Electromagnetic absorption cross section of Reissner-Nordstro ¨ m black holes Luı ´s C. B. Crispino * Faculdade de Fı ´sica, Universidade Federal do Para ´, 66075-110, Bele ´m, PA, Brazil Ednilton S. Oliveira + Faculdade de Fı ´sica, Universidade Federal do Para ´, 66075-110, Bele ´m, PA, Brazil and Instituto de Fı ´sica, Universidade de Sa ˜o Paulo, CP 66318, 05315-970, Sa ˜o Paulo, SP, Brazil (Received 13 May 2008; published 7 July 2008) The absorption cross section of Reissner-Nordstro ¨m black holes for the electromagnetic field is computed numerically for arbitrary frequencies. The numerical results are in excellent agreement with the low- and high-frequency limits, which are obtained with analytical methods. Special emphasis is given to the extreme Reissner-Nordstro ¨m black hole case. DOI: 10.1103/PhysRevD.78.024011 PACS numbers: 04.40.b, 04.70.s, 11.80.m The study of wave scattering and absorption by black holes (see, e.g., Ref. [1] and references therein) has gained additional importance after Hawking realized that black holes should evaporate thermally [2]. One reason for this is that the power emitted by the black hole is related to the rate of absorption. The power spectrum of black holes was largely analyzed soon after Hawking published his results. For instance, we can mention the set of works developed by Page, who computed Hawking emission for different kinds of black holes and particles [3]. Nevertheless, until recently the absorption spectrum of black holes has not been studied so extensively as the black hole emission spectrum. Today, the Schwarzschild black hole absorption cross section for arbitrary frequencies is known for fields with spin 0, 1=2, 1, and 2. The results for the scalar field were first obtained by Sanchez [4], Doran et al. studied the fermion case [5], the electromagnetic field has been studied by the present authors together with Higuchi and Matsas [6], and Dolan has obtained the results for gravitational waves [7]. The absorption and emission spectra of Reissner-Nordstro ¨m black holes have been analyzed for the scalar field by Jung, Kim, and Park [8]. In this paper, we compute the electromagnetic absorp- tion cross section of Reissner-Nordstro ¨m black holes for arbitrary frequencies. We obtain the electromagnetic ab- sorption cross section for different values of the black hole charge including both the Schwarzschild and extreme Reissner-Nordstro ¨m cases. We will use natural units with @ ¼ c ¼ G ¼ 1 and the metric signature ( þ). The solutions of the electromagnetic field equations in charged black hole spacetimes cannot be expressed in terms of well-known special functions for all frequency values. Therefore, we solve the electromagnetic field equa- tions by numerical computation to find the absorption cross section for arbitrary frequencies. We start with the Reissner-Nordstro ¨m line element [9] ds 2 ¼ fðrÞdt 2 ½fðrÞ 1 dr 2  r 2 ðd 2 þ sin 2 d 2 Þ; where fðrÞ¼ð1  r þ =rÞð1  r  =rÞ, with r  ¼ M  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 2  Q 2 p , and M and Q are the black hole mass and charge, respectively. The Lagrangian density of the electromagnetic field in a modified Feynman gauge is [10] L F ¼ ffiffiffiffiffiffiffi g p ½ 1 4 F  F   1 2 H 2 ; (1) where H ¼r  A  þ K  A  . The equations of motion re- lated to the Lagrangian density (1) are r  F  þr  H  K  H ¼ 0: (2) Here we choose K  ¼ð0; df=dr; 0; 0Þ, which makes the equation for the component A t of the electromagnetic potential to decouple from the equations for A r , A  , and A  . Using the fact that Reissner-Nordstro ¨m spacetime is static, we look for solutions of Eq. (2) of the form A ð"n!lmÞ  ¼  "n!lm ðr; ; Þe i!t ; !> 0; where the label " stands for the four different mode polar- izations, namely, G for the pure gauge modes,  ¼ I, II for the physical modes and NP for the nonphysical modes. n ¼! is used for the modes incoming from the past horizon while n ¼ is used for the modes incoming from the past infinity. For the absorption cross section calculations, we only deal with physical modes incoming from the past infinity. (For a discussion about the other modes, see Ref. [10].) The physical modes in the modified Feynman gauge can be written as * crispino@ufpa.br + ednilton@fma.if.usp.br PHYSICAL REVIEW D 78, 024011 (2008) 1550-7998= 2008=78(2)=024011(6) 024011-1 Ó 2008 The American Physical Society