Quasinormal Boundary Conditions and Resonances

A black-hole quasinormal frequency is a resonance of a specified retarded problem: an isolated pole of its analytically continued Green operator. Under regularity assumptions stated below, it is equivalently a frequency at which a nonzero homogeneous solution satisfies both continued radiation conditions. We derive this definition for a massless test scalar on Schwarzschild, then explain what must change for rotation, a massive field, or different outer geometry. The definition does not make quasinormal profiles finite-energy standing waves or prove that a sum of modes gives the entire response.

Required background. PDEs and Green Functions for Gravitation supplies retarded Green functions, operator domains, and contour deformation.

Helpful background. Black-Hole Geometry and Mechanics supplies horizons and asymptotic regions; Formulations of General Relativity supplies perturbative variables and gauge; Computational Relativity supplies numerical boundary-value methods; Black-Hole Perturbation Theory supplies the separated equations.

Schwarzschild scalar resonances

Fix a four-dimensional Schwarzschild exterior with mass $M\gt0$. Write

$$f(r)=1-\frac{2GM}{r}, \qquad r\gt2GM, \qquad \frac{dx}{dr}=\frac1{f(r)},$$

where $x=r_*$ is the tortoise coordinate. The horizon is at $x\to-\infty$ and spatial infinity at $x\to+\infty$. Separate a scalar obeying $\Box\Phi=0$ as

$$\Phi(t,r,\theta,\phi) =\frac{\Psi_\ell(t,x)}{r}Y_{\ell m}(\theta,\phi), \qquad \Psi_\ell=e^{-i\omega t}X_\ell(x).$$

Here $Y_{\ell m}$ is a spherical harmonic, $\ell\geq0$ and $-\ell\leq m\leq\ell$. Spherical symmetry makes the radial equation independent of $m$. The resulting operators are

$$\begin{aligned} P_\ell&=-\partial_t^2+\partial_x^2-V_\ell(r),\\ L_\omega&=\partial_x^2+\omega^2-V_\ell(r),\\ V_\ell(r)&=f(r)\left[\frac{\ell(\ell+1)}{r^2} +\frac{2GM}{r^3}\right]. \end{aligned}$$

Thus $P_\ell\Psi_\ell=0$ and $L_\omega X_\ell=0$. This potential is real, nonnegative, bounded and frequency-independent. Those properties are specific hypotheses here, not properties of every stationary black-hole perturbation equation.

Cauchy data and resonant profiles

One precise spatial realization is $H_\ell=-\partial_x^2+V_\ell$ on $L^2(\mathbb R, dx)$, with domain $H^2(\mathbb R)$, the functions with square-integrable weak derivatives through order two. A bounded real multiplication potential preserves the free operator’s self-adjoint domain. In this realization $L_\omega=\omega^2-H_\ell$; its ordinary eigenvectors must belong to that domain.

For smooth compactly supported Cauchy data, put $\Pi_\ell=\partial_t\Psi_\ell$. The positive conserved energy on a complete Schwarzschild-time slice is

$$E_\ell=\frac12\int_{\mathbb R} \left( |\Pi_\ell|^2+|\partial_x\Psi_\ell|^2 +V_\ell|\Psi_\ell|^2 \right)dx.$$

Differentiating and using $P_\ell\Psi_\ell=0$ gives the endpoint term $\operatorname{Re}[\Pi_\ell^*\partial_x\Psi_\ell]_{-\infty}^{+\infty}=0$ at each finite time. Energy can leave a fixed compact interval while the whole-slice energy remains conserved. An energy completion of Cauchy data and an $L^2$ spatial domain are not interchangeable definitions.

For a damped resonance, $\omega=\omega_R-i\gamma$ with $\gamma\gt0$, the continued radiation profiles instead have

$$|e^{-i\omega t}|=e^{-\gamma t}, \qquad |e^{-i\omega x}|=e^{-\gamma x}, \qquad |e^{+i\omega x}|=e^{+\gamma x}.$$

The ingoing profile grows at the left endpoint and the outgoing profile grows at the right. Such modes are neither ordinary $L^2$ eigenvectors nor finite-energy data on these slices. There is no conflict with finite-energy evolution: the resonant contribution is part of a continued representation of the retarded response, not a globally admissible separated initial state. See Berti–Cardoso–Starinets 2009, §3.1, author-manuscript pp. 17–18, PDF.

Radiation conditions and horizon regularity

At real $\omega\gt0$, the endpoint solutions have unit desired-wave amplitudes,

$$U_H\sim e^{-i\omega x}\quad(x\to-\infty), \qquad U_\infty\sim e^{+i\omega x}\quad(x\to+\infty).$$

The full horizon phase is $e^{-i\omega v}$, with advanced time $v=t+x$; the outer phase is $e^{-i\omega u}$, with retarded time $u=t-x$. They propagate toward the future event horizon and toward future null infinity, respectively. The scattering-basis diagram shows these two directions alongside the incident waves.

The retarded transform starts in an upper half-plane where these endpoint solutions decay spatially. Continue the selected solutions in frequency, with the path and any branch cuts specified. Keep that continued selection at a complex frequency; do not choose anew by asking only which local functions look smooth.

For Schwarzschild, the surface gravity is $\kappa=1/(4GM)$. With an arbitrary positive reference length $\rho_0$, the local tortoise behavior gives

$$x=\frac1{2\kappa}\log\frac{r-2GM}{\rho_0}+O(1), \qquad X\sim\left(\frac{r-2GM}{\rho_0}\right)^{\mp i\omega/(2\kappa)}.$$

After extracting $e^{-i\omega v}$, the leading ingoing radial factor is constant, whereas the outgoing factor has exponent $i\omega/\kappa$. Generically this excludes a smooth outgoing extension. At exceptional integer exponents it is not enough: full Frobenius series can also involve logarithms, and the originally selected branch still matters. The exceptional power is visible in Berti–Cardoso–Starinets 2009, §3.1, author-manuscript p. 17, Eqs. (27)–(30), PDF.

An exact smoothness counterexample

Consider the exact flat Rindler metric

$$ds^2=-2\kappa\rho\,dv^2+2\,dv\,d\rho+dy^2+dz^2, \qquad \kappa\gt0.$$

The exterior has $\rho\gt0$; the future horizon is $\rho=0$ at finite $v$. Take a scalar independent of $y,z$, with $\Phi=e^{-i\omega v}F(\rho)$. The inverse metric has $g^{v\rho}=1$, $g^{\rho\rho}=2\kappa\rho$ and constant determinant, so $\Box\Phi=0$ becomes

$$2\kappa\rho F''+(2\kappa-2i\omega)F'=0.$$

For $\omega\ne0$, its two solutions are

$$F_{\rm in}=1, \qquad F_{\rm out}=\left(\frac{\rho}{\rho_0}\right)^{i\omega/\kappa}.$$

In the upper half-plane, the real part of the second exponent is negative: that branch diverges as $\rho\to0$. Continuing the selected unit ingoing solution keeps $F_{\rm in}=1$. But at

$$\omega=-iN\kappa, \qquad N=1,2,3,\ldots,$$

the other solution is the smooth polynomial $(\rho/\rho_0)^N$. Both branches are now smooth and independent. Local smoothness alone would therefore admit an extra solution that the continued ingoing condition does not select.

This example establishes a limitation of a boundary test, not a list of black-hole quasinormal frequencies: no global outer condition has been imposed. At $\omega=0$, the powers coalesce and an independent pair is $1,\log(\rho/\rho_0)$. On a general black hole, inspect the full local equation at exceptional frequencies instead of inferring smoothness from a leading power.

A real-frequency flux check

For this real Schwarzschild potential, define

$$\mathcal J[X] =\frac{X^*X'-XX'^*}{2i}, \qquad '=\partial_x.$$

At real positive frequency, $\mathcal J[Ae^{+i\omega x}]=+\omega|A|^2$ and $\mathcal J[Be^{-i\omega x}]=-\omega|B|^2$. These signs confirm rightward outer propagation and leftward horizon propagation. At complex frequency the equation instead gives

$$\mathcal J' =\frac{\omega^{*2}-\omega^2}{2i}|X|^2 =-2\operatorname{Re}\omega\,\operatorname{Im}\omega\,|X|^2.$$

It is generally not conserved; purely imaginary frequencies are an exception. In neither case does this expression turn an exponentially growing resonant profile into a positive finite-energy stationary wave. Radiation labels are inherited from the continued scattering problem.

Wronskian zeros and Green-function poles

Fix the source measure as $dx'$ and the frequency-domain source equation as

$$L_\omega g_\ell(\omega;x,x')=\delta(x-x').$$

Continue this kernel between compact sources and local observations; this is not an assertion that the continued resolvent remains bounded on the original $L^2$ space.

Let $x_< =\min(x,x')$ and $x_> =\max(x,x')$. The endpoint solutions give

$$\begin{aligned} W(\omega)&=U_H U_\infty'-U_H'U_\infty,\\ g_\ell(\omega;x,x') &=\frac{U_H(x_<,\omega)U_\infty(x_>,\omega)}{W(\omega)}. \end{aligned}$$

The homogeneous equation makes $W$ independent of $x$. The kernel is continuous at $x=x'$, and its right-minus-left derivative jump is $W/W=1$, fixing the source sign. The mode Green-function derivation explains the matching in detail.

Away from branch points, choose a neighborhood on the declared continuation where both endpoint solutions are holomorphic and neither is the zero solution. Then $W(\omega_n)=0$ means they are proportional: one nonzero solution satisfies both selected conditions. If this zero is isolated and simple, the residue is

$$\operatorname*{Res}_{\omega=\omega_n}g_\ell =\frac{U_H(x_<,\omega_n)U_\infty(x_>,\omega_n)} {\partial_\omega W(\omega_n)}.$$

The full kernel has a pole, although its residue can vanish at particular observation/source points or after a projection. A repeated zero, singular endpoint normalization, or threshold needs its own analysis. The admissible-zero construction and residue appear in Leaver 1986, §II.B, author version p. 7, Eqs. (20)–(23), PDF.

What normalization can change

Multiply the endpoint solutions by holomorphic functions $a(\omega)$ and $b(\omega)$ that are nonzero near the pole. Both numerator and Wronskian acquire the same factor $ab$. At a zero of $W$,

$$\partial_\omega(abW)\big|_{\omega_n} =a(\omega_n)b(\omega_n)\partial_\omega W(\omega_n),$$

because the term $(ab)'W$ vanishes there. Thus the fixed source-normalized Green kernel and its residue are unchanged. Separately normalized mode amplitudes or excitation factors can change; their combination into the same physical response cannot. Leaver makes the corresponding cancellation explicit in Leaver 1986, author version p. 9, Eq. (27) and following paragraph, PDF. A rescaling that vanishes at the candidate frequency is not an admissible change of endpoint basis; the exercise below shows the resulting false zero.

The continuation is local to its analytic sheet, not necessarily meromorphic on the entire complex plane. Long-range propagation produces branch structure, discussed in poles, cuts, and thresholds. The pole condition alone says nothing about when a contour can be replaced by a convergent sum of residues.

Frequency-pair symmetry

For a real frequency-independent potential, conjugating the radial equation replaces $\omega^2$ by $\omega^{*2}$. Define $\widetilde\omega=-\omega^*$, which has the same squared value. The conjugated endpoint phases become

$$\begin{aligned} (e^{-i\omega x})^*&=e^{-i\widetilde\omega x} &&\text{at the horizon},\\ (e^{+i\omega x})^*&=e^{+i\widetilde\omega x} &&\text{at infinity}. \end{aligned}$$

Consequently the Schwarzschild resonances occur in pairs $\omega_{\ell n}\leftrightarrow-\omega_{\ell n}^*$ on correspondingly reflected continuations of the radiation solutions. Both members have the same damping sign. The integer $n$ denotes an overtone label, not an orthogonal basis index. For a full complex spherical harmonic the conjugation also reverses $m$, which does not change the Schwarzschild radial spectrum. This argument neither assumes an $L^2$ inner product nor asserts the same fixed-$m$ symmetry for Kerr.

When the endpoints change

Rotation and the radial variable

For subextremal Kerr, let $a=J/M$ with $|a|\lt GM$, $r_\pm=GM\pm\sqrt{(GM)^2-a^2}$ and $\Delta=(r-r_+)(r-r_-)$. The horizon angular velocity and shifted frequency are

$$\Omega_H=\frac{a}{r_+^2+a^2}, \qquad k_H=\omega-m\Omega_H, \qquad \frac{dr_*}{dr}=\frac{r^2+a^2}{\Delta}.$$

The horizon phase is set by the generator $\partial_t+\Omega_H\partial_\phi$. For the raw spin-weight-$s$ Teukolsky radial variable in the Kinnersley tetrad, the ingoing behavior is

$$R_s\sim\Delta^{-s}e^{-ik_Hr_*}.$$

Dropping $\Delta^{-s}$ changes that variable’s boundary condition. Horizon regularity is assessed in suitable coordinates and a regular field or tetrad, not by treating raw $R_s$ as the Schwarzschild $X_\ell$. Moreover the massless angular eigenvalue $A_{s\ell m}(a\omega)$ changes with the frequency being sought. See Berti–Cardoso–Starinets 2009, author-manuscript p. 16, Eqs. (24)–(26), PDF and the radial Teukolsky equation.

For a real-frequency scalar, horizon Killing-energy flux has the sign of $\omega k_H$. Superradiance can make it negative without reversing the selected ingoing direction. The scalar current calculation above is not a physical flux formula for an arbitrary raw higher-spin Teukolsky amplitude.

A massive asymptotically flat field

For a Schwarzschild scalar with inverse-length mass parameter $\mu\gt0$, the potential becomes $V_\ell+f\mu^2$. Put $k^2=\omega^2-\mu^2$. Initially choose the analytic upper-half-plane branch with $\operatorname{Im}k\gt0$ and $k\sim\omega$ at high frequency, then continue that chosen branch. At fixed $k\ne0$,

$$\omega^2-(V_\ell+f\mu^2) =k^2+\frac{2GM\mu^2}{r}+O(r^{-2}).$$

The inverse-radius term requires a Coulomb phase:

$$X_{\rm out}\sim e^{ikr_*} \left(\frac r{r_0}\right)^{iGM\mu^2/k}, \qquad r_0\gt0.$$

To check its exponent, insert $X=e^{ikx}(r/r_0)^{i\alpha}$ and use $\partial_x=f\partial_r$. The coefficient of $1/r$ in $X''/X+\omega^2-V_\ell-f\mu^2$ is $2(GM\mu^2-k\alpha)$, hence $\alpha=GM\mu^2/k$. A pure exponential leaves an uncancelled long-range term. The equivalent radial asymptotics and outer-condition distinction are in Dolan 2007, §§II.C–D, author-manuscript pp. 8–9, Eqs. (28)–(31), PDF, with $X=rR$ here.

This expansion is not uniform at the thresholds $k=0$. A quasi-bound state imposes decay at infinity; its continuation must be distinguished from an outgoing resonance across the propagating continuum. Do not reselect a square-root sign at each trial frequency or assign all quasi-bound states to one unspecified “other sheet.” For a damped mode, outer decay also does not remove the horizon-end growth on the time slices used above.

Cosmological and timelike boundaries

In a de Sitter static region between two simple horizons, the outer condition is propagation through the future cosmological horizon, with phase $e^{+i\omega r_*}$ as $r_*\to+\infty$. See the two-horizon boundary problem.

An AdS conformal boundary is timelike. Its allowed field falloffs and boundary data depend on the field, mass and chosen theory; Dirichlet, Neumann and Robin conditions are not universally admissible alternatives. Once an admissible condition is selected it is part of the spectral problem. See AdS boundary conditions for those distinctions.

Exercises

Is the smooth branch at a regular horizon?

In the flat Rindler example, set $U=-\rho e^{-\kappa v}$ and $V=e^{\kappa v}/\kappa$. Show that the two-dimensional metric becomes $-2\,dU\,dV$. At $\omega=-iN\kappa$, express both full scalar solutions in these coordinates. Which part of the horizon is covered by the smoothness claim?

Solution

Differentiating gives $dU=-e^{-\kappa v}d\rho+\kappa\rho e^{-\kappa v}dv$ and $dV=e^{\kappa v}dv$, hence

$$-2\,dU\,dV=2\,dv\,d\rho-2\kappa\rho\,dv^2.$$

The two full solutions are

$$\Phi_{\rm in}=e^{-N\kappa v}=(\kappa V)^{-N}, \qquad \Phi_{\rm out}=e^{-N\kappa v} \left(\frac{\rho}{\rho_0}\right)^N =\left(-\frac U{\rho_0}\right)^N.$$

They are both smooth near $U=0$ with $V\gt0$, the future horizon at finite $v$. The ingoing solution is not smooth at the bifurcation corner $U=V=0$. The example claims future-horizon regularity locally, not extension through that corner. Only the constant radial branch is selected by continuation from the upper half-plane.

A false zero of the Wronskian

For the free equation $L_\omega=\partial_x^2+\omega^2$ on the whole line, take $U_H=e^{-i\omega x}$ and $U_\infty=e^{+i\omega x}$. Calculate the positive-source kernel. Then replace $U_H$ by $(L\omega+i)U_H$, where $L\gt0$ is a reference length. The new Wronskian vanishes at $\omega_*=-i/L$. Is this an additional resonance?

Solution

The Wronskian and kernel are

$$W_0=2i\omega, \qquad g_0(\omega;x,x')=\frac{e^{i\omega|x-x'|}}{2i\omega}.$$

The derivative jump is $+1$. After rescaling, numerator and denominator both acquire $L\omega+i$, so the same kernel remains. With $d=|x-x'|$,

$$g_0(\omega_*;x,x')=\frac L2e^{d/L}, \qquad \operatorname*{Res}_{\omega=\omega_*}g_0=0.$$

There is no additional pole. The rescaled endpoint solution is identically zero at $\omega_*$ and therefore fails the nonzero-basis hypothesis. This is a removable normalization zero, not a frequency at which a nonzero free solution satisfies both conditions. The separate zero-frequency behavior of $g_0$ is not the artificial zero considered here.

To compute genuine resonance roots after specifying the endpoint problem, continue to Frobenius series and continued fractions or spectral and collocation methods.

References

• Berti, E., Cardoso, V., and Starinets, A. O. (2009). “Quasinormal modes of black holes and black branes.” Classical and Quantum Gravity 26, 163001. DOI. Open PDF, arXiv:0905.2975v2. Page locators above use the author manuscript.
• Dolan, S. R. (2007). “Instability of the massive Klein-Gordon field on the Kerr spacetime.” Physical Review D 76, 084001. DOI. Open PDF, arXiv:0705.2880v2. Page locators above use the author manuscript.
• Leaver, E. W. (1986). “Spectral decomposition of the perturbation response of the Schwarzschild geometry.” Physical Review D 34, 384–408. DOI. Open PDF, author version. Page locators above use the separately paginated author version.