A set of infinitesimal Virasoro L ⊗ Virasoro R diffeomorphisms are presented which act non-trivially on the horizon of a generic Kerr black hole with spin J. The covariant phase space formalism provides a formula for the Virasoro charges as surface integrals on the horizon. Integrability and associativity of the charge algebra are shown to require the inclusion of ‘Wald-Zoupas’ counterterms. A counterterm satisfying the known consistency requirement is constructed and yields central charges cL = cR = 12J. Assuming the existence of a quantum Hilbert space on which these charges generate the symmetries, as well as the applicability of the Cardy formula, the central charges reproduce the macroscopic area-entropy law for generic Kerr black holes.

/pdf/black-hole-entropy-and-soft-hair-4lchcg1zx6.pdf

Black hole entropy and soft hair

We explore the spacetime structure near nonextremal horizons in any spacetime dimension greater than two and discover a wealth of novel results: (i) Different boundary conditions are specified by a functional of the dynamical variables, describing inequivalent interactions at the horizon with a thermal bath. (ii) The near horizon algebra of a set of boundary conditions, labeled by a parameter $s$, is given by the semidirect sum of diffeomorphisms at the horizon with ``spin-$s$ supertranslations.'' For $s=1$ we obtain the first explicit near horizon realization of the Bondi-Metzner-Sachs algebra. (iii) For another choice, we find a nonlinear extension of the Heisenberg algebra, generalizing recent results in three spacetime dimensions. This algebra allows us to recover the aforementioned (linear) ones as composites. (iv) These examples allow us to equip not only black holes, but also cosmological horizons with soft hair. We also discuss implications of soft hair for black hole thermodynamics and entropy.

https://link.aps.org/pdf/10.1103/PhysRevLett.124.041601

Spacetime Structure near Generic Horizons and Soft Hair

We consider on the bulk side extensions of the Sachdev-Ye-Kitaev (SYK) model to Yang-Mills and higher spins. To this end we study generalizations of the Jackiw-Teitelboim (JT) model in the BF formulation. Our main goal is to obtain generalizations of the Schwarzian action, which we achieve in two ways: by considering the on-shell action supplemented by suitable boundary terms compatible with all symmetries, and by applying the Lee-Wald-Zoupas formalism to analyze the symplectic structure of dilaton gravity. We conclude with a discussion of the entropy (including log-corrections from higher spins) and a holographic dictionary for the generalized SYK/JT correspondence.

/pdf/towards-a-bulk-description-of-higher-spin-syk-3gpi86xfn8.pdf

Towards a bulk description of higher spin SYK

A set of infinitesimal ${\rm Virasoro_{\,L}}\otimes{\rm Virasoro_{\,R}}$ diffeomorphisms are presented which act non-trivially on the horizon of a generic Kerr black hole with spin J. The covariant phase space formalism provides a formula for the Virasoro charges as surface integrals on the horizon. Integrability and associativity of the charge algebra are shown to require the inclusion of `Wald-Zoupas' counterterms. A counterterm satisfying the known consistency requirement is constructed and yields central charges $c_L=c_R=12J$. Assuming the existence of a quantum Hilbert space on which these charges generate the symmetries, as well as the applicability of the Cardy formula, the central charges reproduce the macroscopic area-entropy law for generic Kerr black holes.

/pdf/black-hole-entropy-and-soft-hair-2cq7ol5buc.pdf

Black Hole Entropy and Soft Hair

The advent of gravitational waves and black hole imaging has opened a new window into probing the horizon scale of black holes. An important question is whether string theory results for black holes can predict interesting and observable features that current and future experiments can probe. In this article I review the budding and exciting research being done on understanding the possibilities of observing signals from fuzzballs, where black holes are replaced by string-theoretic horizon-scale microstructure. In order to be accessible to both string theorists and black hole phenomenologists, I give a brief overview of the relevant observational experiments as well as the fuzzball paradigm in string theory and its explicitly constructable solutions called microstate geometries.

Fuzzballs and observations

Out-of-time-order correlators (OTOCs) that capture maximally chaotic properties of a black hole are determined by scattering processes near the horizon. This prompts the question to what extent OTOCs display chaotic behaviour in horizonless microstate geometries. This question is complicated by the fact that Lyapunov growth of OTOCs requires nonzero temperature, whereas constructions of microstate geometries have been mostly restricted to extremal black holes. In this paper, we compute OTOCs for a class of extremal black holes, namely maximally rotating BTZ black holes, and show that on average they display “slow scrambling”, characterized by cubic (rather than exponential) growth. Superposed on this average power-law growth is a sawtooth pattern, whose steep parts correspond to brief periods of Lyapunov growth associated to the nonzero temperature of the right-moving degrees of freedom in a dual conformal field theory. Next we study the extent to which these OTOCs are modified in certain “superstrata”, horizonless microstate geometries corresponding to these black holes. Rather than an infinite throat ending on a horizon, these geometries have a very deep but finite throat ending in a cap. We find that the superstrata display the same slow scrambling as maximally rotating BTZ black holes, except that for large enough time intervals the growth of the OTOC is cut off by effects related to the cap region, some of which we evaluate explicitly.

/pdf/slow-scrambling-in-extremal-btz-and-microstate-geometries-3mgd65ise4.pdf

Slow scrambling in extremal BTZ and microstate geometries

We reconsider warped black hole solutions in topologically massive gravity and find novel boundary conditions that allow for soft hairy excitations on the horizon. To compute the associated symmetry algebra we develop a general framework to compute asymptotic symmetries in any Chern-Simons-like theory of gravity. We use this to show that the near horizon symmetry algebra consists of two $$ \mathfrak{u} $$
 (1) current algebras and recover the surprisingly simple entropy formula S = 2π(J
 0
 +
  + J
 0
 −
 ), where J
 0
 ±
 are zero mode charges of the current algebras. This provides the first example of a locally non-maximally symmetric configuration exhibiting this entropy law and thus non-trivial evidence for its universality.

/pdf/soft-hairy-warped-black-hole-entropy-1pq9pb1ihn.pdf

Soft hairy warped black hole entropy

Out-of-time-order correlators (OTOCs) that capture maximally chaotic properties of a black hole are determined by scattering processes near the horizon. This prompts the question to what extent OTOCs display chaotic behaviour in horizonless microstate geometries. This question is complicated by the fact that Lyapunov growth of OTOCs requires nonzero temperature, whereas constructions of microstate geometries have been mostly restricted to extremal black holes. 
In this paper, we compute OTOCs for a class of extremal black holes, namely maximally rotating BTZ black holes, and show that on average they display "slow scrambling", characterized by cubic (rather than exponential) growth. Superposed on this average power-law growth is a sawtooth pattern, whose steep parts correspond to brief periods of Lyapunov growth associated to the nonzero temperature of the right-moving degrees of freedom in a dual conformal field theory. 
Next we study the extent to which these OTOCs are modified in certain "superstrata", horizonless microstate geometries corresponding to these black holes. Rather than an infinite throat ending on a horizon, these geometries have a very deep but finite throat ending in a cap. We find that the superstrata display the same slow scrambling as maximally rotating BTZ black holes, except that for large enough time intervals the growth of the OTOC is cut off by effects related to the cap region, some of which we evaluate explicitly.

/pdf/slow-scrambling-in-extremal-btz-and-microstate-geometries-4f1dd5og60.pdf

We address the difference between integrable and chaotic motion in
quantum theory as manifested by the complexity of the corresponding
evolution operators. Complexity is understood here as the shortest
geodesic distance between the time-dependent evolution operator and the
origin within the group of unitaries. (An appropriate
'complexity metric' must be used that takes into account the relative difficulty of performing 'nonlocal’
operations that act on many degrees of freedom at once.) While simply
formulated and geometrically attractive, this notion of complexity is
numerically intractable save for toy models with Hilbert spaces of very
low dimensions. To bypass this difficulty, we trade the exact definition
in terms of geodesics for an upper bound on complexity, obtained by
minimizing the distance over an explicitly prescribed infinite set of
curves, rather than over all possible curves. Identifying this upper
bound turns out equivalent to the closest vector problem (CVP)
previously studied in integer optimization theory, in particular, in
relation to lattice-based cryptography. Effective approximate algorithms
are hence provided by the existing mathematical considerations, and they
can be utilized in our analysis of the upper bounds on quantum evolution
complexity. The resulting algorithmically implemented complexity bound
systematically assigns lower values to integrable than to chaotic
systems, as we demonstrate by explicit numerical work for Hilbert spaces
of dimensions up to \sim 10^4∼104.

Bounds on quantum evolution complexity via lattice cryptography

We reconsider warped black hole solutions in topologically massive gravity and find novel boundary conditions that allow for soft hairy excitations on the horizon. To compute the associated symmetry algebra we develop a general framework to compute asymptotic symmetries in any Chern-Simons-like theory of gravity. We use this to show that the near horizon symmetry algebra consists of two u(1) current algebras and recover the surprisingly simple entropy formula $S=2\pi (J_0^+ + J_0^-)$, where $J_0^\pm$ are zero mode charges of the current algebras. This provides the first example of a locally non-maximally symmetric configuration exhibiting this entropy law and thus non-trivial evidence for its universality.

/pdf/soft-hairy-warped-black-hole-entropy-2ghx17fshj.pdf

Philip Hacker

Papers

Slow scrambling in extremal BTZ and microstate geometries

Soft hairy warped black hole entropy

Slow scrambling in extremal BTZ and microstate geometries

Bounds on quantum evolution complexity via lattice cryptography

Soft hairy warped black hole entropy