No dynamics in the extremal Kerr throat

Abstract

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].

Three dimensional origin of AdS 2 quantum gravity

“…The (t, U ) term in the square brackets is locally AdS 2 with radius of curvature ℓ but the global structure is modified into a black hole geometry with horizon located at U = ǫ. The near extreme limit described here is the same considered in [32,33] (and a Kerr analogue of "Limit 2" in [22]). …”

Gravity waves from the Kerr/CFT correspondence

“…Two linearly independent solutions to the homogeneous radial equation (T ℓm = 0) are given by the Whittaker (i.e. confluent hypergeometric) functions [41]:…”

“…Two linearly independent solutions to the homogeneous radial equation (T ℓm = 0) are given by [41]: 19) where h is given by (3.15). These have simple behaviors at infinity and the horizon respectively: The corresponding solution of the radial equation (4.17) is given by:…”

Kerr/CFT, dipole theories and nonrelativistic CFTs

“…The first example is that of the Kerr/CFT correspondence [11] and its generalizations [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. In its generalized form, this correspondence explains the entropy of all extremal black holes (in four and five dimensions at least) in terms of state counting in a conformal field theory.…”