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Subregion algebras in classical and quantum gravity

Venkatesa Chandrasekaran, Éanna É. Flanagan

Abstract

We study the kinematics and dynamics of subregion algebras in classical and perturbative quantum gravity associated with portions of null surfaces such as event horizons and finite causal diamonds. We construct half-sided supertranslation generators by extending subregion phase spaces of the event horizon to include doubled pairs of corner edge modes obtained from splitting the horizon, namely relative boosts and null translations of the respective corners. These edge modes carry a corner symplectic form and give rise to canonical charges generating half-sided boosts and translations. We show that the null translation generator is necessarily two-sided in the complementary translation edge modes. The charges act nontrivially on gravitationally dressed local observables on the horizon, such that the horizon subalgebra naturally takes the form of a crossed product by the associated automorphism group. Quantizing the extended phase space after linearizing around a black hole background, we obtain for each horizon cut a Type II$_{\infty}$ von Neumann algebra equipped with a trace, whose von Neumann entropy coincides with the generalized entropy of that cut. The integrability of the half-sided null translation generator lifts to the existence of a self-adjoint operator that implements null time evolution on the Type II$_\infty$ horizon subalgebras. The area operator is identified as the bulk implementation of the Connes cocycle flow for one-sided observables in excited states. The nesting property of the resulting one-parameter family of horizon subalgebras implies a generalized second law for non-stationary linearized perturbations of Killing horizons. Lastly, we use gravitational half-sided modular inclusion algebras to prove the quantum focusing conjecture in the perturbative quantum gravity regime.

Subregion algebras in classical and quantum gravity

Abstract

We study the kinematics and dynamics of subregion algebras in classical and perturbative quantum gravity associated with portions of null surfaces such as event horizons and finite causal diamonds. We construct half-sided supertranslation generators by extending subregion phase spaces of the event horizon to include doubled pairs of corner edge modes obtained from splitting the horizon, namely relative boosts and null translations of the respective corners. These edge modes carry a corner symplectic form and give rise to canonical charges generating half-sided boosts and translations. We show that the null translation generator is necessarily two-sided in the complementary translation edge modes. The charges act nontrivially on gravitationally dressed local observables on the horizon, such that the horizon subalgebra naturally takes the form of a crossed product by the associated automorphism group. Quantizing the extended phase space after linearizing around a black hole background, we obtain for each horizon cut a Type II von Neumann algebra equipped with a trace, whose von Neumann entropy coincides with the generalized entropy of that cut. The integrability of the half-sided null translation generator lifts to the existence of a self-adjoint operator that implements null time evolution on the Type II horizon subalgebras. The area operator is identified as the bulk implementation of the Connes cocycle flow for one-sided observables in excited states. The nesting property of the resulting one-parameter family of horizon subalgebras implies a generalized second law for non-stationary linearized perturbations of Killing horizons. Lastly, we use gravitational half-sided modular inclusion algebras to prove the quantum focusing conjecture in the perturbative quantum gravity regime.
Paper Structure (45 sections, 1 theorem, 440 equations, 9 figures, 1 table)

This paper contains 45 sections, 1 theorem, 440 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Classical gravity: edge modes and subregion phase spaces
  3. Quantum gravity: the generalized entropy of subregions
  4. Relation to previous work
  5. Notational conventions
  6. Phase spaces of gravitational subregions: preamble
  7. Phase space definitions
  8. Corner charges of gravitational subregions
  9. Explicit corner charges
  10. Canonical half-sided supertranslations
  11. General theories of gravity
  12. General relativity
  13. Half-sided diffeomorphisms and null shocks
  14. Distributional corrections to the symplectic form
  15. Integrability and the null gravitational constraints
  16. ...and 30 more sections

Key Result

Theorem 1

Assume $\{\widehat{\mathcal{M}}_u\}$ forms a half-sided modular inclusion w.r.t. $|\hat{\Omega} \rangle$, and let $S_{\mathrm{rel}}(u)$ be Eq. eq:Srel_u_def. Then, for almost every $u$, where, as discussed in sec:hsm_inclusion_bridge, the half-sided translation generator $\hat{\mathscr{P}}$ is the same as the positive semi-definite operator $G$ implementing half-sided modular inclusions in the th

Figures (9)

  • Figure 1: Penrose diagram of a two-sided eternal black hole. The right future event horizon $\mathscr{H}$ with bifurcation surface $\mathscr{B}$ is shown, together with a cut at affine parameter $u_0$ that splits the horizon into a past portion $\mathscr{H}_{<u_0}$ (dashed red line) and a future portion $\mathscr{H}_{>u_0}$. The subregion whose algebra we study is $\mathscr{H}_{>u_0} \cup \mathscr{I}^+$ (dashed blue line), whose domain of dependence in the exterior region is shaded in blue.
  • Figure 2: Interior version of the setup in \ref{['fig:penrose-eternal-exterior']}. Here the null Cauchy surface $\mathscr{H}' \cup \mathscr{H}_{<u_0}$ (the dashed red line on the left and right future horizons) has future domain of dependence shaded in red in the black hole interior, which terminates at the spacelike singularity, and gives rise to a gravitational subregion phase space and algebra of observables.
  • Figure 3: Penrose diagram for a black hole collapse spacetime. The event horizon forms at the point $\mathcal{P}$, and a cut at affine parameter $u_0$ on the horizon splits it into a future subregion $\mathscr{H}_{>u_0}$ and a past portion $\mathscr{H}_{<u_0}$ determined by the collapse geometry. As before, the cut is associated with both the subregion phase space of the region outside the horizon shaded in blue, and the subregion phase space of the region in the interior of the black hole shaded in red.
  • Figure 4: Half-sided null translation generated by the corner charge $\mathscr{P}_\alpha$. A cut $S_0$ at affine parameter $u_0$ on the future horizon is shifted to $u_0 + \delta u$, moving the subregion $\mathscr{H}_{>u_0}$ relative to its complement. The deformation can be visualized as inserting an impulsive null shock at $S_0$.
  • Figure 5: (a) Cauchy principal value vs. (b) on-shell prescriptions for computing the full symplectic form. The principal value prescription misses the shock (blue wiggly line) resulting from the null gravitational constraint equations. The on-shell prescription accounts for the constraints, thus allowing for consistent definitions of subregion phase spaces for the two complementary subregions while retaining connectedness of spacetime across the corner.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Theorem : Variational formula for the null shape derivative