Abstract
We consider five-dimensional, vacuum Einstein equations with negative cosmological constant within cohomogenity-two biaxial Bianchi IX ansatz. This model allows to investigate the stability of AdS without adding any matter to the energy-momentum tensor, thus analyzing instability of genuine gravitational degrees of freedom. We derive the resonant system and identify vanishing secular terms. The results resemble those obtained for Einstein equations coupled to a spherically-symmetric, massless scalar field, backing the evidence that the scalar field model captures well the relevant features of AdS instability problem. We also list recurrence relations for the interaction coefficients of the resonant system, which might be useful in both numerical simulations and further analytical studies.
Highlights
We derive the resonant system and identify vanishing secular terms. The results resemble those obtained for Einstein equations coupled to a spherically-symmetric, massless scalar field, backing the evidence that the scalar field model captures well the relevant features of AdS instability problem
A toy model of the spherically symmetric massless scalar field minimally coupled to gravity with a negative cosmological constant in four [1] and higher dimensions [2] the numerical simulations showed that there is a class of arbitrarily small perturbations of AdS that evolve into a black hole on the time-scale O(ε−2), where ε measures the amplitude of the perturbation
The arguments of [1, 2] and following works were based on extrapolation of the observed scaling O(ε−2) in time of resonant energy transfers between the modes and collapse times for finite small values of ε, cf. figure 2 in [1], but the limit ε → 0, with the instability time scale ε−2, is obviously inaccessible to numerical simulation
Summary
We consider d + 1 dimensional vacuum Einstein equations with a negative cosmological constant. It is convenient to define the mass function sin x m(t, x) = cos x (1 − A(t, x)). Where the free functions b∞(t), δ∞(t), and mass M uniquely determine the power series. It follows from (2.8) that the asymptotic behaviour of fields at infinity is completely fixed by the assumptions of smoothness and finiteness of total mass, there is no freedom of imposing the boundary data. The pure AdS spacetime corresponds to B = 0, A = 1, δ = 0 This equation is the = 2 gravitational tensor case of the master equation describing the evolution of linearized perturbations of AdS spacetime, analyzed in detail by Ishibashi and.
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