Originally posted by MagnonYup, Q in that is the electric charge of the charged black hole. And yes, Reissner-Nordstrom geometry depends on Q and not q.
I can now see where [B]your confusion comes from: you think that the Q appearing in the Reissner-Nordström metric refers to the charge of some test particle (such as a proton or an electron) moving near the black hole. It does not: Q refers to the electric charge of the (charged) black hole.If you don't have access to a GR book such as MTW, here's a pdf document detailing this:
https://www.diva-portal.org/smash/get/diva2:912393/FULLTEXT01.pdfSee page 31 of this document for the metric tensor of the Reissner-Nordström geometry. The g_tt and g_rr components depend on Q, which is the electric charge of the black hole.
If we then consider motion of a test particle with charge q (note: now we use small q as opposed to the capital Q above) around the black hole, we can do it e.g. as shown in the above document in Chapter 5.3 (pp. 38). As you can see, in the resulting Euler-Lagrange equations of motion, both Q (the charge of the black hole) and q (the charge of the test particle) now appear.
The event horizon, as a purely geometrical feature of the Reissner-Nordström geometry, depends on Q (but not on q). I have already given the expression r+ for the location of the horizon earlier (which, just like the metric tensor itself, depends on Q but not on q). [/B]