Henry Maxfield

A bstract: Hilbert spaces in theories of gravity are notoriously subtle due to the Hamiltonian constraints, particularly regarding the inner product. To demystify this subject, we review and extend a collection of ideas in canonical gravity, and connect to the sum-over-histories approach by clarifying the Hilbert space interpretation of various gravitational path integrals. We use one-dimensional (or mini-superspace) models as the simplest context to exemplify the conceptual ideas. We emphasise that a physical Hilbert space can be defined either by requiring states to be annihilated by constraint operators (e.g., the Wheeler-DeWitt equation) or by equivalence relations between wavefunctions, and explain that these two approaches are related by an inner product. We advocate that the group averaging procedure constructs the correct physical inner product. The Klein-Gordon inner product is not positive-definite, which we explain as arising from a bad gauge choice; nonetheless, it agrees with group averaging when such a problem is absent. These concepts are all embedded in the BRST/BFV formalism, which provides a systematic way to construct these and other physically equivalent inner products (e.g., from maximal-volume gauge and Gaussian averaged gauges). Finally we discuss the application of these ideas in the semi-classical approximation, including non-perturbative gravitational effects.

A bstract: Entanglement islands have played a key role in the recent derivation of the Page curve and other progress on the black hole information problem. Arising from the inclusion of connected wormhole saddles in a gravitational replica trick, islands signal that degrees of freedom in the black hole interior are not microscopically independent of the exterior Hawking radiation. Islands were originally discovered in the context of AdS/CFT coupled to an external, nongravitating reservoir, where the coupling gives graviton excitations an anomalous boundary scaling dimension (or “mass”). It has been claimed in the literature that this mass is crucial for the existence of islands and even the Page curve itself. In this paper, however, we explain how entanglement islands can also appear in setups with massless gravitons and no external reservoir, giving a number of examples including the entanglement wedges of boundary CFT regions, of radiation at null infinity in asymptotically flat spacetimes, and of radiation inside a semiclassical but gravitating spacetime. In each case, the Page curve is physically observable and can be determined with sufficiently careful experiments on many copies of the black hole. We give general arguments for the existence of gauge-invariant operators in gravity which are compactly supported to all orders in perturbation theory (whenever no isometries of the background spacetime exist) and refine a recently-proposed explicit construction of such operators. When applied to islands, these results — together with entanglement wedge reconstruction — guarantee that semiclassical operators in the island can be approximated by nonperturbative operators on the Hawking radiation.

Entanglement islands have played a key role in the recent derivation of the Page curve and other progress on the black hole information problem. Arising from the inclusion of connected wormhole saddles in a gravitational replica trick, islands signal that degrees of freedom in the black hole interior are not microscopically independent of the exterior Hawking radiation. Islands were originally discovered in the context of AdS/CFT coupled to an external, nongravitating reservoir, where the coupling gives graviton excitations an anomalous boundary scaling dimension (or "mass"). It has been claimed in the literature that this mass is crucial for the existence of islands and even the Page curve itself. In this paper, however, we explain how entanglement islands can also appear in setups with massless gravitons and no external reservoir, giving a number of examples including the entanglement wedges of boundary CFT regions, of radiation at null infinity in asymptotically flat spacetimes, and of radiation inside a semiclassical but gravitating spacetime. In each case, the Page curve is physically observable and can be determined with sufficiently careful experiments on many copies of the black hole. We give general arguments for the existence of gauge-invariant operators in gravity which are compactly supported to all orders in perturbation theory (whenever no isometries of the background spacetime exist) and refine a recently-proposed explicit construction of such operators. When applied to islands, these results -- together with entanglement wedge reconstruction -- guarantee that semiclassical operators in the island can be approximated by nonperturbative operators on the Hawking radiation.

We study a non-relativistic limit of physics in AdS which retains the curvature through a harmonic Newtonian potential. This limit appears in a CFT dual through the spectrum of operators of large dimension and correlation functions of those operators with appropriate kinematics. In an additional flat spacetime limit, the spectrum is determined by scattering phase shifts (proportional to anomalous dimensions), and a CFT correlation function is proportional to the S-matrix. In particular, we describe the effect of resonances on the spectrum and correlation functions. As an example, we discuss the Coulomb potential (describing solar systems and hydrogen atoms in AdS) in detail.

We define bit models of evaporating black holes which incorporate the effects of replica wormholes. These enter as non-perturbative corrections to the gravitational inner product, arising from spacetime wormholes that connect replica black holes. The resulting models have exactly unitary evolution, predict measurements on Hawking radiation in accord with a Page curve for entropy, and are compatible with a smooth horizon for infalling observers.

We study a non-relativistic limit of physics in AdS which retains the curvature through a harmonic Newtonian potential. This limit appears in a CFT dual through the spectrum of operators of large dimension and correlation functions of those operators with appropriate kinematics. In an additional flat spacetime limit, the spectrum is determined by scattering phase shifts (proportional to anomalous dimensions), and a CFT correlation function is proportional to the S-matrix. In particular, we describe the effect of resonances on the spectrum and correlation functions. As an example, we discuss the Coulomb potential (describing solar systems and hydrogen atoms in AdS) in detail.

In the AdS/CFT correspondence, single trace operators of large-N gauge theories at large spin J can be described by classical spinning strings, giving a geometric and classical description of their spectrum at strong coupling. We observe that in AdS₃ these strings have significant gravitational back-reaction at sufficiently large spin, since the gravitational force does not decay at long distances. We construct solutions for folded spinning strings coupled to gravity in AdS₃ and compute their spectrum, corresponding to the leading Regge trajectory of Virasroro primary operators. These solutions exist only below a maximal spin J < J_max, and as J → J_max the solution approaches an extremal rotating BTZ black hole.