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Negative modes and the thermodynamics of Reissner-Nordström black holes
We analyse the problem of negative modes of the Euclidean section of the
Reissner-Nordstr\"om black hole in four dimensions. We find analytically that a
negative mode disappears when the specific heat at constant charge becomes
positive. The sector of perturbations analysed here is included in the
canonical partition function of the magnetically charged black hole. The result
obeys the usual rule that the partition function is only well-defined when
there is local thermodynamical equilibrium. We point out the difficulty in
quantising Einstein-Maxwell theory, where the so-called conformal factor
problem is considerably more intricate. Our method, inspired by hep-th/0608001,
allows us to decouple the divergent gauge volume and treat the metric
perturbations sector in a gauge-invariant way
Black funnels
The Hartle-Hawking state of SYM at strong coupling and large
on a fixed black hole background has two proposed gravitational duals: a
black funnel or a black droplet. We construct the black funnel solutions that
are dual to the Hartle-Hawking state on a Schwarzschild black hole and on a
class of three-dimensional asymptotically flat black hole backgrounds. We
compute their holographic stress tensor and argue for the stability of these
solutions
Rotating black droplet
We construct the gravitational dual, in the Unruh state, of the "jammed"
phase of a CFT at strong coupling and infinite N on a fixed five-dimensional
rotating Myers-Perry black hole with equal angular momenta. When the angular
momenta are all zero, the solution corresponds to the five-dimensional
generalization of the solution first studied by Figueras, Lucietti, and
Wiseman. In the extremal limit, when the angular momenta of the Myers-Perry
black hole are maximum, the Unruh, Boulware and Hartle-Hawking states
degenerate. We give a detailed analysis of the corresponding holographic stress
energy tensor for all values of the angular momenta, finding it to be regular
at the horizon in all cases. We compare our results with existent literature on
thermal states of free field theories on black hole backgrounds
Non-axisymmetric instability of rotating black holes in higher dimensions
We calculate the scalar-gravitational quasi-normal modes of equal angular
momenta Myers-Perry black holes in odd dimensions. We find a new bar-mode
(non-axisymmetric) classical instability for . These black holes were
previously found to be unstable to axisymmetric perturbations for spins very
near extremality. The bar-mode instability we find sets in at much slower
spins, and is therefore the dominant instability of these black holes. This
instability has important consequences for the phase diagram of black holes in
higher dimensions
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AdS nonlinear instability: Breaking spherical and axial symmetries
Considerable effort has been dedicated to study the nonlinear instability of
Anti-de Sitter (AdS) within spherical symmetry, but little is known about this
nonlinear instability in the purely gravitational sector, where spherical
symmetry is necessarily broken. In \cite{Bizon:2011gg} the onset of such
nonlinear instability was associated with the existence of irremovable secular
resonances at third order in perturbation theory. Furthermore, it was also
conjectured in \cite{Bizon:2011gg} that certain very fine tuned initial data
would not collapse. Such solutions, upon linearisation, correspond to
individual normal modes of AdS, which can be consistently backreacted to all
orders in perturbation theory. However, the analysis of \cite{Bizon:2011gg} was
restricted to spherical symmetry. The perturbative arguments of
\cite{Bizon:2011gg} were then generalised to gravitational perturbations in
\cite{Dias:2011ss}, and in particular certain time-periodic solutions were also
conjectured to exist - these were coined geons. However, in \cite{Dias:2011ss},
only a certain class of perturbations was considered, for which the
perturbative analysis considerably simplifies. In this manuscript we present
details of the systematic computational formalism and an exhaustive and
complementary analysis of physical properties of the geons and gravitational
AdS instability that were absent in our companion Letter \cite{Dias:2016ewl}.
In particular, we find that, unlike in spherical symmetry, a (single)
gravitational normal mode of AdS can be backreacted to generate a nonlinear
solution only in very exceptional circumstances. We also show that weak
turbulent perturbative theory predicts the existence of direct and inverse
cascades, and give evidence suggesting that the former dominates the latter for
equal energy two-mode seeds
AdS Euclidean wormholes
We explore the construction and stability of asymptotically anti-de Sitter
Euclidean wormholes in a variety of models. In simple ad hoc low-energy models,
it is not hard to construct two-boundary Euclidean wormholes that dominate over
disconnected solutions and which are stable (lacking negative modes) in the
usual sense of Euclidean quantum gravity. Indeed, the structure of such
solutions turns out to strongly resemble that of the Hawking-Page phase
transition for AdS-Schwarzschild black holes, in that for boundary sources
above some threshold we find both a `large' and a `small' branch of wormhole
solutions with the latter being stable and dominating over the disconnected
solution for large enough sources. We are also able to construct two-boundary
Euclidean wormholes in a variety of string compactifications that dominate over
the disconnected solutions we find and that are stable with respect to
field-theoretic perturbations. However, as in classic examples investigated by
Maldacena and Maoz, the wormholes in these UV-complete settings always suffer
from brane-nucleation instabilities (even when sources that one might hope
would stabilize such instabilities are tuned to large values). This indicates
the existence of additional disconnected solutions with lower action. We
discuss the significance of such results for the factorization problem of
AdS/CFT
Black droplets
Black droplets and black funnels are gravitational duals to states of a large N, strongly coupled CFT on a fixed black hole background. We numerically construct black droplets corresponding to a CFT on a Schwarzchild background with finite asymptotic temperature. We find two branches of such droplet solutions which meet at a turning point. Our results suggest that the equilibrium black droplet solution does not exist, which would imply that the Hartle-Hawking state in this system is dual to the black funnel constructed in [1]. We also compute the holographic stress energy tensor and match its asymptotic behaviour to perturbation theory. © 2014 The Author(s).J.E.S.’s work is partially supported by the John Templeton Foundation. B.W. was supported by European Research Council grant no. ERC-2011-StG 279363-HiDGR
Optical conductivity with holographic lattices
We add a gravitational background lattice to the simplest holographic model
of matter at finite density and calculate the optical conductivity. With the
lattice, the zero frequency delta function found in previous calculations
(resulting from translation invariance) is broadened and the DC conductivity is
finite. The optical conductivity exhibits a Drude peak with a cross-over to
power-law behavior at higher frequencies. Surprisingly, these results bear a
strong resemblance to the properties of some of the cuprates
Numerical methods for finding stationary gravitational solutions
The wide applications of higher dimensional gravity and gauge/gravity duality have fuelled the search for new stationary solutions of the Einstein equation (possibly coupled to matter). In this topical review, we explain the mathematical foundations and give a practical guide for the numerical solution of gravitational boundary value problems. We present these methods by way of example: resolving asymptotically flat black rings, singly spinning lumpy black holes in anti-de Sitter (AdS), and the Gregory-Laflamme zero modes of small rotating black holes in AdS. We also include several tools and tricks that have been useful throughout the literature
Black holes with a single Killing vector field: black resonators
We numerically construct asymptotically anti-de Sitter (AdS) black holes in
four dimensions that contain only a single Killing vector field. These
solutions, which we coin black resonators, link the superradiant instability of
Kerr-AdS to the nonlinear weakly turbulent instability of AdS by connecting the
onset of the superradiance instability to smooth, horizonless geometries called
geons. Furthermore, they demonstrate non-uniqueness of Kerr-AdS by sharing
asymptotic charges. Where black resonators coexist with Kerr-AdS, we find that
the black resonators have higher entropy. Nevertheless, we show that black
resonators are unstable and comment on the implications for the endpoint of the
superradiant instability
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