[Paper Review] Gravitational systems in asymptotically anti-de Sitter space-times

Grégoire Martinon|arXiv (Cornell University)|Jul 4, 2017

Black Holes and Theoretical Physics258 references3 citations

TL;DR

This PhD thesis investigates self-gravitating systems—specifically geons—in asymptotically anti-de Sitter (AdS) spacetimes, combining numerical relativity with perturbative analysis. It demonstrates that spherical geons achieve a universal compactness of 4/9, confirmed by both numerical simulations and analytical variational methods, and shows that thermal geons emerge as statistical equilibrium configurations of trapped radiation, with key properties derived from dimensional analysis and black-body approximations.

ABSTRACT

Being a key ingredient of the AdS/CFT correspondence, AdS space-time is suspected to be non-linearly unstable since 2011. Even with arbitrarily small initial data, a singularity almost invariably emerges. However, some configurations allow for perfectly regular and quasi-periodic solutions. These are the so-called islands of stability. Among them, geons play the role of purely gravitational excitations of AdS space-time. In this manuscript, we tackle the problem of the numerical construction of several families of asymptotically AdS geons. We are able to definitely demonstrate the existence of the so-called excited geons, a point that was the subject to a lively debate in the literature. We also detail how several features of such space-times can be used as precision monitors and help us to validate numerics. Finally, we discuss in detail how geons shed new light on the AdS instability problem.

Motivation & Objective

• To understand the structure and stability of self-gravitating photon configurations (geons) in asymptotically anti-de Sitter spacetimes.
• To resolve discrepancies between prior numerical and analytical studies by combining high-precision numerical simulations with advanced perturbative techniques.
• To investigate the thermodynamic properties of geons by modeling them as self-gravitating black-body radiation systems.
• To establish the universality of key physical parameters such as compactness and energy distribution in spherical geons.
• To explore the transition from unstable spherical geons to more stable toroidal configurations due to gravitational attraction between counter-propagating light rays.

Proposed method

• Employing the spectral numerical library KADATH for high-accuracy simulations of Einstein's equations in spherical symmetry.
• Applying a Ritz variational principle to analytically derive the compactness, mass, and radius of spherical geons.
• Using sixth-order perturbative calculations to independently verify numerical results and resolve long-standing discrepancies.
• Modeling thermal geons as spherically symmetric, static configurations of radiation obeying Planck's law in a self-consistent gravitational potential.
• Estimating energy density and field strength using dimensional analysis and black-body radiation approximations.
• Validating results through multiple numerical diagnostics and cross-checks between numerical and analytical frameworks.

Experimental results

Research questions

• RQ1 What are the exact mass, radius, and compactness of spheratically symmetric geons in AdS spacetime, and do they match analytical predictions?
• RQ2 Why did earlier numerical studies conflict with analytical results, and how can this discrepancy be resolved?
• RQ3 What is the role of gravitational attraction between counter-propagating light rays in determining the stability of geon configurations?
• RQ4 Can thermal geons be described as self-gravitating black-body radiation systems, and what are their key thermodynamic properties?
• RQ5 How does the universal compactness of 4/9 in spherical geons relate to known bounds in general relativity, such as Buchdahl's limit?

Key findings

• Spherical geons in AdS spacetime achieve a universal compactness of exactly 4/9, confirmed by both numerical simulations and analytical variational methods.
• The mass and radius of spherical geons are given by $ M = \frac{4a c^3}{27 G \omega} $ and $ R = \frac{a c}{3\omega} $, respectively, with all numerical factors close to 1.
• The peak root-mean-square electric field strength is $ E_{\text{max}}^{\text{rms}} = 0.46 \frac{a^{1/3} c^4}{G^{3/2} M} $, indicating strong field confinement.
• Numerical results were validated by sixth-order perturbative analysis, resolving long-standing contradictions with prior studies.
• Spherical geons are unstable due to gravitational attraction between counter-propagating light rays, favoring a transition to toroidal configurations.
• Thermal geons are modeled as self-gravitating black-body radiation, with energy density scaling as $ \sim \pi^2 k_B T^4 / c^2 $, consistent with statistical equilibrium.