Waves and particles in Kaluza–Klein theory

Abstract

We examine three overlapping problems in the application of five-dimensional ͑5D͒ manifolds to physics. First, we linearize the 5D theory along the lines of the four-dimensional ͑4D͒ theory, using the harmonic gauge condition. The resulting wave equations have sources, and can in principle describe gravitons and scalar particles with finite masses, but the natural choice of gauge parameters makes both massless. Second, we generalize the 5D metric by including separate conformal factors on its 4D and extra parts. Then the 5D harmonic gauge gives back the 4D harmonic gauge of gravitational waves and the Lorentz gauge of electromagnetic waves, but both particles are massless. Third, we again take a conformally rescaled metric, but rewrite the field equations in a novel manner. This allows us to interpret the finite masses of ordinary particles in terms of the wavelengths associated with 4D spaces embedded in a 5D space.

On higher-dimensional dynamics

“…It is therefore natural to ask if other aspects of particles, including their wave nature, can be understood as manifestations of an N(Ͼ4)-dimensional space. 30,46,64 In this section we will therefore proceed to see if it is possible to set up a consistent framework for 4D wave mechanics in N(Ͼ4)D theory, concentrating as before on the 5D case. 56 -58 And exact solutions of Einstein's equations are known which can describe nongravitational waves.…”

“…30,46,[59][60][61][62][63][64] It would be inappropriately long to discuss these here; but to show that there is a match between the dynamics and the field equations, let us take the metric ͑6.1͒ in the Planck gauge (aϭ1, bϭ1) and consider the 5D field equations R AB ϭ0 (A,Bϭ0,1,2,3,4). Both are statements about dynamics, and neither uses the field equations.…”

“…Both are statements about dynamics, and neither uses the field equations. Solutions of the latter of relevant type were mentioned above [26,42,[55][56][57][58][59][60]. It would be inappropriately long to discuss these here; but to show that there is a match between the dynamics and the field equations, let us take the metric (6.1) in the Planck gauge (a = 1, b = 1) and consider the 5D field equations R AB = 0 (A, B = 0, 123, 4).…”

Gravitational waves generated during inflation from a 5D vacuum theory of gravity in a de Sitter expansion

“…ν [x(N )] denote the first and second kind Hankel functions respectively, and A 1 , A 2 are integration constants. Analogously, the dynamical equation that depends of the fifth coordinate (12), by using the transformation ξ m (ψ) = (ψ 0 /ψ) 2 P m (ψ), acquires the form…”

“…In our notation conventions henceforth capital Latin indices A, B, ... = run from 0 to 4, whereas small Latin indices i, j, ..., from 1 to 3. In the context of this theory it has been shown that gravitational waves in 5D can be approached as in 4D, by using a linearized metric and the harmonic gauge condition [12]. With the same spirit, using ideas of the induced matter theory a novel formalism for describing inflation from a 5D apparent vacuum has been recently introduced [13].…”