Relativistic rotation and the anholonomic object

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Abstract
The purpose of this communication is to call attention to the conceptual economy provided by the object of anholonomity for the theory of relativity. This geometric object expresses certain consequences of relativity theory and provides a single, simple framework for discussing a variety of phenomena. It particularly clarifies the description of relativistic rotation. The relativistic rotational transformation of the four coordinate differentials of flat space–time generates a set of anholonomic, or inexact differentials, whose duals are an orthogonal set of basis vectors. How should a rotating observer interpret physical events referred to such orthogonal, but anholonomic frames? The answer to this question rests upon the origin and physical significance of the object of anholonomity. It is demonstrated that not only is the rotational Lorentz transformation an anholonomic transformation, but that the intrinsic anholonomic effects are essential to interpreting rotational phenomena. In particular, the Sagnac effect may be interpreted as the physical manifestation of temporal anholonomity under rotation. The Thomas precession of a reference axis may be interpreted as a consequence of the spatial anholonomity of the rotating frame. Further, the full four-dimensional covariance of Maxwellian electrodynamics, under a relativistic Lorentz rotation, is possible only with the inclusion of anholonomic effects. Schiff attempted to explain the vanishing of the electromagnetic field tensor in the frame of an observer rotating about a charged spherical capacitor, in terms of an interaction with the distant stars. However, if one employs a Lorentzian rotation and the ensuing orthogonal anholonomic reference frames, an explanation is possible without gravitationally induced currents. The anholonomic approach clarifies the distinction between the physically different operations of source rotation and observer rotation in a flat space–time. It is finally concluded that a consistant theory of relativistic rotation, satisfying the principle of general covariance, inherently requires the presence of the object of anholonomity.

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