Abstract
A physical framework has been proposed which describes manifestly covariant relativistic evolution using a scalar time τ. Studies in electromagnetism, measurement, and the nature of time have demonstrated that in this framework, electromagnetism must be formulated in terms of τ-dependent fields. Such an electromagnetic theory has been developed. Gravitation must also use of τ-dependent fields, but many references do not take the metric's dependence on τ fully into account. Others differ markedly from general relativity in their formulation. In contrast, this paper outlines steps towards a τ-dependent classical intrinsic formulation of gravitation, patterned after general relativity, which we call parametrized general relativity (PGR). Given the existence of a preferred foliation, the Hamiltonian constraint is removed. We find that some nonmetricity in the connection is allowed, unlike in general relativity. Conditions on the allowable nonmetricity are found. Consideration of the initial value problem confirms that the metric signature should normally be O(3, 2) rather than O(4, 1). Following the lead of earlier works, we argue that concatenation (integration over τ) is unnecessary for relating parametrized physics to experience, and propose an alternative to it. Finally, we compare and contrast PGR with other relevant gravitational theories.
Similar content being viewed by others
REFERENCES
E. C. G. Stueckelberg, “Remarque à propos de la création de paires de particules en théorie de relativité” and “La Signification du temps propre en mécanique ondulatorie,” Helv. Phys. Acta 14, 588–594 (1941).
J. L. Cook, “Solutions of the relativistic two-body problem, I. Classical mechanics” and “II. Quantum mechanics,” Austr. J. Phys. 25, 2, 322–323 (1972).
L. P. Horwitz and C. Piron, “Relativistic dynamics,” Helv. Phys. Acta 46, 316–326 (1973).
L. P. Horwitz, R. I. Arshansky, and A. C. Elitzur, “On the two aspects of time: The distinction and its implications,” Found. Phys. 18, 12 (1988).
D. Saad, L. P. Horwitz, and R. I. Arshansky, “Off-shell electromagnetism in manifestly covariant relativistic quantum mechanics,” Found. Phys. 19, 10 (1989).
N. Shnerb and L. P. Horwitz, “Canonical quantization of four-and five-dimensional U(1) gauge theories,” Phys. Rev. A 48, 4068–4074 (1993).
M. C. Land, “Particles and events in classical off-shell electrodynamics,” Found. Phys. 27, 1 (1997).
C. Land and L. P. Horwitz, “The Lorentz force and energy-momentum for off-shell electromagnetism,” Found. Phys. Lett. 4, 1 (1991).
C. Land and L. P. Horwitz, “Green's functions for off-shell electromagnetism and spacelike correlations,” Found. Phys. 21(1991).
J. Frastai and L. P. Horwitz, “Off-shell fields and Pauli-Villars regularization,” Found. Phys. 25, 10 (1995).
Kubo, “Five-dimensional formulation of quantum field theory with invariant parameter,” Nuovo Cimento 85, 4, 293–309 (1985).
L. P. Horwitz, “On the definition and evolution of states in relativistic classical and quantum mechanics,” Found. Phys. 22, 3 (1992).
J. R. Franchi, Parametrized Relativistic Quantum Theory (Kluwer, Dordrecht, 1993).
L. Burakovksy and L. P. Horwitz, “5D generalized inflationary cosmology,” Gen. Relativ. Grav. 27, 10, 1043–1070 (1995).
M. Pavsic, “On the resolution of time problem in quantum gravity induced from unconstrained membranes,” Found. Phys. 26, 2 (1996). “The classical and quantum theory of relativistic p-branes without constraints,” Nuovo Cimento 108A, 2 (1995). “Relativistic p-branes without constraints and their relation to the wiggly-extended objects,” Found. Phys. 25, 6 (1995).
F. H. Gaioli and E. T. Garcia-Alvarez, “The problem of time in parametrized theories,” Gen. Relativ. Gravit. 26, 12 (1994).
C. Misner, K. Thorne, and J. Wheeler, Gravitation (Freeman, New York, 1973).
R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th revised English edn., translated by M. Hamermesh (Butterworth-Heinemann, Oxford, 1975).
D. Christodoulou and M. Francaviglia, “Some dynamical properties of Einstein space times admitting a Gaussian foliation,” Gen. Relativ. Gravit. 10, 6 (1979).
J. A. Schouten, Ricci-Calculus, 2nd edn. (Springer, Berlin, 1954).
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II ( Interscience, New York, 1962).
M. Tegmark, “On the dimensionality of spacetime,” Class. Quantum Gravit. 14, 97 (1997).
H.-H. von Borzeszkowski and H.-J. Treder, “The Weyl-Cartan space problem in purely affine theory,” Gen. Relativ. Gravit. 29, 4 (1997).
Rights and permissions
About this article
Cite this article
Pitts, J.B., Schieve, W.C. On Parametrized General Relativity. Foundations of Physics 28, 1417–1424 (1998). https://doi.org/10.1023/A:1018801126703
Issue Date:
DOI: https://doi.org/10.1023/A:1018801126703