Beyond the horizon

Abstract

This paper investigates holographic reconstructions of black hole interiors. It begins by presenting the historical evolution of the black hole information paradox, highlighting the main challenges in achieving a coherent description. It emphasizes that the interior is causally inaccessible to asymptotic observers, making direct observation impossible, and thereby motivating the search for alternative descriptions. The discussion then motivates AdS/CFT duality as an appropriate framework for addressing black hole problems. It provides a review of the standard holographic toolkit: the AdS/CFT dictionary; the duality between an eternal AdS black hole and the thermofield-double state |TFD⟩; and bulk reconstruction, namely the Hamilton-Kabat-Lifschytz-Lowe (HKLL) construction, and how this procedure fails to extend beyond the future horizon. To overcome these limitations, we introduce algebraic quantum field theory (AQFT) and present the classification of von Neumann algebra types I, II, and III, illustrating their physical meaning via trace properties and projections. This material furnishes the prerequisite background for Tomita-Takesaki modular theory, which is central to our discussion. The analysis is structured around two approaches to reconstructing the interior. The first is the Papadodimas-Raju mirror operator proposal (PR proposal), which defines boundary operators that mimic interior modes behind the horizon. However, these operators are built within a low-energy subspace of a given black hole microstate, making them inherently state-dependent. The second approach was introduced by Leutheusser and Liu (LL proposal) and invokes an emergent type III_1 algebra in the large-N limit, together with half-sided modular inclusions, to evolve exterior operators into the interior. The paper concludes with a detailed, side-by- side comparative analysis of the two approaches and addresses several immediate questions: (i) To what extent do the Papadodimas-Raju and Leutheusser-Liu constructions describe the same physical observables behind the horizon? (ii) What are the precise technical differences between them in terms of algebraic structure, modular flow, and state dependence? In our quest to answer the first question, we uncover a link between the two proposals that involves an analytic continuation in the modular translation operator. We explore these questions in detail and comment on future prospects for black hole interior reconstruction.