Complementarity of kinematics and geometry in general relativity
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Relations between the kinematics, geometry (metric) and the law of motion for reference frames are considered in the context of general relativity. More specifically, we analyze the question, to what extent each pair from the three above-mentioned objects determines the third one. It is well known that the metric and the reference frame uniquely determine all kinematic tensors (the acceleration, deformation and spin forms). We show that the kinematic tensors together with the motion of the reference frame determine the geometry up to arbitrary functions of the spatial coordinates if a single integrability condition for the angular velocity tensor is satisfied. In the case where the motion of the reference frame is specified, there emerges rather a complicated set of consistency conditions of both algebraic and differential nature. Some aspects of the geometrization principle, Poincare's geometric conventionalism and a relativistic analogue of the quantum-mechanical complementarity principle are discussed in the light of the results obtained.
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- V. I. Antonov, V. N. Efremov, and Yu. S. Vladimirov, Gen. Rel. Grav. 9 9 (1978).
- Yu. S. Vladimirov, Reference Frames in Gravitation Theory (Energoizdat, Moscow, 1982, in Russian).
- N. V. Mitskievich, Relativistic physics in an arbitrary reference frame, arxiv: gr-qc/9606051.
- J. L. Synge, Relativity: The General Theory (Amsterdam, North-Holland Publishing Company, 1960).
- V. I. Arnold, Ordinary Differential Equations (UGU, RCHD, Izhevsk, 2000, in Russian).
- B. O'Neill, Semi-Riemannian Geometry (Akad. Press, San-Diego, California, 1983).
- H. Poincare, Science and Hypothesis (Walter Scott Publishing, London, 1905).
- F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups (Scott, Foresman & Co., 1971).
- P.-A. Griffiths, Exterior Differential Systems and the Calculus of Variations (Birkhauser, Boston-Basel-Stuttgart, 1983).
- P. K. Rashevski, Uch. Zapiski Mosk. Ped. Inst. im. Libknehta (seriya fiz-mat.) No. 2, 83-94 (1938) (In Russian).
- W. L. Chow, Math. Ann. 117 (1), 98 (1940).
- S. S. Kokarev, Introduction to General Relativity (YarSU, Yaroslavl, 2010, in Russian).
- S. S. Kokarev, Lectures "Elements of the Theory of Smooth Manifolds", (I): Lie Derivatives and Their Applications, in: Coll. Papers of RSEC "Logos", v.4 (Yaroslavl, 2009), pp. 77-166 (In Russian).
- G. Lochak, La Geometrization de la Physique (Flammarion, 1994).
- S. S. Kokarev, Three Lectures on Newton's Laws, in: Coll. Papers of RSEC "Logos", v.1 (Yaroslavl, 2006), pp. 45-72; arXiv: 0905.3285.
- S. Nojiri and S. D. Odintsov, Phys. Rev. D 68 123512 (2003), hep-th/0307288.
- S. S. Kokarev, Gen. Rel. Grav. 41 1777 (2009).
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