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Perturbative semiclassical entropy of dynamical black holes

Avinandan Mondal, Kartik Prabhu

Abstract

We consider perturbative quantum gravity as a quantum field theory of linearized metric perturbation on an asymptotically flat spacetime with a bifurcate Killing horizon. We include the perturbative gravitational constraints into the algebra of observables restricted to the right half of the future horizon of the spacetime. We use the boundary charge, associated to the horizon Killing field, as an auxiliary "observer" degree of freedom. The observables "dressed" with the additional charge are invariant under the Killing symmetry and generate a Type-$\text{II}_{\infty}$ von Neumann factor. We compute the von Neumann entropy of the reduced density matrix of a classical-quantum coherent state constructed from the metric perturbations and the "observer wavefunction". This von Neumann entropy satisfies an analogue of the first law of thermodynamics. We further show that this entropy is related to Hollands-Wald-Zhang entropy of the (second order) perturbed dynamical black hole through the flux of perturbations through the horizon and future null infinity.

Perturbative semiclassical entropy of dynamical black holes

Abstract

We consider perturbative quantum gravity as a quantum field theory of linearized metric perturbation on an asymptotically flat spacetime with a bifurcate Killing horizon. We include the perturbative gravitational constraints into the algebra of observables restricted to the right half of the future horizon of the spacetime. We use the boundary charge, associated to the horizon Killing field, as an auxiliary "observer" degree of freedom. The observables "dressed" with the additional charge are invariant under the Killing symmetry and generate a Type- von Neumann factor. We compute the von Neumann entropy of the reduced density matrix of a classical-quantum coherent state constructed from the metric perturbations and the "observer wavefunction". This von Neumann entropy satisfies an analogue of the first law of thermodynamics. We further show that this entropy is related to Hollands-Wald-Zhang entropy of the (second order) perturbed dynamical black hole through the flux of perturbations through the horizon and future null infinity.
Paper Structure (7 sections, 96 equations, 1 figure)

This paper contains 7 sections, 96 equations, 1 figure.

Table of Contents

  1. Bifurcate Killing horizons, metric perturbations and gauge conditions
  2. Classical phase space and quantization of metric perturbations at future horizon
  3. Quantization on the horizon
  4. Tomita-Takesaki modular theory
  5. Dressed observables and crossed product algebra
  6. Entropy of classical-quantum coherent states of linearized quantum gravity
  7. Calculation of von Neumann entropy for classical-quantum coherent states on the horizon

Figures (1)

  • Figure 1: An asymptotically flat spacetime diagram depicting a bifurcate Killing horizon. The flows of the Killing field locally around the bifurcation surface are shown by the curved dashed arrows. The Killing horizon is $\mathcal{H} = \mathcal{H}^+ \cup \mathcal{H}^-$ and divides the spacetime into four regions: left wedge ($\mathcal{L}$), right wedge ($\mathcal{R}$), past wedge ($\mathcal{P}$) and future wedge ($\mathcal{F}$) as shown. Furthermore the Killing horizons are divided into left ($\mathcal{H}_L^+, \mathcal{H}_L^-$) and right ($\mathcal{H}_R^+, \mathcal{H}_R^-$) sections. $i^+$ and $i^-$ are future and past timelike infinities respectively, $\mathscr{I}^+$ and $\mathscr{I}^-$ are future and past null infinities and finally $i^0$ is spacelike infinity with a subscript of $L$ and $R$ on all of them to differentiate between the left and right wedges respectively. The dotted lines connecting the timelike infinities in past and future can be singularity for Schwarzschild or it might not be the end of the spacetime, e.g. for Kerr black hole. Note that we require $\mathcal{P} \cup \mathcal{R}$ to be globally hyperbolic.

Theorems & Definitions (4)

  • remark 3.1: No maximal entropy state
  • remark 4.1: Gravitational flux through null infinity
  • remark 4.2: Matter fields
  • remark 4.3: Memory and soft radiation