Rotational Consequences of Lorentz Symmetries in Hyperbolic Phase Space

A hyperbolically curved position space when combined with a relativistic velocity-momentum space forms a phase space with the symmetries of a direct product double Lorentz group, expressed by $8\times 8$ matrices. (See an adjoining Abstract, ``Lorentz Symmetries of a Doubly Hyperbolic Phase Space.'') Its rotational subgroup too is a direct product, combining rotations in position space with those in momentum space to form a total angular momentum $\bf J$ and an unfamiliar contra-angular momentum $\bf Q$, a pseudovector whose coupling with other vectors vanishes in the absence of spatial and velocity space curvature. Its quantum numbers and properties may label some particle states. The second-order couplings populating states of $\bf Q$ arise from a combination of both position and velocity space curvatures; they are comparable in nature to the Thomas precession process, which can itself be looked upon as a second-order effect of curvature in relativistic velocity space. Processes altering $\bf Q$ values will therefore occur preferentially at relativistic velocities in regions of high gravitational curvature.