Phase-Space Contractions of Carrollian Black-Hole Thermodynamics

Abstract: We study Carrollian limits of Schwarzschild-AdS black-hole thermodynamics using covariant phase space. Allowing the cosmological constant to vary, we derive the extended Iyer-Wald identity and identify the renormalized bulk term proportional to (δΛ) with the generator-normalized thermodynamic volume contribution (V_ξ\,δP). We show that the Carroll limit contracts the full thermodynamic phase space together with the metric. For fixed Newton constant, the Lorentzian generator (\partial_t) collapses to a zero-norm direction as (c\to0), yielding a degenerate sector with vanishing Hamiltonian variation, temperature and volume. Introducing (ξλ=c^{-α}\partial_t) and (G=c^γG_C), we find that the extended first law scales as (c^{1-α-γ}), so finite phase-space contractions require (α+γ=1). The endpoint ((α,γ)=(1,0)), obtained by (τ=ct), is the ordinary non-degenerate Lorentzian finite-clock normalization. Carrollian finite first laws lie on the segment (α<1), hence (γ=1-α>0), and give (T\to0), (S\to\infty), with finite (T\,δS) and (Vξ\,δP). We test the scaling principle for fixed-charge and fixed-rotation AdS black holes, and extend it to arbitrary spacetime dimension within the Schwarzschild-AdS family.