|A pair of docs|
• An observer in the water would say: the submarine Lorentz contracts (unofficially the well known "shrinkage" problem in the ocean) so its density increases and the sub sinks.
• An observer on the submarine would say: the ocean is coming at me at high speed. The water Lorentz contracts, thereby increasing its density, so the buoyancy force increases, and the sub rises.
Ruh roh. As weird as relativity is, it cannot account for the sub crashing into the sea floor in one reference frame and bursting through the surface like a sperm whale in another. No, that just will not do.
Here is the result of a General Relativity calculation that concludes the sub will sink. I'm not sure I believe it is robust. A Special Relativity calculation also (in a more round-about way) also concludes that the sub will sink.
These agree, of course, with the engineering calculation that says in pushing to such speeds either the engines will explode or the sub will crash, in either case the sub will sink.