Exponentially-growing Mode Instability on the Reissner–Nordström-Anti-de-Sitter black holes

The study of the Klein–Gordon equation on asymptotically AdS spacetimes has been initiated in a series of works [34, 35]. In [34], Holzegel proved the local well-posedness results for Klein–Gordon equations on general asymptotically AdS spacetimes under the Dirichlet boundary conditions with $-\frac{9}{4}<\alpha<0$. For the Neumann boundary conditions, the local well-posedness results are still available [55] for $-\frac{9}{4}<\alpha<-\frac{5}{4}$. [21] establishes the local well-posedness under the dissipative boundary conditions for some certain range of $\alpha$. In [35], Holzegel and Smulevici proved the local well-posedness results for the spherically symmetric Einstein–Klein–Gordon equations with a negative cosmological constant $\Lambda$ under Dirichlet boundary conditions with $-2\leq\alpha<0$.

The pure AdS spacetime takes the form of

$$g_{AdS}=-\left(1+\bigl{(}-\frac{\Lambda}{3}\bigr{)}r^{2}\right)dt^{2}+\left(1+\bigl{(}-\frac{\Lambda}{3}\bigr{)}r^{2}\right)^{-1}dr^{2}+r^{2}d\sigma^{2}.$$

In [36], the following conjecture was made for the pure AdS with optimally dissipative boundary conditions.

Holzegel–Luk–Smulevici–Warnick showed the standard energy boundedness and integrated decay result for the Klein–Gordon equation with $\alpha=-2$ under optimally dissipative boundary condition on the pure AdS spacetime in [36], which is the first step toward the proof of Conjecture 1.6.

In [11, 12], the following instability conjecture about the pure AdS spacetime with reflecting boundary conditions was made by Holzegel and Dafermos.

The above conjecture relies on the expectation that, for any small perturbations to pure AdS initial data, the solutions to the Einstein vacuum equations with a negative cosmological constant $\Lambda$ will form a trapped surface region. In a series of works [47, 46, 48, 49], Moschidis proved the instability of AdS for the Einstein-null dust system and Einstein-massless Vlasov system with reflecting boundary conditions under the spherically symmetric setting.

In [38, 40], Holzegel and Smulevici showed a sharp inverse logarithmic decay rate for the Klein–Gordon equation on the Schwarzschild-AdS spacetime ¹¹1In fact, under the spherically symmetric assumption of $\phi$, they can further show $\phi$ decays exponentially in time. under the Dirichlet boundary condition. Moreover, their sharp logarithmic decay result also holds for the Kerr-AdS spacetime satisfying the Hawking–Reall bound. One should note that the Hawing–Reall bound is crucial in obtaining the decay. In fact, Dold [19] constructed growing mode solutions of the Klein–Gordon equation on the Kerr-AdS spacetime violating the Hawking–Reall bound.

Once the decay result for the Klein–Gordon equation is obtained, the next step is to study the (in)stability of linearized gravity. For the Kerr spacetime, Teukolsky derived two decoupled equations for the linearized curvature components in [53]. Most recently, Graf and Holzegel proved mode stability for the analogous Teukolsky equations on the Kerr-AdS spacetime satisfying the Hawking–Reall bound in their work [23], thus ruling out the possibility of growing mode solutions.

However, the existence of growing mode solutions to the Klein–Gordon equation on the Reissner–Nordström-AdS spacetime constructed in this paper leads to the expectation of mode instability of the analogous Teukolsky equations, in contrast to the cases of Schwarzschild-AdS and Kerr-AdS. We will come back to this problem in our future work.

Now we discuss nonlinear (in)stability results for asymptotically AdS spacetimes. In [39], Holzegel and Smulevici showed that, for any small spherical perturbation of the Schwarzschild-AdS initial data, under the Dirichlet boundary conditions, solutions of the Einstein–Klein–Gordon equations with a negative cosmological constant will converge to the Schwarzschild-AdS spacetime exponentially in time, demonstrating the spherical stability of the Schwarzschild-AdS spacetime.

Most recently, in a series of works [24, 25] establishing the linear stability of Schwarzschild-AdS, Holzegel and Graf showed that under Dirichlet-type boundary conditions, the solutions to the Teukolsky equations with perturbed Schwarzschild-AdS initial data converge to Kerr-AdS with an inverse logarithmic rate in time.

Despite decay results for the Klein–Gordon equations and the Teukolsky equations, this inverse logarithmic decay rate is believed to be too slow to ensure the nonlinear stability of the Kerr-AdS spacetime. Hence, in [38], Holzegel and Smulevici made the following conjecture:

In contrast, the growing mode solutions we construct in this paper show instability even at the linear level. We naturally expect the nonlinear instability of the Reissner–Nordström-AdS spacetimes as solutions to the Einstein–Maxwell equations with a negative cosmological constant $\Lambda$ under reflecting boundary conditions.

Growing mode solutions on the Kerr and Kerr-AdS spacetimes violating the Hawking–Reall bound have been constructed in [51, 19] respectively. Both spacetimes exhibit superradiance induced by rotation, namely, negative energy at the event horizon, which gives rise to the growing mode solutions²²2For the Klein–Gordon equation on Kerr, growing mode solutions are due to the combination of the superradiance and massive character of the equation. Heuristically, the Klein–Gordon mass serves as a reflecting mirror that would reflect the superradiance and result in a “black-hole bomb”; see [51, 15, 17, 59]. Note on the contrary, the massless wave equation on the Kerr spacetime does not admit exponentially growing mode solutions [52, 56]. Moreover, the solutions to the wave equation on Kerr are bounded and decay in time [13, 14, 1].. See [51, 19] for a detailed discussion of this superradiant instability. See also [50, 15, 17, 59] for some previous heuristic discussions.

While the rotation-induced superradiance phenomenon affects the Klein–Gordon equation on the Kerr/Kerr-AdS spacetimes, there is no such phenomenon on Reissner–Nordström/Reissner–Nordström-AdS since they are spherically symmetric. However, if one considers the charged Klein–Gordon equation on the charged spacetime, there is a charged analog of the superradiance induced by the coupling of the black hole charge $e$ and the scalar field charge $q_{0}$; see [18, 8, 16, 2] and references therein. Growing mode solutions on Kerr-Newman-dS and Reissner–Nordström-dS have been constructed in [5], for spacetimes where the product of the angular momentum and the Klein–Gordon mass is small compared to the product of the black hole charge $e$ and the scalar field charge $q_{0}$, which is a rigourous mathematical realization of this charge-induced superradiant instability. However, if the coupling is not strong, namely, the product of the black hole charge $e$ and the scalar field charge $q_{0}$ is small, then the exponential decay of the local energy is obtained in [4] for the Klein–Gordon equation on Reissner–Nordström-dS.

The Breitenlohner–Freedman bound $\alpha>-\frac{9}{4}$ is crucial to provide a positive scalar field energy near the infinity. This is exploited to prove the local well-posedness of the asymptotically AdS spacetime; see [6] for the original physics argument and [35, 55] for a rigorous mathematics proof. The charged Klein–Gordon equation (1.1) takes the form of the uncharged Klein–Gordon equation with a effective mass

$$\alpha_{eff}=\alpha-\frac{q_{0}^{2}A_{t}^{2}}{\bigl{(}-\frac{\Lambda}{3}\bigr{)}\Omega^{2}},$$

for which the relevant Breitenlohner–Freedman bound $\alpha_{eff}>-\frac{9}{4}$ is satisfied when $r$ approaches infinity. Hence, we still have the local well-posedness result. However, for large charge $q_{0}$, it is possible to violate the analogous condition of the Breitenlohner–Freedman bound

$$\alpha_{eff}\leq-\frac{9}{4}$$

for a bounded value of $r$. We exploit this fact in the proof of Lemma 6.2, which plays an important role in the construction. This superradiant mechanism has been identified as tachyonic instability in the context of physics literatures; see for instance [8, 29].

There is, however, no analogous superradiant mechanism for the uncharged Klein–Gordon equation on the Reissner–Nordström-AdS spacetime. Instead, the existence of growing mode solutions results from the black hole being too close to the extremality, which is called the near-extremal instability [29].

We try to illustrate the mathematical meaning of this new type of instability by comparing the uncharged Klein–Gordon equations on Schwarzschild-AdS and Reissner–Nordström-AdS. Using the $(t,r,\theta,\varphi)$ coordinates as in (1.2), let $\Sigma_{t}=\{t=constant\}$ be the foliation of spacetimes. Putting the Dirichlet boundary conditions at timelike infinity, the standard energy estimate gives

$$\int_{\Sigma_{t_{1}}}T_{\mu\nu}X^{\mu}n^{\nu}dvol=\int_{\Sigma_{t_{2}}}T_{\mu\nu}X^{\mu}n^{\nu}dvol,$$

where $X=\partial_{t}$ is the unique timelike Killing vector field on the spacetimes, $n$ is the normal vector field on $\Sigma_{t}$, $dvol$ is the volume form of $\Sigma_{t}$, and the energy momentum tensor $T_{\mu\nu}$ for the Klein–Gordon equation is

$$T_{\mu\nu}=\Re\left({\nabla_{\mu}\phi\overline{\nabla_{\nu}\phi}}\right)-\frac{1}{2}g_{\mu\nu}\left(g^{\alpha\beta}\nabla_{\alpha}\phi\overline{\nabla_{\beta}\phi}+\bigl{(}-\frac{\Lambda}{3}\bigr{)}\alpha|\phi|^{2}\right)$$

(1.13)

The boundedness of the solution follows from a standard argument if the energy on the spacelike slice $\Sigma_{t=0}$ is positive. The conservation of energy shows that the quantity

$$\int_{r_{+}}^{\infty}\int_{\mathbb{S}^{2}}\frac{T_{tt}}{\Omega^{2}}r^{2}\sin{\theta}drd\theta d\varphi$$

is independent of $t$. However, since the Klein–Gordon mass $\alpha$ in our setting is negative, the positivity of $T_{tt}$

$$T_{tt}=\frac{1}{2}\left(|\partial_{t}\phi|^{2}+\Omega^{2}\left(\Omega^{2}|\partial_{r}\phi|^{2}+|\not{\nabla}\phi|^{2}+\bigl{(}-\frac{\Lambda}{3}\bigr{)}\alpha|\phi|^{2}\right)\right)$$

is not guaranteed at first glance. On the Schwarzschild-AdS spacetime, the integral of the negative term $\bigl{(}-\frac{\Lambda}{3}\bigr{)}\alpha|\phi|^{2}$ can be absorbed by the integral of the derivative term $\Omega^{2}|\partial_{r}\phi|^{2}$ using the Hardy inequality. The following Hardy inequality can indeed be obtained:

$$\int_{r_{+}}^{\infty}\int_{\mathbb{S}^{2}}|\phi|^{2}r^{2}\sin{\theta}drd\theta d\varphi\leq\frac{4}{9}\int_{r_{+}}^{\infty}\int_{\mathbb{S}^{2}}|\partial_{r}\phi|^{2}(r-r_{+})^{2}\left(r+r_{+}+\frac{r_{+}^{2}}{r}\right)^{2}drd\theta d\varphi.$$

(1.14)

Using the above Hardy inequality, for the Schwarzschild-AdS spacetime, we have

		$\displaystyle\int_{r_{+}}^{\infty}\int_{\mathbb{S}^{2}}\frac{T_{tt}}{\Omega^{2}}r^{2}\sin{\theta}drd\theta d\varphi=\int_{r_{+}}^{\infty}\int_{\mathbb{S}^{2}}\Omega^{2}|\partial_{r}\phi|^{2}r^{2}\sin\theta+\bigl{(}-\frac{\Lambda}{3}\bigr{)}\alpha|\phi|^{2}r^{2}\sin{\theta}drd\theta d\varphi$		(1.15)
	$\displaystyle\geq$	$\displaystyle\int_{r_{+}}^{\infty}\int_{\mathbb{S}^{2}}\bigl{(}-\frac{\Lambda}{3}\bigr{)}(r-r_{+})\frac{1}{r^{2}}\left(\frac{r^{3}}{\bigl{(}-\frac{\Lambda}{3}\bigr{)}}+r_{+}^{3}r^{2}+r_{+}^{4}r+r_{+}^{5}\right)|\phi|^{2}\sin{\theta}drd\theta d\varphi>0.$

Moreover, for the Reissner–Nordström-AdS spacetime, using the argument as above, one can in fact show that if

$$\frac{e^{2}}{r_{+}}<1+\bigl{(}-\frac{\Lambda}{3}\bigr{)}r_{+}^{2},$$

(1.16)

the energy is still positive. Hence by the standard commuting vector field and redshift methods, one can still show the boundedness of the solutions. Combining (1.16) and (1.5), we can show that growing mode solutions to the uncharged Klein–Gordon equation (1.1) can only exist if

$$\frac{e^{2}}{r_{+}^{2}}>1+\frac{1}{4\left(-\frac{3}{2}-\alpha\right)}.$$

(1.17)

In the above computation, the integral of the negative term $\bigl{(}-\frac{\Lambda}{3}\bigr{)}\alpha r^{2}|\phi|^{2}\sin{\theta}$ in the energy is controlled by the weighted $L^{2}$ norm of $\partial_{r}\phi$ by the Hardy inequality $\eqref{nav hardy}$. This weight in (1.14) decays quadratically like $(r-r_{+})^{2}$ towards the event horizon, whereas the weight $\Omega^{2}$ of the $L^{2}$ norm of $\partial_{r}\phi$ in the original energy of the Schwarzschild-AdS spacetime decays at a linear rate $C(r-r_{+})$ with $C$ bounded away from $0$. This fact is crucial in showing the absence of the negative energy. Hence, near the horizon, the integrand on the right-hand side of $\eqref{after hardy}$ is positive, giving the hope to absorb the negative term in the whole black hole exterior region $r>r_{+}$. However, on the extremal Reissner–Nordström-AdS spacetime, $\Omega^{2}$ decays exactly quadratically like $(r-r_{+})^{2}$. A simple computation shows that the analogous integrand on the right-hand side of $\eqref{after hardy}$ might be negative near the event horizon. Hence, for the near-extremal Reissner–Nordström-AdS spacetime, if the initial data $(\phi,\partial_{t}\phi){\bigl{|}_{t=0}}$ is supported near the event horizon, the energy $\int_{\Sigma_{0}}T_{\mu\nu}X^{\mu}n^{\nu}$ might be negative; see Section 6.2 for the proof of the existence of negative energy in near-extremal Reissner–Nordström-AdS spacetimes. This phenomenon is the key observation allowing us to construct the growing mode solution in our main theorem.

The well-known no-hair conjecture in General Relativity asserts that all stationary black hole solutions to the Einstein (coupled with reasonable matter fields) equations are uniquely determined by the mass, charge, and angular momentum; for instance see the review [33] and references therein.

In [51], Shlapentokh-Rothman constructed solutions of the form $e^{-i\omega t}e^{im\varphi}S_{\kappa ml}(\theta)R(r)$ to the Klein–Gordon equation on the Kerr spacetime, which is interpreted as a “linear hair”, in violation to at least the spirit of the conjecture. He showed the existence of growing mode solutions ($\omega$ has a negative imaginary part) and time-periodic solutions ($\omega\in\mathbb{R}$). Later, with Chodosh [9], they further constructed one-parameter families of the stationary axisymmetric asymptotically flat solutions $(\mathcal{M},g_{\delta},\phi_{\delta})$ to the Einstein–Klein–Gordon equations bifurcating off the Kerr solution. Their work gives a counter-example to the no-hair conjecture.³³3In fact, their result violates the no-hair conjecture to the extend that the metric is stationary but the scalar field is time-periodic. See also [32, 30, 7, 3, 31, 10] for previous physics and numerics results in this direction.

In the works [54, 43], the mathematical study of the hairy black hole interiors has been initiated. Putting the characteristic initial data $(g,\phi)$ on the event horizon, corresponding to a stationary non-zero scalar field $\phi$, Van de Moortel [54] investigated the interior of the stationary Einstein–Maxwell–Klein–Gordon hairy black holes and their singular structures, assuming the scalar field is uncharged. Later, Van de Moortel and Li studied the analogous problem with the charged scalar field [43]. However, the existence of a stationary black hole exterior corresponding to the solutions in [54, 43] was open; see Open Problem iv in [54]. We resolve this problem in the companion work [58], taking advantage of the method in this paper, which gives the affirmative answer to Open Problem iv and provides examples of asymptotically AdS hairy black holes whose interior is governed by the results of [54, 43].

Asymptotically AdS black holes have also emerged as an object of interest in the celebrated AdS/CFT correspondence [44, 45, 57]. Growing efforts have been spent in finding holographic analogs of superconductors, a problem for which hairy black holes appear as natural candidates. In fact, the existence of non-trivial stationary asymptotically AdS hairy black holes has been predicted in the paper [29], using heuristics relying on the so-called near-horizon geometry. Indeed, it is possible to “map” the near-extremal Reissner–Nordström-AdS spacetime to $AdS_{2}\times\mathbb{S}^{2}$ by a limiting procedure. The heuristics of [29] argue that the $2$-dimensional Breitenlohner–Freedman bound is relevant to the existence of growing mode solutions to the uncharged Klein–Gordon equation on the extremal Reissner–Nordström-AdS spacetime, specifically for $\alpha$ in the range

$$-\frac{9}{4}<\alpha<-\frac{3}{2}.$$

Our result actually shows the growing mode solutions exist within this range for sub-extremal Reissner–Nordström-AdS black holes. We also identify explicitly a sufficient condition (1.5) for the growing mode solutions to exist, which is not previously identified in [29]. While our method in principle also works to construct growing mode solutions on the extremal Reissner–Nordström-AdS spacetime, we leave this question to future works.