Hawking radiation by spherically-symmetric static black holes for all spins: Teukolsky equations and potentials

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 quasinormal 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–Nordström) 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.

Figure 1

Comparison of the potentials for a charged BH with $r_{\mathrm{Q}}=r_{\mathrm{S}}/3$ (solid lines) and for the Schwarzschild metric (dashed lines). The vertical black line represents the BH horizon.

Figure 2

Comparison of the potentials for a higher-dimensional BH with $n=2$, $M={10}^{10}{M}_{\mathrm{P}}$, and $M_{*}=10\mathrm{TeV}$ (solid lines) and for the Schwarzschild metric (dashed lines). The vertical black line denotes the BH horizon.

Figure 3

Comparison of the potentials for a polymerized BH with $\epsilon =0.8$ and $a_0={10}^{-10}{r}_{\mathrm{S}}^2$ (solid lines) and for the Schwarzschild metric (dashed lines). The vertical black line denotes the BH horizon.