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On the Absorption by Near-Extremal Black Branes

G. Policastro, A. Starinets

TL;DR

This work develops a reliable analytic framework to compute scalar absorption (greybody factors) by near-extremal black branes, unifying D3-, M2-, M5-, and Dp-brane backgrounds. It introduces a low-temperature uniform expansion using Langer-Olver's method, yielding a recursive, power-series correction to the extremal absorption P_0(l) through f_l(λ) with leading terms in 1/λ^4 and explicit ν=l+2 dependence, and it separately explores a hypergeometric reduction for cross-checking and for capturing certain regimes. A complementary high-temperature expansion is provided, and the D1/D5 system is used as a checkpoint where gravity and finite-temperature field theory results largely agree for the s-wave but diverge for higher partial waves, especially at high temperature; the authors also extend the methodology to non-extremal M-branes and general black p-branes, establishing a broad, adaptable approach to greybody factors in strongly coupled finite-temperature gauge theories. The results illuminate how finite-temperature effects modify absorption and offer a framework to compare gravity predictions with field-theoretic expectations, contributing to the broader understanding of AdS/CFT at finite temperature and transport properties of the dual gauge theories.

Abstract

We study the absorption of a minimally coupled scalar in the gravitational background created by a stack of near-extremal black three-branes, and more generally by M2, M5 and Dp branes. The absorption probability has the form P(l) = P_0(l) f_l(λ), where P_0(l) is the partial wave's absorption probability in the extremal case, and the thermal factor f_l(λ) depends on the ratio of the frequency of the incoming wave and the Hawking temperature, λ= ω/πT. Using Langer-Olver's method, we obtain a low-temperature (λ\gg 1) asymptotic expansion for P(l) with coefficients determined recursively. This expansion, which turns out to be a fairly good approximation even for λ\sim 1, accounts for all power-like finite-temperature corrections to P_0(l), and we calculate a few terms explicitly. We also show that at low temperature the absorption probability contains exponentially suppressed terms, and attempt to develop an approximation scheme to calculate those. The high-temperature expansion is also considered. For the s-wave, the low-temperature gravity result is consistent with the free finite-temperature field theory calculation, while for high temperature and higher partial waves we find a disagreement. As a check of the approximation methods used, we apply them to the D1-D5-brane system, and compare results to the known exact solution.

On the Absorption by Near-Extremal Black Branes

TL;DR

This work develops a reliable analytic framework to compute scalar absorption (greybody factors) by near-extremal black branes, unifying D3-, M2-, M5-, and Dp-brane backgrounds. It introduces a low-temperature uniform expansion using Langer-Olver's method, yielding a recursive, power-series correction to the extremal absorption P_0(l) through f_l(λ) with leading terms in 1/λ^4 and explicit ν=l+2 dependence, and it separately explores a hypergeometric reduction for cross-checking and for capturing certain regimes. A complementary high-temperature expansion is provided, and the D1/D5 system is used as a checkpoint where gravity and finite-temperature field theory results largely agree for the s-wave but diverge for higher partial waves, especially at high temperature; the authors also extend the methodology to non-extremal M-branes and general black p-branes, establishing a broad, adaptable approach to greybody factors in strongly coupled finite-temperature gauge theories. The results illuminate how finite-temperature effects modify absorption and offer a framework to compare gravity predictions with field-theoretic expectations, contributing to the broader understanding of AdS/CFT at finite temperature and transport properties of the dual gauge theories.

Abstract

We study the absorption of a minimally coupled scalar in the gravitational background created by a stack of near-extremal black three-branes, and more generally by M2, M5 and Dp branes. The absorption probability has the form P(l) = P_0(l) f_l(λ), where P_0(l) is the partial wave's absorption probability in the extremal case, and the thermal factor f_l(λ) depends on the ratio of the frequency of the incoming wave and the Hawking temperature, λ= ω/πT. Using Langer-Olver's method, we obtain a low-temperature (λ\gg 1) asymptotic expansion for P(l) with coefficients determined recursively. This expansion, which turns out to be a fairly good approximation even for λ\sim 1, accounts for all power-like finite-temperature corrections to P_0(l), and we calculate a few terms explicitly. We also show that at low temperature the absorption probability contains exponentially suppressed terms, and attempt to develop an approximation scheme to calculate those. The high-temperature expansion is also considered. For the s-wave, the low-temperature gravity result is consistent with the free finite-temperature field theory calculation, while for high temperature and higher partial waves we find a disagreement. As a check of the approximation methods used, we apply them to the D1-D5-brane system, and compare results to the known exact solution.

Paper Structure

This paper contains 19 sections, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Absorption by non-extremal black three-branes
  4. Low-temperature expansion I: Langer - Olver method
  5. Low-temperature expansion II: Reduction to a hypergeometric equation
  6. High-temperature expansion
  7. Absorption by the $D1/D5$-branes
  8. Low-temperature asymptotic expansion
  9. High-temperature expansion
  10. Field theory calculation
  11. D1/D5 system
  12. Non-extremal three-branes
  13. Black $p$-branes and non-extremal $M$-branes
  14. Non-extremal $M$-branes
  15. Black $p$-branes:
  16. ...and 4 more sections

Figures (2)

  • Figure 1: $D1/D5$ system: exact solution Re $\phi (x)$ (solid line) and a partial sum (with all $A_s$, $B_s$ equal to zero for $s>0$) of the asymptotic series for $\lambda = 1$ (dashed line).
  • Figure 2: $D1/D5$ system: ratio $P(l)/P_0(l)$ versus $1/\lambda$ for $l=1$: exact (solid line), high-temperature approximation (\ref{['d1_d5_T_1']}) (dashed line), sum of the first three terms of the low-temperature asymptotic expansion (\ref{['p_00']}) (dotted line).