Residual Diffeomorphisms and Symplectic Softs Hairs: The Need to Refine Strict Statement of Equivalence Principle

TL;DR

The essay argues that general covariance in GR should be refined because a measure-zero set of diffeomorphisms, the residual diffeomorphisms, can carry well-defined symplectic charges that distinguish diffeomorphic geometries. Using the Covariant Phase Space Method, the author constructs a solution phase space and defines symplectic charges $Q_χ$ and $Q_η$, with a possible central extension in their algebra. This leads to the concept of symplectic soft hair on black holes, where near-horizon residual symmetries label horizon microstates (horizon fluffs) invisible to asymptotic charges but potentially relevant to black hole thermodynamics and information. The framework connects to known symmetry structures like BMS and Brown-Henneaux and suggests a unified view of soft hairs, horizon physics, and holography, with implications for black hole microstate counting and the information paradox.

Abstract

General covariance is the cornerstone of Einstein's General Relativity (GR) and implies that any two metrics related by diffeomorphisms are physically equivalent. There are, however, many examples pointing to the fact that this strict statement of general covariance needs refinement. There are a very special (measure-zero) subset of diffeomorphisms, the residual diffeomrphisms, to which one can associate well-defined conserved charges. This would hence render these diffeomorphic geometries physically distinct. We discuss that these symmetries may be appropriately called "symplectic symmetries". Existence of residual diffeomorphisms and sympelctic symmetries can be a quite general feature and not limited to the examples discussed so far in the literature. We propose that, in the context of black holes, these diffeomorphic, but distinct, geometries may be viewed as "symplectic soft hair" on black holes. We comment on how this may remedy black hole microstate problem, which in this context are dubbed as "horizon fluffs".