[Paper Review] Event horizons and holography

S. Avery, David A. Lowe|arXiv (Cornell University)|Oct 30, 2013

Black Holes and Theoretical Physics3 citations

TL;DR

The paper challenges the conclusion that typical black hole states in AdS spacetime violate general covariance at the horizon, arguing instead that the standard semiclassical analysis overestimates the Hilbert space dimension. By showing that only special superpositions of $b$-modes—specifically those related to $d$-modes—correspond to physical states in the exact quantum theory, the authors demonstrate that general covariance can be preserved. The key result is that the typicality argument for firewalls fails due to a mismatch between the semiclassical Hilbert space and the true quantum state space.

ABSTRACT

We consider the microcanonical ensemble of black holes in gravitational theories in asymptotically anti-de Sitter spacetime with a conformal field theory dual. We argue that typical quantum black hole states show no violations of general covariance on the horizon.

Motivation & Objective

• To resolve the apparent conflict between general covariance and the firewall paradox in AdS black holes.
• To address the issue of divergent stress energy tensors in semiclassical field theory near the horizon.
• To clarify the mismatch between the infinite-dimensional semiclassical Hilbert space and the finite-dimensional exact quantum state space in the microcanonical ensemble.
• To argue that only specific superpositions of $b$-modes (related to $d$-modes) represent physical states in the exact theory.
• To provide a consistent framework where general covariance is preserved at the horizon by using a cutoff on $a$-modes and entangled $d$-mode states.

Proposed method

• Analyzes the field decomposition in terms of $b$-modes (associated with timelike Killing vector at infinity) and $a$-modes (Kruskal modes analytic across the horizon).
• Uses Bogoliubov transformations to relate $b$-modes to $d$-modes, showing that $|0 angle_K = \prod_k \sum_{n_k} e^{-\beta \omega_k n_k / 2} |n_k\rangle_L |n_k\rangle_R$.
• Applies a cutoff to the $a$-mode spectrum to match the finite entropy of the CFT, ensuring the Hilbert space is discrete and finite-dimensional.
• Argues that only states with entanglement between $b_{L,k}$ and $b_{R,k}$ modes (via $d_{L,k}^\dagger, d_{R,k}^\dagger$) are physical, not generic $b$-mode number eigenstates.
• Uses the lattice black hole analysis of Corley and Jacobson to show $b$-mode eigenstates are not preserved under time evolution, while $d$-mode states are.
• Constructs an effective local field theory outside the horizon using $d_{R,k}^\dagger$ modes, suppressed by $e^{-\beta \omega_k}$ inside the horizon, ensuring smoothness.

Experimental results

Research questions

• RQ1 Why do typical $b$-mode number eigenstates lead to divergent stress energy tensors near the horizon in semiclassical theory?
• RQ2 How can general covariance be preserved at the black hole horizon if the standard semiclassical analysis suggests violations?
• RQ3 What is the correct Hilbert space structure for the exact quantum theory of black holes in AdS with a CFT dual?
• RQ4 Why does the typicality argument for firewall states fail in the microcanonical ensemble?
• RQ5 How do $d$-mode operators and their entanglement structure resolve the tension between semiclassical and exact quantum descriptions?

Key findings

• The typicality argument of Marolf and Polchinski fails because the Hilbert space of physical states is much smaller than the infinite-dimensional subspace spanned by $b$-mode number eigenstates.
• Only specific superpositions of $b$-modes, such as the Kruskal vacuum $|0\rangle_K$, correspond to physical states in the exact theory, not generic $b$-mode eigenstates.
• The $d$-mode number operator $N_{d,k} = d_{L,k}^\dagger d_{L,k} + d_{R,k}^\dagger d_{R,k}$ has expectation value $\geq 1$ in the microcanonical ensemble, but this does not imply a firewall due to the restricted state space.
• The $d_{R,k}^\dagger$ operators create states with a component outside the horizon that dominates by a factor $e^{-\beta \omega_k}$, enabling a consistent effective field theory outside the horizon.
• The $b$-mode eigenstates are not preserved under time evolution due to non-commutativity with the Hamiltonian, while $d$-mode states are, supporting their use as physical states.
• A cutoff on the $a$-mode spectrum ensures the Hilbert space matches the finite entropy of the CFT, resolving the divergence in entropy and stress energy tensor.