommutative} Heisenberg algebra. This is the generalization of the earlier work (Phys. Rev. D95, 044020 (2017)) \cite{Majhi:2017fua}, done for the Rindler horizon. The present analysis revels that the algebra is very general as it is obtained for a generic null surface and is applicable for any spacetime horizon. Finally using these results, the entropy of the null surface is derived in the form of the Cardy formula. Our analysis is completely {\it off-shell} as no equation of motion is used. We believe present discussion can illuminate the paradigm of `gravity as an emergent phenomenon' and could be a candidate to probe the origin of gravitational entropy.