Motivated by the Kerr/CFT conjecture, we explore solutions of vacuum general relativity whose asymptotic behavior agrees with that of the extremal Kerr throat, sometimes called the Near-Horizon Extreme Kerr (NHEK) geometry. We argue that all such solutions are diffeomorphic to the NHEK geometry itself. The logic proceeds in two steps. We first argue that certain charges must vanish at all times for any solution with NHEK asymptotics. We then analyze these charges in detail for linearized solutions. Though one can choose the relevant charges to vanish at any initial time, these charges are not conserved. As a result, requiring the charges to vanish at all times is a much stronger condition. We argue that all solutions satisfying this condition are diffeomorphic to the NHEK metric.1 A more complete argument notes that the horizon-generating Killing field of a non-extreme Kerr black hole is timelike near the horizon, so that timelike observers sufficiently close to a nonextreme Kerr black hole can co-rotate with the black hole, out to the so-called velocity of light surface where such co-rotating observers must become null. For positive-energy matter, this timelike Killing field defines a positive conserved quantity for excitations in the near-horizon region, ruling out instabilities. In contrast, the horizon-generating Killing field of extreme Kerr is spacelike at all points near the equator outside the horizon, no matter how far one goes down the throat. As a result, the Frolov-Thorne vacuum [9] for linear fields discussed in [3] is not well-defined in the extreme Kerr throat [10,11,12].

We consider asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound in d ≥ 4 spacetime dimensions. The boundary conditions in these "designer gravity" theories are defined in terms of an arbitrary function W .We give a general argument that the Hamiltonian generators of asymptotic symmetries for such systems will be finite, and proceed to construct these generators using the covariant phase space method. The direct calculation confirms that the generators are finite and shows that they take the form of the pure gravity result plus additional contributions from the scalar fields. By comparing the generators to the spinor charge, we derive a lower bound on the gravitational energy when i) W has a global minimum, ii) the Breitenlohner-Freedman bound is not saturated, and iii) the scalar potential V admits a certain type of "superpotential."

The physical process version of the first law for black holes states that the passage of energy and angular momentum through the horizon results in a change in area κ 8π ∆A = ∆E − Ω∆J, so long as this passage is quasi-stationary. A similar physical process first law can be derived for any bifurcate Killing horizon in any spacetime dimension d ≥ 3 using much the same argument. However, to make this law non-trivial, one must show that sufficiently quasi-stationary processes do in fact occur. In particular, one must show that processes exist for which the shear and expansion remain small, and in which no new generators are added to the horizon. Thorne, MacDonald, and Price considered related issues when an object falls across a d = 4 black hole horizon. By generalizing their argument to arbitrary d ≥ 3 and to any bifurcate Killing horizon, we derive a condition under which these effects are controlled and the first law applies. In particular, by providing a non-trivial first law for Rindler horizons, our work completes the parallel between the mechanics of such horizons and those of black holes for d ≥ 3. We also comment on the situation for d = 2.

We prove that the only four dimensional, stationary, rotating, asymptotically flat (analytic) vacuum black hole with a single degenerate horizon is given by the extremal Kerr solution. We also prove a similar uniqueness theorem for the extremal Kerr-Newman solution. This closes a longstanding gap in the black hole uniqueness theorems.

Boundary conditions for massive fermions are investigated in AdS d for d ≥ 2. For fermion masses in the range 0 ≤ |m| < 1/2ℓ with ℓ the AdS length, the standard notion of normalizeability allows a choice of boundary conditions. As in the case of scalars at or slightly above the BreitenlohnerFreedman (BF) bound, such boundary conditions correspond to multi-trace deformations of any CFT dual. By constructing appropriate boundary superfields, for d = 3, 4, 5 we identify joint scalar/fermion boundary conditions which preserve either N = 1 supersymmetry or N = 1 superconformal symmetry on the boundary. In particular, we identify boundary conditions corresponding via AdS/CFT (at large N ) to a 595-parameter family of double-trace marginal deformations of the low-energy theory of N M2-branes which preserve N = 1 superconformal symmetry. We also establish that (at large N and large 't Hooft coupling λ) there are no marginal or relevant multi-trace deformations of 3+1 N = 4 super Yang-Mills which preserve even N = 1 supersymmetry. Contents