NDLI: Reciprocal Relativity of Noninertial Frames and the Quaplectic Group

Content Provider	Springer Nature Link
Author	Low, Stephen G.
Copyright Year	2006
Abstract	The frame associated with a classical point particle is generally noninertial. The point particle may have a nonzero velocity and force with respect to an absolute inertial rest frame. In time–position–energy–momentum-space {t, q, p, e}, the group of transformations between these frames leaves invariant the symplectic metric and the classical line element ds2 = d t2. Special relativity transforms between inertial frames for which the rate of change of momentum is negligible and eliminates the absolute rest frame by making velocities relative but still requires the absolute inertial frame. The Lorentz group leaves invariant the symplectic metric and the line elements $$ds^{2} = -dt^{2} + \frac{1}{c^{2}} dq^{2}$$ and $$d \mu^{2} = dp^{2}-\frac{1}{c^{2}}de^{2}$$ . General relativity for particles under only the influence of gravity avoids the issue of noninertial frames as all particles follow geodesics and hence have locally inertial frames. For other forces, the question of the absolute inertial frame remains.) Born conjectured that the line element should be generalized to the pseudo-orthogonal metric $$ds^{2} = -d t^{2}+\frac{1}{c^{2}} dq^{2} + \frac{1}{b^{2}}(dp^{2}-\frac{1}{c^{2}}de^{2})$$ . The group leaving this metrics and the symplectic metric invariant is the pseudo-unitary group of transformations between noninertial frames. We show that these transformations also eliminate the need for an absolute inertial frame by making forces relative and bounded by b and so embodies a relativity that is shape reciprocal in the sense of Born. The inhomogeneous version of this group is naturally the semidirect product of the pseudo-unitary group with the nonabelian Heisenberg group. This is the quaplectic group.
Starting Page	1036
Ending Page	1069
Page Count	34
File Format	PDF
ISSN	00159018
Journal	Foundations of Physics
Volume Number	36
Issue Number	7
e-ISSN	15729516
Language	English
Publisher	Kluwer Academic Publishers-Plenum Publishers
Publisher Date	2006-04-22
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	noninertial maximal acceleration noncommutative geometry Heisenberg Born reciprocity Biophysics/Biomedical Physics Condensed Matter Relativity and Cosmology Mechanics Quantum Physics Physics
Content Type	Text
Resource Type	Article
Subject	Physics and Astronomy

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