Abstract
In a framework describing manifestly covariant relativistic evolution using a scalar time τ, consistency demands that τ-dependent fields be used. In recent work by the authors, general features of a classical parametrized theory of gravitation, paralleling general relativity where possible, were outlined. The existence of a preferred “time” coordinate τ changes the theory significantly. In particular, the Hamiltonian constraint for τ is removed From the Euler-Lagrange equations. Instead of the 5-dimensional stress-energy tensor, a tensor comprised of 4-momentum density mid flux density only serves as the source. Building on that foundation, in this paper we develop a linear approximate theory of parametrized gravitation in the spirit of the flat spacetime approach to general relativity. Using a modified form of Kraichnan's flat spacetime derivation of general relativity, we extend the linear theory to a family of nonlinear theories in which the flat metric and the gravitational field coalesce into a single effective curved metric.
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REFERENCES
L. P. Horwitz and C. Piron, “Relativistic dynamics,” Helv. Phys. Acta 46, 316 (1973). See also J. R. Fanchi, Parametrized Relativistic Quantum Theory (Kluwer Academic, Dordrecht, 1993).
L. P. Horwitz, R. I. Arshansky, and A. C. Elitzur, “On the two aspects of time: The distinction and its implications,” Found. Phys. 18, 1159 (1988).
D. Saad, L. P. Horwitz, and R. I. Arshansky, “Off-shell electromagnetism in manifestly covariant relativistic quantum mechanics,” Found Phys. 19, 1125 (1989).
L. P. Horwitz, “On the definition and evolution of states in relativistic classical and quantum mechanics,” Found Phys. 22, 421 (1992).
M. C. Land, N. Shnerb, and L. P. Horwitz, “On Feynman's approach to the foundations of gauge theory,” J. Math. Phys. 36, 3263 (1995).
N. Shnerb and L. P. Horwitz, “Canonical quantization of four-and five-dimensional U(1) gauge theories,” Phys. Rev. A 48, 4068 (1993).
M. C. Land, “Particles and events in classical off-shell electrodynamics,” Found Phys. 27, 19 (1997).
M. C. Land and L. P. Horwitz, “The Lorentz force and energy-momentum for off-shell electromagnetism,” Found. Phys. Lett. 4, 61 (1991).
J. B. Pitts and W. C. Schieve, “On parametrized general relativity,” Found. Phys. 28, 1417 (1998).
R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
C. Misner, K. Thorne, and J. Wheeler, Gravitation (Freeman, New York, 1973).
M. Pavsic, “On the resolution of time problem in quantum gravity induced from unconstrained membranes,” Found. Phys. 26, 159 (1996).
A. Papapetrou, “Einstein's theory of gravitation and flat space,” Proc. R. Ir. Acad. A 52, 11 (1948).
S. Gupta, “Gravitation and electromagnetism,” Phys. Rev. 96, 1683 (1954).
R. H. Kraichnan, “Special relativistic derivation of generally covariant theory,” Phys. Rev. 98, 1118 (1955).
W. E. Thirring, “An alternative approach to the theory of gravitation,” Ann. Phys. (NY) 16, 96 (1961).
L. Halpern, “On the structure of the gravitation self-interaction,” Bull. Cl. Sci. Acad R. Belg., 5e série, 49, 226 (1963).
S. Deser, “Self-interaction and gauge invariance,” Gen. Rel. Gravit. 1, 9 (1970).
R. P. Feynman, F. B. Morinigo, and W. G. Wagner, Feynman Lectures on Gravitation, B. Hatfield, ed. (Addison-Wesley, Reading, Mass., 1995).
N. Rosen, “General Relativity and flat space. I and II,” Phys. Rev. 57, 147 (1940). Also, N. Rosen, “Flat-space metric in general relativity theory,” Ann. Phys. (NY) 22, 1 (1963).
W. G. Unruh, “Unimodular theory of canonical quantum gravity,” Phys. Rev. D 40, 1048 (1989).
M. C. Land and L. P. Horwitz, “Green's functions for off-shell electromagnetism and spacelike correlations,” Found. Phys. 21, 299 (1991).
C. Will, Theory and Experiment in Gravitational Physics, rev. edn. (Cambridge University Press, Cambridge, 1993).
H. Ohanian and R. Ruffini, Gravitation and Spacetime, 2nd edn. (Norton, New York, 1994).
J. L. Anderson, Principles of Relativity Physics (Academic, New York, 1967).
S. V. Babak and L. P. Grishchuk, “The Energy-momentum tensor for the gravitational field,” Phys. Rev. D 61, 024038 (2000).
W. Israel, Differential Forms in General Relativity, 2nd edn. (Dublin Institute for Advanced Studies, Dublin, 1979).
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Pitts, J.B., Schieve, W.C. On the Form of Parametrized Gravitation in Flat Spacetime. Foundations of Physics 29, 1977–1985 (1999). https://doi.org/10.1023/A:1018802719705
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DOI: https://doi.org/10.1023/A:1018802719705