AdS Normal Modes and Black-Hole Quasinormal Modes

Pure anti-de Sitter space is a conservative cavity once a positive self-adjoint zero-flux boundary domain is selected, so its regular modes have real, discrete frequencies. A negative spatial eigenvalue instead produces exponential growth. Inserting a black-hole horizon replaces center regularity by absorption. When a positive energy remains available in the actual black-hole domain, this opens a decay channel. Neither that positivity nor a one-to-one continuation of every normal mode follows from reflection alone.

Helpful background. Black-Hole Geometry and Mechanics supplies AdS horizons and temperature; Partial Differential Equations and Green Functions supplies normal and resonant spectra; Formulations of General Relativity supplies perturbation sectors; Gravitational Actions and Boundary Terms supplies boundary flux; Causality and Global Structure supplies timelike conformal infinity; Black-Hole Perturbation Theory supplies QNM conventions.

Global AdS as a normal-mode cavity

For global AdS$_{d+1}$,

$$ds^2=-\left(1+\frac{r^2}{L^2}\right)dt^2 +\left(1+\frac{r^2}{L^2}\right)^{-1}dr^2+r^2d\Omega_{d-1}^2.$$

Take $d\geq2$ and a scalar with real effective mass squared $m_{\rm eff}^2=m_0^2+\xi\mathcal R$, where $\mathcal R=-d(d+1)/L^2$. Separate $\phi=e^{-i\omega t}R(r)Y_\ell$ with angular eigenvalue $\ell(\ell+d-2)$, smooth-center regularity, and fast falloff at infinity. Write

$$\nu=\sqrt{\frac{d^2}{4}+m_{\rm eff}^2L^2},\qquad \Delta_\pm=\frac d2\pm\nu,\qquad \Omega=\omega L.$$

Assume the BF bound $\nu^2\geq0$. The complete radial equation is

$$\frac1{r^{d-1}}\frac{d}{dr} \left(r^{d-1}f\frac{dR}{dr}\right) +\left[\frac{\omega^2}{f} -\frac{\ell(\ell+d-2)}{r^2}-m_{\rm eff}^2\right]R=0, \qquad f=1+\frac{r^2}{L^2}.$$

Set $r=L\tan\rho$ and $y=\sin^2\rho$. Factoring out the regular-center and fast boundary powers, $R=y^{\ell/2}(1-y)^{\Delta_+/2}F$, gives

$$y(1-y)F''+[c-(a+b+1)y]F'-abF=0,$$

where the primes now mean $d/dy$, and

$$a=\frac{\ell+\Delta_+-\Omega}{2},\qquad b=\frac{\ell+\Delta_++\Omega}{2},\qquad c=\ell+\frac d2.$$

The center-regular solution is $F={}_2F_1(a,b;c;y)$; its alternative is singular or logarithmic at the center. At noninteger $\nu\gt0$, the connection formula at $y=1$ gives

$$\begin{aligned} R&\sim\alpha(L/r)^{\Delta_-}+\beta(L/r)^{\Delta_+},\\ \alpha&=\frac{\Gamma(c)\Gamma(\nu)}{\Gamma(a)\Gamma(b)},\qquad \beta=\frac{\Gamma(c)\Gamma(-\nu)} {\Gamma(c-a)\Gamma(c-b)}. \end{aligned}$$

Each power multiplies its own asymptotic series in $(L/r)^2$; $\alpha$ and $\beta$ are branch coefficients, not a two-term truncation. In particular, slow-branch descendants can precede the fast power when $\nu\gt1$.

Gamma functions have no zeros. For positive $\Omega$, $b\gt0$, so eliminating the slow coefficient requires $a=-n$, $n=0,1,\ldots$. The hypergeometric series then terminates at degree $n$ and yields

$$\omega_{n\ell}L=2n+\ell+\Delta_+, \qquad n=0,1,2,\ldots.$$

The negative-frequency partners have the opposite sign. Polynomial termination also proves the fast spectrum at positive integer $\nu$, where the displayed noninteger connection formula for $\beta$ must not be used term by term. At $\nu=0$, select the log-free Friedrichs boundary condition; the same polynomial solutions have $\Delta_+=d/2$. An arbitrary logarithmic or Robin domain is a different problem.

The reduced operator and Hilbert measure are derived on the boundary-domain page; the general global-AdS solution is also given in Ishibashi and Wald 2004, §3, pp. 25–31, Open PDF. These are normal modes: the selected positive domain has no horizon loss. A different admissible Robin domain changes the coefficient relation and therefore the spectrum.

A direct check of two domains

For the conformal AdS$_4$ s-wave, $m_{\rm eff}^2L^2=-2$ and $\Delta_+=2$. The fast ground state $R=(1+r^2/L^2)^{-1}$ has $\omega L=2$. Direct substitution into the complete radial equation gives zero. The slow ground state $R=(1+r^2/L^2)^{-1/2}$ instead has $\omega L=1$. Both belong to positive domains, but they are not the same boundary problem. For the minimally coupled massless scalar, $\Delta_+=3$ and the fast ground-state frequency is $3/L$.

Adding a Schwarzschild–AdS horizon

In $d+1\geq4$ dimensions the spherical Schwarzschild–AdS metric has

$$f(r)=1+\frac{r^2}{L^2}-\frac{\mu_B}{r^{d-2}}, \qquad f(r_+)=0.$$

The center is replaced by a future event horizon. A scalar QNM is ingoing there,

$$R\sim e^{-i\omega r_*},$$

and obeys the same declared falloff at timelike infinity. Horizon absorption excludes growth only when the actual black-hole domain has a nonnegative energy controlling the field. In a mode comparison with positive mode energy, $2\omega_I E=-\mathcal F_H$ gives damping for a nonzero absorbed flux. It does not establish positivity for every real Robin parameter. For the standard massless scalar with fast boundary data, Horowitz and Hubeny formulate and compute this QNM problem Horowitz and Hubeny 2000, §§II–IV.

The real-frequency current explains the sign of the loss. Under the positive energy hypotheses, no energy escapes through the outer boundary, while positive boundary-frame $\partial_t$ energy enters a nonrotating horizon. For a complex mode, use a regular slicing and the full energy identity, not a real-frequency Wronskian with $\omega$ simply made complex. Rotation can reverse the $\partial_t$-energy flux; nonnegativity of the horizon-generator flux is a separate matter-energy statement.

Small black holes: deformation and absorption

When $r_+/L\ll1$, a low normal mode has wavelength of order $L$ and samples a horizon occupying a small central region. For branches continuously connected to global-AdS modes,

$$\operatorname{Re}\omega_{n\ell}\longrightarrow \frac{2n+\ell+\Delta_+}{L}, \qquad \operatorname{Im}\omega_{n\ell}\longrightarrow0^-$$

as $r_+\to0$, at fixed $n$, $\ell$, field parameters, and boundary domain. The imaginary part is controlled by the small absorption probability and is typically much smaller for higher angular momentum because the centrifugal barrier suppresses horizon penetration.

A concrete source-backed case is the four-dimensional massless scalar with fast falloff, at fixed $n,\ell$, with $r_+\ll L$ and $|\omega|r_+\ll1$. Near and far solutions overlap in $r_+\ll r\ll L$. Their coefficient matching approaches the center-regular AdS condition as the horizon shrinks. The nonrotating limit of the small, slowly rotating Kerr–AdS calculation gives the damped approach above Cardoso and Dias 2004, §III, pp. 3–5, Open PDF. This named regime substantiates the limiting example; it is not a proof for all masses, Robin parameters, or branches.

This is a singular geometric limit: the horizon endpoint disappears and regular-center data return. One must track a mode continuously and verify its eigenfunction localization. Highly damped branches, modes whose overtone index scales while $r_+\to0$, or gravitational sectors with changing gauge content need not obey the fixed-mode limit above.

Large black holes and planar scaling

For $r_+/L\gg1$, the near-horizon and thermal scales dominate. In four bulk dimensions, for example,

$$T=\frac{1+3r_+^2/L^2}{4\pi r_+} \sim\frac{3r_+}{4\pi L^2}.$$

At fixed dimensionless momentum and overtone, many large-hole frequencies scale as

$$\omega=T\,\widehat\omega(k/T,m_{\rm eff}^2L^2,\ldots).$$

This is not a small perturbation of the global-AdS spectrum. The spherical boundary may approach a planar brane regime locally, hydrodynamic branches can appear for conserved fields, and different perturbation sectors scale differently. The field, horizon topology, ensemble, and outer boundary condition must accompany any quoted number.

A reliable mode-tracking test

Choose one simple eigenvalue at small $r_+/L$ and continue it in the horizon radius using the previous frequency as the next initial guess. At every step check both endpoint residuals, compare eigenfunction nodes, and increase resolution. A nearest-frequency rule alone can jump branches at an avoided crossing. In the small-hole direction verify the analytic normal-mode limit; in the large-hole direction plot $\omega/T$, not only $\omega L$.

This page does not report a newly computed continuation sequence. Its endpoint limits are analytic and source-backed; a quantitative mode track would need to publish the horizon-radius grid, continuous labels, residual and resolution sweeps, and an independent benchmark before any numerical interpolation is treated as a result.

The page on AdS boundary conditions explains why the outer domain must remain fixed during this comparison. Changing a Robin parameter while shrinking the horizon mixes deformation of the geometry with deformation of the operator itself.

Common pitfalls

Calling every complex AdS frequency a deformed normal mode. A continuous fixed-label branch and converged eigenfunction are required. New horizon or hydrodynamic families can have no fixed normal-mode ancestor.

Comparing dimensional frequencies across regimes. Small-hole modes are naturally measured in $L^{-1}$; large-hole modes often scale with $T$. State which ratio is held fixed.

Exercises

Check both ends of the interpolation

For a massless scalar in global AdS$_4$ with fast falloff, find the lowest $\ell=0$ frequency. Then show how the Schwarzschild–AdS temperature scales for $r_+/L\ll1$ and $r_+/L\gg1$. Which scale should organize each spectral limit?

Solution

Here $d=3$, $m_{\rm eff}^2=0$, and $\Delta_+=3$, so $n=\ell=0$ gives $\omega L=3$. The four-dimensional temperature is

$$T=\frac{1+3r_+^2/L^2}{4\pi r_+}.$$

For $r_+/L\ll1$, $T\sim(4\pi r_+)^{-1}$ diverges even though the exterior tends to global AdS away from the vanishing horizon; fixed low modes are therefore organized by $L^{-1}$, not by $T$. For $r_+/L\gg1$, $T\sim3r_+/(4\pi L^2)$ and the planar thermal scale organizes fixed dimensionless-momentum QNMs. This contrast is why one scaling cannot describe both ends.

References

• Cardoso, V., and Dias, Ó. J. C. (2004). “Small Kerr–anti-de Sitter black holes are unstable.” Physical Review D 70, 084011. DOI. Open PDF.
• Horowitz, G. T., and Hubeny, V. E. (2000). “Quasinormal modes of AdS black holes and the approach to thermal equilibrium.” Physical Review D 62, 024027. DOI. Open PDF.
• Ishibashi, A., and Wald, R. M. (2004). “Dynamics in non-globally-hyperbolic static spacetimes. III. Anti-de Sitter spacetime.” Classical and Quantum Gravity 21, 2981–3014. DOI. Open PDF.