Carroll stories

TL;DR

This work analyzes the Carroll limit c -> 0, identifying fundamental pathologies in Carroll quantum field theories such as infinite spectrum degeneracies and ill-defined partition functions, and then introduces a controlled Carroll regime via a leading-order small-c expansion to recover meaningful thermodynamics. It develops Carroll geometry from ultra-local gravity, derives Carroll energy-momentum tensors, and studies gravitational solutions and geodesics in both electric and magnetic Carroll frameworks, including wormhole-like constructions and cosmological spacetimes. The paper also explores two complementary routes to Carroll hydrodynamics: a Carrollian expansion of relativistic fluids and hydrostatic partition functions, obtaining consistent Carroll fluid energy-momentum tensors and highlighting the difficulties of realizing true Carroll thermodynamics in the strict c -> 0 limit. Together these results illuminate how Carrollian physics connects flat space holography, non-Lorentzian geometry, and novel thermodynamic and gravitational structures. The findings emphasize that while the strict Carroll limit yields pathological thermodynamics, its regulated Carroll regime can yield physically meaningful insights and potential holographic interpretations.

Abstract

We study various aspects of the Carroll limit in which the speed of light is sent to zero. A large part of this paper is devoted to the quantization of Carroll field theories. We show that these exhibit infinite degeneracies in the spectrum and may suffer from non-normalizable ground states. As a consequence, partition functions of Carroll systems are ill-defined and do not lead to sensible thermodynamics. These seemingly pathological properties might actually be a virtue in the context of flat space holography. Better defined is the Carroll $\textit{regime}$, in which we consider the leading order term in an expansion around vanishing speed of light without taking the strict Carroll limit. Such an expansion may lead to sensible notions of Carroll thermodynamics. An interesting example is a gas of massless particles with an imaginary chemical potential conjugate to the momentum. In the Carroll regime we show that the partition function of such a gas leads to an equation of state with $w=-1$. As a separate story, we study aspects of Carroll gravity and couplings to Carrollian energy-momentum tensors. We discuss many examples of solutions to Carroll gravity, including wormholes, Maxwell fields, solutions with a cosmological constant, and discuss the structure of geodesics in a Carroll geometry. The coupling of matter to Carroll gravity also allows us to derive energy-momentum tensors for hypothetical Carroll fluids from expanding relativistic fluids as well as directly from hydrostatic partition functions.