# Charge dependence on time

From the CT or CPT invariance, we know that charge is reversed upon the reversal of time. Consider the following hypothetical scenario. We put a charge in out rest chamber where we have the ability to manipulate time at will. Then we make time in the chamber run slower, stop, reverse, and run in the opposite direction.

Before and after the moment of the reversal, charge conservation should apply. Therefore our test charge should first remain the same, as time is slowing down, then reverse when time changes the direction, and finally remain the same reversed, as time is speeding up in the opposite direction.

What remains unclear is the value of the charge at the very moment of the time reversal (as observed from outside the chamber). It cannot be both positive and negative, so it has to be either zero or undefined. However, it also unclear what an "undefined charge" actually means.

This becomes interesting in General Relativity where the speed of time is relative. If a local observer is falling to a black hole, his proper time remains unchanged while crossing the horizon. However, for a remote observer, the time of the falling observer stops at the horizon (although only in the infinite future). If the falling observer is charged, what happens to his charge exactly at the horizon in the view of the remote observer?

I know that black holes conserve charge per the "no hair" theorem, but how is this resolved with the CPT consideration above? Does this have anything to do with the behavior of the Parity part of CPT at the horizon? Thanks for your insight.