Retarded Green Functions and Quasinormal Modes

The dynamical meaning of a horizon

Equilibrium thermodynamics uses only a few pieces of horizon data. The entropy is area, the temperature is surface gravity, and the free energy is the Euclidean on-shell action. Real-time physics is more demanding. To ask how a thermal state reacts to a small disturbance, we must solve a Lorentzian initial-boundary-value problem.

The central lesson is simple:

Dissipation in the boundary theory is absorption by the black-hole horizon.

More precisely, the retarded Green function of a thermal CFT operator is obtained by solving the dual bulk fluctuation equation with a prescribed source at the AdS boundary and an infalling condition at the future horizon. The poles of this retarded Green function are the quasinormal frequencies of the black brane.

Schematically,

$$\text{source at boundary} +\text{infalling at horizon} \quad\Longrightarrow\quad G_R(\omega,k),$$

and

$$\text{infalling at horizon} +\text{no source at boundary} \quad\Longrightarrow\quad \omega=\omega_n(k).$$

The same boundary value problem therefore gives both linear response and relaxation. A boundary perturbation excites a tower of damped collective modes,

$$\delta\langle \mathcal O(t,\vec k)\rangle \sim \sum_n R_n(\vec k)e^{-i\omega_n(\vec k)t}, \qquad \operatorname{Im}\omega_n<0,$$

where the mode closest to the real axis controls the latest-time exponential decay, unless a hydrodynamic pole or a branch cut dominates.

The retarded holographic problem has two ends. Near the boundary, the field has source and response coefficients $A(\omega,k)$ and $B(\omega,k)$, with $G_R\sim B/A$ up to local counterterms. Near the future horizon, regularity in ingoing coordinates selects the infalling wave. Quasinormal modes satisfy the same infalling condition but have $A(\omega_n,k)=0$, so they appear as poles of $G_R$ in the lower half $\omega$-plane.

This page completes the finite-temperature module. Later, in the transport module, we will use exactly this technology to compute conductivities, diffusion constants, shear viscosity, sound attenuation, and hydrodynamic dispersion relations.

Retarded correlators in thermal field theory

Let $\mathcal O$ be a bosonic Heisenberg operator in a thermal state

$$\rho_\beta={e^{-\beta H}\over Z}, \qquad Z=\operatorname{Tr} e^{-\beta H}, \qquad \beta={1\over T}.$$

The retarded Green function is

$$G_R(t,\vec x) = -i\theta(t) \langle[\mathcal O(t,\vec x),\mathcal O(0,\vec 0)]\rangle_\beta.$$

Its Fourier transform is defined by

$$G_R(\omega,\vec k) = \int dt\,d^{d-1}x\, e^{i\omega t-i\vec k\cdot\vec x}G_R(t,\vec x).$$

The step function makes $G_R$ causal: it vanishes for $t<0$. As a consequence, for a stable thermal state, $G_R(\omega,\vec k)$ is analytic in the upper half of the complex $\omega$-plane. Singularities below the real axis encode the decay of disturbances.

The spectral density is

$$\rho(\omega,\vec k) =-2\operatorname{Im}G_R(\omega,\vec k),$$

up to the sign convention used for Fourier transforms and operator ordering. For Hermitian bosonic operators one often chooses conventions for which $\rho(\omega,\vec k)$ is positive for $\omega>0$ in suitable channels. The exact normalization is less important than the analytic structure: poles, cuts, residues, and low-frequency slopes are physical.

A small source deformation

$$\delta S_{\mathrm{CFT}} = -\int dt\,d^{d-1}x\,J(t,\vec x)\mathcal O(t,\vec x)$$

produces the linear response

$$\delta\langle\mathcal O(\omega,\vec k)\rangle = G_R(\omega,\vec k)J(\omega,\vec k).$$

The retarded function is therefore the object that answers the physical question: after a small perturbation, what does the plasma do?

Why Euclidean data are not enough by themselves

At finite temperature, Euclidean correlators are defined on a thermal circle. Their Fourier frequencies are discrete Matsubara values,

$$\omega_n=2\pi nT \quad\text{for bosons}, \qquad \omega_n=(2n+1)\pi T \quad\text{for fermions}.$$

The retarded correlator is related to Euclidean data by analytic continuation, but this statement hides a real difficulty. Knowing a function only at discrete Matsubara frequencies is not the same as knowing its analytic continuation. In a holographic computation, Lorentzian signature gives a direct prescription: solve the bulk equations at complex $\omega$ with the correct horizon condition.

The practical rule is:

Desired boundary object	Bulk prescription
Euclidean thermal correlator	smooth Euclidean field on the Euclidean black hole
retarded correlator $G_R$	Lorentzian solution infalling at the future horizon
advanced correlator $G_A$	Lorentzian solution outgoing at the future horizon
quasinormal frequency	infalling at horizon and source-free at boundary
normal mode of thermal AdS	regular in the interior and normalizable at boundary

The difference between a Euclidean regularity condition and a Lorentzian infalling condition is not cosmetic. It is the difference between equilibrium data and causal response.

The scalar black-brane problem

Consider a scalar field $\Phi$ of mass $m$ in the planar AdS$_{d+1}$ black-brane geometry

$$ds^2 = {L^2\over z^2} \left[ -f(z)dt^2+d\vec x^{\,2}+{dz^2\over f(z)} \right], \qquad f(z)=1-\left({z\over z_h}\right)^d.$$

The boundary is at $z=0$, the horizon is at $z=z_h$, and

$$T={d\over4\pi z_h}.$$

Take a Fourier mode

$$\Phi(t,\vec x,z) =e^{-i\omega t+i\vec k\cdot\vec x}\phi_{\omega,\vec k}(z).$$

The Klein-Gordon equation is

$${1\over\sqrt{-g}}\partial_z \left(\sqrt{-g}\,g^{zz}\partial_z\phi\right) - \left(g^{tt}\omega^2+g^{ij}k_i k_j+m^2\right)\phi =0.$$

This is a second-order radial equation. The holographic computation of $G_R$ is a boundary value problem with two asymptotic conditions:

1. near $z=0$, specify the non-normalizable coefficient, interpreted as the boundary source;
2. near $z=z_h$, impose infalling behavior, interpreted as absorption into the future horizon.

Near the AdS boundary, the scalar behaves as

$$\phi(z) = A(\omega,k)z^{d-\Delta}\left(1+\cdots\right) + B(\omega,k)z^{\Delta}\left(1+\cdots\right),$$

where

$$\Delta(\Delta-d)=m^2L^2.$$

For standard quantization, $A$ is the source and $B$ is proportional to the expectation value. With a conventional bulk normalization,

$$G_R(\omega,k) = \mathcal N_\Delta{B(\omega,k)\over A(\omega,k)} +G_{\mathrm{local}}(\omega,k),$$

where

$$\mathcal N_\Delta\propto 2\Delta-d$$

and $G_{\mathrm{local}}$ denotes contact terms fixed by counterterms and scheme choices. The ratio $B/A$ is the important nonlocal part.

The infalling condition

The horizon condition is easiest to derive using the tortoise coordinate. Near the horizon,

$$f(z) \simeq 4\pi T(z_h-z),$$

up to a positive constant depending on the precise radial coordinate convention. Define $r_*$ by

$$dr_*=-{dz\over f(z)}.$$

The minus sign is convenient because the coordinate $z$ increases inward. As $z\to z_h$, one then has $r_*\to -\infty$, as in the usual Schwarzschild radial coordinate. The wave equation locally becomes

$$\left(\partial_{r_*}^2+\omega^2\right)\phi\simeq0,$$

so

$$\Phi\sim e^{-i\omega t}e^{\pm i\omega r_*}.$$

Introduce the ingoing Eddington-Finkelstein coordinate

$$v=t+r_*.$$

The mode

$$e^{-i\omega(t+r_*)}=e^{-i\omega v}$$

is regular on the future horizon. This is the infalling solution. In Schwarzschild time coordinates, it appears as

$$\phi(z) \sim \left(1-{z\over z_h}\right)^{-i\omega/(4\pi T)}.$$

The outgoing solution behaves as

$$\phi(z) \sim \left(1-{z\over z_h}\right)^{+i\omega/(4\pi T)}.$$

The outgoing solution is regular on a past horizon, not on the future horizon. It computes the advanced correlator rather than the retarded correlator. This is one of the most common places to lose a sign.

There is also a cleaner modern phrasing: use ingoing Eddington-Finkelstein coordinates from the start. Then the retarded solution is simply the solution that is regular at the future horizon.

Canonical momentum and the Green function

The ratio $B/A$ is the simplest way to state the scalar result, but the variational formula is more general and more robust.

For a quadratic bulk action

$$S_2 =-{1\over2}\int d^{d+1}x\sqrt{-g} \left(g^{MN}\partial_M\Phi\partial_N\Phi+m^2\Phi^2\right),$$

the radial canonical momentum is

$$\Pi(z;\omega,k) ={\delta S_2\over\delta(\partial_z\Phi)} =-\sqrt{-g}\,g^{zz}\partial_z\Phi.$$

After solving the equation of motion, the on-shell action reduces to a boundary term,

$$S_2^{\mathrm{os}} ={1\over2}\int{d\omega\,d^{d-1}k\over(2\pi)^d} \Phi(-\omega,-\vec k,z)\Pi(\omega,\vec k,z)\bigg|_{z=\epsilon}^{z=z_h}.$$

The horizon term is handled by the causal prescription; the UV term is renormalized by adding local counterterms. The retarded Green function is then obtained from the renormalized momentum-to-field ratio,

$$G_R(\omega,k) = -\lim_{z\to0} {\Pi_{\mathrm{ren}}(z;\omega,k)\over\Phi(z;\omega,k)}.$$

The overall sign depends on the action and Fourier conventions. The invariant statement is that differentiating the renormalized Lorentzian on-shell action with infalling boundary conditions gives the retarded correlator.

A powerful observation, used constantly in transport computations, is that the imaginary part of the retarded correlator is related to radial flux. For a real-frequency solution, define the conserved Wronskian-like flux

$$\mathcal F_z \propto \sqrt{-g}\,g^{zz} \operatorname{Im}\left(\phi^*\partial_z\phi\right).$$

Absorption by the horizon gives a nonzero imaginary part of $G_R$. This is the bulk origin of dissipation.

Quasinormal modes as poles

A quasinormal mode is a solution of the linearized bulk equations satisfying:

$$\text{infalling at the future horizon}, \qquad \text{normalizable/source-free at the AdS boundary}.$$

For the scalar expansion

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

the source-free condition is

$$A(\omega,k)=0.$$

But the retarded Green function has the form

$$G_R(\omega,k) \sim {B(\omega,k)\over A(\omega,k)}.$$

Therefore, when $A(\omega,k)$ vanishes and $B(\omega,k)$ is nonzero, $G_R$ has a pole. The quasinormal spectrum is the pole spectrum of the retarded correlator:

$$A(\omega_n,k)=0 \quad\Longleftrightarrow\quad \omega_n(k)\in\operatorname{Poles}G_R(\omega,k).$$

In a stable black-brane background, these poles lie in the lower half of the complex frequency plane:

$$\operatorname{Im}\omega_n(k)<0.$$

The sign has a direct physical meaning. With the convention $e^{-i\omega t}$,

$$e^{-i\omega_n t} = e^{-i\operatorname{Re}\omega_n t}e^{\operatorname{Im}\omega_n t}.$$

Thus $\operatorname{Im}\omega_n<0$ gives decay for $t>0$.

If a pole crosses into the upper half-plane, the background is linearly unstable. In holographic superconductors, for example, a charged scalar quasinormal mode becomes unstable below the critical temperature. The instability is not a pathology of AdS/CFT; it is the bulk signal that the thermal state wants to reorganize into a new phase.

Time-domain response and thermalization

Suppose a small source perturbs the thermal state and is then turned off. The late-time response can be written by inverse Fourier transform:

$$\delta\langle\mathcal O(t,\vec k)\rangle = \int_{-\infty}^{\infty}{d\omega\over2\pi} e^{-i\omega t}G_R(\omega,k)J(\omega,k).$$

For $t>0$, one closes the contour in the lower half $\omega$-plane. If the retarded Green function is meromorphic, the answer is a sum over residues of poles:

$$\delta\langle\mathcal O(t,\vec k)\rangle \sim \sum_n \operatorname{Res}_{\omega=\omega_n} \left[G_R(\omega,k)J(\omega,k)\right] e^{-i\omega_n t}.$$

The mode with smallest $|\operatorname{Im}\omega_n|$ dominates the latest exponential relaxation. This is why quasinormal modes are often called the ringdown spectrum of the black brane.

There are important qualifications:

• If the operator overlaps with a conserved density, hydrodynamic poles approach the origin as $k\to0$ and dominate long-time, long-distance response.
• At finite $N$ or finite volume, exact CFT evolution is unitary and can include recurrences; the classical black-brane computation captures the large-$N$, thermodynamic, coarse-grained response.
• Beyond the classical supergravity limit, branch cuts and multiparticle effects can modify the late-time analytic structure.
• Very late-time behavior can be sensitive to effects that are invisible in the leading $N\to\infty$ saddle.

The safe statement is that classical quasinormal modes describe the leading large-$N$ linear relaxation of strongly coupled thermal states with a black-hole dual.

Hydrodynamic and nonhydrodynamic modes

Not all quasinormal modes are alike. The most important distinction is between hydrodynamic and nonhydrodynamic modes.

Hydrodynamic modes are forced by conservation laws. Their frequencies vanish as $k\to0$:

$$\omega(k)\to0 \qquad \text{as}\qquad k\to0.$$

Examples include charge diffusion, shear diffusion, and sound:

Channel	Hydrodynamic pole
conserved charge density	$\omega=-iDk^2+\cdots$
transverse momentum density	$\omega=-iD_\eta k^2+\cdots$
longitudinal energy-momentum	$\omega=\pm c_s k-i\Gamma_s k^2+\cdots$

For a neutral conformal plasma in $3+1$ boundary dimensions with a two-derivative Einstein gravity dual,

$$D_\eta={\eta\over\epsilon+p}={1\over4\pi T}, \qquad c_s^2={1\over3}, \qquad \omega_{\mathrm{sound}} = \pm {k\over\sqrt3}-{i k^2\over6\pi T}+\cdots.$$

These formulas will be derived in the transport module. The point here is structural: hydrodynamic poles are quasinormal modes protected by conservation laws.

Nonhydrodynamic modes remain at complex frequencies of order the temperature when $k\to0$:

$$\omega_n(k=0) \sim 2\pi T\times(\text{complex number}).$$

They are not captured by ordinary hydrodynamics. They encode microscopic relaxation at strong coupling. In a weakly coupled plasma one might look for quasiparticle poles near the real axis. In many classical holographic plasmas, by contrast, nonhydrodynamic poles are typically far from the real axis, reflecting rapid damping and the absence of long-lived quasiparticles.

Gauge-invariant perturbations

For scalar fields, the boundary value problem is straightforward. For currents and stress tensors, one must handle gauge redundancy carefully.

A bulk gauge field $A_M$ has gauge transformations

$$A_M\to A_M+\partial_M\Lambda.$$

Metric perturbations have infinitesimal diffeomorphism redundancy,

$$h_{MN}\to h_{MN}+\nabla_M\xi_N+\nabla_N\xi_M.$$

Individual components such as $h_{tx}$ or $A_z$ are therefore not automatically physical. The reliable procedure is to build gauge-invariant master fields or fix a gauge and then impose the constraint equations consistently.

For a perturbation with momentum along $x$, the black-brane rotational symmetry decomposes fluctuations into channels:

Channel	Typical boundary operators	Typical physics
scalar field channel	scalar operator $\mathcal O$	nonconserved relaxation
vector/current channel	$J^\mu$	charge diffusion, conductivity
shear metric channel	$T^{ty}$, $T^{xy}$	momentum diffusion, shear viscosity
sound metric channel	$T^{tt}$, $T^{tx}$, $T^{xx}$, trace combinations	sound propagation

The radial constraint equations are the bulk versions of Ward identities. For example, gauge invariance leads to current conservation, and diffeomorphism invariance leads to stress-tensor conservation. This is why hydrodynamic modes appear in current and stress-tensor channels but not for a generic nonconserved scalar operator.

A canonical example: massless scalar in AdS$_5$ black brane

For the AdS$_5$ black brane, a convenient radial coordinate is

$$u= \left({z\over z_h}\right)^2, \qquad f(\nu)=1-\nu^2, \qquad \nu=0\ \text{boundary}, \qquad \nu=1\ \text{horizon}.$$

For a massless scalar, dual to an operator of dimension $\Delta=4$, define

$$\mathfrak w={\omega\over2\pi T}, \qquad \mathfrak q={k\over2\pi T}.$$

The radial equation takes the standard form

$$\phi''(\nu) - {1+\nu^2\over\nu f(\nu)}\phi'(\nu) + {\mathfrak w^2-\mathfrak q^2 f(\nu)\over \nu f(\nu)^2}\phi(\nu)=0,$$

where primes denote derivatives with respect to $\nu$. The infalling behavior at $\nu=1$ is

$$\phi(\nu) = (1-\nu)^{-i\mathfrak w/2}F(\nu),$$

with $F(\nu)$ regular at $\nu=1$. Near the boundary,

$$\phi(\nu) =A(\omega,k) \left(1+\cdots\right) + B(\omega,k)\nu^2 \left(1+\cdots\right) +\text{logarithmic terms}.$$

The logarithmic terms occur because $\Delta-d=0$ for a massless scalar in AdS$_5$; they contribute local contact terms. The nonlocal retarded correlator is still read from the ratio $B/A$ after holographic renormalization.

The numerical computation is conceptually simple:

1. factor out the infalling behavior;
2. impose regularity of $F(\nu)$ at $\nu=1$;
3. integrate to the boundary;
4. extract $A(\omega,k)$ and $B(\omega,k)$;
5. compute $G_R\sim B/A$ or find zeros of $A$ to obtain quasinormal frequencies.

The practical computation can be done by shooting, Frobenius series, spectral collocation, or determinant methods. In serious applications, one checks convergence by increasing resolution, verifies gauge-invariant variables, and confirms that residual gauge modes are absent.

Poles, peaks, and spectral functions

A pole of the retarded Green function near the real axis often produces a peak in the spectral density. But a peak is not the same thing as a pole, and absence of a narrow peak does not mean absence of a mode.

Near a simple pole,

$$G_R(\omega,k) \simeq {R_n(k)\over \omega-\omega_n(k)} +\text{regular}.$$

If

$$\omega_n=\Omega_n-i\Gamma_n, \qquad \Gamma_n>0,$$

then the time-domain contribution is

$$e^{-i\omega_n t}=e^{-i\Omega_n t}e^{-\Gamma_n t}.$$

A small $\Gamma_n$ produces a long-lived excitation and a sharp spectral feature. A large $\Gamma_n$ produces rapid decay and a broad response. Strongly coupled holographic plasmas often have broad spectral functions because their excitations decay on timescales of order $1/T$.

For conserved currents and stress tensors, the small-frequency part of the spectral function determines transport coefficients through Kubo formulas. For example, shear viscosity is extracted from

$$\eta = - \lim_{\omega\to0}{1\over\omega} \operatorname{Im}G_R^{T^{xy}T^{xy}}(\omega,\vec k=0).$$

The next module will turn this equation into a computation.

Quasinormal modes versus normal modes

At zero temperature in global AdS, regularity in the interior and normalizability at the boundary lead to real normal-mode frequencies. These modes correspond to stable oscillations of the CFT on the cylinder.

At finite temperature with a black-hole horizon, the boundary condition changes. The future horizon absorbs energy. The spectrum becomes complex:

$$\omega_n=\Omega_n-i\Gamma_n.$$

This is the real-time version of the difference between a box and an absorber. AdS still reflects waves at the boundary, but the black hole supplies a dissipative sink in the interior.

For thermal AdS without a horizon, there is no classical absorption and no black-hole quasinormal spectrum. For an AdS black brane, the horizon is essential. This is why the Hawking-Page transition is not merely thermodynamic; it changes the analytic structure of real-time correlators at leading order in large $N$.

What QNMs do and do not prove

Quasinormal modes are extraordinarily useful, but their interpretation should be precise.

They do tell us:

• the pole spectrum of leading large-$N$ retarded correlators;
• linear relaxation times of small perturbations around a thermal state;
• hydrodynamic dispersion relations in conserved channels;
• stability or instability of a chosen black-hole background;
• how horizon absorption appears as boundary dissipation.

They do not by themselves tell us:

• the full nonlinear thermalization process;
• finite-$N$ recurrence physics;
• the complete late-time behavior when branch cuts or nonperturbative effects matter;
• that a narrow quasiparticle exists;
• that every broad spectral feature should be interpreted as a particle.

A mature use of holography keeps the hierarchy clear: QNMs are the correct language for linear response about a black-hole saddle.

Common mistakes

Mistake 1: using the outgoing mode for a retarded correlator. The retarded prescription uses the future horizon, hence infalling or regular in ingoing coordinates. The outgoing condition gives the advanced correlator.

Mistake 2: confusing source-free with field-free. A quasinormal mode is not zero at the boundary in every coefficient. It has zero non-normalizable/source coefficient and generally nonzero normalizable/response coefficient.

Mistake 3: reading physical poles from gauge-dependent variables. For currents and stress tensors, use gauge-invariant combinations or impose constraints carefully.

Mistake 4: treating contact terms as relaxation physics. Local polynomial terms in $\omega$ and $k$ can shift real parts of correlators but do not create dissipative poles.

Mistake 5: assuming Euclidean smoothness automatically gives the retarded answer. Euclidean data require analytic continuation. The Lorentzian prescription directly selects the causal Green function.

Mistake 6: overinterpreting the leading saddle. Classical black-hole QNMs are leading large-$N$ objects. Exact finite-$N$ CFT evolution is unitary.

Exercises

Exercise 1: The infalling exponent

Consider a black-brane horizon with

$$f(z)\simeq4\pi T(z_h-z).$$

Using the tortoise coordinate $dr_*=-dz/f(z)$, show that an infalling scalar mode behaves as

$$\phi(z)\sim\left(1-{z\over z_h}\right)^{-i\omega/(4\pi T)}.$$

Solution

Near the horizon,

$$dr_*=-{dz\over f(z)} \simeq -{dz\over4\pi T(z_h-z)}.$$

Therefore

$$r_* \simeq {1\over4\pi T}\log(z_h-z)+\text{constant}.$$

Since $z_h-z$ is proportional to $1-z/z_h$, the infalling Eddington-Finkelstein coordinate is

$$v=t+r_*.$$

A regular infalling wave is

$$\Phi\sim e^{-i\omega v}=e^{-i\omega t}e^{-i\omega r_*}.$$

Thus the radial factor is

$$\phi(z) \sim (z_h-z)^{-i\omega/(4\pi T)} \sim \left(1-{z\over z_h}\right)^{-i\omega/(4\pi T)}.$$

This is exactly the exponent used in the retarded prescription. The outgoing mode has the opposite exponent.

Exercise 2: Why QNMs are poles

Suppose the infalling solution near the boundary has the expansion

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

Assume standard quantization and no degeneracy. Explain why $A(\omega_n,k)=0$ implies that $G_R(\omega,k)$ has a pole at $\omega=\omega_n$.

Solution

In standard quantization, the coefficient $A$ is the source and $B$ is proportional to the response. The retarded Green function is the linear response coefficient,

$$G_R(\omega,k) \sim {B(\omega,k)\over A(\omega,k)} +\text{local terms}.$$

A quasinormal mode is an infalling solution with no source at the boundary, so

$$A(\omega_n,k)=0.$$

If $B(\omega_n,k)\neq0$ and the zero of $A$ is simple, then near $\omega_n$,

$$A(\omega,k) \simeq A'(\omega_n,k)(\omega-\omega_n),$$

so

$$G_R(\omega,k) \sim {B(\omega_n,k) \over A'(\omega_n,k)(\omega-\omega_n)}.$$

This is a simple pole.

Exercise 3: Decay from a complex pole

Let a retarded correlator have a pair of poles at

$$\omega_\pm=\pm\Omega-i\Gamma, \qquad \Gamma>0.$$

What is the qualitative time dependence of the corresponding contribution to the response?

Solution

A pole at $\omega_+=\Omega-i\Gamma$ contributes

$$e^{-i\omega_+t} =e^{-i\Omega t}e^{-\Gamma t}.$$

A pole at $\omega_-=-\Omega-i\Gamma$ contributes

$$e^{-i\omega_-t} =e^{+i\Omega t}e^{-\Gamma t}.$$

For a real perturbation, the two contributions combine into a damped oscillation,

$$\delta\langle\mathcal O(t)\rangle \sim e^{-\Gamma t}\cos(\Omega t+\varphi),$$

where $\varphi$ depends on the residues and the source. The damping time is $\tau=1/\Gamma$.

Exercise 4: The shear diffusion pole

For a neutral conformal plasma with a two-derivative Einstein gravity dual in $3+1$ boundary dimensions, use

$${\eta\over s}={1\over4\pi}, \qquad \epsilon+p=sT$$

to show that the shear diffusion constant is

$$D_\eta={1\over4\pi T}.$$

Solution

The shear diffusion constant is

$$D_\eta={\eta\over\epsilon+p}.$$

For a thermal state with no chemical potential,

$$\epsilon+p=sT.$$

Therefore

$$D_\eta ={\eta\over sT} ={1\over T}{\eta\over s} ={1\over4\pi T}.$$

The corresponding hydrodynamic quasinormal mode is

$$\omega=-iD_\eta k^2+\cdots =-i{k^2\over4\pi T}+\cdots.$$

Exercise 5: Normal modes versus quasinormal modes

Explain why a scalar field in global thermal AdS has real normal-mode frequencies at leading classical order, while a scalar field on an AdS black-brane background has complex quasinormal frequencies.

Solution

In global thermal AdS there is no horizon. The radial problem is like a reflecting box: impose regularity in the interior and normalizability at the boundary. The resulting linear operator has a self-adjoint structure under suitable boundary conditions, so the frequencies are real. The dual CFT on the cylinder has stable oscillatory modes at leading large $N$.

In an AdS black-brane background there is a future horizon. The retarded problem imposes infalling boundary conditions at the horizon. Energy can be absorbed by the black hole, so the boundary perturbation decays. The radial problem is not a self-adjoint normal-mode problem with reflecting conditions at both ends. The frequencies are complex,

$$\omega_n=\Omega_n-i\Gamma_n, \qquad \Gamma_n>0,$$

for a stable black brane.

Further reading

• G. T. Horowitz and V. E. Hubeny, Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium.
• D. Birmingham, I. Sachs, and S. N. Solodukhin, Conformal Field Theory Interpretation of Black Hole Quasinormal Modes.
• D. T. Son and A. O. Starinets, Minkowski-space Correlators in AdS/CFT Correspondence: Recipe and Applications.
• P. K. Kovtun and A. O. Starinets, Quasinormal Modes and Holography.
• C. P. Herzog and D. T. Son, Schwinger-Keldysh Propagators from AdS/CFT Correspondence.
• K. Skenderis and B. C. van Rees, Real-time Gauge/Gravity Duality.