Emergent times in holographic duality

TL;DR

The work develops a boundary-theory construction for emergent infalling time evolutions in holography, showing that large-$N$ boundary operator algebras acquire a type III$_1$ von Neumann structure through half-sided modular translations. This algebraic framework yields explicit boundary realizations of bulk horizons and interiors, including how infalling time can be defined globally across the horizon and how causal structure is encoded in left-right operator relations. The authors connect boundary subregions to bulk reconstructions, discuss RT-surface factorization and its relation to horizon physics, and provide detailed analyses in generalized free fields and AdS-Rindler/BTZ contexts, including a large-mass limit where transformations become point-wise. These results offer a novel, non-geometric route to understanding bulk time inside black holes and suggest broad implications for subregion duality, entanglement structure, and emergent symmetries in holography.

Abstract

In holographic duality an eternal AdS black hole is described by two copies of the boundary CFT in the thermal field double state. In this paper we provide explicit constructions in the boundary theory of infalling time evolutions which can take bulk observers behind the horizon. The constructions also help to illuminate the boundary emergence of the black hole horizons, the interiors, and the associated causal structure. A key element is the emergence, in the large $N$ limit of the boundary theory, of a type III$_1$ von Neumann algebraic structure from the type I boundary operator algebra and the half-sided modular translation structure associated with it.

Figures (17)

• Figure 1: The Penrose diagram of an eternal black hole. The dashed lines are event horizons and the wavy lines are the singularities.
• Figure 2: Left: Evolution of the ${\rm t} =0$ bulk slice under $H_R - H_L$, where $H_{R, L}$ denote the Hamiltonians of the boundary theories. Center: Evolution of the ${\rm t} =0$ slice under $H_R + H_L$, the action of which is singular at the horizon. In fact any linear combination of $H_R$ and $H_L$ other than $H_R - H_L$ is expected to have a singular action at the horizon. Right: A smooth Kruskal-like evolution. If such an evolution can be described in a diffeomorphism invariant way, it must be emergent in the boundary theory.
• Figure 3: Kruskal diagram for an eternal black hole. The dashed lines are event horizons, the solid red lines are the singularities, and the solid black lines are the boundaries.
• Figure 4: Below $T_{HP}$ the bulk theory is two separate global AdS spacetimes whose small excitations are entangled in the thermal field double state.
• Figure 5: The single-trace algebras ${{\mathcal{A}}}_1$ and ${{\mathcal{A}}}_2$ associated with the two different Cauchy slices shown are inequivalent, even though they share a causal diamond, since single-trace operators do not obey any equation of motion among themselves (standard Heisenberg evolution takes a single-trace operator outside of the algebra). The same statements apply to algebras generated by generalized free fields (e.g. subalgebras of ${\mathcal{Y}}$) which do not obey any equations of motion.