# Asymptotic Properties of Linear Field Equations in Anti-de Sitter Space

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## Abstract

We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

## 1 Introduction

*g*. The past few decades have seen fundamental progress in understanding the global dynamics of solutions to (1). In particular, a satisfactory answer – asymptotic stability – has been given for the dynamics of (1) with \(\Lambda =0\) near Minkowski space [1]; the dynamics of (1) with \(\Lambda >0\) near de Sitter space [2, 3], the maximally symmetric solution of (1) with \(\Lambda >0\); as well as for the (\(\Lambda >0\)) Kerr-dS black hole spacetimes [4]. Today, the dynamics near black hole solutions of (1) for \(\Lambda =0\) is a subject of intense investigation [5, 6], with the current state-of-the-art being the results in [7] establishing the linear stability of Schwarzschild, and [8] establishing nonlinear stability for Schwarzschild within a restricted symmetry class.

In contrast to the above, the global dynamics of (1) with \(\Lambda <0\) near anti-de Sitter space (AdS), the maximally symmetric solution of the vacuum Einstein equations with \(\Lambda <0\), is mostly unknown. Part of the problem is that in the case of \(\Lambda <0\), the PDE problem associated with (1) takes the form of an initial boundary value problem. Therefore, even to construct local in time solutions, one needs to understand what appropriate (well-posed, geometric) boundary conditions are. It also suggests that the global behaviour of solutions starting initially close to the anti-de Sitter geometry may depend crucially on the choice of these boundary conditions [9, 10].

*g*satisfying (1). As is well known [1], the equations (2) can be used to estimate the curvature components of a dynamical metric

*g*. The boundary data required for well-posed evolution of (2) will generally imply a condition on the energy-flux of curvature through the timelike boundary. Two “extreme” cases seem particularly natural and interesting: The case when this flux vanishes, corresponding to reflecting (Dirichlet or Neumann) conditions, and the case when this flux is “as large as possible”, corresponding to “optimally dissipative” conditions. While any such boundary data for (2) will have to be complemented with other data (essentially the choice of a boundary defining function and various gauge choices) to estimate the full spacetime metric [9], it is nevertheless reasonable, in view of the strong non-linearities appearing in the Einstein equations, to conjecture the following loose statement for the global dynamics of perturbations of AdS under the above “extreme” cases of boundary conditions:

### Conjecture 1

Anti-de Sitter spacetime is non-linearly unstable for reflecting and asymptotically stable for optimally dissipative boundary conditions.

The instability part of Conjecture 1 was first made in [12, 13] in connection with work on five-dimensional gravitational solitons. See also [14]. By now there exist many refined versions of this part of the conjecture as well as strong heuristic and numerical support in its favour [15, 16, 17]. In a series of papers [18, 19, 20], Moschidis has considered the stability problem for the AdS spacetime within spherical symmetry, in the presence of null dust or massless Vlasov matter, culminating in a proof of the instability of AdS for the Einstein–Massless Vlasov system in [21].

The present paper is the first of a series of papers establishing the stability part of Conjecture 1. Here we contribute the first fundamental ingredient, namely robust decay estimates for the associated linear problem. Our interest in the case of dissipative boundary conditions in part goes back to the original work of Friedrich in [10], which establishes that local well posedness does not single out a preferred boundary condition at null infinity. It is then reasonable to ask how questions of global existence depend on the choice of boundary condition. We also observe that to date the only solutions to the vacuum Einstein equations (regardless of boundary conditions) which are future-complete are necessarily stationary. See for example [22] for general constructions of such spacetimes. A positive resolution of the stability part of Conjecture 1 would necessarily imply the existence of truly dynamical, future-complete, solutions to (1).

### 1.1 Linear field equations on AdS

*linear*problem is robustly controlled [23]. In the present (dissipative) context, this means that the mechanisms and obstructions for the decay of linear waves in the

*fixed*AdS geometry should be understood and decay estimates with constants depending on the initial data available. We accomplish this by giving a complete description of the decay properties of three fundamental field equations of mathematical physics:

- (W)The conformal wave equation for a scalar function on the AdS manifold,
^{1}$$\begin{aligned} \Box _{g_{AdS}} u + 2u = 0. \end{aligned}$$(3) - (M)Maxwell’s equations for a two-form
*F*on the AdS manifold,$$\begin{aligned} dF = 0 \ \ \ \text {and} \ \ \ d \star _{g_{AdS}} F = 0. \end{aligned}$$(4) - (B)The Bianchi equations for a Weyl field
*W*(see Definition 1 below) on the AdS manifold$$\begin{aligned} \left[ \nabla _{g_{AdS}}\right] ^a W_{abcd} = 0. \end{aligned}$$(5)

*state*the latter, one requires a choice of boundary defining function for AdS and a choice of timelike vectorfield (or, alternatively, the choice of an outgoing null-vector) at each point of the AdS boundary. For us, it is easiest to state these conditions in coordinates which suggest a canonical choice for both these vectors. We write the AdS metric in spherical polar coordinates on \(\mathbb {R}^4\), where it takes the simple familiar form

^{2}Note that 1 /

*r*is a boundary defining function and thatwith Open image in new window (\(A=1,2\)) an orthonormal frame on the sphere of radius

*r*defines an orthonormal frame for AdS. Finally, the vector \(\partial _t\) singles out a preferred timelike direction.

^{3}

That the above boundary conditions are indeed correct, naturally dissipative, boundary conditions leading to a well-posed boundary initial value problem will be a result of the energy identity. While this is almost immediate in the case of the wave equation and Maxwell’s equations, we will spend a considerable amount of time on the “derivation” of (9) in the Bianchi case, see Sect. 4.3.

*u*is more complicated [24], but a well posedness theory with dissipative boundary conditions is still available [25] for

*a*in a certain range. We restrict attention to the case \(a=2\) for two reasons. Firstly, for this choice of

*a*the equation has much in common with the systems (M), (B), owing to their shared conformal invariance. This is the correct ‘toy model’ for the Einstein equations. Secondly, the problem appears to be considerably more challenging for \(a \ne 2\). In particular, it does not appear that the methods of this paper can be directly applied, even once allowance is made using the formalism of [24] for the more complicated boundary behaviour of solutions.

### 1.2 The main theorems

We now turn to the results.

### Theorem 1.1

- (W)
- (M)\(\Psi \) is a Maxwell-two-form and a smooth solution of (4) subject to dissipative boundary conditions (8). We associate with \(\Psi \) the energy densitywhere$$\begin{aligned} \varepsilon \left[ \Psi \right] = \sqrt{1+r^2} \left( |E|^2 + |H|^2 \right) \end{aligned}$$
*E*and*H*denote the electric and magnetic part of \(\Psi \) respectively. - (B)\(\Psi \) is a Weyl-field and a smooth solution of (5) subject to dissipative boundary conditions (9). We associate with \(\Psi \) the energy densitywhere$$\begin{aligned} \varepsilon \left[ \Psi \right] = \left( 1+r^2\right) ^\frac{3}{2} \left( |E|^2 + |H|^2 \right) \end{aligned}$$
*E*and*H*denote the electric and magnetic part of the Weyl-field respectively.

- (1)Uniform Boundedness: For any \(0<T<\infty \) we havewhere the implicit constant is independent of$$\begin{aligned} \int _{\Sigma _T} \frac{\varepsilon \left[ \Psi \right] }{\sqrt{1+r^2}}r^2 dr d\omega \lesssim \int _{\Sigma _0} \frac{\varepsilon \left[ \Psi \right] }{\sqrt{1+r^2}}r^2 dr d\omega , \end{aligned}$$
*T*. - (2)Degenerate (near infinity) integrated decay without derivative loss:$$\begin{aligned} \int _0^\infty dt \int _{\Sigma _t} \frac{\varepsilon [\Psi ]}{1+r^2} r^2 dr d\omega \lesssim \int _{\Sigma _0} \frac{\varepsilon \left[ \Psi \right] }{\sqrt{1+r^2}}r^2 dr d\omega . \end{aligned}$$
- (3)Non-degenerate (near infinity) integrated decay with derivative loss:$$\begin{aligned} \int _0^\infty dt \int _{\Sigma _t} \frac{\varepsilon [\Psi ]}{\sqrt{1+r^2}} r^2 dr d\omega \lesssim \int _{\Sigma _0} \frac{\varepsilon \left[ \Psi \right] +\varepsilon \left[ \partial _t \Psi \right] }{\sqrt{1+r^2}}r^2 dr d\omega . \end{aligned}$$

### Remark 1

Similar statements hold for higher order energies by commuting with \(\partial _t\) and doing elliptic estimates. As this is standard we omit the details.

### Corollary 1.2

What is remarkable about the above theorem is that the derivative loss occurring in (3) allows one to achieve integrated decay of the energy without loss in the asymptotic weight *r*. While it is likely that more refined methods can reduce the loss of a full derivative in (3), we shall however establish that *some* loss is necessary and in fact reflects a fundamental property of the hyberbolic equations on AdS: the presence of trapping at infinity.

### Theorem 1.3

With the assumptions of Theorem 1.1, the term \(\varepsilon [\partial _t \Psi ]\) on the right hand side of estimate (3) of Theorem 1.1 is necessary: The estimate fails (for general solutions) if it is dropped.

We will prove Theorem 1.3 only for the case of the wave equation (W), see (45) and Corollary 5.8. The proof is based on the Gaussian beam approximation for the wave equation.^{4} In particular, we construct a solution of the conformal wave equation in AdS which contradicts estimate (3) of Theorem 1.1 without the \(\varepsilon [\partial _t \Psi ]\)-term on the right hand side. Similar constructions can be given for the Maxwell and the Bianchi case.^{5}

### 1.3 Overview of the proof of Theorem 1.1 and main difficulties

The proof of Theorem 1.1 is a straightforward application of the vectorfield method once certain difficulties have been overcome. Let us begin by discussing the proof in case of the wave- (W) and Maxwell’s equation (M) as it is conceptually easier.

In the case of (W) and (M), statement (1) follows immediately from integration of the divergence identity \(\nabla ^a \left( \mathbb {T}_{ab} \left( \partial _t\right) ^b\right) =0\) with \(\mathbb {T}\) being the energy momentum tensor of the scalar or Maxwell field respectively. In addition, in view of the dissipative condition, this estimate gives control over certain derivatives of *u* (certain components of the Maxwell field, namely \(E_A\) and \(H_A\)) integrated along the boundary.

The statement (2) of the main theorem is then obtained by constructing a vectorfield *X* (see (17)) which is almost conformally Killing near infinity. The key observations are that firstly, the right hand side of the associated divergence identity \(\nabla ^a \left( \mathbb {T}_{ab} X^b\right) ={}^X\pi \cdot \mathbb {T}\) controls all derivatives of *u* in the wave equation case (components of *F* in the Maxwell case). Secondly, when integrating this divergence identity, the terms appearing on the boundary at infinity come either with good signs (angular derivatives in the case of the wave equation, \(E_{{\overline{r}}}\) and \(H_{{\overline{r}}}\) components in the Maxwell case) or are components already under control from the previous \(\mathbb {T}_{ab} \left( \partial _t\right) ^b\)-estimate. The integrated decay estimate thus obtained comes with a natural degeneration in the *r*-weight, as manifest in the estimate (2) of Theorem 1.1.

To remove this degeneration we first note that in view of the fact that the vectorfield \(\partial _t\) is Killing, the estimates (1) and (2) also hold for the \(\partial _t\)-commuted equations. In the case of the wave equation, controlling \(\partial _t \partial _t \psi \) in \(L^2\) on spacelike slices implies an estimate for all spatial derivatives of \(\psi \) through an elliptic estimate. The crucial point here is that *weighted* estimates are required. Similarly, one can write Maxwell’s equation as a three-dimensional div-curl-system with the time derivatives of *E* and *H* on the right hand side. Again the crucial point is that *weighted* elliptic estimates are needed to prove the desired results.

Once all (spatial) derivatives are controlled in a weighted \(L^2\) sense on spacelike slices one can invoke Hardy inequalities to improve the weight in the lower order terms and remove the degeneration in the estimate (2).

For the case of the spin 2 equations (B), the proof follows a similar structure. However, the divergence identity for the Bel-Robinson tensor (the analogue of the energy momentum tensor in cases (W) and (M)) alone *will not* generate the estimate (1). In fact, the term appearing on the boundary after integration does not have a sign unless one imposes an additional boundary condition! On the other hand, one can show (by proving energy estimates for a reduced system of equations, see Sect. 4.3.1) that the boundary conditions (9) already uniquely determine the solution.^{6} The resolution is that in the case of the Bianchi equations one needs to prove (1) and (2) at the same time: Contracting the Bel-Robinson tensor with a suitable combination of the vectorfields \(\partial _t\) and *X* ensures that the boundary term on null-infinity does have a favorable sign and so does the spacetime-term in the interior. Once (1) and (2) are established, (3) follows from doing elliptic estimates for the reduced system of Bianchi equations similar to the Maxwell case (M).

### 1.4 Remarks on Theorem 1.1

The estimates of Theorem 1.1 remain true for a class of \(C^k\) perturbations of AdS which preserve the general properties of the deformation tensor of the timelike vectorfield \(\sqrt{3}\partial _t +X\), cf. the proof of Proposition 5.6. This includes perturbations which may be dynamical. This fact is of course key for the non-linear problem [27].

The estimates of Theorem 1.1 are stable towards perturbations of the optimally dissipative boundary conditions. In the cases of (W) and (M) one can in fact establish these estimates for any (however small) uniform dissipation at the boundary, cf. Sect. 6. Whether this is possible also in case of (B) is an open problem and (if true) will require a refinement of our techniques. In Sect. 6.2 we discuss the case of Dirichlet conditions for (B) which fix the conformal class of the induced metric at infinity to linear order. The Dirichlet boundary conditions may be thought of as a limit of dissipative conditions in which the dissipation vanishes. We outline a proof of boundedness for solutions of the Bianchi equations in this setting. We also briefly discuss the relation of our work to the Teukolsky formalism in Sect. 6.3, and argue that the proposed boundary conditions of [28] may not lead to a well posed dynamical problem. We state a set of boundary conditions for the Teukolsky formulation that *does* lead to a well posed dynamical problem.

Finally, one may wonder whether and how the derivative loss in Theorem 1.1 manifests itself in the non-linear stability problem. In ongoing work [27], we shall see that the degeneration is sufficiently weak in the sense that the degenerate estimate (2) will be sufficient to deal with the non-linear error-terms.

### 1.5 Main ideas for the proof of Theorem 1.3

*conformal*or

*conformally invariant*because if

*u*is a solution of (3), then \(v := \Omega u\) is a solution of a wave equation with respect to the conformally transformed metric \(g = \Omega ^2 g_{AdS}\). This suggests that understanding the dynamics of (3) for \(g_{AdS}\) is essentially equivalent to that of understanding solutions of

- (1)
Problem 1: The wave equation (12) on \(\mathbb {R}_t \times \mathbb {S}_h^3\) (with the natural product metric of the Einstein cylinder) where \(\mathbb {S}_h^3\) is the (say northern) hemisphere of the 3-sphere \(\mathbb {S}^3\) with boundary at \(\psi =\frac{\pi }{2}\), where (say optimally) dissipative boundary conditions are imposed. We contrast this problem with

- (2)
Problem 2: the wave equation (3) on \(\mathbb {R}_t \times \mathbb {B}^3\) (with the flat metric) where \(\mathbb {B}^3\) is the unit ball with boundary \(\mathbb {S}^2\) where dissipative boundary conditions are imposed.

^{7}

This phenomenon can be explained in the geometric optics approximation for the wave equation. Recall that in this picture, the optimally dissipative boundary condition says that the energy of a ray is fully absorbed if it hits the boundary orthogonally. For rays which graze the boundary, the fraction of the energy that is absorbed upon reflection depends on the glancing angle: the shallower the incident angle, the less energy is lost in the reflection.

Now let us fix a (large) time interval \(\left[ 0,T\right] \) for both Problem 1 and Problem 2. To construct a solution which decays very slowly, we would like to identify rays which a) hit the boundary as little as possible and b) if they do hit the boundary, they should do this at a very shallow angle (grazing rays).

*does not depend on the incident angle*! This goes back to the fact that all geodesics emanating from point on the three sphere refocus at the antipodal point. As a consequence, in Problem 1 we can keep the number of reflections in \(\left[ 0,T\right] \) fixed while choosing the incident angle as small as we like. This observation is at the heart of the Gaussian beam approximation invoked to prove Theorem 1.3.

### 1.6 Structure of the paper

We conclude this introduction providing the structure of the paper. In Sect. 2 we define the coordinate systems, frames and basic vectorfields which we are going to employ. Section 3 introduces the field equations of spin 0, 1 and 2 fields, together with their energy momentum tensors. The well-posedness under dissipative boundary conditions for each of these equations is discussed in Sect. 4 with particular emphasis on the importance of the reduced system in the spin 2 case. Section 5 is at the heart of the paper proving the global results of Theorem 1.1, first for the spin 0 (Sect. 5.1), then the spin 1 (Sect. 5.2) and finally the spin 2 case (Sect. 5.3). Corollary 1.2 is proven in Sect. 5.4 and Theorem 1.3 in Sect. 5.5. We conclude the paper outlining generalizations of our result. Some elementary computations have been relegated to the appendix.

## 2 Preliminaries

### 2.1 Coordinates, Frames and Volume forms

*g*. It will be convenient to introduce an orthonormal basis \(\{e^a\}\):for \(A=1,2\), where Open image in new window are an orthonormal basis for the round sphere

^{8}of radius

*r*. Throughout this paper, we will use capital Latin letters for indices on the sphere while small Latin letters are reserved as spacetime indices. The dual basis of vector fields is denoted \(\{e_a\}\):We introduce the surfaces \(\Sigma _T = \{t = T\}\) and \(\tilde{\Sigma }^{[T_1, T_2]}_R = \{r =R, T_1\le t \le T_2\}\), which have respective unit normals:

### 2.2 Vectorfields

*t*.

*T*, we will exploit the properties of the vectorfield

*X*is not Killing, it is almost conformally Killing near infinity. More importantly, it generates terms with a definite sign. To see this, note that

### 2.3 Differential operators

*g*. Define also \(\Box _g\) as the standard Laplace-Beltrami operator:

### 2.4 The divergence theorem

We denote by \(\text {Div }\) the spacetime divergence associated to the metric *g* for a vector field. In coordinates, it takes the following form:

### Lemma 2.1

*K*is a suitably regular vector field defined on \(\mathbb {R}^4\). Then we have

## 3 The Field Equations

### 3.1 Spin 0: The wave equation

^{9}by:

*K*is any vector field then we can define the current

*K*is Killing and \(K(r) = 0\), then \(K^a \tilde{\nabla }_a 1 = 0\) and we can see that \({}^K\mathbb {J}[u]\) is a conserved current when

*u*solves the conformal wave equation (20).

### 3.2 Spin 1: Maxwell’s equations

*F*on \(\mathbb {R}^4\):

^{10}The dual Maxwell \(2-\)form is:

*H*. Since

*F*is a smooth \(2-\)form, and the basis vectors \(e_\mu \) are bounded (but not continuous) at the origin, we deduce that the functions \(E_i\), \(H_i\) are bounded, but not necessarily continuous, at \(r = 0\). With respect to this decomposition, Maxwell’s vacuum equations (3.2) split into six evolution equations:and two constraints:Here Open image in new window is the covariant derivative on the

*unit*sphere, which commutes with \(\nabla _{\partial _r}\) (note that our conventions are that Open image in new window are orthonormal vector fields on the sphere of radius

*r*). The evolution equations (25–28) form a symmetric hyperbolic system. If the evolution equations hold, and assuming sufficient differentiability, it is straightforward to verify that

#### 3.2.1 The energy momentum tensor

*F*satisfies Maxwell’s equations, \(\mathbb {T}[F]\) is divergence free and traceless. We define in the obvious fashion the current

*K*.

### 3.3 Spin 2: The Bianchi equations

The equations for a spin 2 field, also called the Bianchi equations, can be expressed as first order differential equations for a Weyl tensor, which is an arbitrary \(4-\)tensor which satisfies the same symmetry properties as the Weyl curvature tensor. More precisely:

### Definition 1

*W*is a Weyl tensor if it satisfies:

- i)
\(W_{abcd} = - W_{bacd} = -W_{abdc}\).

- ii)
\(W_{abcd} + W_{acdb} + W_{abdc} = 0\).

- iii)
\(W_{abcd} = W_{cdab}\).

- iv)
\(W^a{}_{bad}=0\).

*F*decomposes as a pair of vectors tangent to \(\Sigma _t\). In the spin 2 case, the Weyl tensor

*W*decomposes as pair of symmetric tensors tangent to \(\Sigma _t\). We define:

*E*,

*H*are symmetric and tangent to \(\Sigma _t\). Moreover, both

*E*and

*H*are necessarily trace-free. In fact, one can reconstruct the whole tensor

*W*from the symmetric trace-free fields

*E*,

*H*, see [1, §7.2].

*E*,

*H*along the orthonormal frame defined in (14). To write the equations of motion in terms of

*E*,

*H*we consider the equations \(0 = \nabla ^a W_{a{\overline{r}}{\overline{r}}0}\), \(0 = \nabla ^a W_{a(A{\overline{r}})0}\) and \(0 = \nabla ^a W_{a(AB)0}\), from which we respectively find the evolution equations for

*E*: From the equations \(0 = \nabla ^a W_{a0A0}\) and \(0 = \nabla ^a W_{a0r0}\) respectively we find the constraint equations: By considering the equivalent equations for \(^{\star }{}W\), we find that the evolution equations for

*H*can be obtained from these by the substitution \((E, H) \rightarrow (H, -E)\): and the constraint equations for

*H*are:

#### 3.3.1 The Bel-Robinson tensor

*W*satisfies the Bianchi equations then

### Lemma 3.1

*p*if and only if

*W*vanishes at

*p*.

### Proof

*B*. By [23, Prop. 4.2], all of the quantities \(Q(e'_a, e'_b, e'_c, e'_d)\) are non-negative, and so we have established the first part. For the second part, we note that, again following [23, Prop. 4.2], the quantity \(Q(t_1, t_1, t_1, t_1)\) controls all components of

*W*. \(\square \)

## 4 Well Posedness

### 4.1 Dissipative boundary conditions

#### 4.1.1 Wave equation.

*u*has the following behaviour:

#### 4.1.2 Maxwell’s equations

#### 4.1.3 Bianchi equations

*trace-free*parts of \(E_{AB}\), \(H_{AB}\)). We could choose more general dissipative boundary conditions. For simplicity we shall just consider those above, which ought in some sense to represent “optimal dissipation” at the boundary, but see §6 for generalisations.

### 4.2 The well-posedness statements

We now state a general well-posedness statement for each of our three models with dissipative boundary conditions. As is well-known, the key to prove these theorems is the existence of a suitable energy estimate under the boundary conditions imposed. In the Wave- and Maxwell case such an estimate is immediate. In the Bianchi case, however, there is a subtlety (discussed and resolved already in [10]). We will dedicate Sect. 4.3 to derive a local energy estimate showing that the condition (42) indeed leads to a well-posed problem.

#### 4.2.1 Spin 0: The wave equation

The following result can be established either directly, or by making use of the conformal invariance of (20):

### Theorem 4.1

*u*, such that:

- i)
*u*solves the conformal wave equation (20) in \(S_{[0, T]}\). - ii)We have the estimate: for \(k, l, m = 0, 1, \ldots \) and a constant \(C_{T, u_0, u_1, K}\) depending on
*K*,*T*and the initial data. This in particular implies a particular asymptotic behaviour for the fields. - iii)The initial conditions hold:$$\begin{aligned} \left. u\right| _{t=0} = u_0, \qquad \left. u_t\right| _{t=0} = u_1. \end{aligned}$$
- iv)
Dissipative boundary conditions (40) hold.

#### 4.2.2 Spin 1: Maxwell’s equations

Working either directly with (25–30), or else by making use of the conformal invariance of Maxwell’s equations, it can be shown that:

### Theorem 4.2

- i)
\(E_i(t), H_i(t)\) solve (25–30) in \(S_{[0, T]}\) and the corresponding Maxwell tensor

*F*is a smooth \(2-\)form on \(S_{[0, T]}\). - ii)We have the estimate: for any \(K\ge 0\), where \(C_{T, E^0, H^0, K}\) depends on
*K*,*T*and the initial data. This in particular implies a particular asymptotic behaviour for the fields. - iii)The initial conditions hold:$$\begin{aligned} E_i(0) = E_i^0, \qquad H_i(0) = H_i^0. \end{aligned}$$
- iv)
Dissipative boundary conditions (41) hold.

The asymptotic conditions on the initial data are corner conditions that come from ensuring that the initial data are compatible with the boundary conditions. It is certainly possible to construct initial data satisfying the constraints and the corner conditions to any order. We could work at finite regularity, and our results will in fact be valid with much weaker assumptions on the solutions, but for convenience it is simpler to assume the solutions are smooth.

#### 4.2.3 Spin2: Bianchi equations

In the case of the Bianchi equation we can prove:

### Theorem 4.3

- i)
\(E_{ab}(t), H_{ab}(t)\) are traceless and the corresponding Weyl tensor

*W*is a smooth \(4-\)tensor satisfying the Bianchi equations on \(S_{[0, T]}\). - ii)We have the asymptotic behaviour: as \(r \rightarrow \infty \), for any \(K \ge 0\).
- iii)The initial conditions hold:$$\begin{aligned} E_{ab}(0) = E_{ab}^0, \qquad H_{ab}(0) = H_{ab}^0 \end{aligned}$$
- iv)
Dissipative boundary conditions (42) hold.

We will spend the remainder of this section to derive the key-energy estimate that is behind the proof of Theorem 4.3.^{11}

### 4.3 The modified system of Bianchi equations

In order to establish a well posedness theorem for the initial-boundary value problem associated to the spin 2 equations, the natural thing to do is to consider the evolution equations (Evol) as a symmetric hyperbolic^{12} system. Having established existence and uniqueness for this system (Step 1), one can then attempt to show that the constraints (Con) are propagated from the initial data (Step 2).

^{13}

*Ar*and the

*AB*components of

*E*(or alternatively

*H*) at infinity. We shouldn’t be so hasty, however. Before declaring victory, we must return to look at the constraints (Step 2).

*E*and

*H*are propagated by the evolution equations. Next, we turn to the differential constraints. We find that if (Evol) and the trace constraints hold, then the functions \(\mathscr {E}_a, \mathscr {H}_a\) defined in Sect. 3.3 satisfy the system of equations: Now, things appear to be working in our favour. This system is symmetric hyperbolic and we can check that by taking:

#### 4.3.1 The modified equations

*before*attempting to solve them as a symmetric hyperbolic system. In the previous calculation, the problematic boundary terms arise due to the radial derivatives appearing on the right hand side of (Evol \(E_{A{\overline{r}}}\)), (Evol \(H_{A{\overline{r}}}\)). We can remove these radial derivatives at the expense of introducing angular derivatives by using the constraint equations (Con \(E_{{\overline{r}}}\)). It is also convenient to eliminate \(E_{{\overline{r}}{\overline{r}}}\) and \(H_{{\overline{r}}{\overline{r}}}\) from our equations using the trace constraints. Doing this, we arrive at the modified set of propagation equations: This again forms a symmetric hyperbolic system. Taking

^{14}

Notice also that, unlike the estimate (43) for the (Evol) equations, we now have a bulk term in the energy estimate (44). For well-posedness this is no significant obstacle, but it will make establishing global decay estimates more difficult. In particular, it is no longer immediate that solutions with Dirichlet boundary conditions remain uniformly bounded globally.

To our knowledge, identifying the above modified system as the correct formulation to prove well-posedness goes back to Friedrich’s work [10]. In particular, Theorem 4.3 above could be inferred from this paper.

## 5 Proof of the Main Theorems

### 5.1 Proof of Theorem 1.1 for Spin 0

The Killing field *T* immediately gives us a boundedness estimate:

### Proposition 5.1

*u*be a solution of (20) subject to dissipative boundary conditions (40) as in Theorem 4.1. Define the energy to be:Then we have for any \(T_1<T_2\):

### Proof

We next show an integrated decay estimate with a loss in the weight at infinity:

### Proposition 5.2

### Proof

*X*is the radial vector field defined in (17). The proof of the theorem is a straightforward corollary of the following two Lemmas.

### Lemma 5.1

### Proof

*X*has the deformation tensor:

*u*solves the conformal wave equation, we have:

### Lemma 5.2

### Proof

*u*on the boundary by Theorem 5.1. \(\square \)

This concludes the proof of Proposition 5.2, and establishes the claimed degenerate integrated decay without derivative loss result. \(\square \)

We next improve the radial weight of the spacetime term in the integrated decay estimate, at the expense of losing a derivative.^{15}

### Proposition 5.3

### Proof

*T*and applying Proposition 5.2, we have

### Lemma 5.3

### Proof

See Appendix 7.1. \(\square \)

Finally, the \((\tilde{\partial }_r\partial _t u)^2\) term appearing in Proposition 5.3 is directly controlled by the *T*-commuted version of the estimates in Proposition 5.2. \(\square \)

We finally improve the weight in the spacetime term of Proposition 5.2 making use of the fact that by Proposition 5.3 we now control radial derivatives of Open image in new window and \(\tilde{\partial }_r u\) which lead to improved zeroth order terms through a Hardy inequality. This is a standard result, but for convenience we include here a proof.

### Lemma 5.4

### Proof

From Lemma 5.4 we establish:

### Theorem 5.1

*u*be a smooth function such that \(\left| r u \right| \) is bounded. Then the estimateholds for some constant \(C>0\) independent of \(T_1\) and \(T_2\). If, moreover,

*u*solves (20) subject to (40), then the right-hand side may be bounded by \(C' (E_{T_1}[u]+E_{T_1}[\partial _t u])\) for some \(C'>0\) independent of \(T_1\) and \(T_2\).

### Proof

*u*supported either on \(r\le 2\) or on \(r\ge 1\). For

*u*supported on \(r\le 2\), the estimate follows immediately, since the first order terms on the right hand side are comparable to those on the left hand side on any finite region. For

*u*supported on \(r\ge 1\), we first apply Lemma 5.4 with \(f = r^2\sqrt{1+r^2} \tilde{\partial }_r u\) and \(a=1\) to deduce

*u*is supported on \(r\ge 1\). A similar calculation gives the estimate for Open image in new window . Finally, making use of Propositions 5.2, 5.3, we see that if

*u*satisfies the equation, then we can bound the right hand side in terms of \(E_{T_1}[u]+E_{T_1}[\partial _t u]\). \(\square \)

Combining Proposition 5.3 with Theorem 5.1 we have established the claimed non-degenerate integrated decay with derivative loss result for the wave equation.

### 5.2 Proof of Theorem 1.1 for Spin 1

The proof of the theorem for the Maxwell field follows a similar pattern to that of the conformal scalar field. There is a simplification owing to the fact that the energy-momentum tensor is trace-free, and the elliptic estimate takes a slightly different form.

### Proposition 5.4

*F*is a solution of Maxwell’s equations (25–30), subject to the dissipative boundary conditions (41) as in Theorem 4.2. Then we have:

### Proof

We next show an integrated decay estimate with a loss in the weight at infinity:

### Proposition 5.5

*F*is a solution of Maxwell’s equations (25–30), subject to the dissipative boundary conditions (41) as in Theorem 4.2. Then we have:

### Proof

*r*. We have:

*E*,

*H*at infinity. Integrating \(\text {Div }{}^X\mathbb {J}\) over \(S_{[T_1, T_2]}\), applying the divergence theorem and using the estimates above for the fluxes completes the proof of the Proposition. \(\square \)

*T*we have:

### Lemma 5.5

### Proof

See Appendix 7.2. \(\square \)

### Theorem 5.2

*F*is a solution of Maxwell’s equations, as in Theorem 4.2. Then there exists a constant \(C>0\), independent of \(T_1\) and \(T_2\) such that we have

### Proof

^{16}a term proportional to \(\left| r^2 H_{\overline{r}} \right| ^2\), integrated over the cylinder, which we control with the estimate in Theorem 5.5. This immediately gives the result for all of the terms except the term Open image in new window , which we obtain from a standard elliptic estimate on the sphere (see for instance Proposition 2.2.1 in [1]) :after noticing that we already control Open image in new window and Open image in new window with a suitable weight from (25) and (30). Finally, we note that the estimate can be derived in an identical manner for

*E*. \(\square \)

### Theorem 5.3

*F*is a solution of Maxwell’s equations, as in Theorem 4.2. Then we have

### 5.3 Proof of Theorem 1.1 for Spin 2

^{17}

### Proposition 5.6

### Proof

*Y*is timelike, since

*J*through a spacelike surface with respect to the future directed normal will be positive. Moreover, since

*Q*is trace-free and divergence free, \(\text {Div } J\) will also be positive. To establish a combined energy and integrated decay estimate, we simply have to verify that the surface term on \(\mathscr {I}\) has a definite sign (and check the weights appearing in the various integrals). We shall require some components of

*Q*, which are summarised in the following Lemma:

### Lemma 5.6

*J*above, we apply the divergence theorem, Lemma 2.1. We now verify that all the terms have a definite sign.

- a)
**Fluxes through**\(\Sigma _t\) We computeNow, defining \(\hat{Y} = (1+ \frac{2}{3}r^2)^{-\frac{1}{2}}Y\), we have that$$\begin{aligned} \int _{\Sigma _t} J_a n^a dS_{\Sigma _t} = \int _{\Sigma _t} Q(e_0, Y, Y, Y) \frac{r^2}{\sqrt{1+r^2}} dr d\omega . \end{aligned}$$so by Lemma 3.1 we deduce$$\begin{aligned} -g(e_0, \hat{Y}) = \sqrt{\frac{1+r^2}{1+\frac{2}{3}r^2}} \le \sqrt{\frac{3}{2}} \end{aligned}$$^{18}:so that$$\begin{aligned} Q(e_0, \hat{Y}, \hat{Y}, \hat{Y}) \sim Q_{0000}, \end{aligned}$$and hence$$\begin{aligned} Q(e_0, Y, {Y}, {Y}) \sim \left( 1+r^2\right) ^{\frac{3}{2}} Q_{0000}, \end{aligned}$$$$\begin{aligned} \int _{\Sigma _t} J_a n^a dS_{\Sigma _t} \sim \int _{\Sigma _t} \left( \left| E_{ab} \right| ^2 + \left| H_{ab} \right| ^2 \right) r^2 (1+r^2) dr. \end{aligned}$$ - b)
**Bulk term**We haveAgain applying Lemma 3.1 to \(\hat{Y}\) and rescaling, we have$$\begin{aligned} \text {Div} J = 3Q_{abcd}({}^Y\pi )^{ab} Y^c Y^d = \frac{\sqrt{3}}{\sqrt{1+r^2}} Q(e_0, e_0, Y, Y). \end{aligned}$$As a result, we have$$\begin{aligned} \text {Div} J \sim \big (1+r^2\big )^{\frac{1}{2}} Q_{0000}. \end{aligned}$$$$\begin{aligned} \int _{S_{[T_1, T_2]}} \text {Div }J d\eta \sim \int _{S_{[T_1, T_2]}} \left( \left| E_{ab} \right| ^2 + \left| H_{ab} \right| ^2 \right) r^2 \sqrt{1+r^2} dt dr d\omega . \end{aligned}$$ - c)
**Boundary term at**\(\mathscr {I}\) Finally, we consider the flux through surfaces \(\tilde{\Sigma }_r\). We haveNow, inserting our expression for$$\begin{aligned} \int _{\tilde{\Sigma }_r} J_a m^a dS_{\tilde{\Sigma }_r} = \int _{\tilde{\Sigma }_r} Q(e_{{\overline{r}}}, Y, Y, Y) r^2 \sqrt{1+r^2} dt d\omega , \end{aligned}$$*Y*and expanding, we haveso that$$\begin{aligned}&Q(e_{{\overline{r}}}, Y, Y, Y) \\&\quad =\left( 1+r^2\right) ^{\frac{3}{2}}\left( Q_{{\overline{r}}000} + \sqrt{3} \frac{r}{\sqrt{1+r^2}} Q_{{\overline{r}}{\overline{r}}00} + \frac{r^2}{1+r^2} Q_{{\overline{r}}{\overline{r}}{\overline{r}}0} + \frac{1}{3 \sqrt{3}} \frac{r^3}{\left( 1+r^2\right) ^{\frac{3}{2}}} Q_{{\overline{r}}{\overline{r}}{\overline{r}}{\overline{r}}} \right) \end{aligned}$$Let us consider the first and third terms on the right hand side.$$\begin{aligned} \lim _{r \rightarrow \infty } Q(e_{{\overline{r}}}, Y, Y, Y) r^2 \sqrt{1+r^2} = \lim _{r \rightarrow \infty } r^6 \left( Q_{{\overline{r}}000} + \sqrt{3} Q_{{\overline{r}}{\overline{r}}00} + Q_{{\overline{r}}{\overline{r}}{\overline{r}}0} + \frac{1}{3 \sqrt{3}} Q_{{\overline{r}}{\overline{r}}{\overline{r}}{\overline{r}}} \right) . \end{aligned}$$^{19}We have:so that$$\begin{aligned} Q_{{\overline{r}}000} + Q_{{\overline{r}}{\overline{r}}{\overline{r}}0}&= 4 E_{AC}\epsilon ^{AB}H_{B}{}^C \\&= 2 \left( E_{AB} -\frac{1}{2} \delta _{AB} E_C{}^C + \epsilon _{(A}{}^CH_{B)C} \right) \\&\quad \times \left( \hat{E}^{AB} -\frac{1}{2} \delta ^{AB} E_D{}^D + \epsilon ^{(A}{}_D H^{B)D} \right) \\&\quad - 2 \left| E_{AB} \right| ^2 - 2 \left| H_{AB} \right| ^2 \end{aligned}$$where we make use of the boundary condition. Taking this together with the expressions for \(Q_{{\overline{r}}{\overline{r}}00}\), \(Q_{{\overline{r}}{\overline{r}}{\overline{r}}{\overline{r}}}\) we have:$$\begin{aligned} \lim _{r \rightarrow \infty } r^6 \left( Q_{{\overline{r}}000} + Q_{{\overline{r}}{\overline{r}}{\overline{r}}0} \right) = -2 \lim _{r \rightarrow \infty } r^6 \left( \left| E_{AB} \right| ^2 + \left| H_{AB} \right| ^2 \right) \end{aligned}$$so that$$\begin{aligned} \lim _{r \rightarrow \infty } Q(e_{{\overline{r}}}, Y, Y, Y) r^2 \sqrt{1+r^2}&= -\lim _{r \rightarrow \infty } r^6 \Bigg [ \left( 2 - \frac{10}{3\sqrt{3}}\right) \left( \left| E_{AB} \right| ^2 + \left| H_{AB} \right| ^2 \right) \\&\quad + \frac{8}{3\sqrt{3}} \left( \left| E_{{\overline{r}}{\overline{r}}} \right| ^2 + \left| H_{{\overline{r}}{\overline{r}}} \right| ^2 \right) \\&\quad + \frac{2}{3\sqrt{3}} \left( \left| E_{A{\overline{r}}} \right| ^2 + \left| H_{A{\overline{r}}} \right| ^2\right) \Bigg ], \end{aligned}$$and$$\begin{aligned} \lim _{r \rightarrow \infty } Q(e_{{\overline{r}}}, Y, Y, Y) r^2 \sqrt{1+r^2} \sim -\lim _{r \rightarrow \infty } r^6\left( \left| E_{ab} \right| ^2 + \left| H_{ab} \right| ^2\right) \end{aligned}$$$$\begin{aligned} \int _{\tilde{\Sigma }_\infty ^{[T_1, T_2]}} J_a m^a dS_{\tilde{\Sigma }_r} \sim - \int _{\tilde{\Sigma }_\infty ^{[T_1, T_2]}} \lim _{r \rightarrow \infty } r^6\left( \left| E_{ab} \right| ^2 + \left| H_{ab} \right| ^2\right) dt d\omega . \end{aligned}$$

As we did in the Maxwell case, we shall now use the structure of the equations to allow us to establish (weighted) integrated decay estimates for all derivatives of the fields *E*, *H*. To control time derivatives we can simply commute with the Killing field *T* and apply Proposition 5.6. To control spatial derivatives we replace the time derivatives by the equations of motion and integrate the resulting cross terms by parts making use also of the constraints equations. The remarkable fact is that in the process we only see spacetime terms with good signs and lower order surface terms that we already control by the estimate before commutation.

We will note the following useful result, which allows the cross term to be integrated by parts:

### Lemma 5.7

*K*be the vector fieldIf

*H*satisfies the constraint equation (Con \(H_{B}\)), we have the identity

### Proof

See Appendix 7.3. \(\square \)

### Proposition 5.7

### Proof

*Z*on \(\mathbb {S}^2\) we have \(Z_{[AB]} = \frac{1}{2} \epsilon _{AB}(\epsilon ^{CD}Z_{CD})\), we deduce that if the constraints hold then (Evol’ \(E_{AB}\)) may be re-written:whence we deduceNow, using the second equality and applying the estimates in Proposition 5.6 for

*H*and the commuted quantity \(\dot{E}\), we can verify that

This gives all the desired estimates for the derivatives of *H*. As in the Maxwell case, similar bounds for the derivatives of *E* can be derived in an identical manner. \(\square \)

Finally, much as in the Maxwell case, we apply the Hardy inequalities to establish integrated decay of the non-degenerate energy with the loss of a derivative:

### Theorem 5.4

*W*is Weyl tensor, satisfying the Bianchi equations with dissipative boundary conditions, as in Theorem 4.3. Then there exists a constant \(C>0\), independent of

*T*such that we have

### Proof

Using the result of Proposition 5.6, 5.7 with the Hardy estimates of Lemma 5.4 to improve the weights near infinity in the integrated decay estimates, making use of a cut-off in much the same way as for the spin 0 and spin 1 problems, we obtain the desired estimates for \(|E_{AB}|\), \(|H_{AB}|\), \(|E_{A\bar{r}}|\) and \(|H_{A \bar{r}}|\). Finally, the bounds for \(|E_{\bar{r}\bar{r}}|\) and \(|H_{\bar{r}\bar{r}}|\) are obtained trivially using the trace-free condition for *E* and *H*. \(\square \)

### 5.4 Proof of Corollary 1.2 (uniform decay)

In this subsection, we show a uniform decay rate for the solutions to the confomal wave, Maxwell and Bianchi equations, hence proving Corollary 1.2. We will in fact prove the uniform decay estimates for all of the equations at once by showing that this is a consequence of the bounds that we have obtained previously. The result below is a combination of relatively standard ideas (for example, see [35] and Prop 3.1 (a) of [36]), but for completeness we include a direct proof.

### Lemma 5.8

*t*which satisfies:

- 1.
\(\mathcal {E}[\Psi ](t)\) is a non-increasing \(C^1\) function of

*t*, - 2.For every \(0\le T_1\le T_2\), \(\mathcal {E}[\Psi ](t)\) satisfies the integrated decay estimate:for some \(C>0\) independent of \(T_1\) and \(T_2\).$$\begin{aligned} \int _{T_1}^{T_2} \mathcal {E}[\Psi ](t) dt \le C \left\{ \mathcal {E}[\Psi ](T_1) + \mathcal {E}[\partial _t \Psi ](T_1) \right\} , \end{aligned}$$

*n*and

*C*.

### Proof

*n*. The \(n=1\) case has just been established. Suppose now that the statement holds for some

*n*. Noticing that the equation commutes with \(\partial _t\), we use the induction hypothesis for both \(\Psi \) and \(\partial _t\Psi \) to obtain

### Proof of Corollary 1.2

### 5.5 Proof of Theorem 1.3: Gaussian beams

It is noteworthy that in the first instance, for all of the integrated decay estimates we obtained above the *r*-weight near infinity is weaker than that for the energy estimate. In particular, in order to show a uniform-in-time decay estimate, we needed to lose a derivative. In this section, we show that without any loss, there cannot be any uniform decay statements for the conformal wave equation. Moreover, an integrated decay estimate with no degeneration in the *r*-weight does not hold.

In order to show this, we will construct approximate solutions to the conformally coupled wave equation for a time interval [0, *T*] with an arbitrarily small loss in energy. We will in fact first construct a Gaussian beam solution on the Einstein cylinder and make use of the fact that (one half of) the Einstein cylinder is conformally equivalent to the AdS spacetime to obtain an approximate solution to the conformally coupled wave equation on AdS.

In the following, we will first study the null geodesics on the Einstein cylinder. We then construct Gaussian beam approximate solutions to the wave equation on the Einstein cylinder. Such construction is standard and in particular we follow closely Sbierski’s geometric approach [26] (see also [37, 38]). After that we return to the AdS case and build solutions that have an arbitrarily small loss in energy.

#### 5.5.1 Geodesics in the Einstein cylinder

*E*and

*L*defined as

*s*. We require from now on that \(0\le |L|<E\). The geodesic equation therefore reduces to the ODE

#### 5.5.2 Constructing the Gaussian beam

*a*on \(\gamma \). More precisely, on \(\gamma \), we require the following conditions:

- (1)
\(\varphi (\gamma (s))=0 \)

- (2)
\(d\varphi (\gamma (s))=\dot{\gamma }(s)_{\flat } \)

- (3)The matrix \(M_{\mu \nu }:= \partial _\mu \partial _\nu \varphi (\gamma (s))\) is a symmetric matrix satisfying the ODEwhere$$\begin{aligned} \frac{d}{ds} M=-A-BM-MB^T-MCM, \end{aligned}$$
*A*,*B*,*C*are matrices given byand obeying the initial conditions$$\begin{aligned} A_{\kappa \rho }= & {} \frac{1}{2}(\partial _\kappa \partial _\rho g^{\mu \nu })\partial _\mu \varphi \partial _\nu \varphi ,\\ B_{\kappa \rho }= & {} \partial _\kappa g^{\rho \mu }\partial _\mu \varphi ,\\ C_{\kappa \rho }= & {} g^{\kappa \rho }, \end{aligned}$$- (a)
*M*(0) is symmetric; - (b)
\(M(0)_{\mu \nu }\dot{\gamma }^\nu =(\dot{\partial _\mu \varphi })(0)\);

- (c)
\(\mathfrak {I}(M(0)_{\mu \nu })dx^\mu \left. \right| _{\gamma (0)}\otimes dx^\nu \left. \right| _{\gamma (0)} \) is positive definite on a three dimensional subspace of \(T_{\gamma (0)}M\) that is transversal to \(\dot{\gamma }\).

- (a)
- (4)\(a(\gamma (0))\ne 0\) and
*a*satisfies the ODEalong \(\gamma \).$$\begin{aligned} 2 \text{ grad } \varphi (a)+\Box \varphi \cdot a=0 \end{aligned}$$

The main result^{20} of Sbierski regarding the approximate solution constructed above is the following theorem^{21} (see Theorems 2.1 and 2.36 in [26]):

### Theorem 5.5

*E*and

*L*as above, let

- (1)
\(\Vert \Box w_{E,L,\lambda ,\mathcal {N}}\Vert _{L^2(\mathcal {S}_{[0,T]})}\le C(T)\);

- (2)
\(\hat{E}_0(w_{E,L,\lambda ,\mathcal {N}})\rightarrow \infty \text{ as } \lambda \rightarrow \infty \);

- (3)
\(w_{E,L,\lambda ,\mathcal {N}}\) is supported in \(\mathcal {N}\), a tubular neighborhood of \(\gamma \);

- (4)Fix \(\mu >0\) and normalize the initial energy of \(w_{E,L,\lambda ,\mathcal {N}}\) byThen for \(\mathcal {N}\) a sufficiently small neighborhood of \(\gamma \) and \(\lambda \) sufficiently large, the following bound holds:$$\begin{aligned} \tilde{w}_{E,L,\lambda ,\mathcal {N}}:=\frac{w_{E,L,\lambda ,\mathcal {N}}}{\sqrt{\hat{E}_0(w_{E,L,\lambda ,\mathcal {N}})}}\cdot E(\gamma ). \end{aligned}$$$$\begin{aligned} \sup _{t\in [0,T]}\left| \hat{E}_t(\tilde{w}_{E,L,\lambda ,\mathcal {N}})-E(\gamma ) \right| <\mu . \end{aligned}$$

We also need another fact regarding the second derivatives of \(\varphi \) which is a consequence of the construction in [26] (see (2.14) in [26]):

### Lemma 5.9

\(\mathfrak {I}(\varphi \left. \right| _\gamma )=\mathfrak {I}(\nabla \varphi \left. \right| _\gamma )=0.\) Moreover, \(\mathfrak {I}(\nabla \nabla \varphi \left. \right| _\gamma )\) is positive definite on a 3-dimensional subspace transversal to \(\dot{\gamma }\).

The fact that \(a_{\mathcal {N}}\) is independent of \(\lambda \) together with Lemma 5.9 imply that the Gaussian beam approximate solution constructed above has bounded \(L^2\) norm independent of \(\lambda \). We record this bound in the following lemma:

### Lemma 5.10

#### 5.5.3 The conformal transformation

*R*(

*g*) is the scalar curvature of the metric

*g*.

*L*. More precisely, we have

### Lemma 5.11

*w*be a function on \(\mathcal {M}_E\) restricted to \(\psi \le \frac{\pi }{2}\). Then we have

### Proof

This can be verified by an explicit computation. \(\quad \square \)

We are now ready to construct the approximate solution. To heuristically explain the construction, consider Fig. 1. On the left we sketch the curve \(\gamma \) along which our approximate solution is concentrated in the Einstein cylinder. To allow this to be plotted, we project onto a surface of constant *t*, \(\theta \) which carries a spherical geometry and can be visualised via its embedding in \(\mathbb {R}^3\). We actually wish to construct an approximate solution on anti-de Sitter, which is conformal to one half of the Einstein cylinder. To do so, we restrict our attention to one hemisphere and arrange that whenever the curve \(\gamma \) strikes the equator of the Einstein cylinder, it is reflected. Each time this occurs, we shall arrange that the Gaussian beam is attenuated by a factor which depends on the angle of incidence (which in turn depends on *E* and *L*). The more shallow the reflection, the weaker the attenuation. Crucially, the time, *t*, between reflections is independent of the angle that we choose. As a result, by taking the angle of incidence to be sufficiently small, we can arrange that an arbitrarily small fraction of the initial energy is lost at the boundary over any given time interval.

To be more precise, let us fix \(T\ge 0\). We will construct an approximate solution for \(t\in [0,T]\). Take \(N\in \mathbb {N}\) be the smallest integer such that \(T<\frac{(4N+3)\pi }{2}\). We then construct an approximate solution for \(t\in (-\frac{\pi }{2}, \frac{(4N+3)\pi }{2})\) starting from the function \(w_{E,L,\lambda ,\mathcal {N}}\) constructed previously. This will correspond to a Gaussian beam which strikes the boundary \(\sim 2N\) times between \(t=0\) and \(t=T\). Notice that the geodesic \(\gamma \) has the property that it lies in the hemisphere \(\{\psi <\frac{\pi }{2}\}\) for \(t\in (2k\pi -\frac{\pi }{2},2k\pi +\frac{\pi }{2})\) (for \(k\in \mathbb {Z}\)) and that it lies in the other hemisphere, i.e., \(\{\psi >\frac{\pi }{2}\}\), for \(t\in ((2k+1)\pi -\frac{\pi }{2},(2k+1)\pi +\frac{\pi }{2})\) (for \(k\in \mathbb {Z}\)). Therefore, we will assume without loss of generality that the neighborhood \(\mathcal {N}\) has been taken sufficiently small such that it lies entirely in \(\{\psi <\frac{\pi }{2}\}\) for \(t\in (2k\pi -\frac{\pi }{4},2k\pi +\frac{\pi }{4})\) (and entirely in \(\{\psi >\frac{\pi }{2}\}\) for \(t\in ((2k+1)\pi -\frac{\pi }{4},(2k+1)\pi +\frac{\pi }{4})\)), where \(k\in \mathbb {Z}\).

*R*is taken to be \(R=-\frac{E-\sqrt{E^2-L^2}}{E+\sqrt{E^2-L^2}}\) and \(\chi \) is the indicator function. Notice in particular that when the time cutoff function is 0 in the first term (resp. in the second term), the support of \(w_{E,L,\lambda ,\mathcal {N}}\) is entirely in \(\{\psi >\frac{\pi }{2}\}\) (resp. \(\{\psi <\frac{\pi }{2}\}\)). We also depict this in Fig. 2, where we denote

The definition above is such that in the time interval \(t\in (-\frac{3\pi }{4},\frac{3\pi }{4}]\), we take \(w_{E,L,\lambda ,\mathcal {N}}\) constructed previously, restrict it to \(\psi \le \frac{\pi }{2}\) and rescale it by \(\frac{1}{\sqrt{1+r^2}}\). Then on the time interval \(t\in (\pi -\frac{3\pi }{4},\pi +\frac{3\pi }{4}]\), we take the part of \(w_{E,L,\lambda ,\mathcal {N}}\) that is supported in \(\psi \ge \frac{\pi }{2}\), reflect it across the \(\psi =\frac{\pi }{2}\) hypersurface, rescale by a factor \(\frac{1}{\sqrt{1+r^2}}\) and then multiply by the factor *R*. As we will see later, the factor *R* is chosen so that the boundary conditions are approximately satisfied. We then continue this successively, taking parts of the solutions in \(\psi \le \frac{\pi }{2}\) and \(\psi \ge \frac{\pi }{2}\), reflecting when appropriate, and multiplying by factors of *R*’s.

### Lemma 5.12

*Proof.*First note that \(\hat{E}_0[w_{E,L,\lambda }]\le CE_{0}[u_{E,L,\lambda }]\), so the first claim follows from Theorem 5.5. Next recall that we have assumed that the neighbourhood \(\mathcal {N}\) has been taken sufficiently small so that for each of the summands, the indicator function \(\chi \) is constant on the support of \(w_{E,L,\lambda ,\mathcal {N}}\) when restricted to a hemisphere. It therefore suffices to consider only the contributions from \(w_{E,L,\lambda ,\mathcal {N}}\) (as no derivatives fall on the indicator functions). We will write

*R*, \(((\partial _t\varphi +\partial _\psi \varphi )+R(\partial _t\varphi -\partial _\psi \varphi ))\) vanishes on \(\gamma \). More precisely, by points (1), (2) in Sect. 5.5.2 and Lemma 5.9, we have

*R*. More precisely, we have

### Lemma 5.13

*f*is a function defined on \(\{\psi =\frac{\pi }{2}\}\) which vanishes to order 0 at \((t=\frac{\pi }{2},\, \theta =\frac{\pi }{2},\, \phi = \int _0^{\frac{E\pi }{2}}\frac{L E^2}{(E^2-L^2)\sin ^2(Es')+L^2})\). Then

### Proof

*f*vanishes to order 0 is equivalent to

#### 5.5.4 Building a true solution

Given the approximate solution constructed above, we are now ready to build a true solution to the homogeneous conformally coupled wave equation with dissipative boundary condition. To this end, we need a strengthening of Theorem 5.1 which includes inhomogeneous terms:

### Theorem 5.6

*u*be a solution of the inhomogeneous conformally coupled wave equation

### Proof

*g*, so we have

### Theorem 5.7

### Proof

In particular, by taking |*L*| sufficiently close to *E* and \(\epsilon \) sufficiently small, this implies that on the time interval [0, *T*], the loss of energy can be arbitrarily small. This also implies that any uniform integrated decay estimates without loss do not hold:

### Corollary 5.8

*u*to the conformal wave equation with finite initial energy subject to dissipative boundary conditions.

*u*to the conformal wave equation with finite initial energy subject to dissipative boundary conditions.

### Remark 2

We are grateful to an anonymous referee, who points out that our construction above can in fact be adapted to establish the stronger statement that the degeneracy in *r* for the integrated decay rate established above is in fact optimal. That is to say for any \(\delta >0\), there can exist no constant *C* such that Proposition 5.2 holds with the weight \(\frac{r^2}{\sqrt{1+r^2}}\) replaced by \(\frac{r^{2+\delta }}{\sqrt{1+r^2}}\). This in particular suggests that the trapping phenomenon present here is different to the normally hyperbolic trapping observed, for example, at the photon sphere of the Schwarzschild black hole.

## 6 Generalizations

### 6.1 Alternative boundary conditions

#### 6.1.1 Conformal Wave

*u*satisfied the boundary condition

*u*to the boundary condition. This can also be handled by the methods above, but this will require combining the energy and integrated decay estimates.

#### 6.1.2 Maxwell

*E*,

*H*to appear in the boundary condition with small coefficients and this can be handled by combining the energy and integrated decay estimates.

#### 6.1.3 Bianchi

Of course, this does not imply that boundary conditions which don’t satisfy this inequality lead to growth, simply that our approach breaks down when the boundary conditions are too far from the “optimally dissipative” ones (64).

### 6.2 The Dirichlet problem for (B)

It is possible to extract from the Bianchi equations a symmetric hyperbolic system on \(\mathscr {I}\) involving only \({r^3E_{A{\overline{r}}}}, {r^3H_{{\overline{r}}{\overline{r}}}}\), where \(r^3\hat{H}_{AB}\) appears as a source term. Using this system it is possible to show that if \({r^3E_{A{\overline{r}}}}\) and \({r^3H_{{\overline{r}}{\overline{r}}}}\) vanish at infinity for the initial data then this condition propagates. Moreover, it is easy to see how to construct a large class initial data satisfying this vanishing condition at infinity, as well as a large class *not* satisfying it illustrating that fixing the conformal class of the metric on the boundary is a more restrictive condition than purely fixing Dirichlet-conditions on the Weyl tensor.

In conclusion, for initial data satisfying the vanishing condition, we may return to the estimate (43) and establish directly that solutions of the Bianchi system representing a linearised gravitational perturbation fixing the conformal class of the boundary metric are bounded. This is in accordance with the results of [39], in which it is shown that metric perturbations obeying the linearised Einstein equations can be decomposed into components which separately obey wave equations admitting a conserved energy.

### 6.3 The relation to the Teukolsky equations

We finally contrast our result with an alternative approach to study the spin 2 equations on AdS, which has a large tradition in the asymptotically flat context and relies on certain curvature components satisfying decoupled wave equations. As we will see below, however, in the AdS context this approach merely obscures the geometric nature of the problem and does not provide any obvious simplification as the resulting decoupled equations couple via the boundary conditions (and moreover do not admit a conserved energy).

^{22}and choose as basis \(e_1 = r^{-1} \partial _\theta \), \(e_2 = (r \sin \theta )^{-1} \partial _\phi \). We then write:

*W*can be recovered by solving an elliptic system coupled to the symmetric hyperbolic system in the boundary discussed in Sect. 6.2. It might appear that one can simply study the decoupled equations for \(\Psi ^\pm \) separately. Unfortunately, in general, the correct boundary conditions couple the equations.

^{23}Inserting this into the Bianchi equations we can derive that

We remark that the Teukolsky approach was recently used in [28] to consider perturbations of the Kerr-AdS family of metrics (which includes anti-de Sitter for \(a=m=0\)). In this paper, the Teukolsky equation is separated and a boundary condition (preserving the conformal class of the metric at infinity) is proposed for the radial part of each mode of \(\Psi ^{\pm }\)*separately*. This appears to contradict our discussion above. When one examines equations (3.9–12) of [28] one sees spectral parameters appearing up to fourth order.^{24} Returning to a physical space picture, these will appear as fourth order operators on the boundary. Accordingly, it is far from clear whether these boundary conditions can be meaningfully interpreted as giving boundary conditions for a dynamical evolution problem. Indeed, our heuristic argument for the coupling strongly suggests that the conditions of [28] understood as boundary conditions for a dynamical problem *cannot* give rise to a well posed evolution. That is not to say that these boundary conditions are not suitable for calculating (quasi)normal modes: any such mode will certainly obey these conditions, providing a useful trick to simplify such computations.

## Footnotes

- 1.
To simplify the algebra we choose \(\Lambda =-3\) throughout the paper.

- 2.
The causal nature of this boundary is clearer in the Penrose picture discussed in Sect. 1.5 below.

- 3.
We use here the nomenclature of the putative AdS/CFT correspondence.

- 4.
See [26] for a discussion of the Gaussian beam approximation on general Lorentzian manifolds.

- 5.
These will be more involved in view of the fact that the equations involve constraints.

- 6.
The introduction of this system in the context of the full non-linear problem goes back to [10].

- 7.
- 8.
Obviously there’s no global orthonormal basis for \(\mathbb {S}^2\). We can either take multiple patches, or else understand the capital Latin indices as abstract indices.

- 9.
When acting on a scalar, we will sometimes write \(\tilde{\nabla }_\mu f =: \tilde{\partial }_\mu f\).

- 10.
\(\epsilon _{12} = 1\) in this basis.

- 11.
Theorem 1.1 of course establishes a stronger (global) estimate. The key point here is to explain why the naive approach using only the Bel-Robinson tensor and the Killing fields fails and also to derive the (reduced) system of equations which will be needed in the second part of the proof of Theorem 1.1.

- 12.
Or at least Friedrichs

*symmetrizable*, but the distinction is not important for our purposes. The key issue is the existence of a good energy estimate. Providing this exists, putting the system into symmetric hyperbolic form is straightforward. - 13.
Here and elsewhere, in expressions like \(\left| E_{A{\overline{r}}} \right| ^2 +\left| E_{AB} \right| ^2\), contraction over the indices

*A*,*B*etc. is implied, i.e. \(\left| E_{A{\overline{r}}} \right| ^2 +\left| E_{AB} \right| ^2 = E_{A{\overline{r}}}E^A{}_{\overline{r}}+ E_{AB}E^{AB}\). - 14.
For instance, specifying \(E_{AB}=0\) on \(\mathscr {I}\) will clearly lead to a unique solution of the modified system. On the other hand, the same boundary condition will not completely fix the boundary term occurring in (43) in the unmodified formulation, illustrating the severe drawback of the latter.

- 15.
We shall take the frugal approach of commuting with the timelike Killing field. If one is happy to exploit the angular symmetries of AdS, our approach can be simplified.

- 16.
Notice that there is no surface term contribution at infinity arising from Open image in new window due to the decay of the angular derivative of

*H*in*r*by the local well-posedness theorem (Theorem 4.2). - 17.
As we will see in the proof, the key difference to the case of the wave- and Maxwell’s equations is that here we will have to establish both these estimates at the same time!

- 18.
We write \(f \sim g\) to mean that there exists a numerical constant \(C>0\) such that \(C^{-1} f \le g \le C f\).

- 19.
The factor \(\frac{1}{\sqrt{3}}\) appearing in the definition of

*Y*was chosen to arrange a cancellation between these terms. - 20.Translated into our notation, the result in [26] requires the following bounds on the geometry of \((\mathcal {M}_E, g_E)\):where \(e_i\) is an orthonormal frame on the \(\mathbb {S}^3\) slice. These estimates are obviously satisfied in our setting.$$\begin{aligned} -C\le g(\partial _t,\partial _t)\le c<0,\,|\nabla \partial _t(\partial _t,\partial _t)|+|\nabla \partial _t(\partial _t,e_i)|+|\nabla \partial _t(e_i,e_j)|\le C, \end{aligned}$$
- 21.
Regarding point (4) in the theorem below, the original work of Sbierski gives a more general characterization of the energy of Gaussian beams in terms of the the energy of geodesics on general Lorentzian manifold. Since in our special setting, the energy of a geodesic is conserved, we will not record the most general result but will refer the readers to [26] for details.

- 22.
The \((r, t,\theta ,\phi )\) do not quite cover AdS, and so some care must be taken at the axis. For the purposes of this section, we shall ignore this difficulty.

- 23.
In the general case, their trace on the boundary can de determined by solving transport equations as outlined in Sect. 6.2.

- 24.
In the presence of rotation things are even worse, as the square root in the boundary conditions implies that the operator on the boundary is

*non-local*.

## Notes

### Acknowledgements

G.H. is grateful for the support offered by a grant of the European Research Council (Grant No. 337488). G.H., J.S. and C.M.W. are grateful to the Newton Institute for support during the programme “Mathematics and Physics of the Holographic Principle”. J.L. thanks the support of the NSF Postdoctoral Fellowship DMS-1204493. J.S. acknowledges funding from the European Research Council under the European Union’s Horizon 2020 research and innovation program (project GEOWAKI, grant agreement 714408).

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