Boundary Conditions for Kerr-AdS Perturbations

The Teukolsky master equation and its associated spin-weighted spheroidal harmonic decomposition simplify considerably the study of linear gravitational perturbations of the Kerr(-AdS) black hole. However, the formulation of the problem is not complete before we assign the physically relevant boundary conditions. We find a set of two Robin boundary conditions (BCs) that must be imposed on the Teukolsky master variables to get perturbations that are asymptotically global AdS, i.e. that asymptotes to the Einstein Static Universe. In the context of the AdS/CFT correspondence, these BCs allow a non-zero expectation value for the CFT stress-energy tensor while keeping fixed the boundary metric. When the rotation vanishes, we also find the gauge invariant differential map between the Teukolsky and the Kodama-Ishisbashi (Regge-Wheeler-Zerilli) formalisms. One of our Robin BCs maps to the scalar sector and the other to the vector sector of the Kodama-Ishisbashi decomposition. The Robin BCs on the Teukolsky variables will allow for a quantitative study of instability timescales and quasinormal mode spectrum of the Kerr-AdS black hole. As a warm-up for this programme, we use the Teukolsky formalism to recover the quasinormal mode spectrum of global AdS-Schwarzschild, complementing previous analysis in the literature.Comment: 33 pages, 6 figure

AdS nonlinear instability: moving beyond spherical symmetry

Anti-de Sitter (AdS) is conjectured to be nonlinear unstable to a weakly turbulent mechanism that develops a cascade towards high frequencies, leading to black hole formation [1,2]. We give evidence that the gravitational sector of perturbations behaves differently from the scalar one studied in [2]. In contrast with [2], we find that not all gravitational normal modes of AdS can be nonlinearly extended into periodic horizonless smooth solutions of the Einstein equation. In particular, we show that even seeds with a single normal mode can develop secular resonances, unlike the spherically symmetric scalar field collapse studied in [2]. Moreover, if the seed has two normal modes, more than one resonance can be generated at third order, unlike the spherical collapse of [2]. We also show that weak turbulent perturbative theory predicts the existence of direct and inverse cascades, with the former dominating the latter for equal energy two-mode seeds.Comment: 7 pages, no figures, 2 table

Localised AdS5×S5\bf{AdS_5\times S^5} Black Holes

We numerically construct asymptotically global $\mathrm{AdS}_5\times \mathrm{S}^5$ black holes that are localised on the $\mathrm{S}^5$. These are solutions to type IIB supergravity with $\mathrm S^8$ horizon topology that dominate the theory in the microcanonical ensemble at small energies. At higher energies, there is a first-order phase transition to $\mathrm{AdS}_5$-Schwarzschild$\times \mathrm{S}^5$. By the AdS/CFT correspondence, this transition is dual to spontaneously breaking the $SO(6)$ R-symmetry of $\mathcal N=4$ super Yang-Mills down to $SO(5)$. We extrapolate the location of this phase transition and compute the expectation value of the resulting scalar operators in the low energy phase.Comment: 11 pages, 6 figure

Lumpy AdS5×\bf{_5\times} S5\bf{^5} Black Holes and Black Belts

Sufficiently small Schwarzschild black holes in global AdS$_5\times$S$^5$ are Gregory-Laflamme unstable. We construct new families of black hole solutions that bifurcate from the onset of this instability and break the full SO$(6)$ symmetry group of the S$^5$ down to SO$(5)$. These new "lumpy" solutions are labelled by the harmonics $\ell$. We find evidence that the $\ell = 1$ branch never dominates the microcanonical/canonical ensembles and connects through a topology-changing merger to a localised black hole solution with S$^8$ topology. We argue that these S$^8$ black holes should become the dominant phase in the microcanonical ensemble for small enough energies, and that the transition to Schwarzschild black holes is first order. Furthermore, we find two branches of solutions with $\ell = 2$. We expect one of these branches to connect to a solution containing two localised black holes, while the other branch connects to a black hole solution with horizon topology $\mathrm S^4\times\mathrm S^4$ which we call a "black belt".Comment: 20 pages (plus 17 pages for Appendix on Kaluza-Klein Holography), 14 figure

Discrete harmonic analysis associated with ultraspherical expansions

We study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted l^p-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by certain difference operator. We also prove weighted l^p-boundedness properties of transplantation operators associated to the system of ultraspherical functions. In order to show our results we previously establish a vector-valued local Calder\'on-Zygmund theorem in our discrete setting

Conical square functions associated with Bessel, Laguerre and Schr\"odinger operators in UMD Banach spaces

In this paper we consider conical square functions in the Bessel, Laguerre and Schr\"odinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hyt\"onen, van Neerven and Portal, in order to define our conical square functions, we use $\gamma$-radonifying operators. We obtain new equivalent norms in the Lebesgue-Bochner spaces $L^p((0,\infty ),\mathbb{B})$ and $L^p(\mathbb{R}^n,\mathbb{B})$, $1<p<\infty$, in terms of our square functions, provided that $\mathbb{B}$ is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions

Riesz transforms, Cauchy-Riemann systems and amalgam Hardy spaces

In this paper we study Hardy spaces $\mathcal{H}^{p,q}(\mathbb{R}^d)$, $0<p,q<\infty$, modeled over amalgam spaces $(L^p,\ell^q)(\mathbb{R}^d)$. We characterize $\mathcal{H}^{p,q}(\mathbb{R}^d)$ by using first order classical Riesz transforms and compositions of first order Riesz transforms depending on the values of the exponents $p$ and $q$. Also, we describe the distributions in $\mathcal{H}^{p,q}(\mathbb{R}^d)$ as the boundary values of solutions of harmonic and caloric Cauchy-Riemann systems. We remark that caloric Cauchy-Riemann systems involve fractional derivative in the time variable. Finally we characterize the functions in $L^2(\mathbb{R}^d) \cap \mathcal{H}^{p,q}(\mathbb{R}^d)$ by means of Fourier multipliers $m_\theta$ with symbol $\theta(\cdot/|\cdot|)$, where $\theta \in C^\infty(\mathbb{S}^{d-1})$ and $\mathbb{S}^{d-1}$ denotes the unit sphere in $\mathbb{R}^d$.Comment: 24 page

Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions

We consider the Weinstein type equation $\mathcal{L}_\lambda u=0$ on $(0,\infty )\times (0,\infty )$, where $\mathcal{L}_\lambda=\partial _t^2+\partial _x^2-\frac{\lambda (\lambda -1)}{x^2}$, with $\lambda >1$. In this paper we characterize the solutions of $\mathcal{L}_\lambda u=0$ on $(0,\infty )\times(0,\infty )$ representable by Bessel-Poisson integrals of BMO-functions as those ones satisfying certain Carleson properties

Motion of buoyant particles and coarsening of solid-liquid mixtures in a random acceleration field

Flow induced by a random acceleration field (g-jitter) is considered in two related situations that are of interest for microgravity fluid experiments: the random motion of an isolated buoyant particle and coarsening of a solid-liquid mixture. We start by analyzing in detail actual accelerometer data gathered during a recent microgravity mission, and obtain the values of the parameters defining a previously introduced stochastic model of this acceleration field. We then study the motion of a solid particle suspended in an incompressible fluid that is subjected to such random accelerations. The displacement of the particle is shown to have a diffusive component if the correlation time of the stochastic acceleration is finite or zero, and mean squared velocities and effective diffusion coefficients are obtained explicitly. Finally, the effect of g-jitter on coarsening of a solid-liquid mixture is considered. Corrections due to the induced fluid motion are calculated, and estimates are given for coarsening of Sn-rich particles in a Sn-Pb eutectic fluid, experiment to be conducted in microgravity in the near future.Comment: 25 pages, 4 figures (included). Also at http://www.scri.fsu.edu/~vinals/ross2.p