Aretakis Growth and Near-Extremal Transients

An exactly extremal horizon can carry conserved scalar quantities that force a field or first derivative not to decay and, under additional hypotheses, produce transverse-derivative blow-up. A near-extremal horizon has no identical conserved charge: positive surface gravity restores redshift, but only on the long time scale $\kappa^{-1}$. Its response can therefore imitate the extremal growth over a parametrically long interval and reach a $\kappa$-dependent large amplitude before decaying. The theorem and the transient are related singular limits, not the same statement.

Helpful background. Black-Hole Geometry and Mechanics supplies extremal and subextremal horizons; PDEs and Green Functions for Gravitation supplies characteristic transport and energy estimates; Formulations of General Relativity supplies horizon-regular field equations; Events, Intervals, and Minkowski Geometry fixes late-time orientation; Black-Hole Perturbation Theory supplies the separated response.

Exact-extremal theorem and hypotheses

Consider a smooth four-dimensional Lorentzian geometry with commuting stationary and axial Killing fields $V,X$, complete line-like $V$ orbits, closed spacelike $X$ orbits away from its two-dimensional axis, and integrable two-planes orthogonal to their orbits. Let the null horizon $\mathcal H^+$ be normal to $V$, tangent to $X$, and degenerate: $\nabla_VV=0$. Assume compact spherical spacelike horizon sections. These are the geometric assumptions A1–A4 of Aretakis 2013, § 2.1, p. 4 (PDF); the work was published in 2015.

In adapted coordinates $(v,\rho,\theta,\phi)$ use $V=\partial_v$, $X=\partial_\phi$, $Y=\partial_\rho$ and $\Theta=\partial_\theta$, with horizon $\rho=0$ and translated sections $S_v$. Here $\rho$ is not the RN areal radius. Consider a smooth solution

$$\Box_g\psi=0$$

from sufficiently regular initial data on a hypersurface intersecting $\mathcal H^+$. At $\kappa=0$, integration of the horizon-restricted wave equation over a cross-section produces a translation-invariant linear functional

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

The measure is induced by the section metric; the bounded coefficients $\alpha,\beta,\gamma$ are fixed by the geometry and independent of $v$. The horizon equation loses its extremal transverse damping term, and angular divergences integrate to zero on the closed section. Hence nonzero $H[\psi]$ prevents

$$\int_{S_v}\left(|\psi|+|Y\psi|+|V\psi|+|\Theta\psi|\right)d\mu_{S_v}$$

from tending to zero. This is a section-norm statement, not nondecay at every angular point. The scalar need not itself be axisymmetric. See Aretakis 2013, Proposition 3.1 and proof, pp. 8–9; Theorem 1 and proof, pp. 11–12 (PDF).

For higher-derivative growth, additionally require $g(Y,\Theta)=0$ on the horizon and constant nonzero $g_{v\rho}\partial_\rho^2g^{\rho\rho}|_{\rho=0}$. If $H[\psi]\ne0$ and the field and first tangential derivatives tend uniformly to zero, the differentiated section identity forces a second derivative to be unbounded in the section supremum. Its time-integrated source has a nonzero linear term; uniform bounds on all second derivatives would bound the endpoint terms and contradict that growth. This is the sufficient implication of Aretakis 2013, proof of Theorem 2, p. 12 (PDF), not an unconditional blow-up theorem for every nonzero horizon charge. The general theorem discussion explains the angular mean/boundedness step.

This is a linear scalar theorem. “Generic” refers to nonvanishing of the relevant continuous linear functional in the specified data class. The result does not assert that $\psi$ itself diverges, that every derivative diverges, or that a bulk norm away from the horizon grows. It also does not establish a nonlinear destruction of the horizon.

Explicit horizon charge in extremal Reissner–Nordström

A spherical example exposes the proof mechanism without angular algebra. Set $c=1$, $\mu=GM$, and use ingoing coordinates

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

For the spherical mean, define $\varphi=r\psi_0$. Its wave equation is

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

At the extremal horizon $r=\mu$, both $D$ and $D'$ vanish. Therefore

$$H_0=\left.\partial_r\varphi\right|_{r=\mu}, \qquad \partial_vH_0=0.$$

If $\psi_0|_{\mathcal H^+}\to0$, then $\partial_r\psi_0\to H_0/\mu$, so the first transverse derivative does not decay when $H_0\ne0$. Differentiating the wave equation once in $r$ introduces $D''(\mu)=2/\mu^2$. Writing $Z=\partial_r^2\varphi|_{r=\mu}$ gives the exact identity $\partial_v Z=(\psi_0(v,\mu)-H_0)/\mu^2$. Since the average of a decaying field also tends to zero, integration yields

$$\left.\partial_r^2\psi_0\right|_{\mathcal H^+} =-\frac{H_0}{\mu^3}v+o(v).$$

Thus $H_0\ne0$ gives linear blow-up with a coefficient fixed by the stated coordinates and charge normalization. Conservation, lower-order decay, and one further differentiated transport equation are the three steps of the proof. The full derivation and its constant-scalar countercheck are in Extremal Horizons and Aretakis Instability. A coordinate derivative is meaningful here because ingoing coordinates and $Y=\partial_r$ are regular at the future horizon.

Positive surface gravity cuts off conservation

For subextremal Reissner–Nordström, $D'(r_+)=2\kappa>0$. Restricting the same spherical equation to the horizon gives

$$\partial_v(\partial_r\varphi) +\kappa\left(\partial_r\varphi-\frac{\varphi}{r_+}\right)=0.$$

Writing $\psi_H=\psi_0(v,r_+)$ and $H_\kappa=\partial_r\varphi|_{r=r_+}$, the exact integrating-factor solution is

$$H_\kappa(v)=e^{-\kappa v}H_\kappa(0) +\kappa\int_0^v e^{-\kappa(v-s)}\psi_H(s)\,ds.$$

If the lower-order field is negligible throughout the relevant window, the would-be charge follows the forcing-neglected transport model

$$H_\kappa(v)\simeq H_\kappa(0)e^{-\kappa v}.$$

For $v\ll\kappa^{-1}$ it is approximately conserved, and the next transverse derivative can accumulate an approximately linear response. At $v=O(\kappa^{-1})$ the redshift term becomes order one and cuts off that extremal-like regime in the model. If $|\psi_H|\le\varepsilon$ throughout the window, the discarded integral is at most $\varepsilon(1-e^{-\kappa v})$. Applying the model uniformly to a near-extremal family requires a corresponding uniform forcing bound and specified data normalization. The model illustrates the time scale and possible noncommutation; it does not establish the full PDE limit. Detailed Kerr exponents require the angular and spin structure.

Kerr components and the observed transient

Fix $\mu=GM$ and take a Kerr family with $a\gt0$ approaching $\mu$. Use

$$\sigma=\frac{r_+-r_-}{r_+}=4\mu\kappa, \qquad x=\frac{r-r_+}{r_+}.$$

For the displayed generic formula choose one co-rotating nonaxisymmetric mode $m\gt0$, $\ell\ge\max(|s|,m)$, and $s=0,\pm1,\pm2$. Gralla, Zimmerman and Zimmerman (GZZ) calculate the near-horizon-mode pole contribution to a source-normalized retarded kernel. It is not the whole retarded solution or a uniform PDE remainder estimate.

Field and angular conventions

The source’s spin label is not directly the label of the site’s Kinnersley field. Write $D=r^2+a^2$, $z=r-ia\cos\theta$ and $\Delta=(r-r_+)(r-r_-)$. Using the Chapter 5 curvature conventions, define the Teukolsky–Press (TP) variable and future-regular Hartle–Hawking (HH) boost by

$$\begin{aligned} \Omega_s&=(2/\Delta)^s\Upsilon_{-s}^{K},\qquad b_H=\frac{\Delta}{2D},\\ \ell_{HH}&=b_H\ell_K,\qquad n_{HH}=b_H^{-1}n_K,\qquad m_{HH}=m_K,\\ \Upsilon_q^{HH}&=b_H^q\Upsilon_q^K,\qquad \Omega_s=D^{-s}\Upsilon_{-s}^{HH}. \end{aligned}$$

The last equality follows by substituting the boost, including its power and factor two. It agrees with Teukolsky and Press 1974, Eqs. (2.5), (4.14)–(4.15), pp. 444 and 449. The inverse $D$ powers printed in GZZ Appendix A do not obey that relation; here the normalization is obtained from the explicit TP boost. Overall Weyl signs are those of Chapter 5 and must also be used in the physical forcing.

Label in the kernel	Physical HH component represented by Ω	Principal field amplitude
$s=0$	$\psi$	$\sigma^{1/2}$
$s=-1$	$D\phi_0^{HH}$	$\sigma^{3/2}$
$s=+1$	$D^{-1}z^2\phi_2^{HH}$	$\sigma^{-1/2}$
$s=-2$	$D^2\Psi_0^{HH}$	$\sigma^{5/2}$
$s=+2$	$D^{-2}z^4\Psi_4^{HH}$	$\sigma^{-3/2}$

At fixed mass the $D,z$ factors are smooth, bounded and nonzero in the layer. Thus the $s=+2$ amplitude describes HH Ψ₄, not the unboosted Kinnersley $\Upsilon_{-2}^K$. The source’s angular function also needs translation: radial multiplication does not change the site’s $-s$ angular profile. With polar unit norm, write

$$\mathscr S_s^{TP}(\theta;c) :={}_{-s}S_{\ell m}^{\mathrm{site}}(\theta;c) =e^{i\zeta}\,{}_sS_{\ell m}^{\mathrm{site}}(\pi-\theta;c).$$

The paired eigenvalues obey $A_{-s,\ell m}=A_{s\ell m}+2s$. Reflection preserves $\sin\theta\,d\theta$; the constant normalization phase cancels between the two endpoints of the kernel. Use $\mathscr S_s^{TP}(\theta)\overline{\mathscr S_s^{TP}(\theta')}$, or reflect both angles when using the site’s $+s$ functions. This translates the source’s angular symbol in Teukolsky and Press 1974, Eqs. (2.6)–(2.8), pp. 444–445.

Fixed source, layer and shifted time

Hold the source point in the far region, $r'-r_+\gg r_+\sigma$, with a fixed, nonzero mode projection of order one. In ingoing Kerr coordinates use the source’s shifted time $V$ (not the Killing-vector symbol of the earlier theorem) by

$$\begin{aligned} V&=\frac{v-v'-2r_*'}{2\mu},\qquad X=\frac{x}{\sigma},\\ T&=\sigma V=2\kappa(v-v'-2r_*'),\qquad \Phi=\varphi-V. \end{aligned}$$

Keep the same tortoise additive convention through the family. This is a time-origin convention for the far-source approximation, not an exact wavefront-arrival calculation. A fixed order-one shift of $V$ does not change the scaling below. Compare $X$ in a fixed compact subset of $[0,\infty)$ and $0\lt T_1\le T\le T_2\lt\infty$. This follows the shrinking layer after the far-source excitation, at an elapsed time of order $\kappa^{-1}$; it does not include the prompt limit $V\to0$ or a fixed exterior observer with $X\to\infty$.

At the limiting synchronous frequency $a\omega=m/2$, set

$$\delta^2=\frac{7m^2}{4}-\left(s+\frac12\right)^2-A_{s\ell m}(m/2), \qquad \alpha_\pm=\frac12+s\pm i\delta-im.$$

For principal modes choose real $\delta\gt0$; for supplementary modes choose $\delta=i\rho$, $\rho\gt0$. The opposite-spin eigenvalue translation gives the same $\delta^2$. The exceptional cases $\delta=0$, $m=0$ and coalescent Gamma parameters require separate limiting solutions, not substitution into separately singular factors. The source’s critical-frequency text omits $m$ in two places; the general-$m$ condition used here is $a\omega=m/2$.

The profile and derivative powers

Put $w(T)=1-e^{-T/2}$. For one mode, the leading small-pole-shift approximation to the summed NHM response has the profile

$$\begin{aligned} G_{\ell m}^{NHM}\simeq{}& \mathcal N_{\ell m}(x',\theta',\varphi')\, \mathscr S_s^{TP}(\theta)e^{im\Phi} \sigma^{1/2-s-i\delta+im}\\ &\times e^{i\delta T/2-T/4} \frac{w(T)^{\alpha_+-1}}{\Gamma(\alpha_+)} [1+Xw(T)]^{-\alpha_-}. \end{aligned}$$

To specify a useful dimensionless normalization of this profile, measure all lengths and times in units of $\mu$, and put $\hat r'=r'/\mu$. The large-radius coefficient in the source’s response convention is

$$\begin{aligned} \mathcal N_{\ell m} &=-\frac{\overline{\mathscr S_s^{TP}(\theta')}e^{-im\varphi'}} {4\hat r'm\,\mathcal C_{\ell m}},\\ \mathcal C_{\ell m} &=-i(-im)^{-1/2-s+i\delta-im} \frac{\Gamma(1-2i\delta)\Gamma(-2i\delta)} {\Gamma(1/2-s-i\delta-im)\Gamma(1/2+s-i\delta-im)}. \end{aligned}$$

Use real logarithms of positive $\sigma,w,1+Xw$ and $\arg(-im)=-\pi/2$. For a source merely in the far region rather than at large radius, $1/\hat r'$ is replaced by its smooth radial source factor. This specifies the dimensionless response convention, not an independently normalized physical Green function for a unit delta source. The latter also requires the radial source density, its integration measure, and restoration of the dimensionful connection coefficients. For example, a coefficient of $x^{-2s-1}$ becomes $r_+^{2s+1}$ times that coefficient when the asymptotic basis is $r^{-2s-1}$. Such fixed-mass, fixed-far-source factors can be absorbed in $\mathcal N_{\ell m}$; the powers below require this projection to have a finite nonzero limit. They do not determine absolute strain or luminosity. The residue sum and source restrictions are given in Gralla et al. 2016, § II, Eqs. (13)–(20) and footnote 3, pp. 3–4 (PDF). The omitted pole shift is small in the approximation but, for principal modes, need not tend to zero as a positive power of $\sigma$; see Zero-Damped and Damped Mode Families.

At fixed source, differentiation of the last factor gives

$$\partial_x^d[1+xw/\sigma]^{-\alpha_-} =(-1)^d(\alpha_-)_d(w/\sigma)^d [1+xw/\sigma]^{-\alpha_--d},$$

where $(a)_d=a(a+1)\cdots(a+d-1)$ and $(a)_0=1$. Thus, wherever the leading projection and profile derivative are nonzero, the complex-mode amplitude has the powers

$$|\partial_x^dG_{\ell m}^{NHM}|\sim \begin{cases} \sigma^{1/2-s-d},&\delta^2\gt0,\\ \sigma^{1/2+\rho-s-d},&\delta^2\lt0. \end{cases}$$

The residual principal factor $\sigma^{i(m-\delta)}$ is a phase, not extra magnitude growth or a convergent coefficient. These powers concern the displayed leading profile, not a bound on the full infinite-mode sum. A source projection of order $\sigma^q$ shifts each power by $q$; a vanishing projection, angular node or cancelling polarization can remove the leading term. A nonzero source need not excite every mode.

In the intermediate interval $1\ll V\ll\sigma^{-1}$, $w\sim\sigma V/2$. The powers of $\sigma$ cancel in the undifferentiated profile, leaving the principal derivative envelope $V^{s-1/2+d}$ until the cutoff $V=O(\sigma^{-1})$. The isolated pole approximation’s early singularity is not a physical initial singularity. At fixed positive $\sigma$ its principal envelope later decays as $e^{-T/4}$; supplementary modes have the extra factor $e^{-\rho T/2}$. Prompt and branch contributions, including an excited late-time asymptotically flat tail, remain separate.

Observers and local energy

A frame regular for exterior infall is not uniformly finite in the scaled throat coordinates. A throat-regular null frame can be chosen as $\ell_N=\sigma^{-1}\ell_{HH}$, $n_N=\sigma n_{HH}$, $m_N=m_{HH}$. Consequently $\Psi_4^N=\sigma^2\Psi_4^{HH}$ and $\Psi_0^N=\sigma^{-2}\Psi_0^{HH}$: the principal undifferentiated radiative amplitudes in this frame are of order $\sigma^{1/2}$. There is no contradiction with the HH amplification; the observer families have an unbounded relative boost. See Gralla et al. 2016, Appendices A–B, pp. 6–7 (PDF).

For a principal scalar mode, a generic infaller with a finite nonzero exterior-limit four-velocity and a nonzero transverse projection measures a derivative of envelope $\sigma^{-1/2}$. The corresponding local stress-energy density has envelope $T_{ab}u^au^b=O(\sigma^{-1})$, not a pointwise lower bound at every angle or phase. Combine conjugate modes before forming the stress tensor of a real field. The shrinking crossing time means that this density need not imply divergent integrated energy or a divergent kick; that requires a specified observer trajectory and interaction, as emphasized in Gralla et al. 2016, § IV, p. 5 (PDF).

Separately from that response convention, a correctly normalized inverse obeys a general field-conversion identity. With the same integration measure and delta-source convention on both sides, if $\Omega=D_s H$, then $L_H=D_s^{-1}L_\Omega D_s$ and

$$G_H(x,x')=D_s(x)^{-1}G_\Omega(x,x')D_s(x').$$

For already fixed forcing in the original equation, only the observer factor is applied to that response; for a unit source in the new equation both factors are needed. They are order one in the present fixed-mass setup, but omitting the source factor changes absolute amplitudes.

The order of limits

For the zero-forcing transport model with a common nonzero initial value $H_0$, the two orders of limits give

$$\lim_{\kappa\to0}\lim_{v\to\infty}H_\kappa(v)=0, \qquad \lim_{v\to\infty}\lim_{\kappa\to0}H_\kappa(v)=H_0.$$

In the first order, each fixed positive damping coefficient removes the model charge. In the second, zero damping is imposed first. Holding $u=\kappa v$ instead produces a finite crossover $H_0e^{-u}$. These are different regimes, not contradictory answers. Identifying them with a full scalar-wave family additionally requires control of the field forcing and initial data; the model alone does not provide that control.

The radial limit is similarly nonuniform. The near-extremal enhancement lives in $x=O(\sigma)$. An observer who holds $\bar x=x/\sigma$ fixed follows the shrinking throat layer; an observer at fixed $x>0$ remains outside it. A transverse derivative introduces $\partial_x=\sigma^{-1}\partial_{\bar x}$, which explains how a smooth near-zone profile becomes large in an exterior frame. Quoting a power of $\sigma$ without the field component, tetrad, derivative order, source normalization, and observation region is therefore incomplete.

What the transient establishes

At every fixed $\kappa>0$, the retained NHM pole contribution eventually decays. It is not a mode with $\operatorname{Im}\omega>0$, and it should not be called an exponential instability. It arises from coherent excitation of lower-half-plane near-horizon resonances over a time that diverges as $\kappa\to0$. The exactly extremal theorem instead gives nondecay for nonzero charge, and unbounded higher-derivative growth under its additional lower-order decay hypotheses.

Several hypotheses can remove a leading effect. Data in the kernel of the Aretakis functional have $H_0=0$, so the first displayed growth conclusion does not follow at that derivative order. A far-zone source with zero projection onto the relevant near-horizon modes can suppress the leading transient. Scaling the source amplitude to zero with $\sigma$ can offset a negative response power. None of these counterexamples disproves the generic result; each shows why data class and normalization belong in its statement.

Nor does linear curvature or energy-density amplification by itself determine a nonlinear endpoint. Once a growing perturbation competes with the background, backreaction leaves the fixed-Kerr calculation. Near-horizon background classification remains in Black-Hole Geometry, while nonlinear endpoints and microscopic duals require further nonlinear and quantum analysis.

The matched origin of the transient is described in Matched Asymptotic Expansions Across the Throat. Its broader limit-order interpretation is assessed in Critical Response and Noncommuting Limits.

Exercises

Let

$$H_\kappa(v)=H_0e^{-\kappa v}, \qquad Y_\kappa(v)=C\int_0^v H_\kappa(v')\,dv'.$$

Find $Y_\kappa$, its early-time behavior, and its value at $v=u/\kappa$. Explain which part models Aretakis growth and which part models the near-extremal cutoff.

Solution

Integration gives

$$Y_\kappa(v)=\frac{CH_0}{\kappa} \left(1-e^{-\kappa v}\right).$$

For $\kappa v\ll1$, $1-e^{-\kappa v}=\kappa v+O((\kappa v)^2)$, so

$$Y_\kappa(v)=CH_0v+O(\kappa v^2).$$

This reproduces the linear extremal growth over the long pre-redshift window. At scaled time $v=u/\kappa$,

$$Y_\kappa(u/\kappa) =\frac{CH_0}{\kappa}(1-e^{-u}),$$

which is parametrically large but finite for every $\kappa>0$. The model then saturates rather than grows forever. It omits damping in the next-derivative equation and does not establish eventual decay of $Y_\kappa$; that behavior belongs to the separately sourced Kerr retarded response, not this integral model. The exact-extremal limit taken first sets $H_\kappa=H_0$ and yields unbounded $CH_0v$; positive surface gravity cuts off that conservation on the scale $\kappa^{-1}$.

For an order-one excited principal scalar mode, compare the first derivative in $x$ with the first derivative in $X=x/\sigma$. For the exponent exercise only, choose a supplementary weight $\rho=3/4$: which of its first two $x$ derivatives grows? What changes if the source projection is itself of order $\sigma$?

Solution

The principal scalar field has amplitude $\sigma^{1/2}$. Its $x$ derivative has amplitude $\sigma^{-1/2}$, while $\partial_X=\sigma\partial_x$ gives $\sigma^{1/2}$. The supplementary first and second derivatives have powers $1/2+3/4-1=1/4$ and $1/2+3/4-2=-3/4$, respectively: only the second grows. This arithmetic choice is not a claim that an actual angular eigenvalue gives exactly $\rho=3/4$. A source projection proportional to $\sigma$ adds one to every power, and removes growth from these examples. Derivative convention, mode class and excitation amplitude all matter.

References

• Aretakis, S. (2015). “Horizon instability of extremal black holes.” Advances in Theoretical and Mathematical Physics 19, 507–530. DOI. Open PDF, 2013 preprint v2.
• Gralla, S. E., Zimmerman, A., and Zimmerman, P. (2016). “Transient instability of rapidly rotating black holes.” Physical Review D 94, 084017. DOI.
• Teukolsky, Saul A., and William H. Press. 1974. “Perturbations of a Rotating Black Hole. III. Interaction of the Hole with Gravitational and Electromagnetic Radiation.” Astrophysical Journal 193, 443–461. DOI.