Quasinormal Modes, Poles, and Spectral Response

Quasinormal frequencies are eigenvalues of a non-self-adjoint boundary problem: infalling at a future horizon and source-free at the AdS boundary. They coincide with poles of the classical retarded Green function and govern linear relaxation while their residues remain appreciable. They are not exact finite-$N$ energy levels and do not imply irreversible decay for arbitrarily late time.

Required background. Lorentzian Holographic Correlators and Infalling Conditions supplies the causal boundary problem.

Helpful background. Spectral Functions and Transport Peaks fixes the boundary interpretation, and Greybody Scattering and Flux Accounting supplies the flux checks.

The quasinormal eigenvalue condition

Quasinormal decay in asymptotically AdS black holes was connected to boundary relaxation by Horowitz and Hubeny 2000. For a scalar mode $\phi=e^{-i\omega t+i\mathbf k\cdot\mathbf x}\psi(r)$, write the boundary expansion as

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

After imposing the infalling horizon solution, $A$ and $B$ are analytic functions of complex $\omega$ away from branch structures. The retarded response is

$$G_R(\omega,\mathbf k) =(2\Delta-d)\frac{B(\omega,\mathbf k)} {A(\omega,\mathbf k)}+G_{\mathrm{contact}}.$$

Therefore

$$A(\omega_n,\mathbf k)=0 \quad\Longrightarrow\quad G_R(\omega,\mathbf k) \simeq\frac{R_n(\mathbf k)}{\omega-\omega_n}.$$

For a stable black brane in the $e^{-i\omega t}$ convention, $\operatorname{Im}\omega_n<0$. Channel decomposition is essential: scalar, shear, sound, charge, and tensor master fields obey different equations and can contain gauge constraints.

Lowest scalar mode in AdS5–Schwarzschild

Use $u=(r_h/r)^2$, factor

$$\psi(u)=(1-u)^{-i\omega/(4\pi T)}F(u),$$

and expand regular $F$ in Chebyshev polynomials. Collocation converts the radial equation into a generalized matrix eigenproblem. For a massless scalar at zero momentum, a standard normalization gives the lowest pair approximately as

$$\frac{\omega_0}{\pi T} \simeq \pm3.11945-2.74668\,i,$$

consistent with the thermal-correlator calculation of Núñez and Starinets 2003. Substituting this complex frequency into the source coefficient makes $A$ vanish; evaluating $B/\partial_\omega A$ gives the pole residue.

The numerical value is not portable without the spacetime dimension, scalar mass, momentum, and frequency convention. The reproducible result includes the radial coordinate map, grid, factored endpoint behavior, precision, and convergence sequence.

Adversarial numerical controls

Repeat the calculation after:

1. changing $u$ to a smooth compact coordinate;
2. increasing spectral resolution and arithmetic precision;
3. solving by an independent shooting or continued-fraction method;
4. checking the conserved radial flux away from the endpoints.

A genuine isolated pole converges under all four tests. A spurious collocation eigenvalue typically drifts with grid size or violates the boundary residual. Replacing the infalling exponent by the outgoing one computes the advanced spectrum. The gauge-invariant master-field formulation and pole/QNM correspondence are developed by Kovtun and Starinets 2005.

Relaxation window and evidence ceiling

For a perturbation in one channel,

$$\delta\langle\mathcal O(t)\rangle \sim\sum_n R_n e^{-i\omega_n t} +\text{cuts and nonlinear terms}.$$

The least damped relevant singularity controls an intermediate late-time regime, not necessarily the earliest transient and never the exact finite-$N$ limit. Nonlinear mode coupling, hydrodynamic branch structure, exponentially small sectors, and discrete recurrences can dominate outside that window.

Thermal and Nonequilibrium QFT owns spectral response; curved-spacetime QFT owns general black-hole modes. The next page extracts hydrodynamic poles, while the chapter’s final page restores finite-$N$ discreteness.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Horowitz, Gary T., and Veronika E. Hubeny. “Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium.” Physical Review D 62, 024027 (2000). doi:10.1103/PhysRevD.62.024027.
• Kovtun, Pavel K., and Andrei O. Starinets. “Quasinormal Modes and Holography.” Physical Review D 72, 086009 (2005). doi:10.1103/PhysRevD.72.086009.
• Núñez, Alvaro, and Andrei O. Starinets. “AdS/CFT Correspondence, Quasinormal Modes, and Thermal Correlators in $\mathcal N=4$ SYM.” Physical Review D 67, 124013 (2003). doi:10.1103/PhysRevD.67.124013.