Universal bound on dynamical relaxation times and black-hole quasinormal ringing

Abstract

From information theory and thermodynamic considerations a universal bound on the relaxation time τ of a perturbed system is inferred, τ ≥h/πT , where T is the system's temperature. We prove that black holes comply with the bound; in fact they actually saturate it. Thus, when judged by their relaxation properties, black holes are the most extreme objects in nature, having the maximum relaxation rate which is allowed by quantum theory.A fundamental problem in thermodynamic and statistical physics is to study the response of a system in thermal equilibrium to an outside perturbation [1][2][3]. In particular, one is typically interested in calculating the relaxation timescale at which the perturbed system returns to a stationary, equilibrium configuration. Can this relaxation time be made arbitrarily small? That the answer may be negative is suggested by the third-law of thermodynamics, according to which the relaxation time of a perturbed system is expected to go to infinity in the limit of absolute zero of temperature. Finite temperature systems are expected to have faster dynamics and shorter relaxation times-how small can these be made? In this Letter we use general results from quantum information theory in order to derive a fundamental bound on the maximal rate at which a perturbed system approaches thermal equilibrium.On another front, deep connections between the world of black-hole physics and the realms of thermodynamics and information theory were revealed by Hawking's theoretical discovery of black-hole radiation [4], and its corresponding temperature and black-hole entropy [5]. These discoveries imply that black holes behave as thermodynamic systems in many respects. Furthermore, black holes have been proven to be very useful in deriving fundamental, static bounds on information storage [6][7][8][9][10][11]. Can one use black holes to obtain deep insights into natural limitations on dynamical relaxation times? Indeed one can, as we shall show below.Quantum information theory and Thermodynamics.-A fundamental question in quantum information theory is what is the maximum rate,İ max , at which information may be transmitted by a signal of duration τ and energy ∆E. The answer to this question was already found in the 1960's (see e.g. [12,13]):(We use units in which k B = G = c = 1.) An outside perturbation to a thermodynamic system is characterized by energy and entropy changes in the system, ∆E and ∆S, respectively. By the complementary relation between entropy and information (entropy as a measure of one's uncertainty or lack of information about the actual internal state of the system [14][15][16]12]), the relation Eq. (1) sets an upper bound on the rate of entropy change [17],where τ is the characteristic timescale for this dynamical process (the relaxation time required for the perturbed system to return to a quiescent state). Taking cognizance of the second-law of thermodynamics, one obtains from Eq. (2)where T is the system's temperature. Thus, according to quantum theory, a thermodynamic system has...

Quasinormal frequencies of asymptotically anti-de Sitter black holes in two dimensions

“…Defining the quantity H = ω F /(πT H ), we obtain that the bound (34) becomes H ≤ 1. The upper bound (34) is valid for the fundamental QNF of several black holes as Schwarzschild, small Schwarzschild anti-de Sitter, Schwarzschild de Sitter, (extreme) Kerr, and Kerr-Newman [64,67,68]. Other gravitational systems saturate the upper bound (34), for example, the Nariai spacetime [69] and it is expected that in the extremal limit the Schwarzschild de Sitter and the Kerr-Newman black holes saturate this bound [68,70].…”

“…For these gravitational objects it is possible to show that the lower bound (33) transforms into an upper bound on the absolute value of the imaginary part for the fundamental QNF (in this Subsection this quantity is denoted by ω F ). Thus Hod finds that for a black hole the bound (33) can be written as [64] …”

“…Thus a perturbed thermodynamic system has at least one relaxation mode that satisfies the inequality (33). Hod calls to the bound (33) as Time Times Temperature bound or briefly TTT bound [64]. See also Refs.…”

Gravitation, thermodynamics, and the bound on viscosity

“…• The universal relaxation bound [11,12]. This bound asserts that the relaxation time of a perturbed thermodynamic system is bounded from below by…”

On the statistical-mechanical meaning of the Bousso bound

“…From equation (5) note that Bousso bound demands the null energy condition be satified (ρ + p ≥ 0, in present case), unphysical negative values for entropy density would be otherwise required. Equation (6), in particular when µ = 0, recalls the expression for the universal bound to the relaxation timescales of perturbed systems found in [15].…”