Quasinormal modes and holography

Pavel K. Kovtun and Andrei O. Starinets

Phys. Rev. D 72, 086009 – Published 14 October 2005

Abstract

Quasinormal frequencies of electromagnetic and gravitational perturbations in asymptotically anti-de Sitter spacetime can be identified with poles of the corresponding real-time Green’s functions in a holographically dual finite temperature field theory. The quasinormal modes are defined for gauge-invariant quantities which obey an incoming-wave boundary condition at the horizon and a Dirichlet condition at the boundary. As an application, we explicitly find poles of retarded correlation functions of $R$ -symmetry currents and the energy-momentum tensor in strongly coupled finite temperature $N = 4$ supersymmetric $S U (N_{c})$ Yang-Mills theory in the limit of large $N_{c}$ .

Received 21 July 2005

DOI:https://doi.org/10.1103/PhysRevD.72.086009

Authors & Affiliations

Pavel K. Kovtun^*

KITP, University of California, Santa Barbara, California 93106-4030, USA

Andrei O. Starinets^†

Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada

^*Electronic address: kovtun@kitp.ucsb.edu
^†Electronic address: starina@perimeterinstitute.ca

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Vol. 72, Iss. 8 — 15 October 2005

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Images

Figure 1
Quasinormal spectrum of electric field fluctuations in the plane of complex $w \equiv ω / 2 π T$ , shown for spatial momentum $q \equiv q / 2 π T = 1$ . Quasinormal frequencies on the left are defined by Eq. (4.5a) for the transverse electric field and coincide with the poles of $Π^{T} (ω, q)$ , as explained in the text. Quasinormal frequencies on the right are defined by Eq. (4.5b) for the longitudinal electric field and coincide with the poles of $Π^{L} (ω, q)$ . As $q$ decreases, all poles stay at a finite distance from the real axis, except for the one marked with a large dot. This pole is purely imaginary and approaches the origin in the limit $q \to 0$ . The presence of this special quasinormal frequency is a manifestation of the diffusive relaxation of $R$ -charge density fluctuations in the dual $N = 4$ SYM theory.Reuse & Permissions
Figure 2
Quasinormal spectrum of gravitational fluctuations in the scalar channel, shown in the plane of complex $w \equiv ω / 2 π T$ , for spatial momentum $q \equiv q / 2 π T = 1$ . The quasinormal frequencies coincide with poles of $G_{3} (ω, q)$ , as explained in the text. As $q \to 0$ , all poles stay at a finite distance from the real axis, as one expects from the absence of hydrodynamic singularities in $G_{3} (ω, q)$ .Reuse & Permissions
Figure 3
Quasinormal spectrum of gravitational fluctuations in the shear channel, shown in the plane of complex $w \equiv ω / 2 π T$ , for spatial momentum $q \equiv q / 2 π T = 1$ . The quasinormal frequencies coincide with poles of $G_{1} (ω, q)$ , as explained in the text. As $q$ decreases, all poles stay at a finite distance away from the real axis, except for the one marked with a large dot. This pole is purely imaginary and approaches the origin in the limit $q \to 0$ . The presence of this special quasinormal frequency is a manifestation of the diffusive relaxation of transverse momentum density fluctuations in the dual $N = 4$ SYM theory.Reuse & Permissions
Figure 4
Quasinormal spectrum of gravitational fluctuations in the sound channel, shown in the plane of complex $w \equiv ω / 2 π T$ , for spatial momentum $q \equiv q / 2 π T = 1$ . The quasinormal frequencies coincide with the poles of $G_{2} (ω, q)$ , as explained in the text. As $q$ decreases, all poles stay at a finite distance away from the real axis, except for the ones marked with large dots, which approach the origin as $q \to 0$ (see Appendix for the corresponding dispersion curves). Such behavior of the lowest quasinormal frequencies is a manifestation of oscillatory relaxation of longitudinal momentum density (as well as energy density) fluctuations in the dual $N = 4$ SYM theory.Reuse & Permissions
Figure 5
Real and imaginary parts of the lowest quasinormal (sound wave) frequency as a function of spatial momentum. Light curves correspond to the analytic approximation (4.43) for small $q$ . The dashed line is $w = q$ . As $q$ grows, the dispersion curve enters the region where $\partial Re (w) / \partial q > 1$ . This, however, happens for $q \sim 1$ , where the corresponding singularity of the energy-momentum correlation function no longer has the interpretation of a sound wave.Reuse & Permissions
Figure 6
Real and imaginary parts of three lowest quasinormal frequencies as functions of spatial momentum. The curves for which $w \to 0$ as $q \to 0$ correspond to hydrodynamic sound mode in the dual finite temperature $N = 4$ SYM theory.Reuse & Permissions

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