Horizon supertranslation and degenerate black hole solutions
Rong-Gen Cai, Shan-Ming Ruan, Yun-Long Zhang
TL;DR
The paper investigates horizon-level analogs of BMS supertranslations and their role in creating infinitely degenerate near-horizon black hole geometries with soft hair. By working in Gaussian null coordinates, it derives the horizon supertranslation Killing vector and its associated charge, and constructs on-shell degenerate solutions parameterized by horizon data such as $\theta(z,\bar z)$ and related tensor data $\lambda_{AB}$. The results show that horizon diffeomorphisms connect a continuum of degenerate black hole states without altering the total mass, with higher-order corrections and decaying modes that preserve consistency with vacuum Einstein equations. This horizon-centered framework links soft hair and gravitational memory to information storage in black holes, offering a horizon-based perspective on the information problem and connecting to the broader BMS/soft theorem landscape.
Abstract
In this note we first review the degenerate vacua arising from the BMS symmetries. According to the discussion in [1] one can define BMS-analogous supertranslation and superrotation for spacetime with black hole in Gaussian null coordinates. In the leading and subleading orders of near horizon approximation, the infinitely degenerate black hole solutions are derived by considering Einstein equations with or without cosmological constant, and they are related to each other by the diffeomorphism generated by horizon supertranslation. Higher order results and degenerate Rindler horizon solutions also are given in appendices.