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Newtonian mechanics has the concept of an absolute inertial rest frame.Special relativity eliminates the absolute rest frame but continues to requirethe absolute inertial frame. General relativity solves this for gravity byrequiring particles to have locally inertial frames on a curved position-timemanifold. The problem of the absolute inertial frame for other forces remains.We look again at the transformations of frames on an extended phase space withposition, time, energy and momentum degrees of freedom. Under nonrelativisticassumptions, there is an invariant symplectic metric and a line element dt^2.Under special relativistic assumptions the symplectic metric continues to beinvariant but the line elements are now -dt^2+dq^2/c^2 and dp^2-de^2/c^2. MaxBorn conjectured that the line element should be generalized to the pseudo-orthogonal metric -dt^2+dq^2/c^2+ (1/b^2)(dp^2-de^2/c^2). The group leavingthese two metrics invariant is the pseudo-unitary group of transformationsbetween noninertial frames. We show that these transformations eliminate theneed for an absolute inertial frame by making forces relative and bounded by band so embodies a relativity that is 'reciprocal' in the sense of Born. Theinhomogeneous version of this group is naturally the semidirect product of thepseudo-unitary group with the nonabelian Heisenberg group. This is thequaplectic group. The Heisenberg group itself is the semidirect product of twotranslation groups. This provides the noncommutative properties of position andmomentum and also time and energy that are required for the quantum mechanicsthat results from considering the unitary representations of the quaplecticgroup.Comment: Substantial revision, Publicon LaTe

Reciprocal Relativity of Noninertial Frames and the Quaplectic Group