Chapter 5. Classical Mechanics with Evolving Local Metrics


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

Part of the book: Future Relativity, Gravitation, Cosmology


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.


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