Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ten Proofs of the Generalized Second Law

Aron C. Wall

TL;DR

The paper surveys ten broad approaches to proving the Generalized Second Law (GSL) in black hole spacetimes, highlighting that most proofs rely on simplifying regimes or additional assumptions. It systematically classifies proofs by the physical regime (classical, hydrodynamic, semiclassical, full quantum gravity) and by the core technique (entropy bounds, S-matrix, time-independent states, covariant bounds, 2D models), critically evaluating limitations such as entropy renormalization, locality of entanglement entropy, and applicability to rotating or confined black holes. A central tension is the treatment of outside entropy $S_{ ext{out}}$ and its renormalization, the choice of horizon (event vs apparent), and the role of ultraviolet divergences near horizons. The review argues that while several proofs are compelling within their narrow domains (e.g., Hawking’s area theorem, certain hydrodynamic proofs, and Frolov–Page S-matrix proof), a universal, regime-independent proof likely requires a deeper quantum-gravity framework and robust renormalization consistent across horizons. The work also highlights promising directions, including quantum-corrected covariant bounds and 2D models as testbeds for new ideas toward a more complete GSL proof in realistic settings.

Abstract

Ten attempts to prove the Generalized Second Law of Thermodyanmics (GSL) are described and critiqued. Each proof provides valuable insights which should be useful for constructing future, more complete proofs. Rather than merely summarizing previous research, this review offers new perspectives, and strategies for overcoming limitations of the existing proofs. A long introductory section addresses some choices that must be made in any formulation the GSL: Should one use the Gibbs or the Boltzmann entropy? Should one use the global or the apparent horizon? Is it necessary to assume any entropy bounds? If the area has quantum fluctuations, should the GSL apply to the average area? The definition and implications of the classical, hydrodynamic, semiclassical and full quantum gravity regimes are also discussed. A lack of agreement regarding how to define the "quasi-stationary" regime is addressed by distinguishing it from the "quasi-steady" regime.

Ten Proofs of the Generalized Second Law

TL;DR

The paper surveys ten broad approaches to proving the Generalized Second Law (GSL) in black hole spacetimes, highlighting that most proofs rely on simplifying regimes or additional assumptions. It systematically classifies proofs by the physical regime (classical, hydrodynamic, semiclassical, full quantum gravity) and by the core technique (entropy bounds, S-matrix, time-independent states, covariant bounds, 2D models), critically evaluating limitations such as entropy renormalization, locality of entanglement entropy, and applicability to rotating or confined black holes. A central tension is the treatment of outside entropy and its renormalization, the choice of horizon (event vs apparent), and the role of ultraviolet divergences near horizons. The review argues that while several proofs are compelling within their narrow domains (e.g., Hawking’s area theorem, certain hydrodynamic proofs, and Frolov–Page S-matrix proof), a universal, regime-independent proof likely requires a deeper quantum-gravity framework and robust renormalization consistent across horizons. The work also highlights promising directions, including quantum-corrected covariant bounds and 2D models as testbeds for new ideas toward a more complete GSL proof in realistic settings.

Abstract

Ten attempts to prove the Generalized Second Law of Thermodyanmics (GSL) are described and critiqued. Each proof provides valuable insights which should be useful for constructing future, more complete proofs. Rather than merely summarizing previous research, this review offers new perspectives, and strategies for overcoming limitations of the existing proofs. A long introductory section addresses some choices that must be made in any formulation the GSL: Should one use the Gibbs or the Boltzmann entropy? Should one use the global or the apparent horizon? Is it necessary to assume any entropy bounds? If the area has quantum fluctuations, should the GSL apply to the average area? The definition and implications of the classical, hydrodynamic, semiclassical and full quantum gravity regimes are also discussed. A lack of agreement regarding how to define the "quasi-stationary" regime is addressed by distinguishing it from the "quasi-steady" regime.

Paper Structure

This paper contains 43 sections, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. What does the Generalized Second Law say?
  3. Boltzmann or Gibbs?
  4. The Choice of Horizon
  5. Types of Regimes
  6. The Quasi-stationary and Quasi-steady Regimes
  7. The First Law
  8. The Adiabatic Limit
  9. Classical Black Hole Thermodynamics
  10. The Hydrodynamic Approximation
  11. The Semiclassical Regime
  12. The Entanglement Entropy Divergence
  13. Full Quantum Gravity
  14. Are the Entropy Bounds necessary for the GSL?
  15. Proofs applying the OSL to the Thermal Atmosphere
  16. ...and 28 more sections

Figures (2)

  • Figure 1: The Penrose diagram of an eternal black hole. The S-matrix is used to evolve the UP and IN modes into the DOWN and OUT modes. In the case of the black hole which forms from collapse, the white hole horizon is replaced by the collapsing star and the UP modes are populated by the Hawking effect.
  • Figure 2: A Penrose diagram of the two dimensional black hole. The point $P$ on the apparent horizon can be traced backwards to $\sigma^{+}_B$ or $\sigma^{+}_H$. The "outside" is the region whose fine-grained entropy is being calculated.