AdS Robin solitons and their stability

We consider the four-dimensional Einstein-Klein-Gordon-AdS system with 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.


Introduction
We consider the four-dimensional Einstein-Klein-Gordon-AdS system with mass µ related to the negative cosmological constant Λ through µ 2 = 2 3 Λ. For this, and only this, value of mass the system is conformally well-behaved at null and spatial infinity and consequently the initial-boundary value problem is wellposed for a variety of different boundary conditions at infinity [1,2]. Here, we focus on the one-parameter family of Robin boundary conditions. It has been known that along this family there is a critical parameter value at which the system undergoes a bifurcation: the (zero energy) anti-de Sitter (AdS) spacetime becomes linearly unstable above that critical value [3] and there emerges a pair of (negative energy) globally regular static solutions (henceforth called AdS Robin solitons) [4]. The main goal of this paper is to establish the existence of AdS Robin solitons rigorously and analyze the structure of the bifurcation in more detail. In preparation of future analysis of the role of solitons in dynamics, we also determine their spectrum of linearized perturbations.

Setup
The Einstein-Klein-Gordon-AdS system is given by where 2 g = g αβ ∇ α ∇ β is the wave operator associated with the metric g αβ , µ is the mass of the scalar field, and Λ is a negative constant. We assume spherical symmetry and write the metric in the form g = 2 cos 2 x −Ae −2δ dt 2 + A −1 dx 2 + sin 2 x dω 2 , where (t, x, ω) ∈ (−∞, ∞) × [0, π/2) × S 2 , dω 2 is the round metric on S 2 and 2 = −3/Λ. The metric functions A, δ and the scalar field φ depend on (t, x).
We choose units such that = 1 and 4πG = 1 and introduce new variables Then the system (1) reduces to is the polar wave operator associated with the conformal metricĝ αβ = cos 2 x g αβ . On the right side of equation (4a) the derivatives of metric functions were eliminated using equations (4b) and (4c). In the following we set µ 2 = −2. 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) vanishes 1 . 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] 2 . In this paper we focus our attention on the one-parameter family of Robin boundary conditions where b is a constant (hereafter referred to as the Robin parameter). For b = 0 the Robin condition reduces to the Neumann condition ∂ x f | x=π/2 = 0.
1 This cancellation is due to the fact that for µ 2 = −2 the left sides of equations (1b) and (4a) are asymptotically conformal, that is (2 g + 2)φ ≈ (cos x) −3 (2ĝ − 1)f near x = π/2. However, the constraint equation (4b) has a singularity at x = π/2 so it does not appear possible to extend the solutions 'beyond infinity' (cf. [5] where an extension of solutions across the conformal boundary at timelike infinity was analyzed for the system (1) with µ 2 = 2 3 Λ > 0). 2 This should be contrasted with the widely studied massless case for which only the Dirichlet boundary condition is compatible with the basic requirement of finite total mass [6].
Assuming (5) and expanding the fields in power series in z = π/2 − x we obtain the following asymptotic behavior near z = 0 where α(t) and δ ∞ (t) are free functions 3 and M is a constant. To see the physical meaning of M , let us define the renormalized mass function The first term on the right side is the Misner-Sharp mass function defined by m M S = r(1 + r 2 − g µν ∂ µ r∂ ν r), where r = tan x is the areal radial coordinate. This function diverges as x → π/2 and the purpose of the second term (called the counterterm) is to cancel this divergence. The leading order behavior of the counterterm is determined by the asymptotics (6) and (7) but otherwise can be chosen freely. Using equation (4b) we get where hence This quantity can be interpreted as the bulk energy. From the asymptotic expansions (6) and (7) it follows that where the second term on the right side can be viewed as the energy stored on the boundary. Although both the bulk and boundary energies are time dependent, their sum M is conserved. In what follows, we will refer to M as the total energy (mass). The exchange of energy between the bulk and the boundary is a characteristic feature of systems subject to the Robin boundary conditions. Note that some of the bulk energy is "lost" to the boundary if b < 0 and "gained" from the boundary if b > 0. We remark that the expression (13) can be obtained in a systematic way within the diffeomorphism covariant Hamiltonian framework of Wald and Zoupas [7] (see section 2.2 in [8]). Nonetheless, we believe that our hands-on approach, based solely on the analysis of the system (4), is helpful in getting insight into not so widely known physics of the Robin boundary conditions.

AdS Robin solitons
For time-independent solutions the system (4) with µ 2 = −2 takes the form where the derivatives of metric functions were eliminated from equation (14a) using equations (14b) and (14c). It is routine to prove that this system has local solutions near x = 0 which behave as follows where c is a free parameter.
Proof. To prove this lemma it is convenient to use the radial coordinate r = tan x and return to the original field variables Then, equations (14a) and (14b) become The local solutions (15) translate to We first observe that the function B = (A−1)(1+r 2 ) is monotone increasing. To see this, suppose that B(r) has a maximum at some point r 0 > 0. Differentiating equation (17b), substituting B (r 0 ) = 0, and eliminating φ (r 0 ) and B(r 0 ) using equations (17a) and (17b), respectively, we get after simplifications which is manifestly positive, contradicting that the point r 0 exists. Since B(r) is positive for small r > 0, this implies that A(r) ≥ 1 for all r.
Next, we define a function Using the system (17) we obtain which is manifestly negative, hence H(r) is a monotonically decreasing Lyapunov function. Since A ≥ 1, it follows that r 2 φ 2 and φ 2 remain bounded for all r.
To determine the asymptotic behavior of solutions for r → ∞ it is convenient to use the logarithmic radial variable τ = log r. In terms of τ the system (17) is asymptotically autonomous for τ → ∞ and the limiting autonomous system is where dot denotes the derivative with respect to τ (by an abuse of notation, we use the same symbols for the original and limiting systems). From the general theory of asymptotically autonomous dynamical system [9,10] (6) and (7) by This completes the proof.
The above Lemma ensures that for each c the solution starting with the initial conditions (15) automatically satisfies the Robin condition f (π/2) = bf (π/2) for some parameter b (which depends on c). We will refer to these globally regular static solutions as the AdS Robin solitons (or just solitons for short) and denote them by (f s , B s , δ s ). The profiles of solitons can be easily determined numerically by integrating the system (14) with the boundary conditions (15). We note in passing that an analogous reasoning leads to a two-parameter family of hairy black holes (where the second parameter is the horizon radius).
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], however, to the best of our knowledge, their existence remained unproven.

Bifurcation analysis
It is illuminating to look at the solitons from the viewpoint of the local bifurcation theory. To this end, consider the perturbation expansion of solitons for small c Inserting this expansion into the system (14) and requiring regularity at x = 0, at the lowest order we get At the third order equation (14a) becomes Substituting f 1 and B 2 , given in (26), into the right hand side and imposing f 3 (0) = 0, we find where ζ is the Riemann zeta function and we defined the functions From (28) we read off At the fourth order equation (14b) becomes Substituting (26) and (28) into the right hand side and requiring regularity at x = 0, we find from which we read off B 4 π 2 = π 48 54 + π 2 (32 ln 2 − 11) ≈ 10.7566 .
Using (30) and (31) and imposing the Robin condition in the expansion (25), we get The fact that b 2 is positive means that at b * we have a supercritical pitchfork bifurcation where the AdS solution bifurcates into a pair of solitons (±f s , B s , δ s ). As usual, this kind of bifurcation is associated with exchange of linear stability and, indeed, in the next section we will show that for b > b * the AdS space becomes linearly unstable whereas the solitons are linearly stable. Using (7) and the expansion (25), we get the approximation for the mass which upon substitution of (30) and (34) yields It is instructive to rederive this result along the lines of designer gravity [4]. Letting α = f (π/2) and β = f (π/2), we get from (25) (in this paragraph '=' means equality up to order O(c 4 )) which can be viewed as the parametric equation of the curve in the (α, β) plane. Eliminating c we get the function where the subscript 's' indicates that the function is associated with solitons. Following the approach used in designer gravity we introduce the effective potential By construction, critical points of the effective potential correspond to solitons.
The key observation, made by Hertog and Horowitz in [4], is that the value of the effective potential at the critical point is equal to the soliton mass. In our case, V (α) = 0 for α 2 s = π 2 4b 2 (b − b * ) (and, of course, for α = 0 corresponding to the AdS space). Substituting this into (39) we get M s = V(α s ) which reproduces the formula (36). Note that V (α s ) > 0.
Further from the bifurcation point, the soliton function β s (α) and the corresponding effective potential V(α) can be determined numerically 4 . We find that for each b > b * the effective potential has the shape of a Mexican hat (see Fig. 2) with exactly three critical points: the local maximum at zero and two global minima at ±α s . This implies that the soliton solution is unique (modulo reflection symmetry) and suggests that it is stable. Remark. It is natural to expect that for any given b > b * the soliton is the ground state, i.e. for any regular initial data satisfying the Robin condition (5) there holds the inequality M ≥ M s which saturates if and only if the data correspond to the soliton [4]. Our numerical constructions of initial data corroborate this conjecture but we have not been able to prove it (see [8] for partial results in this direction).

Linear stability analysis
Linearizing the system (4) around the AdS solution (f = B = δ = 0) and separating time f (t, x) = e iωt v(x) we get the eigenvalue problem 5 The operator L (which is just the polar conformal Laplacian on the 3-sphere) is symmetric on the Hilbert space L 2 [0, π/2], sin 2 x dx and the Robin boundary condition v − bv| x= π 2 = 0 (41) provides a one-parameter family of its self-adjoint extensions.
For ω 2 > 0 the regular solution of (40) is Imposing the Robin condition (41) we obtain the quantization condition for the eigenfrequencies ω = b tan (ωπ/2) . From the graphical analysis shown in Fig. 2a we see that for each non-negative integer n there is exactly one eigenfrequency ω n such that For large n the quantization condition (43) gives the asymptotically resonant spectrum The lowest eigenvalue ω 2 0 vanishes at b = b * = 2/π; the corresponding eigenfunction is the linearized static solution f 1 given in (26). An elementary perturbative calculation gives near b * For general b > b * there is an exponentially growing mode e λ 0 t v 0 (x), where the exponent λ 0 = −ω 2 0 is given by the unique positive root of the equation (see Fig. 2b) λ = b tanh (λπ/2) (46) and the corresponding eigenfunction is v 0 (x) = sinh(λ 0 x)/ sin x.
Next, we look at the linear stability of solitons. Linearizing the system (4) around the soliton and separating time, we get the eigenvalue problem where where the operator P can be calculated using the expansions (25). To calculate the perturbations of eigenvalues we assume the following ansatz v n = cv * n +c 3 u n +O(c 5 ),ω 2 where ω * n 2 and v * n are the eigenvalues and normalized eigenfunctions of the operator L at the bifurcation point and b 2 is given in (35). Substituting this ansatz into the Robin boundary condition we get at the first and third order in c v * n (π/2) = b * v * n (π/2), u n (π/2) = b * u n (π/2) + b 2 v * n (π/2) .
Substituting the ansatz (51) into (47), we get at the third order in c Lu n + P v * n = ω * n 2 u n + γ n v * n .
The positivity of γ 0 confirms the expectation that the solitons are linearly stable near the bifurcation point. Solving the eigenvalue problem numerically, we find that the eigenvaluesω 2 n grow monotonically with c. The numerical values of the first few eigenfrequencies (as measured by the central observer) for a small parameter c = 0.1 (corresponding to b ≈ 0.6467) are displayed in Table 1. From the leading order WKB approximation [12] it follows that for large ñ ω n = 2n + 1 a + O 1 n , a = 2 π π/2 0 A −1 s e δs dx, which compares well with numerical results even if n is not very large.

Discussion
The Einstein-Klein-Gordon-AdS system with mass µ 2 = 2 3 Λ < 0 is wellbehaved at the conformal boundary which makes it a good toy model for studying the role of boundary conditions in dynamics of asymptotically AdS spacetimes [13]. In this paper we focused on the Robin boundary conditions and proved existence of a one-parameter family of solitons for b > b * . We also demonstrated that the linearized perturbations around these solitons have no growing modes. A natural question is: are the AdS Robin solitons nonlinearly stable? Numerical simulations, to be reported in [14], indicate a positive answer and provide evidence for existence of plethora of time-periodic and quasiperiodic solutions, not only in the perturbative regime (which is expected in view of the non-resonant spectrum) but also, somewhat surprisingly, for large perturbations.
Of course, the analogous question of nonlinear stability arises for the AdS spacetime for b < b * . However here, in contrast to the Dirichlet case [6], the numerical simulations are as yet not conclusive and we leave this question to future investigations.