Relativistic submarine with GA and gauge gravity

A submarine moving with a relativistic velocity v can be modeled by a liquid volume with the same intrinsic density q as the liquid medium.

The apparent paradox is that the sub either sinks or rises due to increased density of either the medium or the sub caused by Lorenz contraction corsponingldung to the 2 different frames of reference.

This is resolved by noting that only the sub moves wrt to the gravitational field and thus is lenght contracted and sinks.

This can be expressed more precisely by the relationship of the energy momentum tensor T and the force on the boundary between the sub and liquid as a contraction of T with a boundary area element dA which gives infinitesimal force dF.

I am interested in how and if GA formulation could apply to this problem.

For example, in gauge gravity,
Hestenes defines momentum tensor as
T(a) = mq(a*v) v, where q and v are proper density and proper velocity.

Does anyone have any idea how exactly would the gauge tensor(metric) look like here, so that the density transforms correctly and solves the paradox?