Extended Hamiltonian thermodynamics and Carroll contractions of AdS black holes

Abstract

We study extended anti-de Sitter black-hole thermodynamics under correlated contractions of the time generator and Newton coupling. We relate the contraction parameter to physical units and distinguish variations within an equilibrium black-hole family from changes along the contraction. Starting from the magnetic Carroll Hamiltonian action, we derive the boundary identity with a variable cosmological constant and evaluate its energy, horizon and pressure contributions for Schwarzschild--anti-de Sitter geometries. We verify the spatial and lapse equations, retaining the boost connection in the coframe formulation. A three-form potential supplies a canonical variable conjugate to the cosmological constant. We compare this static boundary mechanics with Lorentzian thermodynamic contractions, whose common scaling of the work terms selects a finite Carroll branch. On this branch, temperature vanishes and entropy diverges while their conjugate product remains finite. The pressure and volume scalings depend on the chosen physical normalization. Charged, rotating and higher-dimensional families provide further thermodynamic tests, including the asymptotic frame and full thermodynamic volume for rotation. We use symbolic identities and numerical differentiation to verify the thermodynamic variations, resolve the conjugate contributions and check their scaling behaviour. These constructions relate extended static Carroll boundary mechanics to Lorentzian thermodynamic contractions and identify the phase-space conditions under which they correspond.