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!Kartik Prabhu
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!Kartik Prabhu
The electric potential (which is the analog of surface gravity) is indeed constant as I showed in my thesis (Sec. 3 Theorem 1). The electric flux E.n is more analogous to the area element/ Noether charge which need not be constant.