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.
Highlights
The problem of late-time dynamics in asymptotically anti–de Sitter (AdS) spacetimes received quite a lot of attention in the past five years and revealed a few surprises
Paper [1] provided numerical and heuristic evidence for two types of possible scenarios in the model EinsteinAdS–massless scalar field system at spherical symmetry: (1) turbulent dynamics leading to concentration of energy on small spatial scales leading to a black hole formation on the time scale Oðε−2Þ, where ε measures the amplitude of initial data and (2) quasiperiodic evolution
We have presented the formalism for nonlinear gravitational perturbations of AdS spacetime and applied it to provide the evidence for the existence and properties of globally regular, asymptotically AdS, time-periodic solutions of vacuum Einstein equations
Summary
The problem of late-time dynamics in asymptotically anti–de Sitter (AdS) spacetimes received quite a lot of attention in the past five years and revealed a few surprises. The spectrum of linear perturbation in 3 þ 1 dimensions reads ωl;j 1⁄4 q þ l þ 2j, where l is the mode angular momentum index, j is the nodal number of the corresponding radial wave function, and q reads 1 or 2 for polar and axial modes, respectively It was noted with a bit of surprise that only in special cases (listed in Sec. VI in [13]) linear eigenmodes do admit a nonlinear extension to a TP solution. Regge-Wheeler (RW) seminal paper [16] and use spherical symmetry of the zero order solution (AdS in our case) to expand the metric perturbations into scalar, vector, and tensor spherical harmonics. In the two following sections we discuss solutions to all problems listed above for axially symmetric perturbations starting from polar- and axial-type perturbations at linear order. The symbols ðiÞζl μ and ðiÞηl μ appear in expansion of polar gauge vectors ðiÞζμ and axial gauge vectors ðiÞημ according to (17)–(19), respectively (see below for the usage of these gauge vectors)
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