Hawking radiation from spherically symmetrical gravitational collapse to an extremal R-N black hole for a charged scalar field

Sijie Gao has recently investigated Hawking radiation from spherically symmetrical gravitational collapse to an extremal R-N black hole for a real scalar field. Especially he estimated the upper bound for the expected number of particles in any wave packet belonging to $\mathcal{H}_{out}$ spontaneously produced from the state $|0>_{in}$, which confirms the traditional belief that extremal black holes do not radiate particles. Making some modifications, we demonstrate that the analysis can go through for a charged scalar field.