Abstract
In the context of the dynamics and stability of black holes in modified theories of gravity, we derive the Teukolsky equations for massless fields of all spins in general spherically-symmetric and static metrics. We then compute the short-ranged potentials associated with the radial dynamics of spin 1 and spin 1/2 fields, thereby completing the existing literature on spin 0 and 2. These potentials are crucial for the computation of Hawking radiation and quasi-normal modes emitted by black holes. In addition to the Schwarzschild metric, we apply these results and give the explicit formulas for the radial potentials in the case of charged (Reissner--Nordstr\"om) black holes, higher-dimensional black holes, and polymerized black holes arising from loop quantum gravity. These results are in particular relevant and applicable to a large class of regular black hole metrics. The phenomenological applications of these formulas will be the subject of a companion paper.
Highlights
Black holes (BHs) are fascinating astrophysical objects
In addition to the Schwarzschild metric, we apply these results and give the explicit formulas for the radial potentials in the case of charged (Reissner–Nordström) black holes, higher-dimensional black holes, and polymerized black holes arising from loop quantum gravity
We focus on spherically symmetric static metrics of the form (2.1) and show how the equations of motion can be written in a form similar to the Regge–Wheeler equation for Schwarzschild BHs, i.e., as a one-dimensional Schrödinger-like radial wave equation with a short-ranged potential
Summary
Black holes (BHs) are fascinating astrophysical objects. As the ultimate stage of the gravitational collapse of stars, they probe the limits of general relativity and our understanding of high energy and high density physics. For example, on charged Bardeen BHs [33], Gauss–Bonnet BHs [34,35], Palatini gravity [36,37], fðRÞ gravity [38], Kerr– de Sitter BHs [39], conformal gravity [40,41], higher derivative gravity [42], and so-called polymerized BHs within loop quantum gravity [43,44,45] The computation of both Hawking radiation and quasinormal modes is related to the response of black holes to perturbations. Understanding the physically measurable consequences of modified gravity on the Hawking radiation and quasinormal modes requires one to work out the equations of motion of the various spin perturbations to black hole metrics. A more detailed application of the formalism to various metrics will appear in the companion paper [50]
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