Universal Lichnerowicz Lifting of Near-Horizon Soft Modes

Abstract

A recurring universality appears in the low-temperature quantum thermodynamics of near-extremal black holes, where distinct parent geometries often lead to the same logarithmic temperature dependence at one loop. In this work, we study the Lichnerowicz spectral origin of this infrared universality and understand why the relevant spectral data become insensitive to the details of the parent geometry. For extremal near-horizon geometries containing a two-dimensional maximally symmetric throat, we construct the normalizable transverse-traceless tensor zero modes associated with near-horizon reparametrizations. Turning on a small temperature lifts these zero modes through the first-order deformation of the Lichnerowicz operator. Although the local matrix element depends on detailed parent-geometry data, these data cancel after projection onto normalized tensor modes, leaving the universal result. For static spherically symmetric backgrounds, the resulting eigenvalue shift is proportional to the Fourier mode number and temperature. For rotating backgrounds, the same structure persists in the near-horizon reparametrization tensor sector, while angular warp factors only modify the overall projection factor. We fix the normalized coefficient explicitly, showing how radial integration and bulk normalization remove the near-horizon Taylor coefficients. We further show that this lifted bulk spectrum is the Lichnerowicz realization of the Schwarzian soft sector. Thus, the universal one-loop temperature dependence in the tensor sector is traced to an infrared bulk-boundary matching between near-horizon tensor zero modes and boundary reparametrization dynamics.