Kubo Formulae and Horizon Response

A transport coefficient can be read from horizon data only when the boundary Kubo observable, fluctuation channel, current normalization, and zero-frequency limit have first been fixed. The simplification follows from a radially conserved canonical momentum in a decoupled low-frequency channel. It is not a rule that every response is local to the horizon.

Required background. Sources, Linear Response, and Kubo Formulae supplies the boundary response definitions. Lorentzian Holographic Correlators and Infalling Conditions supplies the retarded horizon prescription.

Helpful background. Renormalized One-Point Functions and the Variational Problem supplies the boundary canonical momenta and counterterms. Transport Sum Rules and Ultraviolet Constraints supplies checks beyond the low-frequency limit.

The observable comes before the horizon

Use Fourier convention $e^{-i\omega t+i\mathbf k\cdot\mathbf x}$ and define

$$G_R^{AB}(\omega,\mathbf k) =-i\int_0^\infty dt\int d^d x\, e^{i\omega t-i\mathbf k\cdot\mathbf x} \langle[A(t,\mathbf x),B(0)]\rangle .$$

For a homogeneous electric perturbation $E_x=i\omega A_x^{(0)}$, the conductivity is

$$\sigma(\omega)=\frac{G_{R,\mathrm{tr}}^{J_xJ_x}(\omega,\mathbf0)}{i\omega}, \qquad G_{R,\mathrm{tr}}(\omega)=G_R(\omega)-G_R(0)$$

when the static local term must be subtracted. The precise subtraction follows from the renormalized variational problem; it is not optional curve processing. For shear viscosity,

$$\eta=-\lim_{\omega\to0^+}\frac{1}{\omega} \operatorname{Im}G_R^{T_{xy}T_{xy}}(\omega,\mathbf0).$$

The order is $\mathbf k=0$ first, then $\omega\to0^+$. Reversing it can instead measure a static susceptibility or hydrostatic response. Charge diffusion uses a different longitudinal scaling limit, with $\omega\sim k^2$; it cannot be inferred by silently recycling the optical limit.

Radial canonical momentum

Suppose a gauge-invariant fluctuation $\phi(r,\omega)$ has quadratic action

$$S^{(2)}=\frac12\int dr\,\frac{d\omega}{2\pi} \left[K(r)\phi'(-\omega)\phi'(\omega) +M(r,\omega)\phi(-\omega)\phi(\omega)\right].$$

Its radial momentum is $\Pi=K\phi'$. The equation of motion gives

$$\partial_r\Pi=M\phi .$$

If $M=O(\omega^2)$ at zero spatial momentum, then $\Pi$ is conserved through $O(\omega)$. The retarded response follows from the renormalized boundary ratio

$$G_R(\omega,0)=-\lim_{r\to\partial}\frac{\Pi(r,\omega)}{\phi(r,\omega)}+G_{\mathrm{ct}}.$$

This infalling source–response prescription was established in Son and Starinets 2002.

Because $\Pi/(i\omega\phi)$ is radially constant at the required order, it may be evaluated where the infalling condition is algebraic: the future horizon. This membrane argument was formulated systematically by Iqbal and Liu 2009.

Maxwell conductivity as the first application

Take

$$S_A=-\frac{1}{4g_F^2}\int d^{d+1}x\sqrt{\lvert g\rvert}\,F_{ab}F^{ab}$$

in a neutral, diagonal, translation-invariant black-brane background. In the inherited $(+,-,\ldots,-)$ convention, $g_{tt}>0$ and $g_{rr},g_{xx}<0$ outside the horizon. For $A_x(r)e^{-i\omega t}$, the radial canonical momentum is the contravariant boundary current

$$J^x(r)=-\frac{1}{g_F^2}\sqrt{\lvert g\rvert}\,g^{rr}g^{xx}A_x'(r).$$

At $k=0$, $\partial_rJ^x=O(\omega^2)$. Future-horizon regularity implies

$$A_x'=-i\omega\sqrt{\frac{-g_{rr}}{g_{tt}}}\,A_x+O(\omega^2),$$

and therefore, using $E_x=i\omega A_x^{(0)}$,

$$\sigma_{\mathrm{dc}} =\left.\frac{1}{g_F^2}\sqrt{\lvert g\rvert}\, (-g^{xx})\sqrt{\frac{-g^{rr}}{g_{tt}}}\right|_{r_h}.$$

This result includes the chosen Maxwell normalization. At nonzero density, $A_x$ generally mixes with $h_{tx}$, and exact momentum conservation supplies a delta function at zero frequency. One must then isolate an incoherent channel or add momentum relaxation before interpreting a finite horizon expression as the full dc conductivity.

Failure of radial conservation

The adversarial control is to retain the term that was dropped. At finite $\omega$, finite $k$, with a bulk mass, or in a coupled channel, $M\phi$ is generally nonzero and the response flows between horizon and boundary. Computing the same ratio at several radii exposes this immediately. A radially varying result cannot be repaired by evaluating it at the horizon more accurately.

Even when the dc flux is conserved, quasinormal poles, contact terms, and optical spectral weight require the full radial solution. Horizon regularity fixes causality; it does not erase ultraviolet normalization or operator mixing.

Exercises

Explain why the formula for $\eta$ does not use the limit $\omega=0$ before forming $\operatorname{Im}G_R/\omega$.

Solution

At exactly zero frequency the dissipative imaginary part vanishes. Viscosity is its first slope as the real-frequency axis is approached from the retarded side. Setting $\omega=0$ first discards the quantity being measured.

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

• Iqbal, Nabil, and Hong Liu. “Universality of the Hydrodynamic Limit in AdS/CFT and the Membrane Paradigm.” Physical Review D 79, 025023 (2009). DOI.
• Son, Dam T., and Andrei O. Starinets. “Minkowski-Space Correlators in AdS/CFT Correspondence: Recipe and Applications.” Journal of High Energy Physics 2002, 042 (2002). DOI.