Extremal charged black holes have a unique property: they neither attract nor repel any object with $Q=M$ (in geometrized units and a positively charged hole). This behavior holds exactly true in both Newtonian gravity and general relativity and motivates this question as a followup.
But wait, doesn't gravity (the proper acceleration needed to stand still) go to infinity near the horizon? The electric field is finite at the horizon, so one would think that close in gravity would win.
Except that the horizon is infinitely far away as measured by stationary rulers!
So here is my understanding of the geometry near an extremal hole and what it is like to fall in:
- Far away it behaves like any charged Newtonian mass. The metric is static so "not moving" is well-defined.
- When you get close, you enter a bottomless "tube". The spatial geometry (as determined by hovering rulers) in this tube is $S^2⨯E$ (spherinders). Inside the tube there is a constant gravity and a constant electric field no matter how deep you go. Time dilation and the blueshift of distant stars increases exponentially with depth.
- Objects with Q=M will neither be pulled in or be pushed away: the electrostatic repulsion cancels out gravity. If given a push inward, they will slowly drift inward but never reach the horizon (not even in their own proper time). This allows you in principle to explore quite deeply and return using very little rocket fuel.
- Uncharged objects dropped into the tube will reach the horizon in finite proper time because they are moving faster and faster as they descend. After a certain head-start it is impossible to catch up to these accelerating objects and save them.
- In practice, even if matter were strong enough to endure such intense electric fields, the tube's depth is quite limited. The tube is only logarithmically deep with how close to extremality the hole is. If the hole is very close to extremality, a tiny amount of extra mass or stray negative charge will vastly shorten the length of the tube and doom any tour guides deep inside of it.
Is my reasoning correct?
