Extremal Horizons and Aretakis Instability

An extremal horizon can support a conserved scalar quantity even while the scalar field decays. The surviving quantity then forces a transverse derivative not to decay and, one derivative higher, produces growth. These are statements about particular derivatives on the horizon, not exponential growth of the field or a conclusion about nonlinear destruction of the black hole. We prove the mechanism explicitly for the spherical mean of a neutral massless scalar on fixed extremal Reissner–Nordström, then explain the broader horizon theorem and the subextremal comparison.

Required background. PDEs and Green Functions for Gravitation supplies characteristic transport, initial data and the distinction between pointwise and energy estimates.

Helpful background. Black-Hole Geometry and Mechanics supplies Killing horizons; Formulations of General Relativity supplies regular coordinates; Mathematical Relativity supplies norm-dependent stability; the volume overview places this example among the different stability questions.

Scalar waves at an extremal horizon

For this sustained RN calculation use geometrized units $G=c=1$, with $M\gt0$ a mass length. To restore $G$ at $c=1$, replace this $M$ by $G M_{\mathrm{phys}}$. In ingoing Eddington–Finkelstein coordinates,

$$ds^2=-D(r)\,dv^2+2\,dv\,dr+r^2d\Omega^2, \qquad D(r)=\left(1-\frac Mr\right)^2.$$

Here $r$ is areal radius, $d\Omega^2$ is the unit-sphere metric, and $v=t+r_*$ with $dr_*/dr=D^{-1}$. The future horizon is $r=M$. The stationary Killing field $V=\partial_v$ has unit time normalization at infinity. The vector $Y=\partial_r$ means differentiation at fixed $v,\theta,\phi$, not at fixed Schwarzschild time. Both vectors are regular at the future horizon, satisfy $[V,Y]=0$, and have $g(V,Y)=1$. Thus $V$ differentiates along the horizon, while $Y$ differentiates across it.

Surface gravity is defined by $\nabla_VV=\kappa V$ on the horizon. For this metric $\kappa=D'(M)/2=0$: the Killing parameter $v$ is also affine there. This is the normalization underlying the usual Killing/affine relation in Wald 1984, § 12.5, p. 331. The important radial facts are

$$D(M)=D'(M)=0,\qquad D''(M)=\frac{2}{M^2}.$$

Consider a real smooth solution of $\Box_g\psi=0$, smooth across the horizon for every finite $v$. Its spherical mean and rescaled mean are

$$\psi_0(v,r)=\frac{1}{4\pi}\int_{S^2}\psi(v,r,\theta,\phi)\,d\Omega, \qquad \Phi(v,r)=r\psi_0(v,r).$$

No spherical symmetry of the full field is required: averaging commutes with the wave operator on this spherical background. For a concrete global data class, take smooth scalar Cauchy data on a standard smooth spherical spacelike hypersurface crossing the future horizon, supported within $r\lt R_0$ for a fixed $R_0\gt2M$. They need not vanish at its horizon section. Use the source’s foliation obtained by stationary translation of a slice that is spacelike near the horizon and completed by an outgoing null branch beyond $R_0$. Finite propagation makes the initial outgoing branch zero; smooth compact data make its required higher initial norms finite. This gives a sufficient subclass of Aretakis 2011, Part II, PDF, §§ 2–2.1, pp. 3–4 and Theorem 4, pp. 6–7. Stationary translation adds the same parameter to $v$ on the horizon, so the theorem implies $\psi_0(v,M)\to0$. We do not reprove that global decay theorem or a general change-of-slicing result here. The local conservation proof below needs no assumption about infinity; the growth conclusion explicitly uses horizon-field decay.

The inverse radial metric has $g^{vv}=0$, $g^{vr}=1$ and $g^{rr}=D$, with $\sqrt{-g}=r^2\sin\theta$. Substituting these into $\Box_g\psi=(-g)^{-1/2}\partial_\mu(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\psi)$ and averaging gives

$$2\partial_v\partial_r\psi_0+\frac2r\partial_v\psi_0 +\frac1{r^2}\partial_r\!\left(r^2D\partial_r\psi_0\right)=0.$$

Multiplication by $r$ and substitution of $\Phi=r\psi_0$ yield the useful transport form

$$2\partial_v\partial_r\Phi +\partial_r(D\partial_r\Phi)-\frac{D'}r\Phi=0.$$

The term $-D'\Phi/r$ must be retained before differentiating. It vanishes on the extremal horizon, but its radial derivative does not.

Conservation and derivative growth

Restrict the last equation to $r=M$. The double zero of $D$ removes every term except the mixed derivative:

$$H_0:=\left.\partial_r\Phi\right|_{r=M} =\psi_0(v,M)+M\partial_r\psi_0(v,M), \qquad \partial_vH_0=0.$$

This is the Aretakis charge in our normalization. It equals $M$ times the charge $\partial_r\psi_0+\psi_0/M$ in Aretakis 2011, Part II, PDF, Proposition 3.0.1, p. 8. If the field on the horizon decays, conservation immediately implies

$$\partial_r\psi_0(v,M)\longrightarrow\frac{H_0}{M}.$$

For $H_0\ne0$, the first transverse derivative therefore does not decay. The field can approach zero while its slope across the horizon approaches a nonzero value.

To see growth rather than nondecay, differentiate the full rescaled equation once in $r$ and only then set $r=M$. Define $\psi_H(v)=\psi_0(v,M)$ and $Z(v)=\partial_r^2\Phi(v,M)$. The result is

$$\frac{dZ}{dv}=-\frac{H_0}{M^2}+\frac{\psi_H(v)}{M^2}.$$

The charge supplies a persistent source. Integrating from a finite $v_0$,

$$Z(v)=Z(v_0)-\frac{H_0}{M^2}(v-v_0) +\frac1{M^2}\int_{v_0}^{v}\psi_H(s)\,ds.$$

If $\psi_H(v)\to0$, its average over $[v_0,v]$ also tends to zero: split the integral at a sufficiently late fixed time, bound the early part by a constant, and bound the remaining average by $\sup_{s\ge v_1}|\psi_H(s)|$. Thus $Z(v)/v\to-H_0/M^2$. No integrability of $\psi_H$ and no separate assumption about $\partial_v\psi_H$ are needed for this leading coefficient.

Finally $Z=M\partial_r^2\psi_0+2\partial_r\psi_0$, so

$$\frac{\partial_r^2\psi_0(v,M)}{v}\longrightarrow-\frac{H_0}{M^3}, \qquad \partial_r^2\psi_0(v,M)=-\frac{H_0}{M^3}v+o(v).$$

For $H_0\ne0$ this is linear blow-up in magnitude. The coefficient is fixed, not an unspecified normalization constant: $v$, $r$ and $H_0$ have already been defined. If $\psi$ has units $[\psi]$, then $[H_0]=[\psi]$ and $H_0v/M^3$ has the required units $[\psi]/\mathrm{length}^2$.

The logical steps are worth keeping distinct:

Horizon quantity	Additional input	Consequence as $v\to\infty$
$H_0=\partial_r(r\psi_0)$	Smooth wave equation and extremal double zero	Exactly conserved
$\partial_r\psi_0$	$\psi_H\to0$	Limit $H_0/M$
$\partial_r^2\psi_0$	$\psi_H\to0$ and $H_0\ne0$	Linear growth with coefficient $-H_0/M^3$

This is an exact transport calculation followed by an asymptotic argument, not a constructed closed-form solution throughout the exterior.

Higher harmonics and data dependence

Write a fixed spherical harmonic component as $\psi_{\ell m}(v,r)Y_{\ell m}(\theta,\phi)$, where $\Delta_{S^2}Y_{\ell m}=-\ell(\ell+1)Y_{\ell m}$, and set $\Phi_{\ell m}=r\psi_{\ell m}$. Its equation is

$$2\partial_v\partial_r\Phi_{\ell m} +\partial_r(D\partial_r\Phi_{\ell m}) -\left(\frac{D'}r+\frac{\ell(\ell+1)}{r^2}\right)\Phi_{\ell m}=0.$$

The angular potential prevents $\partial_r\Phi_{\ell m}$ alone from being conserved when $\ell\gt0$. After $k$ radial differentiations of the unrescaled wave equation, the coefficient of $\partial_r^k\psi_{\ell m}$ on the horizon is

$$\frac{k(k+1)-\ell(\ell+1)}{M^2}.$$

For $k\lt\ell$ it is nonzero, allowing the lower radial derivatives to be expressed through $v$ derivatives. At $k=\ell$ it vanishes. Substitution of those lower identities then produces a conserved linear combination

$$\mathcal H_{\ell m} =\left[ \partial_r^{\ell+1}\psi_{\ell m} +\sum_{j=0}^{\ell}b_{\ell j}(M)\partial_r^j\psi_{\ell m} \right]_{r=M}, \qquad \partial_v\mathcal H_{\ell m}=0.$$

This is the induction mechanism, rather than a claim that the bare highest derivative is conserved. Here $\mathcal H_{00}$ denotes the appropriately normalized harmonic coefficient, not our spherical-mean charge $H_0$. For example, the dipole combination is $\mathcal H_{1m}=\psi_{1m,rr}+3\psi_{1m,r}/M+\psi_{1m}/M^2$ at $r=M$. Its conservation is checked in the exercises. See Aretakis 2011, Part II, PDF, Propositions 3.0.2–3.0.3 and the proof of Theorem 1, pp. 8–9.

For smooth compact horizon-crossing data, the fixed-harmonic decay and growth theorem gives decay through transverse order $\ell$, a charge-controlled nonzero limit at order $\ell+1$, and polynomial lower bounds for higher orders when $\mathcal H_{\ell m}\ne0$. For the full fixed-$\ell$ angular field, the conserved profile is $\sum_{m=-\ell}^{\ell}\mathcal H_{\ell m}Y_{\ell m}(\theta,\phi)$; the coefficients obey the reality condition when the field is real. The corresponding pointwise lower bound is forced only where this profile is nonzero, not at its angular zeros. These statements require the lower-order estimates as well as conservation; see Aretakis 2011, Part II, PDF, Theorems 5–6, p. 7.

For the spherical mean, “generic” can be made precise in the space of smooth Cauchy data whose support is contained in a fixed compact set $K$ containing a neighborhood of the horizon section, equipped with the $C^\infty_K$ topology (uniform convergence of every derivative on $K$). The map from data to $H_0$ is a nontrivial continuous linear functional. Its kernel is closed and has empty interior, so $H_0\ne0$ is open and dense in that space. This is not a probability statement. If one restricts instead to data vanishing near the horizon section, $H_0$ vanishes identically. Then this proof does not establish the displayed linear growth; it also does not prove stability of all higher derivatives.

The general horizon theorem

Spherical symmetry is not essential for a section-integrated conservation law. The geometric hypotheses are: a smooth four-dimensional Lorentzian manifold with commuting Killing fields $V$ and $X$; complete line-like orbits of $V$ and closed spacelike axial orbits of $X$ away from its two-dimensional axis; a null horizon normal to $V$, tangent to $X$, with $\nabla_VV=0$; spherical spacelike horizon sections; and integrability of the two-planes orthogonal to the Killing orbits. These are the actual assumptions A1–A4 in Aretakis 2015, PDF, § 2.1, p. 4.

Use the source’s adapted coordinates $(v,\rho,\theta,\phi)$, with horizon $\rho=0$, $V=\partial_v$, $X=\partial_\phi$, $Y=\partial_\rho$, and $\Theta=\partial_\theta$; $\rho$ is not the RN areal radius. The horizon sections $S_v$ are translates of $S_0$. For smooth scalar waves there exist bounded geometric coefficients $\alpha,\beta,\gamma$, independent of $v$, such that

$$\mathcal H[\psi] =\int_{S_v}(Y\psi+\alpha\psi+\beta V\psi+\gamma\Theta\psi)\,d\mu_{S_v}, \qquad \frac{d\mathcal H}{dv}=0.$$

The measure is induced by the section metric. The proof integrates the horizon-restricted wave equation: extremality removes the undifferentiated transverse transport term, and angular divergences integrate to zero on the closed section. Consequently $\mathcal H\ne0$ prevents $\int_{S_v}(|\psi|+|Y\psi|+|V\psi|+|\Theta\psi|)\,d\mu_{S_v}$ from tending to zero. This is a section-norm statement, not a pointwise conclusion at every angular position. The scalar need not itself be axisymmetric for this conservation/nondecay result. See Aretakis 2015, PDF, Proposition 3.1 and proof, pp. 8–9; Theorem 1 and proof, pp. 11–12.

For the stronger conclusion, the proof additionally uses $g(Y,\Theta)=0$ on the horizon and a constant nonzero $g_{v\rho}\partial_\rho^2g^{\rho\rho}|_{\rho=0}$. Assume $\mathcal H\ne0$ and that the field and its first tangential derivatives tend uniformly to zero on the transported compact sections. The differentiated section identity then forces a second derivative to be unbounded in the supremum over the sections, not necessarily at every angular position. Indeed, its time-integrated transverse source has a nonzero linear term. Uniform bounds on all second derivatives would also bound $Y\psi$: its angular variation is bounded and its section integral tends to $\mathcal H$. They would therefore bound the endpoint terms, contradicting that linear term. This is the sufficient implication of the argument in Aretakis 2015, PDF, proof of Theorem 2, p. 12, not a universal RN growth coefficient for arbitrary extremal geometries. The local argument does not, by itself, construct global solutions or prove decay in the exterior.

Subextremal horizons and limits

For subextremal RN the metric function is $D=(1-r_+/r)(1-r_-/r)$, with $r_+\gt r_-$ and outer horizon $r_+$. With the same normalization of $V$,

$$\kappa=\frac{D'(r_+)}2=\frac{r_+-r_-}{2r_+^2}\gt0.$$

For the spherical mean set $H_\kappa(v)=\partial_r(r\psi_0)|_{r_+}$ and now $\psi_H(v)=\psi_0(v,r_+)$. The exact horizon equation is

$$\frac{dH_\kappa}{dv}+\kappa H_\kappa=\kappa\psi_H(v).$$

Positive surface gravity restores a damping term, but there is also forcing from the field. Thus $H_\kappa$ is not an exactly conserved charge, nor is it generally a pure exponential. Its integrating-factor solution is

$$H_\kappa(v)=e^{-\kappa(v-v_0)}H_\kappa(v_0) +\kappa\int_{v_0}^{v}e^{-\kappa(v-s)}\psi_H(s)\,ds.$$

If $\psi_H\to0$ for a fixed $\kappa\gt0$, this convolution tends to zero; split it into an early exponentially suppressed part and a late part with small field amplitude. Uniformity as $\kappa\to0$ is a different question, because the kernel remembers an interval of size $\kappa^{-1}$.

In the zero-forcing transport model, with a common initial value $h$ and $v_0=0$, $H_\kappa=he^{-\kappa v}$. For $h\ne0$,

$$\lim_{\kappa\to0^+}\lim_{v\to\infty}H_\kappa(v)=0, \qquad \lim_{v\to\infty}\lim_{\kappa\to0^+}H_\kappa(v)=h.$$

This exhibits noncommuting limits for a specified model and quantity; it is not a construction of a family of full scalar-wave solutions with fixed compact Cauchy data. To transfer the model to such a family one must control its forcing and data normalization uniformly on the claimed time scale. The separate Aretakis Growth and Near-Extremal Transients page treats the relation to near-extremal response.

What this instability does not imply

Field decay is not derivative decay. The RN calculation explicitly shows the difference. For a nonspherical field, growth of its spherical mean derivative also obstructs a uniform bound on the full angular field derivative, but does not assert growth at every angle.

Horizon growth is not a growing normal mode. Polynomial growth of a transverse derivative does not require a separated solution with $\operatorname{Im}\omega\gt0$. Mode stability alone cannot control the relevant time-domain norms. Conversely, horizon pointwise growth alone does not prove divergence of a spacelike bulk-energy integral: a trace and a volume integral are different quantities. The latter requires additional estimates.

The differentiating frame matters. In the RN calculation, $Y$ is a fixed translation-invariant, horizon-regular transverse vector, with its scale fixed by areal radius and the normalized $V$. A coordinate singularity or a time-dependent rescaling of $Y$ would not establish the same claim.

A test-field theorem is not a nonlinear endpoint. The Einstein equations have not been evolved with scalar backreaction. No conclusion about horizon destruction or cosmic censorship follows. Compare the precise norm and region in Boundedness and Decay of Linear Fields and the additional requirements in From Linear Decay to Nonlinear Stability. Nonlinear evolution belongs to Black-Hole Dynamics; proof architecture belongs to Mathematical Relativity.

Exercises

Why the decay assumption matters

Use the exact solution $\psi=C$, with $C\ne0$ constant, to test both the conservation law and the differentiated horizon identity. Does a nonzero $H_0$ alone prove derivative blow-up?

Solution

All derivatives of $\psi$ vanish, so $\Box_g\psi=0$. Nevertheless $\Phi=Cr$, $H_0=C$ and $Z=0$. The two terms in $Z'=(-H_0+\psi_H)/M^2$ cancel exactly. Dropping the field source would therefore predict a false instability. This solution fails $\psi_H\to0$ and is not in the selected compact-data, decaying class. It is an exact local countercheck on the logical implication, not a counterexample to the conditional theorem.

Derive the dipole charge

For $\ell=1$, put $F_j(v)=\partial_r^j\Phi_{1m}(v,M)$. Restrict the full harmonic equation and its first radial derivative to the horizon. Find a conserved combination of $F_1,F_2$ and translate it back to $\psi_{1m}$.

Solution

The angular potential is $2/r^2$. Using $D''(M)=2/M^2$ gives

$$\dot F_1=\frac{F_0}{M^2},\qquad \dot F_2=-\frac{F_0}{M^3},\qquad \frac{d}{dv}\left(F_2+\frac{F_1}{M}\right)=0.$$

Since $F_1=\psi_{1m}+M\psi_{1m,r}$ and $F_2=2\psi_{1m,r}+M\psi_{1m,rr}$ on the horizon,

$$\frac1M\left(F_2+\frac{F_1}{M}\right) =\psi_{1m,rr}+\frac3M\psi_{1m,r}+\frac1{M^2}\psi_{1m} =\mathcal H_{1m}.$$

The first derivative alone is forced by $F_0$; conservation appears only after the angular cancellation and the lower-derivative correction.

When may the forcing be neglected?

Suppose $|\psi_H(s)|\le\varepsilon$ for $v_0\le s\le v$ in the subextremal transport equation. Bound the error in replacing $H_\kappa$ by its homogeneous exponential, and explain the requirement for a near-extremal family.

Solution

The integrating-factor formula gives

$$\left|H_\kappa(v)-e^{-\kappa(v-v_0)}H_\kappa(v_0)\right| \le\varepsilon\left(1-e^{-\kappa(v-v_0)}\right).$$

On an early interval $\kappa(v-v_0)\ll1$ this is at most $\varepsilon\kappa(v-v_0)$; on a scaled interval $v-v_0=u/\kappa$ it is at most $\varepsilon(1-e^{-u})$. A useful approximation requires this error to be small compared with the retained amplitude. A statement that each field eventually becomes small at fixed $\kappa$ does not give a common onset time or a uniform $\varepsilon$ as $\kappa\to0$. Those are additional PDE estimates, not consequences of the integrating factor.

References

• Aretakis, S. (2011). “Stability and instability of extreme Reissner–Nordström black hole spacetimes for linear scalar perturbations II.” Annales Henri Poincaré 12, 1491–1538. DOI. Open PDF, arXiv:1110.2009v1.
• Aretakis, S. (2015). “Horizon instability of extremal black holes.” Advances in Theoretical and Mathematical Physics 19, 507–530. DOI. Open PDF, arXiv:1206.6598v2.
• Wald, R. M. (1984). General Relativity. University of Chicago Press. Publisher.