## Abstract

Following a bi-cylindrical model of geometrical dynamics, our study shows that a 6D-gravitational equation leads to geodesic description in an extended symmetrical time–space, which fits Hubble-like expansion on a microscopic scale. As a duality, the geodesic solution is mathematically equivalent to the basic Klein–Gordon–Fock equations of free massive elementary particles, in particular, the squared Dirac equations of leptons. The quantum indeterminism is proved to have originated from space–time curvatures. Interpretation of some important issues of quantum mechanical reality is carried out in comparison with the 5D space–time–matter theory. A solution of lepton mass hierarchy is proposed by extending to higher dimensional curvatures of time-like hyper-spherical surfaces than one of the cylindrical dynamical geometry. In a result, the reasonable charged lepton mass ratios have been calculated, which would be tested experimentally.

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## Acknowledgements

The author is grateful to Dr. Tran Chi Thanh (Vietnam Atomic Energy Institute) for VINATOM’s support. We thank Nguyen Anh Ky and Nguyen Ai Viet (Hanoi Institute of Physics), Do Quoc Tuan (Physics Faculty, Hanoi University of Natural Sciences) for useful discussions. A heart-felt thanks is also extended to N. B. Nguyen (Thang Long University) for valuable technical assistance, T. Thu and M. L. Le-Vo for their continuous support and encouragement.

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## Appendix A: A Review of the 5D-STM Theory

### Appendix A: A Review of the 5D-STM Theory

It is to recall that Kaluza–Klein theory was the first to introduce the induced-matter approach to link higher-dimensional general relativity with some properties of elementary particles. The K–K interval consists of 4D element *ds* and another part of the ED axis *l* as: \(dS^2=ds^2-\left( dl+A_{\mu } dx^{\mu } \right) ^2 \), where \(A_{\mu }\) is a covariant vector added for unifying electro-magnetic field. Among the modern K–K theories the problem of consistency between general relativity and quantum mechanics was investigated under the space–time–matter theory (5D-STM). From the first formulation by Wesson and Ponce de Leon [9] there was proposed later by Mashhoon, Liu and Wesson for 5D-STM to have a canonical metric form [10] corresponding to quadratic 5D-geometry as:

where \(A,B=0-4; ds^2=g_{\alpha \beta } (x_\gamma ,l)dx_\alpha dx_\beta \) is 4D space–time sub-geometry; *L* is a length scale parameter; \(\epsilon =\pm 1\) determining that ED is a space-like \((\epsilon =-\,1)\) or time-like \((\epsilon =+\,1)\) and \(\Phi (x_\gamma ,l)\) is a new scalar field, then \(g_{44}=\epsilon \Phi ^2\). More general, it is possible to replace ED by a shift \(l\rightarrow (l-l_0)\) for a constant distance \(l_0\). As a modern version of 5D relativity the introduction of the fifth independent dimension drops the cylindrical condition of the traditional K–K theory which STM co-workers consider may be daunting algebraically, but this would gain in being richer physically. In 5D-STM theory there was proposed a null geodesic model with \(dS^2=0\) of extended general relativity in 5D manifolds [10] which is to start from a 5D-Ricci vacuum gravitational equation:

where \(R_{AB}\) is Ricci tensor. Applying Campbell theory [26, 27] to derive the 4D-solutions leads to 4D-Einstein gravitational equation and 4D-Maxwell equations of electro-magnetism as it was done in the classical K–K theory. Moreover, new sub-solutions are found additionally in terms of the scalar field \(\Phi \) which would have wide applicability. For example, suggesting \(\Phi \) slowly evolving with time, it is able to determine a cosmological pseudo-constant which is able to interpret the dark matter decay [28]. In the simplest case, when \(g_{44}=-\Phi ^2=-\,1\), it leads general solutions of (A.2) to 4D-gravitational and Maxwell equations, exacting solutions of the classical K–K theory. Therefore, the 5D-STM theory is a generalization of the traditional 5D K–K theory.

Considering the cylindrical condition as too strong a constraint, 5D-STM colleagues managed to propose more physical consequences by non-compactification of ED and flexible modeling of time–space curvature. In practice, the STM-theory is often simplified by applying a diagonal metric signature \(\{+---\pm \}\) in 5D-geometry (A.1). In a link with 4D space–time physics of elementary particles the 5D-STM theory would lead to a qualitative interpretation of important issues of quantum mechanics, such as Heisenberg indeterminism [12, 13]. For example, it was shown that the indeterministic inequality should depend on the fifth dimension, instead of being constrained only by the Planck constant:

In 4D space–time when ED is quantized as \(l=n.l_{min}\) then \({|}dp_{\alpha }dx^{\alpha }{|}=\frac{\hbar }{c}\frac{dn^2}{n}\) which resembles Heisenberg inequalities. Therefore, the indeterminism in 4D space–time would be governed by a deterministic variation of the fifth dimension in 5D manifold.

An interpretation of wave-particle duality and a wave-like origin of proper mass can be found in [12]. There is also a derivation of Klein–Gordon equation from the wave solution of 4D-gravitational Eq. [21] and the meaning of quantization [29]. In particular, Klein–Gordon equation having originated from the 5D geodesic equation, contains an extra-term with constant extra-length *L*. On replacing latter with \(\frac{1}{m}\) as the divert mass of particle, this leads to:

where \(\Delta =g^{\alpha \beta } \nabla _\alpha \nabla _\beta \) labels d’Alembert operator. Another method has been mentioned briefly in [12] based on splits of the second covariant derivatives of a higher-dimensional complex wave function \(\psi \) into a real part and an imaginary part, that the former of which led to Klein–Gordon equation and the latter led to a conservation law of the energy-momentum current of elementary particles. In fact, it coincides with an application of Klein–Fock reduction formalism in our early study [14].

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Vo Van, T. A Time–Space Symmetry Based Cylindrical Model for Quantum Mechanical Interpretations.
*Found Phys* **47, **1559–1581 (2017). https://doi.org/10.1007/s10701-017-0123-2

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### Keywords

- Time–space symmetry
- Higher-dimensional general relativity
- Wave-like solution
- Klein–Gordon–Fock equation
- Heisenberg inequality
- Lepton mass hierarchy