Linear Stability of Schwarzschild-Anti-de Sitter spacetimes I:
The system of gravitational perturbations

From a PDE perspective, non-linear stability results typically rely on a robust understanding of the underlying linearised problem including quantitative estimates on the rates of decay in the geometry under consideration. Such an analysis also seems a prerequisite for non-linear instability results in order to gain control on potential blow-up or growth mechanisms. Estimating the linearisation of the equations (1) requires choosing a gauge and, independently of the specifics of the gauge, already results in a complicated coupled system of equations. A good first intuition can often be gained from the study of the scalar wave equation $\Box_{g}\psi=0$ on the background under consideration. This removes the problem of gauge as well as the coupled tensorial character of the problem. At the same time, the scalar equation (due to its Lagrangian structure) inherits natural coercive conservation laws from the symmetries of the background which can be exploited in the analysis.

In the case of pure AdS, one thus discovers the existence of time-periodic (i.e. non-decaying) solutions of $\Box_{g_{AdS}}\psi=0$, which lie at the heart of the non-linear instability exploited in [Mos18]. In the case of asymptotically AdS black holes, the corresponding scalar problem was studied in [HS13, HS14], where it is shown that solutions of $\Box_{g_{KAdS}\psi}+\mu\psi=0$ with Dirichlet boundary conditions decay inverse logarithmically and not faster on Kerr-AdS black holes whose parameters satisfy the Hawking-Reall bound.²²2Beyond that bound, one has exponentially growing solutions. See [Dol17]. In view of the slow decay, the authors of [HS14] conjectured non-linear instability of these black holes. Recently, a concrete instability mechanism (related to weak turbulence and the growth of higher order Sobolev norms) has been suggested for a non-linear scalar toy-model on Schwarzschild-AdS [KM].

The goal of this series of works is to show that the logarithmic decay established for the scalar toy problem also holds for the linearised Einstein equations on Schwarzschild-AdS. More precisely we will prove the following statement:

For a precise statement of the theorem, see already Theorem 4.7 below.

We will only very briefly comment on the global structure of the proof of the main theorem here. Afterwards, we immediately provide the formal set-up for the problem in Section 2, which includes a derivation of the linearised Einstein equations in double null gauge. Section 3 is concerned with the construction of appropriate initial data and solutions to this linearised system, i.e. well-posedness of the linearised system. A formal version of the main theorem is then formulated in Section 4 and proven in Section 5. The impatient reader wishing to take the existence of solutions of the linearised system (Theorem 3.9) for granted may turn immediately to the main theorem in Section 4, which concerns the global properties of such solutions.

At the highest level, our strategy follows closely that of [DHR19] in the asymptotically flat ($\Lambda=0$) case and starts by expressing the linearised Einstein equations in a double null gauge. A first key ingredient of the analysis, carried out in our companion paper [GH24], is to prove boundedness and decay estimates for the so-called Teukolsky quantities, denoted $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}},\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$. These are certain linearised null-curvature components of the linearised system, which (a) do not depend on the specific gauge in which the equations (1) are linearised and (b) satisfy decoupled wave equations.³³3For $\Lambda=0$ these observations go back to Bardeen–Press [BP73] and Teukolsky [Teu72] in the physics literature and are easily generalised to $\Lambda\neq 0$, see [Kha83]. In our case, the two equations couple through the boundary condition imposed at the conformal boundary. The second key ingredient is to exploit the hierarchical structure of the double null gauge to prove boundedness of all geometric quantities in a gauge normalised with respect to initial data using the bounds for the Teukolsky quantities. As already mentioned, in contrast to the asymptotically flat case, all Ricci coefficients and null curvature components can be shown to decay without adding a residual pure gauge solution. The reason can be understood as follows. As in [DHR19], one first proves boundedness and decay of the linearised shear $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$ from the estimates for $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$. This relies on the (commuted) redshift effect for $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$. The quantity $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$ then inherits this decay through the boundary conditions and we can hence integrate $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$ in the ingoing direction from the boundary (using the estimates for $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$) to establish boundedness and decay for $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$.⁴⁴4This second step is not possible in the asymptotically flat case as there is no boundary and the $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$ equation cannot be integrated directly from data in the ingoing direction either because of unfavourable $r$-weights in the integrating factor. Decay for the other Ricci-coefficients and curvature components then follows by going hierarchically through the system analogous to [DHR19], except that here the boundary condition and its consequences need to be exploited at various stages. While one does not have to add a pure gauge solution to establish decay, one can improve the radial decay of certain geometric quantities and ensure that also the metric on the double null spheres converges to the round metric in standard form by adding one. Such a pure gauge solution is constructed from the trace of the original solution at the conformal boundary and controlled uniformly by initial data. See Theorem 4.14 below.

G.H. acknowledges support by the Alexander von Humboldt Foundation in the framework of the Alexander von Humboldt Professorship endowed by the Federal Ministry of Education and Research as well as ERC Consolidator Grant 772249. Both authors acknowledge funding through Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics-Geometry-Structure.

Let $\mathcal{Q}\subset\mathbb{R}^{2}_{U,V}$ be the $2$-dimensional submanifold with (piecewise smooth) boundary defined by

We define the associated $4$-dimensional manifold

equipped with coordinates $(U,V,\theta^{1},\theta^{2})$, which we will refer to as Kruskal coordinates on $\mathcal{M}$. We denote the boundary components

which will be referred to as null infinity, the future event horizon and the initial data hypersurface respectively. Observe that all boundary components are topologically $\mathbb{R}\times\mathbb{S}^{2}$. We denote by $S^{2}_{U,V}$ the $2$-spheres $(U,V)\times S^{2}$ in $\mathcal{M}$.

Given a fixed parameter $M>0$ we can define coordinates $(u,v,\theta^{1},\theta^{2})$ on $\mathcal{M}\setminus\mathcal{H}^{+}$ by

Note $v\geq 0$ on $\mathcal{M}\setminus\mathcal{H}^{+}$ and $v=u$ on $\mathcal{I}$. Defining also $r^{\star}(u,v)=v-u$ and $t(u,v)=v+u$ we have another coordinate system $(t,r^{\star},\theta^{1},\theta^{2})$ on $\mathcal{M}\setminus\mathcal{H}^{+}$. We observe that $r^{\star}$ is a boundary defining function for $\mathcal{I}$ in that it vanishes at $\mathcal{I}$ and $\partial_{u}r^{\star}=-1$, $\partial_{v}r^{\star}=1$. One may parametrise the boundary $\mathcal{I}=[0,\infty)_{t}\times S^{2}_{t,t}$ by the coordinate $t$. Finally, we can define on $\mathcal{M}\setminus\mathcal{H}^{+}$ a function $r(u,v)=r(r^{\star}=v-u)$ by the relation

where $-3k^{2}=\Lambda$. Note that the function $r$ depends on $M$ and that we have the asymptotics

Let us denote

and fix $M>0$, which we may think of as the mass of a Schwarzschild-AdS background metric we are about to install on $\mathcal{M}_{int}$. Given $\mathcal{M}_{int}$, equipped with local coordinates $(u,v,{\theta}^{1},{\theta}^{2})$ on $\mathcal{M}_{int}\setminus\mathcal{H}^{+}$, we consider a $1$-parameter family of metrics $\boldsymbol{g}(\epsilon)$ expressed in double null gauge⁵⁵5Note that the $\boldsymbol{b}$ is on $du$ here instead of on $dv$ as in [DHR19]. As is well-known, this does not change the null-structure and Bianchi equations (collected in Section 2.3.5 below). The only change is in the propagation equation for the metric component $\boldsymbol{b}$, equation (25), which is now in the outgoing direction and with an additional minus on the right. An outgoing transport equation for $\boldsymbol{b}$ is desirable as it can be integrated from data, where $\boldsymbol{b}$ is normalised. Note also that it is $\boldsymbol{\Omega}^{-2}\boldsymbol{b}$ which extends regularly to the horizon $\mathcal{H}^{+}$ as can be seen by transforming (9) to the regular Kruskal coordinates.

such that

1. 1.

   The $\boldsymbol{g}\left(\epsilon\right)$ satisfy on $\mathcal{M}_{int}\setminus\mathcal{H}^{+}$ the Einstein equations with negative cosmological constant $\Lambda=-3k^{2}$,

   $\displaystyle\mathrm{Ric}(\boldsymbol{g})=-3k^{2}\boldsymbol{g}\,.$

   (10)

2. 2.

   We have that $\boldsymbol{g}\left(0\right)$ is the Schwarzschild-AdS metric of mass $M$ and cosmological constant $\Lambda=-3k^{2}$

   $\displaystyle\boldsymbol{g}\left(0\right):=-4\left(1-\frac{2M}{r(u,v)}+k^{2}\cdot r^{2}({u},{v})\right)d{u}d{v}+r^{2}({u},{v}){\gamma}_{AB}d{\theta}^{A}d{\theta}^{B}\,,$

   (11)

   where $r(u,v)$ is defined as in (6) and ${\gamma}$ denotes the round metric on the unit sphere.

3. 3.

   The $\boldsymbol{g}\left(\epsilon\right)$ are asymptotically Anti-de Sitter in that the function $(v-u)^{2}\boldsymbol{\Omega}^{2}$ as well as the conformally rescaled metric $\boldsymbol{\Omega}^{-2}\boldsymbol{g}$ extend regularly to $\mathcal{I}$, i.e. in particular they can be defined on the larger manifold $\mathcal{M}\setminus\mathcal{H}^{+}$. Specifically, recalling $t=u+v$, the family

   $\displaystyle\boldsymbol{g}^{(3)}_{\mathcal{I}}(\epsilon)=-d{t}^{2}+\frac{\not{\boldsymbol{g}}_{AB}(\epsilon)}{\boldsymbol{\Omega}^{2}(\epsilon)}\left(d\theta^{A}-\boldsymbol{b}^{A}(\epsilon)\frac{d{t}}{2}\right)\left(d\theta^{B}-\boldsymbol{b}^{B}(\epsilon)\frac{d{t}}{2}\right)\,$

   (12)

   defines a smooth family of $3$-dimensional Lorentzian metrics on $\mathcal{I}$.

4. 4.

   The $\boldsymbol{g}^{(3)}_{\mathcal{I}}(\epsilon)$ are all conformal to the Lorentzian cylinder $(\mathbb{R}\times S^{2},-k^{2}\mathrm{d}t^{2}+\gamma)$, the latter being the metric induced on $\mathcal{I}$ by the conformally rescaled Schwarzschild-AdS metric $\boldsymbol{\Omega}^{-2}\boldsymbol{g}(0)$. In particular, the $\boldsymbol{g}^{(3)}_{\mathcal{I}}(\epsilon)$ are all locally conformally flat.

5. 5.

   In regular Kruskal coordinates, the $\boldsymbol{g}\left(\epsilon\right)$ and regular derivatives thereof extend smoothly to the boundary $\mathcal{H}^{+}$ of $\mathcal{M}_{int}$ in the sense of Section 5.1.1 of [DHR19]. In particular, arbitrary concatenations of frame vectors from the set $\{\boldsymbol{\Omega}^{-2}(\epsilon)(\partial_{u}+\boldsymbol{b}^{A}\partial_{A}),\partial_{v},\partial_{\theta^{1}},\partial_{\theta^{2}}\}$ applied to $\boldsymbol{g}\left(\epsilon\right)$ extend regularly to $\mathcal{H}^{+}$.

The existence of such families is of course an implicit assumption. Locally in time, i.e. for a finite $v$-interval this can be fully justified by a local-well-posedness theorem in double null gauge. Such a theorem could either be inferred from the literature [Fri95, EK19] or be proven directly by combining the (linear) estimates obtained in this paper with an appropriate contraction mapping argument. Since our main theorem is formulated directly as a statement concerning the linearised system (for which we will prove well-posedness directly) we will not address this issue further.

For the reader’s convenience we briefly recall the basic geometric notions in a double null gauge. The familiar reader can jump immediately to Section 2.6 while the reader unfamiliar with the double null gauge can consult Section 3 of [DHR19] or the original [Chr09] for many more details.

Associated with a double null gauge (9) on $\mathcal{M}_{int}\setminus\mathcal{H}^{+}$ is a double null frame consisting of the null vectorfields

satisfying ${\bf g}(e_{3},e_{4})=-2$, ${\bf g}(e_{3},e_{3})={\bf g}(e_{4},e_{4})=0$, which is complemented with a local coordinate frame on $S^{2}_{u,v}$, $e_{A}=\partial_{A}$ for $A\in\{1,2\}$, satisfying ${\bf g}(e_{A},e_{B})=\not{\boldsymbol{g}}_{AB}$. Note ${\bf g}(e_{3},e_{A})=0$ and ${\bf g}(e_{4},e_{A})$ for $A\in\{1,2\}$.

We collect the asymptotic behaviour towards the conformal boundary for the geometric quantities on $\mathcal{M}_{int}$ from the assumption that (by $3.$ of Section 2.2) the metric $(u-v)^{2}{\boldsymbol{g}}$ extends regularly to the larger manifold $\mathcal{M}$. The proof of Proposition 2.1 is postponed to Appendix A.

From the assumption that the metric is conformal to the Anti-de Sitter metric on the boundary at infinity ($4.$ of Section 2.2), one has the following boundary conditions for the null curvature components. The proof is postponed to Appendix A.

In this section we discuss the Schwarzschild-AdS manifold $(\mathcal{M}_{int},g=\boldsymbol{g}(0))$. In complete analogy with [DHR19] we use unbolded notation to indicate $\epsilon=0$ quantities, for instance we write $\Omega$, $\not{g}$ and $b$ for the metric components, $\hat{\chi}$ for the outgoing shear etc.

Moreover, all the constructions and definitions of Sections 2.3.1–2.3.4 may be repeated in unbolded notation. As this is done in detail in Section 4.3.1 of [DHR19] we only give brief summary.