We construct two types of phase spaces for asymptotically de Sitter Einstein-Hilbert gravity in each spacetime dimension d 3. One type contains solutions asymptotic to the expanding spatially-flat (k = 0) cosmological patch of de Sitter space while the other is asymptotic to the expanding hyperbolic (k = 1) patch. Each phase space has a non-trivial asymptotic symmetry group (ASG) which includes the isometry group of the corresponding de Sitter patch. For d = 3 and k = 1 our ASG also contains additional generators and leads to a Virasoro algebra with vanishing central charge. Furthermore, we identify an interesting algebra (even larger than the ASG) containing two Virasoro algebras related by a reality condition and having imaginary central charges . Our charges agree with those obtained previously using dS/CFT methods for the same asymptotic Killing fields showing that (at least some of) the dS/CFT charges act on a well-defined phase space. Along the way we show that, despite the lack of local degrees of freedom, the d = 3, k = 1 phase space is non-trivial even in pure Λ > 0 Einstein-Hilbert gravity due to the existence of a family of wormhole solutions labeled by their angular momentum, a mass-like parameter θ 0, the topology of future infinity (I +), and perhaps additional internal moduli. These solutions are Λ > 0 analogues of BTZ black holes and exhibit a corresponding mass gap relative to empty de Sitter. © 2012 IOP Publishing Ltd.

Abstract: Recent works have related the bulk first law of black hole mechanics to the first law of entanglement in a dual CFT. These are first order relations, and receive corrections for finite changes. In particular, the latter is naively expected to be accurate only for small changes in the quantum state. But when Newton’s constant is small relative to the AdS scale, the former holds to good approximation even for classical perturbations that contain many quanta. This suggests that — for appropriate states — corrections to the first law of entanglement are suppressed by powers of N in CFTs whose correlators satisfy ’t Hooft large-N power counting. We take first steps toward verifying that this is so by studying the large-N structure of the entropy of spatial regions for a class of CFT states motivate dby those created from the vacuum by acting with real-time single-trace sources. We show that 1/N counting matches bulk predictions, though we require the effect of the source on the modular hamiltonian to be non-singular. The magnitude of our sources is ϵN with ϵ fixed-but-small as N → ∞. Our results also provide a perturbative derivation — without relying on the replica trick — of the subleading Faulkner-Lewkowycz-Maldacena correction to the Ryu-Takayagi and Hubeny-Rangamani-Takayanagi conjectures at all orders in 1/N.

The gravitational Dirichlet problem - in which the induced metric is fixed on boundaries at finite distance from the bulk - is related to simple notions of UV cutoffs in gauge/gravity duality and appears in discussions relating the low-energy behavior of gravity to fluid dynamics. We study the Einstein-Maxwell version of this problem, in which the induced Maxwell potential on the wall is also fixed. For flat walls in otherwise asymptotically flat spacetimes, we identify a moduli space of Majumdar-Papapetrou-like static solutions parametrized by the location of an extreme black hole relative to the wall. Such solutions may be described as balancing gravitational repulsion from a negative-mass image source against electrostatic attraction to an oppositely signed image charge. Standard techniques for handling divergences yield a moduli space metric with an eigenvalue that becomes negative near the wall, indicating a region of negative kinetic energy and suggesting that the Hamiltonian may be unbounded below. One may also surround the black hole with an additional (roughly spherical) Dirichlet wall to impose a regulator whose physics is more clear. Negative kinetic energies remain, though new terms do appear in the moduli space metric. The regulator dependence indicates that the adiabatic approximation may be ill-defined for classical extreme black holes with Dirichlet walls.

Dirichlet walls - time-like boundaries at a finite distance from the bulk on which the induced metric is held fixed - have been used to model AdS spacetimes with a finite cutoff. In the context of gauge/gravity duality, such models are often described as dual to some novel UV-cutoff version of a corresponding CFT that maintains local Lorentz invariance. We study linearized gravity in the presence of such a wall and find it to differ significantly from the seemingly analogous case of Dirichlet boundary conditions for fields of spins zero and one. In particular, using the Kodama-Ishibashi formalism, the boundary conditions that must be imposed on the scalar-sector master field with harmonic time dependence depend explicitly on their frequency. That this feature first arises for spin-2 appears to be related to the second-order nature of the equations of motion. It gives rise to a number of novel instabilities, though both global and planar anti-de Sitter remain (linearly) stable in the presence of large-radius Dirichlet cutoffs. The instabilities arise on the outside of spherical Dirichlet walls, and also inside sufficiently large spherical walls in de Sitter space. We analyze both the inside and outside of flat and spherical walls in Minkowski, de Sitter, and anti-de Sitter space, as well as in certain black hole spacetimes, and find stability for the cases not mentioned above. In particular, we find no linear instabilities in the presence of flat walls. We also find evidence supporting the conjecture that neutral black holes are repelled by Dirichlet walls.