Higher order perturbations of Anti-de Sitter space and time-periodic solutions of vacuum Einstein equations

Abstract

Motivated by the problem of stability of Anti-de Sitter (AdS) spacetime, we discuss nonlinear gravitational perturbations of maximally symmetric solutions of vacuum Einstein equations in general and the case of AdS in particular. We present the evidence that, similarly to the self-gravitating scalar field at spherical symmetry, the negative cosmological constant allows for the existence of globally regular, asymptotically AdS, time-periodic solutions of vacuum Einstein equations whose frequencies bifurcate from linear eigenfrequencies of AdS. Interestingly, our preliminary results indicate that the number of one parameter families of time-periodic solutions bifurcating from a given eigenfrequency equals the multiplicity of this eigenfrequency.

16 pages, v2: minor changes / corrections in text and equations; in particular missing $\theta$ dependence in eqs. (110,111,114,117,120) added (thanks to Gyula Fodor for pointing it out!), misprints in eqs. (28,55) corrected, implicit limits in the integrals in eqs. (71-73,77-81,95-97) made explicit, misplaced Lorentz indices in eqs. (17-19) lowered, 2 references added

References in corpus (6)

• Renormalization group, secular term resummation and AdS (in)stability
• Renormalization, averaging, conservation laws and AdS (in)stability
• A complete gauge-invariant formalism for arbitrary second-order perturbations of a Schwarzschild black hole
• What drives AdS unstable?
• Gravitational geons in asymptotically anti-de Sitter spacetimes
• Comment on "AdS nonlinear instability: moving beyond spherical symmetry" [Class. Quantum Grav. 33 23LT01 (2016)]

Cited by in corpus (16)

• Collapse and Nonlinear Instability of AdS with Angular Momenta
• Gravitational geons in asymptotically anti-de Sitter spacetimes
• A proof of the instability of AdS for the Einstein--null dust system with an inner mirror
• Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes
• Black resonators and geons in AdS${}_5$
• Non-Spherically Symmetric Collapse in Asymptotically AdS Spacetimes
• Anti-de Sitter geon families
• Towards a theory of nonlinear gravitational waves: a systematic approach to nonlinear gravitational perturbations in vacuum
• AdS nonlinear instability: breaking spherical and axial symmetries
• Breathing modes, quartic nonlinearities and effective resonant systems
• On turbulence for spacetimes with stable trapping
• Time-periodicities in holographic CFTs
• Gauge invariant description of weak gravitational field on a spherically symmetric background with cosmological constant
• On periodic solutions of nonlinear wave equations, including Einstein equations with a negative cosmological constant
• Gravitational Perturbations in a cavity: Nonlinearities
• Nonlinear perturbations of higher dimensional anti-de Sitter spacetime