Abstract
We perform a systematic study of the Maxwell quasinormal spectrum in a mirror-like cavity following the generic Robin type vanishing energy flux principle, by starting with the Schwarzschild black holes in this paper. It is shown that, for black holes in a cavity, the vanishing energy flux principle leads to two different sets of boundary conditions. By solving the Maxwell equations with these two boundary conditions both analytically and numerically, we observe two distinct sets of modes. This indicates that the vanishing energy flux principle may be applied not only to asymptotically anti-de Sitter (AdS) black holes but also to black holes in a cavity. In the analytic calculations, the imaginary part of the Maxwell quasinormal modes are derived analytically for both boundary conditions, which match well with the numeric results. While in the numeric calculations, we complete a thorough study on the two sets of the Maxwell spectrum by varying the mirror radius r_m, the angular momentum quantum number ell , and the overtone number N. In particular, we proclaim that the Maxwell spectrum may bifurcate for both modes when the mirror is placed around the black hole event horizon, which is analogous to the spectrum bifurcation effects found for the Maxwell fields on asymptotically AdS black holes. This observation provides another example to exhibit the similarity between black holes in a cavity and the AdS black holes.
Highlights
(QNMs) play vital roles in both of the aforementioned observations
We confirm that the vanishing energy flux principle is applicable to a BHmirror system and leads to two different branches of modes
We have studied the Maxwell quasinormal spectrum on Schwarzschild Black holes (BHs) in a mirror-like cavity, by imposing the vanishing energy flux boundary conditions
Summary
(QNMs) play vital roles in both of the aforementioned observations. For the former case, QNMs determine the ringdown phase of gravitational waves [11,12]; and for the latter case, in the eikonal limit, the real part of QNMs is related to the radius of the BH shadow and the imaginary part is related to the amplitude ratio between the nth image and the (n + 2)th image [13,14]. For asymptotically anti-de Sitter (AdS) spacetimes, on the other hand, recently we have proposed the vanishing energy flux principle to impose boundary conditions These boundary conditions are Robin type, and lead to novel and interesting results, for example, two distinct sets of QNMs are present for both the Maxwell [19,20,21,22] and the Dirac fields [23,24], and the Maxwell spectrum may bifurcate [21]. As the first paper in this direction, here we focus on the Maxwell fields interacting with the Schwarzschild-mirror system To this end, we first formulate the Maxwell equations on the Schwarzschild background by using the Regge–Wheeler–Zerilli approach [30,31] and based on the vanishing energy flux principle, derive two sets of explicit boundary conditions at the mirror’s location and obtain two branches of quasinormal spectrum. Some analytic calculations, including normal modes calculations in the Regge– Wheeler–Zerilli formalism and QNMs calculations in the Teukolsky formalism with low frequency approximations, are left in the Appendices
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