Quasinormal Modes and Retarded Poles

A holographic quasinormal mode is a bulk solution that is ingoing at the future horizon and normalizable at the AdS boundary with no applied source. Those are exactly the conditions for the boundary response-to-source ratio to diverge. The correspondence is therefore not a slogan: it follows from the same radial boundary-value problem that defines the retarded Green function. This page derives the pole condition, its residue, and the caveats introduced by gauge constraints, contact terms, and branch cuts.

Required background. Gauge Theory and String-Theory Preparation provides bulk-field and source normalization; Quantum Information provides the response-function language used below.

Helpful background. Quantum Gravity supplies the semiclassical holographic regime, while Black-Hole Perturbation Theory provides master variables and quasinormal boundary conditions.

Ingoing response from the radial problem

Take a gauge-invariant bulk fluctuation with Fourier dependence $e^{-i\omega t+i\mathbf{k}\cdot\mathbf{x}}$. Near a nonextremal horizon,

$$\phi(r)\sim (r-r_h)^{-i\omega/(4\pi T)}$$

is ingoing in regular advanced coordinates. With the site convention, stable modes therefore have $\operatorname{Im}\omega\lt0$. Near an asymptotically AdS boundary, a scalar behaves schematically as

$$\phi(r)=A(\omega,\mathbf{k})\,r^{\Delta-d} +B(\omega,\mathbf{k})\,r^{-\Delta}+\cdots .$$

Standard quantization identifies $A$ as the source and $B$ as the response, after local counterterms are added. The retarded function takes the form

$$G_R(\omega,\mathbf{k})={\cal N}\frac{B(\omega,\mathbf{k})} {A(\omega,\mathbf{k})}+G_{\rm contact}(\omega,\mathbf{k}).$$

The real normalization $\mathcal N$ depends on the bulk action and operator convention. Son and Starinets 2002, §§3.2 and 4.1, pp. 8–14 derive this Lorentzian prescription from the on-shell action; the ingoing choice implements retarded analyticity rather than an arbitrary absorption rule.

Source-free modes are poles

A quasinormal frequency $\omega_n(\mathbf{k})$ satisfies

$$A(\omega_n,\mathbf{k})=0,\qquad B(\omega_n,\mathbf{k})\ne0 .$$

It is normalizable and source-free, yet carries a nonzero response. If the zero of $A$ is simple, expansion around it gives

$$G_R(\omega,\mathbf{k})= \frac{{\cal N}B(\omega_n,\mathbf{k})} {[\partial_\omega A(\omega_n,\mathbf{k})](\omega-\omega_n)} +\text{regular terms}.$$

Thus the residue is fixed by the normalizable amplitude and the derivative of the source coefficient. A simultaneous numerator/denominator zero requires care. In a nondegenerate second-order scalar problem, setting both independent boundary coefficients to zero gives the trivial solution; a common zero may therefore reflect a vanishing normalization rather than a new physical mode. Pole skipping instead involves an exceptional loss of uniqueness in the ingoing problem and a path-dependent limit in $(\omega,k)$. Gauge modes and singular choices of basis must be removed before making that test.

The analytic BTZ example in Son and Starinets 2002, §§4.2–4.3, pp. 14–17 shows the same frequencies in the thermal CFT correlator and in the bulk quasinormal spectrum. It is a stringent sign check: changing $e^{-i\omega t}$ to $e^{+i\omega t}$ without changing the horizon branch reflects every stable pole into the wrong half-plane.

Hydrodynamic and nonhydrodynamic poles

Conservation laws force some poles toward the origin as momentum vanishes, such as $\omega=-iDk^2+\cdots$. Other quasinormal poles remain at frequencies of order $T$ and are nonhydrodynamic. The full retarded correlator contains both, and their collision at complex momentum can limit the convergence of a hydrodynamic series.

Poles are not the only possible nonanalyticities. At finite temperature and in a classical large-N black brane, the response is often meromorphic in the domain of interest. Zero-temperature limits, extremal throats, loop corrections, or continua can generate branch cuts. Local counterterms change contact terms but cannot move genuine poles. Alternative boundary quantization swaps the source-response identification and must be declared before the spectrum is interpreted.

Coupled channels add another layer. A metric, gauge field, and scalar may mix, so the source and response are vectors and the pole condition becomes the vanishing of a source matrix determinant. A zero in one matrix entry is not enough. One must choose independent ingoing solutions, form the full boundary source matrix, and check that its rank drops. The corresponding null vector identifies the operator combination excited by the mode, while the residue matrix determines which boundary probes can see it.

Independent checks and limitations

Three checks are practical. First, integrate the same gauge-invariant equation from the horizon and from the boundary and verify that the Wronskian vanishes at $\omega_n$. Second, compute $B/A$ near the root and confirm the expected inverse-linear scaling. Third, use reality of the time-domain equation to check the appropriate complex-conjugation relation between modes at opposite momentum.

The pole-to-mode statement assumes a well-posed linear boundary problem and a specified operator quantization. A quasinormal calculation alone does not determine nonlinear thermalization, and a boundary pole alone may mix several bulk fields in a coupled channel. Residues, not only frequencies, decide how strongly a mode appears in a chosen operator.

Exercises

Derive the residue of a simple retarded pole from the near-boundary coefficients and explain what changes if both coefficients vanish at the same frequency.

Solution

Write $A(\omega)=A'(\omega_n)(\omega-\omega_n)+\cdots$ and $B(\omega)=B_n+\cdots$. Substitution into $G_R={\cal N}B/A$ gives residue $\mathcal N B_n/A'(\omega_n)$. At fixed $k$, equal simple zeros give the unique finite limit $\mathcal N B'(\omega_n)/A'(\omega_n)$; they do not by themselves give direction dependence. Unequal orders can leave a pole or a zero. To diagnose pole skipping, vary both $\omega$ and $k$ and check the exceptional ingoing solution space as well as the boundary ratio. The physical conclusion requires a gauge-invariant variable and fixed boundary quantization.

References

• Son, Dam T., and Andrei O. Starinets. “Minkowski-Space Correlators in AdS/CFT Correspondence: Recipe and Applications.” Journal of High Energy Physics 09 (2002) 042. arXiv:hep-th/0205051.