arXiv subjects
Publications and source records attributed to Toby Wiseman.
This book chapter gives an introductory review of a numerical framework for finding static and stationary vacuum black hole solutions. Such methods may be applied to explore the exotic space of higher dimensional black holes that are thought to exist.
The Harmonic Einstein equation is the vacuum Einstein equation supplemented by a gauge fixing term which we take to be that of DeTurck. For static black holes analytically continued to Riemannian manifolds without boundary at the horizon this equation has previously been shown to be elliptic, and Ricci flow and Newton's method provide good numerical algorithms to solve it. Here we extend these techniques to the arbitrary cohomogeneity stationary case which must be treated in Lorentzian signature. For stationary spacetimes with globally timelike Killing vector the Harmonic Einstein equation is elliptic. In the presence of horizons and ergo-regions it is less obviously so. Motivated by the Rigidity theorem we study a class of stationary black hole spacetimes, considered previously by Harmark, general enough to include the asymptotically flat case in higher dimensions. We argue the Harmonic Einstein equation consistently truncates to this class of spacetimes giving an elliptic problem. The Killing horizons and axes of rotational symmetry are boundaries for this problem and we determine boundary conditions there. As a simple example we numerically construct 4D rotating black holes in a cavity using Anderson's boundary conditions. We demonstrate both Newton's method and Ricci flow to find these Lorentzian solutions.
We show how to construct low energy solutions to the Randall Sundrum II (RSII) model using an associated AdS_5-CFT_4 problem. The RSII solution is given in terms of a perturbation of the AdS-CFT solution, with the perturbation parameter being the radius of curvature of the brane metric compared to the AdS length \ell. The brane metric is then a specific perturbation of the AdS-CFT boundary metric. For low curvatures the RSII solution reproduces 4d GR on the brane. The leading correction is from local higher derivative curvature terms. The subleading correction is derived from similar terms and also the dual CFT stress tensor. Recently AdS-CFT solutions with 4d Schwarzschild boundary metric were numerically constructed. We modify the boundary conditions to introduce the RSII brane, and use elliptic numerical methods to solve the resulting boundary value problem. We construct large RSII static black holes with radius up to ~ 20 \ell. For large radius the RSII solutions are indeed close to the associated AdS-CFT solution. In this case the local curvature corrections vanish, and we confirm the leading correction is given by the AdS-CFT solution stress tensor. We also follow the black holes to small radius << \ell, where as expected they transition to a 5d behaviour. Our numerical solutions indicate the RSII black holes are dynamically stable for axisymmetric perturbations for all radii.
The elliptic Einstein-DeTurck equation may be used to numerically find Einstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics are considered by analytically continuing to Euclidean time. Ricci-DeTurck flow is a constructive algorithm to solve this equation, and is simple to implement when the solution is a stable fixed point, the only complication being that Ricci solitons may exist which are not Einstein. Here we extend previous work to consider the Einstein-DeTurck equation for Riemannian manifolds with boundaries, and those that continue to static Lorentzian spacetimes which are asymptotically flat, Kaluza-Klein, locally AdS or have extremal horizons. Using a maximum principle we prove that Ricci solitons do not exist in these cases and so any solution is Einstein. We also argue that Ricci-DeTurck flow preserves these classes of manifolds. As an example we simulate Ricci-DeTurck flow for a manifold with asymptotics relevant for AdS_5/CFT_4. Our maximum principle dictates there are no soliton solutions, and we give strong numerical evidence that there exists a stable fixed point of the flow which continues to a smooth static Lorentzian Einstein metric. Our asymptotics are such that this describes the classical gravity dual relevant for the CFT on a Schwarzschild background in either the Unruh or Boulware vacua. It determines the leading O(N^2) part of the CFT stress tensor, which interestingly is regular on both the future and past Schwarzschild horizons.
We model the large scale late time universe as a Lambda-CDM cosmology driven by cosmological constant and perfect dust fluid. Our aim is to find new solutions in the matter and Lambda epoch consistent with inflationary initial conditions, namely that to the far past in the matter era the cosmology tends to a flat FLRW solution. We identify the moduli degrees of freedom that parametrize the flat Lambda-dust FLRW solution and then promote these moduli to slowly varying functions of the spatial coordinates and show how to solve the Einstein equations in a comoving gradient expansion, controlled by the cosmological constant length scale. Our initial conditions ensure that the approximation remains under control to the far past of the matter era, and to the far future of Lambda domination. The solution is fully non-perturbative in the amplitude of the metric deformation, and we explicitly construct it to fourth order in derivatives. A general Lambda-dust universe dominated by Lambda in the future is characterized by a 3-metric and a stress tensor (with positive trace) defined on the future conformal boundary. The new cosmologies with inflationary initial conditions are characterized only by the boundary 3-metric, the stress tensor being locally determined entirely in terms of that metric.
We study the black hole - black string phase transitions of gravitational theories compactified on a circle using the holographic duality conjecture. The gauge theory duals of these theories are maximally supersymmetric and strongly coupled 1 + 1 dimensional SU(N) Yang-Mills theories compactified on a circle, in the large $N$ limit. We perform the strongly coupled finite temperature gauge theory calculations on a lattice, using the recently developed exact lattice supersymmetry methods based on topological twisting and orbifolding. The spatial Polyakov line serves as relevant order parameter of the confinement - deconfinement phase transitions in the gauge theory duals.
Assuming homogeneous isotropic Lambda-CDM cosmology allows Lambda, spatial curvature and dark matter density to be inferred from large scale structure observations such as supernovae. The purpose of this paper is to extend this to allow observations to measure or constrain inhomogeneity and anisotropy. We obtain the general inhomogeneous anisotropic Lambda-CDM solution which is locally asymptotic to an expanding de Sitter solution as a late time expansion using Starobinsky's method (analogous to the `holographic renormalization' technique in AdS/CFT) together with a resummation of the series. The dark matter is modeled as perfect dust fluid. The terms in the expansion systematically describe inhomogeneous and anisotropic deformations of an expanding FLRW solution, and are given as a spatial derivative expansion in terms of data characterizing the solution - a 3-metric and a perturbation of that 3-metric. Leading terms describe inhomogeneity and anisotropy on the scale set by the cosmological constant, approximately the horizon scale today. Higher terms in the expansion describe shorter scale variations. We compute the luminosity distance-redshift relation and argue that comparison with current and future observation would allow a partial reconstruction of the characterizing data. We also comment on smoothing these solutions noting that geometric flows (such as Ricci flow) applied to the characterizing data provide a canonical averaging method.
We present a consistent Kaluza-Klein truncation of D=11 supergravity on an arbitrary seven-dimensional Sasaki-Einstein space (SE_7) to a D=4 theory containing a metric, a gauge-field, a complex scalar field and a real scalar field. We use this D=4 theory to construct various black hole solutions that describe the thermodynamics of the d=3 CFTs dual to skew-whiffed AdS_4 X SE_7 solutions. We show that these CFTs have a rich phase diagram, including holographic superconductivity with, generically, broken parity and time reversal invariance. At zero temperature the superconducting solutions are charged domain walls with a universal emergent conformal symmetry in the far infrared.
We propose a general framework for solving the Einstein equation for static and Euclidean metrics. First, we address the issue of gauge-fixing by borrowing from the Ricci-flow literature the so-called DeTurck trick, which renders the Einstein equation strictly elliptic and generalizes the usual harmonic-coordinate gauge. We then study two algorithms, Ricci-flow and Newton's method, for solving the resulting Einstein-DeTurck equation. We illustrate the use of these methods by studying localized black holes and non-uniform black strings in five-dimensional Kaluza-Klein theory, improving on previous calculations of their thermodynamic and geometric properties. We study spectra of various operators for these solutions, in particular finding negative modes of the Lichnerowicz operator. We classify the localized solutions into two branches that meet at a minimum temperature. We find good evidence for a merger between the localized and non-uniform solutions. We also find a narrow window of localized solutions that possess negative modes yet appear to have positive specific heat.
Using seven-dimensional Sasaki-Einstein spaces we construct solutions of D=11 supergravity that are holographically dual to superconductors in three spacetime dimensions. Our numerical results indicate a new zero temperature solution dual to a quantum critical point.
We perform lattice simulations of N D0-branes at finite temperature in the decoupling limit, namely 16 supercharge SU(N) Yang-Mills quantum mechanics in the 't Hooft limit. At low temperature this theory is conjectured to be dual to certain supergravity black holes. We emphasize that the existence of a non-compact moduli space renders the partition function of the quantum mechanics theory divergent, and we perform one loop calculations that demonstrate this explicitly. In consequence we use a scalar mass term to regulate this divergence and argue that the dual black hole thermodynamics may be recovered in the appropriate large N limit as the regulator is removed. We report on simulations for N up to 5 including the Pfaffian phase, and N up to 12 in the phase quenched approximation. Interestingly, in the former case, where we may calculate this potentially difficult phase, we find that it appears to play little role dynamically over the temperature range tested, which is certainly encouraging for future simulations of this theory.
TeVeS uses a dynamical vector field with timelike unit norm constraint to specify a preferred local frame. When matter moves slowly in this frame - the so-called quasi-static regime - Modified Newtonian Dynamics (MoND) results. Theories with such vectors (such as Einstein-aether) are prone to the vector dynamics forming singularities which render their classical evolution problematic. Here we analyse the dynamics of the vector in TeVeS in various situations. We find that, quite generically, the vector field develops caustic singularities on time scales of order the gravitational in-fall time. Having shown singularity formation is generic with or without matter, Bekenstein's original formulation of TeVeS appears dynamically problematic. We argue that by modifying the vector field kinetic terms to the more general form used by Einstein-Aether this problem may be avoided.
We consider holographic renormalization for the decoupling limit of coincident Dp-branes. We truncate the theory to the supergravity sector which is homogeneous on the (8-p)-sphere and carries only RR electric (p+2)-flux, leaving a graviton and two scalar degrees of freedom associated to the dilaton and the sphere radius. We non-linearly construct the asymptotic graviton and dilaton deformations - the analog of the Graham-Fefferman expansion for AdS/CFT - and compute counterterms to give a finite renormalized bulk action and dual one point functions. Restricting to linear deformations we find additional counterterms to include the remaining sphere deformations which strongly deform the asymptotic behaviour.
We report on lattice simulations of 16 supercharge SU(N) Yang-Mills quantum mechanics in the 't Hooft limit. Maldacena duality conjectures that in this limit the theory is dual to IIA string theory, and in particular that the behavior of the thermal theory at low temperature is equivalent to that of certain black holes in IIA supergravity. Our simulations probe the low temperature regime for N <= 5 and the intermediate and high temperature regimes for N <= 12. We observe 't Hooft scaling and at low temperatures our results are consistent with the dual black hole prediction. The intermediate temperature range is dual to the Horowitz-Polchinski correspondence region, and our results are consistent with smooth behavior there. We include the Pfaffian phase arising from the fermions in our calculations where appropriate.
We describe the conjectured holographic duality between Yang-Mills quantum mechanics and type IIa string theory. This duality allows us to use lattice Monte Carlo simulations to probe the physics of the gravitational theory - for example, at low energies it provides a computation of black hole entropy in terms of a sum over microstates of the dual gauge theory. Numerical results are presented of the 4 supercharge theory at finite temperature
A generalization of the AdS/CFT conjecture postulates a duality between IIA string theory and 16 supercharge Yang-Mills quantum mechanics in the large N 't Hooft limit. At low temperatures string theory describes black holes, whose thermodynamics may hence be studied using the dual quantum mechanics. This quantum mechanics is strongly coupled which motivates the use of lattice techniques. We argue that, contrary to expectation, the theory when discretized naively will nevertheless recover continuum supersymmetry as the lattice spacing is sent to zero. We test these ideas by studying the 4 supercharge version of this Yang-Mills quantum mechanics in the 't Hooft limit. We use both a naive lattice action and a manifestly supersymmetric action. Using Monte Carlo methods we simulate the Euclidean theories, and study the lattice continuum limit, for both thermal and non-thermal periodic boundary conditions, confirming continuum supersymmetry is recovered for the naive action when appropriate. We obtain results for the thermal system with N up to 12. These favor the existence of a single deconfined phase for all non-zero temperatures. These results are an encouraging indication that the 16 supercharge theory is within reach using similar methods and resources.
The second del Pezzo surface is known by work of Tian-Zhu and Wang-Zhu to admit a unique Kaehler-Ricci soliton. Applying a method described in hep-th/0703057, we use Ricci flow to numerically compute that soliton metric. We numerically compute the value of its Perelman entropy (or Gaussian density).
Based on holographic arguments Tanaka and Emparan et al have claimed that large localized static black holes do not exist in the one-brane Randall-Sundrum model. If such black holes are time-dependent as they propose, there are potentially significant phenomenological and theoretical consequences. We revisit the issue, arguing that their reasoning does not take into account the strongly coupled nature of the holographic theory. We claim that static black holes with smooth metrics should indeed exist in these theories, and give a simple example. However, although the existence of such solutions is relevant to exact and numerical solution searches, such static solutions might be dynamically unstable, again leading to time dependence with phenomenological consequences. We explore a plausible instability, suggested by Tanaka, analogous to that of Gregory and Laflamme, but argue that there is no reliable reason at this point to assume it must exist.