[Paper Review] Space-Time--Time: Five-dimensional Kaluza--Weyl Space

Homer G. Ellis|ArXiv.org|Jul 6, 2001

Quantum Mechanics and Applications10 references3 citations

TL;DR

This paper proposes a five-dimensional space-time-theory unifying gravity and electromagnetism by merging Kaluza's compactified fifth dimension with Weyl's conformal geometry, replacing the cylinder condition with an exponential expansion constraint. The resulting space-time-time geometry dynamically generates both electromagnetic and metrical gauge transformations, enabling test particles to exhibit mass, charge, and momentum evolution governed by four apparent forces, with particle creation/annihilation at events tied to scalar field extrema—offering a classical alternative to quantum atomic structure.

ABSTRACT

Space-time--time couples Kaluza's five-dimensional geometry with Weyl's conformal space-time geometry to produce an extension that goes beyond what either of those theories can achieve by itself. Kaluza's ``cylinder condition'' is replaced by an ``exponential expansion constraint'' that causes translations along the secondary time dimension to induce both the electromagnetic gauge transformations found in the Kaluza and the Weyl theories and the metrical gauge transformations unique to the Weyl theory, related as Weyl had postulated. A space-time--time geodesic describes a test particle whose rest mass, space-time momentum, and electric charge q, all defined kinematically, evolve in accord with definite dynamical laws. Its motion is governed by four apparent forces: the Einstein gravitational force, the Lorentz electromagnetic force, a force proportional to the electromagnetic potential, and a force proportional to a scalar field's gradient d(ln phi). The test particles exhibit quantum behavior: (1) they appear and disappear in full-blown motion at definite events; (2) all that share an event E of appearance or disappearance do so with the same charge magnitude |q| = phi(E); (3) conservation of space-time--time momentum at such an event entails conservation of electric charge in addition to conservation of space-time momentum, among the participating particles; (4) at such events the d(ln phi) force infinitely dominates the other three --- this strongly biases the appearance and disappearance events to be concentrated deep in the discretely spaced potential wells of ln phi, and sparse elsewhere.

Motivation & Objective

• To unify general relativity and Maxwell's electromagnetism through a geometric extension of spacetime beyond Kaluza–Klein theory.
• To resolve the incompatibility between Kaluza’s isometric fifth dimension and Weyl’s conformal geometry by introducing a hybrid five-dimensional manifold.
• To provide a classical framework for particle creation and charge dynamics without invoking quantum mechanics, particularly for atomic-scale phenomena.
• To reformulate gauge invariance such that both electromagnetic and metrical (Weyl-type) transformations emerge naturally from isometries along a secondary time dimension.
• To explore the physical interpretation of the fifth dimension as a secondary temporal dimension, rather than a spatial compact dimension, via a modified Lie derivative condition.

Proposed method

• Replace Kaluza’s cylinder condition (∇ξĜ = 0) with an exponential expansion constraint (∇ξĜ = 2Ĝ), inducing conformal transformations on spacetime cross-sections.
• Define the five-dimensional metric Ĝ as a conformal rescaling of a four-dimensional metric G̃ and a gauge potential Â, with a scalar field φ controlling the conformal factor.
• Introduce a complexified coordinate ζ and field φ to absorb both ε = +1 and ε = −1 metric solutions into a single complexified geometry, preserving real spacetime metrics under gauge transformations.
• Use the Lie derivative along the fifth-dimensional vector field ξ to generate both electromagnetic gauge transformations (Â → Â + dλ) and metrical gauge transformations (Ĝ → e^{2λ}Ĝ), as postulated by Weyl.
• Construct space-time-time geodesics whose projections onto four-dimensional spacetime exhibit four effective forces: gravitational, electromagnetic, potential-force (proportional to d(ln φ)), and a scalar force.
• Demand momentum and charge conservation at particle creation/disappearance events E₁ and E₂, where q = −φ(E₁) and q = φ(E₂), respectively, to model particle interactions without quantum statistics.

Experimental results

Research questions

• RQ1 Can a five-dimensional spacetime geometry be constructed that unifies gravity and electromagnetism while preserving both Kaluza’s and Weyl’s key geometric insights?
• RQ2 How can the fifth dimension be reinterpreted as a secondary time dimension rather than a spatial compact dimension, and what are the dynamical consequences?
• RQ3 Can particle creation and annihilation be modeled as geodesic motion in a higher-dimensional manifold without invoking quantum field theory?
• RQ4 Does the interplay between the scalar field φ and the electromagnetic potential  lead to a natural emergence of both electromagnetic and metrical gauge invariance?
• RQ5 Can classical dynamics in this framework reproduce features of atomic structure, such as electron transitions, through extremal behavior of the scalar field φ?

Key findings

• The exponential expansion constraint (∇ξĜ = 2Ĝ) replaces Kaluza’s cylinder condition and induces conformal transformations on spacetime cross-sections, unifying Kaluza’s and Weyl’s geometries.
• Test particle geodesics in space-time-time exhibit four effective forces in four-dimensional spacetime: Einstein gravity, Lorentz force, potential force (proportional to d(ln φ)), and a scalar force from the gradient of ln φ.
• Particles appear at events E₁ with q = −φ(E₁) and disappear at E₂ with q = φ(E₂), where the gradient force dominates, suggesting a classical mechanism for particle creation and annihilation.
• Particles with m̃ = 0 and q ≠ 0 must follow paths where φ(p) = const and q = ±φ(p), and can be transported via φ-wave forms with null propagation vectors at light speed.
• Conservation of total space-time momentum and electric charge at common events E can be enforced by requiring the sum of momenta to vanish, enabling classical particle interactions.
• Complexification of the fifth dimension and fields φ and  allows both ε = +1 and ε = −1 metric solutions to be unified, with φ → φe^{−iν} under gauge transformations, echoing London’s phase shift in superconductivity.