Abstract
We consider the four-dimensional Einstein–Klein–Gordon–AdS system with the conformal mass subject to the Robin boundary conditions at infinity. Above a critical value of the Robin parameter, at which the AdS spacetime goes linearly unstable, we prove existence of a family of globally regular static solutions (that we call AdS Robin solitons) and discuss their properties.
Highlights
We consider the four-dimensional Einstein–Klein–Gordon–anti-de Sitter (AdS) system with mass μ related to the negative cosmological constant Λ through μ2 = Λ.For this, and only this, value of mass the system is conformally well-behaved at null and spatial infinity and the initial-boundary value problem is well-posed for a variety of different boundary conditions at infinity [1, 2]
It has been known that along this family there is a critical parameter value at which the system undergoes a bifurcation: the anti-de Sitter (AdS) spacetime becomes linearly unstable above that critical value [3] and there emerges a pair of globally regular static solutions [4]
In this paper we focused on the Robin boundary conditions and proved existence of a one-parameter family of solitons for b > b∗
Summary
For this value of mass the wave equation (4a) is regular at x = π/2 because the first term on the right side (which is the only singular term) vanishes3 Thanks to this fact, the initial-boundary value problem for the system (4) is well posed for a variety of boundary conditions at the conformal boundary (both reflective and dissipative) [1, 2].4. The first term on the right side is the Misner–Sharp mass function defined by mMS = r(1 + r2 − gμν∂μr∂νr), where r = tan x is the areal radial coordinate. From the general theory of asymptotically autonomous dynamical system [9, 10] and the existence of the Lyapunov function H, it follows that the asymptotic behavior of solutions of the system (17) for τ → ∞ is governed by the above limiting system. As far as we know, the AdS Robin solitons and hairy black holes were first studied in the literature in the context of so called ‘designer gravity’ [4, 11], to the best of our knowledge, their existence remained unproven
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