Martin Land
Department of Computer Science, Hadassah College, Jerusalem, Israel

Part of the book: Future Relativity, Gravitation, Cosmology

Abstract

We extend the canonical approach of Stueckelberg, Horwitz, and Piron in classical relativistic mechanics and electrodynamics to general relativity. Classical spacetime events x µ (τ), evolving as τ grows monotonically, trace out particle worldlines dynamically in geodesic motion determined by a 5D local metric gαβ(x, τ), for α, β = 0, 1, 2, 3, 5. The 5D metric obeys extended Einstein equations. To obtain a reasonable equivalence principle, the formal 5D tensor symmetries must break to tensor and scalar representations of O(3,1) by a consistent prescription. We take as an example the field produced by a τ-dependent mass M(τ), first as a perturbation in the Newtonian approximation and then for a Schwarzschild-like metric. The extended Einstein equations imply a flow of energy into spacetime corresponding to the changing source mass. The Hamiltonian, driven by terms proportional to dM/dτ, is not generally conserved, but relaxes to a generalized Schwarzschild solution with vanishing Einstein tensor in τ-equilibrium.

References

[1] Stueckelberg E. (1941) Helv. Phys. Acta 14 321–322 (In French).
[2] Stueckelberg E. (1941) Helv. Phys. Acta 14 588–594 (In French).
[3] Horwitz L and Piron C. (1973) Helv. Phys. Acta 48 316–326.
[4] Saad D, Horwitz L and Arshansky R. (1989) Found. Phys. 19 11251149.
[5] Horwitz L P. (2015) Relativistic Quantum Mechanics (Dordrecht, Netherlands: Springer).
[6] Land M. (2018) Classical Stueckelberg-Horwitz-Piron Electrodynamics Rela tivity, Gravitation, Cosmology: Beyond Foundations Contemporary Fundamental
Physics ed Dvoeglazov V (Nova Science Publishers).
[7] Horwitz L, Arshansky R and Elitzur A. (1988) Found. Phys. 18 1159.
[8] Land M and Horwitz L. (1991) Found. Phys. Lett. 4 61.
[9] Horwitz L P. (2019) Journal of Physics: Conference Series 1239 012014 URL
https://doi.org/10.1088%2F1742-6596%2F1239%2F1%2F012014.
[10] Horwitz L P. (2019) The European Physical Journal Plus 134 313 ISSN 2190-5444
URL https://doi.org/10.1140/epjp/i2019-12689-7.
[11] Pitts J B and Schieve W C 1998 Foundations of Physics 28 1417–1424 ISSN 1572-
9516 URL https://doi.org/10.1023/A:1018801126703.
[12] Pitts J B and Schieve W C. (2001) Foundations of Physics 31 1083–1104 ISSN
1572-9516 URL https://doi.org/10.1023/A:1017578424131.
[13] Weinberg S. (1972) Gravitation and Cosmology: Principles and Applications of the
General Theory of Relativity (New York, NY: Wiley) URL
https://cds.cern.ch/record/100595.
[14] Wald R M. (1984) General relativity (Chicago, IL: Chicago Univ. Press) URL
https://cds.cern.ch/record/106274.
[15] Land M. (2019) Journal of Physics: Conference Series 1239 012005 URL
https://doi.org/10.1088%2F1742-6596%2F1239%2F1%2F012005.
[16] Misner C W, Thorne K S and Wheeler J A. (1973) Gravitation (San Francisco:
W.H. Freeman and Co.).
[17] Land M and Horwitz L. (1991) Found. Phys. 21 299–310