btw: for Einstein-Maxwell the electric potential being constant is a result by Carter in some old-ish Les Houches publication that I can’t find right now.
I also expect static to imply that the electric flux/area element is constant, but I really have no idea how to show that without using global results like Birkhoff’s theorem or uniqueness (in the stationary case). Somehow our guts know about global results that our brains don’t!
Yeh, that’s sort of what I’m thinking as well — Some weird stuff can happen if you allow for strange asymptotics, and this is harder to grasp intuitively. For example, there can be toroidal horizons, which could give counterexamples (?) 🤷🏻♂️
anything violating the uniqueness theorems would be fair game: asymptotics, higher dimensions, modified gravity… But getting an explicit counter-example seems pretty hard. Would be nice to have some local existence proof though not sure how to even approach that!
Yeah, I might ask around. I am beginning to feel like it’s not actually true without assuming global properties though. If it were something you could prove locally, then there aren’t really *that* many ways you could approach it. So I feel if it were true, it’d be known by now