Axially-symmetric, stationary fields , gravito-electromagnetic waves and a possibility to travel in higher dimentions in the Nonsymmetric Kaluza-Klein Theory

TL;DR

This paper explores axially-symmetric and stationary gravito-electromagnetic fields and waves within the Nonsymmetric Kaluza-Klein Theory, proposing potential mechanisms for higher-dimensional travel.

Contribution

It formulates field equations for symmetric and wave solutions in the Nonsymmetric Kaluza-Klein framework and introduces a generalized Kerner-Wong-Kopczynski equation.

Findings

Derived key quantities using symbolic manipulation.

Presented partial solutions to the field equations.

Discussed implications for higher-dimensional travel possibilities.

Abstract

We consider in the paper axially-symmetric and stationary fields and cylindrically symmetric gravito-electromagnetic waves in the Nonsymmetric Kaluza-Klein Theory.Using symbollic manipulations we write down all important quantities in the theory. We write field equations for both cases and partially integrate them.We consider also a generalized Kerner-Wong-Kopczynski equation in GSW model.

Equations

$$g_{\alpha \beta }=g_{(\alpha \beta )}+g_{[\alpha \beta ]}$$

$${\overline{W}{}^{\alpha }}_{\beta }={\overline{\omega }{}^{\alpha }}_{\beta }-\frac{2}{3}\overline{W}{\delta {}^{\alpha }}_{\beta }$$

$$\overline{W}={\overline{W}}_{\mu }{\overline{\theta }}^{\mu },$$

$$\overline{W}{}_{\mu }=\frac{1}{2}({\overline{W}{}^{\sigma }}_{\mu \sigma }-{\overline{W}{}^{\sigma }}_{\sigma \mu }).$$

$$\begin{array}{c}\overline{D}{g}_{\alpha \beta }-g_{\alpha \delta }{\overline{Q}{}^{\delta }}_{\beta \gamma }(\overline{\Gamma })\overline{\theta }{}^{\gamma }=0\\ {\overline{Q}{}^{\alpha }}_{\beta \alpha }(\overline{\Gamma })=0\end{array}$$

$${\overline{\omega }{}^{\alpha }}_{\beta }={\overline{\Gamma }{}^{\alpha }}_{\beta \gamma }\overline{\theta }{}^{\gamma }.$$

$$\sqrt{-g}R(W)=\sqrt{-g}(\text{vbox}to9.5pt\text{n@space}\overline{R}(\overline{W})+(2(g^{[\mu \nu ]}{F}_{\mu \nu }{)}^2-H^{\alpha \mu }{F}_{\alpha \mu })\text{vbox}to9.5pt)\text{n@space}$$

$$H^{\alpha \mu }=g^{\alpha \beta }{g}^{\mu \nu }{H}_{\beta \nu }$$

$$H_{\beta \nu }=-H_{\nu \beta }$$

$$g_{\delta \beta }{g}^{\gamma \delta }{H}_{\gamma \sigma }+g_{\alpha \delta }{g}^{\delta \gamma }{H}_{\beta \gamma }=2g_{\alpha \delta }{g}^{\delta \gamma }{F}_{\beta \gamma }$$

$$\overline{R}(\overline{W})=\overline{R}(\overline{\Gamma })+\frac{2}{3}{g}^{[\mu \nu ]}{W}_{[\mu ,\nu ]}.$$

$${\overline{R}}_{\alpha \beta }(\overline{W})-\frac{1}{2}{g}_{\alpha \beta }\overline{R}(\overline{W})=8\pi {\stackrel{\mathrm{em}}{T}}_{\alpha \beta },$$

$${\text{lower}5.0pt\sim g{}^{[\mu \nu ]}}_{,\nu =0}$$

$$g_{\mu \nu ,\sigma }-g_{\xi \nu }{\overline{\Gamma }{}^{\xi }}_{\mu \sigma }-g_{\mu \xi }{\overline{\Gamma }{}^{\xi }}_{\sigma \nu }=0,$$

$${\partial }_{\nu }({\text{lower}4.0pt\sim H}^{\alpha \mu })=2{\text{lower}5.0pt\sim g}^{[\sigma \beta ]}{\partial }_{\beta }(g^{[\mu \nu ]}{F}_{\mu \nu }),$$

$${\overline{\nabla }}_{\mu }(H^{\alpha \mu })=2g^{\alpha \beta }{\overline{\nabla }}_{\beta }(g^{[\mu \nu ]}{F}_{\mu \nu }).$$

$${\stackrel{\mathrm{em}}{T}}_{\alpha \beta }=\frac{1}{4\pi }(\text{vbox}to14.5pt\text{n@space}{g}_{\gamma \beta }{g}^{\tau \mu }{g}^{\epsilon \gamma }{H}_{\mu \alpha }{H}_{\tau \epsilon }-2g^{[\mu \nu ]}{F}_{\mu \nu }{F}_{\alpha \beta }-\frac{1}{4}{g}_{\sigma \beta }(\text{vbox}to11.5pt\text{n@space}{H}^{\mu \nu }{F}_{\mu \nu }-2(g^{[\mu \nu ]}{F}_{\mu \nu }{)}^2\text{vbox}to11.5pt)\text{n@space}\text{vbox}to14.5pt)\text{n@space},$$

$${\text{lower}5.0pt\sim g}^{[\mu \nu ]}=\sqrt{-g}{g}^{[\mu \nu ]},$$

$${\text{lower}4.0pt\sim H}^{\mu \alpha }=\sqrt{-g}{g}^{\beta \mu }{g}^{\gamma \alpha }{H}_{\beta \gamma }.$$

$${\overline{R}}_{(\alpha \beta )}(\overline{\Gamma })=8\pi {\stackrel{\mathrm{em}}{T}}_{(\alpha \beta )},$$

$${\overline{R}}_{[[\alpha \beta ]},\gamma ](\overline{\Gamma })-8\pi {\stackrel{\mathrm{em}}{T}}_{[[\alpha \beta ],\gamma ]}=0,$$

$${\overline{\Gamma }}_{\mu }=0,$$

$$g_{\mu \nu ,\sigma }-g_{\xi \nu }{\overline{\Gamma }{}^{\xi }}_{\mu \sigma }-g_{\mu \xi }{\overline{\Gamma }{}^{\xi }}_{\sigma \nu }=0,$$

$${\partial }_{\mu }(\text{vbox}to11.5pt\text{n@space}{\text{lower}4.0pt\sim H}^{\sigma \mu }-2g^{[\sigma \mu ]}(g^{[\nu \beta ]}{F}_{\nu \beta })\text{vbox}to11.5pt)\text{n@space}=0,$$

$$\begin{array}{c}{\overline{\omega }{}^{\alpha }}_{\beta }={\overline{\Gamma }{}^{\alpha }}_{\beta \gamma }\overline{\theta }{}^{\gamma }\\ {\overline{\Gamma }}_{\mu }={\overline{\Gamma }{}^{\alpha }}_{[\mu \alpha ]}.\end{array}$$

$${\stackrel{\mathrm{em}(1)}{T}}_{\alpha \beta }={\stackrel{\mathrm{em}}{T}}_{\alpha \beta }\text{(from Eq. eqref2.16)}$$

$${\stackrel{\mathrm{em}(2)}{T}}_{\mu \nu }=\frac{1}{4\pi }\{\text{vbox}to11.5pt\text{n@space}{g}^{\alpha \beta }{H}_{\beta \nu }{H}_{\sigma \mu }-2g^{[\alpha \beta ]}{F}_{\alpha \beta }{F}_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }[\text{vbox}to9.5pt\text{n@space}{H}^{\alpha \beta }{H}_{\alpha \beta }-2(g^{[\alpha \beta ]}{F}_{\alpha \beta }^2\text{vbox}to9.5pt]\text{n@space}\text{vbox}to11.5pt\}\text{n@space}-\frac{1}{8\pi }{J}_{\mu \nu },$$

$$J_{\mu \nu }=4H_{\alpha \mu }{H}_{\beta \nu }{g}^{[\alpha \beta ]}-4H_{\alpha \mu }{H}_{\tau \epsilon }{g}^{\tau \sigma }{g}_{\beta \nu }{g}^{[\epsilon \beta ]},$$

$${\stackrel{\mathrm{em}(3)}{T}}_{\alpha \beta }=\frac{1}{4\pi }\{\text{vbox}to11.5pt\text{n@space}{g}_{\gamma \beta }{H}^{\mu \sigma }{F}_{\mu \sigma }-2g^{\mu \nu }{F}_{\mu \nu }{F}_{\alpha \beta }-\frac{1}{4}{g}_{\alpha \beta }[\text{vbox}to9.5pt\text{n@space}{H}^{\mu \nu }{F}_{\mu \nu }-2(g^{[\mu \nu ]}{F}_{\mu \nu }{)}^2\text{vbox}to9.5pt]\text{n@space}\text{vbox}to11.5pt\}\text{n@space}.$$

$$g^{\alpha \beta }{\stackrel{\mathrm{em}(1)}{T}}_{\alpha \beta }=g^{\alpha \beta }{\stackrel{\mathrm{em}(2)}{T}}_{\alpha \beta }=g^{\alpha \beta }{\stackrel{\mathrm{em}(3)}{T}}_{\alpha \beta }=0.$$

$$g^{[\mu \nu ]}{H}_{\mu \nu }=g^{[\mu \nu ]}{F}_{\mu \nu },$$

$$g^{\alpha \omega }{g}^{\beta \mu }{H}_{\alpha \beta }{H}_{\omega \mu }=g^{\alpha \omega }{g}^{\beta \mu }{H}_{\alpha \beta }{F}_{\omega \mu },$$

$$g^{\sigma \nu }{g}^{\alpha \mu }{H}_{\sigma \alpha }{F}_{\mu }+g^{\mu \sigma }{g}^{\nu \beta }{H}_{\beta \sigma }{F}_{\mu \nu }=2g^{\mu \sigma }{g}^{\nu \beta }{F}_{\mu \nu }{F}_{\beta \sigma }$$

$$H_{\nu \mu }=F_{\nu \mu }-\tilde{g}{}^{(\tau \alpha )}{F}_{\alpha \nu }{g}_{[\mu \tau ]}-\tilde{g}{}^{\tau \alpha }{F}_{\alpha \mu }{g}_{[\nu \tau ]}.$$

$${\mathcal{L}}_{\mathrm{em}}=-\frac{1}{8\pi }(\text{vbox}to9.5pt\text{n@space}{H}^{\mu \nu }{F}_{\mu \nu }-2(g^{[\mu \nu ]}{F}_{\mu \nu }{)}^2\text{vbox}to9.5pt)\text{n@space}$$

$${\mathcal{L}}_{\mathrm{em}}=-\frac{1}{8\pi }(\text{vbox}to11.5pt\text{n@space}(g^{\mu \alpha }{g}^{\nu \beta }-g^{\nu \beta }\tilde{g}{}^{(\mu \alpha )}+g^{\nu \beta }{g}^{\mu \omega }\tilde{g}{}^{(\mu \alpha )}{g}_{\omega \tau })F_{\alpha \beta }{F}_{\mu \nu }-2(g^{[\mu \nu ]}{F}_{\mu \nu }{)}^2\text{vbox}to11.5pt)\text{n@space}$$

$${\mathcal{L}}_{\mathrm{em}}=-\frac{1}{8\pi }(\text{vbox}to11.5pt\text{n@space}{F}^{\mu \nu }{F}_{\mu \nu }-2(g^{[\mu \nu ]}{F}_{\mu \nu }{)}^2+(\text{vbox}to9.5pt\text{n@space}{g}^{\nu \beta }{g}^{\mu \omega }\tilde{g}{}^{(\tau \alpha )}{g}_{\omega \tau }-g^{\nu \beta }\tilde{g}{}^{(\mu \alpha )}\text{vbox}to9.5pt)\text{n@space}{F}_{\alpha \beta }{F}_{\mu \nu }\text{vbox}to11.5pt)\text{n@space}$$

$$F^{\mu \nu }=g^{\mu \alpha }{g}^{\nu \beta }{F}_{\alpha \beta }.$$

Full text

1 Introduction

In the paper we consider stationary and axially-symmetric fields and field equations in the Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory. We consider only 5-dimensional, electromagnetic case. A difference between Nonsymmetric Kaluza–Klein Theory and Nonsymmetric Jordan–Thiry Theory consists in introducing a scalar field in Nonsymmetric Jordan–Thiry Theory. This is the same as in the classical Kaluza–Klein (Jordan–Thiry) Theory. The Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory satisfies a Bohr correspondence principle to the classical one.

The Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory has been created and developed (see Refs [1], [2], [3], [4], also [5], [6], [7]). The theory uses a nonsymmetric metrization of a fibre bundle. On a space-time theory uses a formalism of Einstein Unified Field Theory with Moffat interpretation as an extended theory of gravitation (see Ref. [8]). The theory has been considered in 5-dimensional case (Refs [1], [3]) and also in $(n+4)$-dimensional case (see Refs [1], [2]), including also spontaneous symmetry breaking and Higgs’ mechanism (see Ref. [1]). In 5-dimensional case the theory is bi-invariant with respect to $\text{U}(1)$-group action on a fibre bundle. In a general non-Abelian case we have only right-invariance of an action of a group $G$ on a bundle. The theory has been applied to a problem of an anomalous acceleration of Pioneer 10 and 11 spacecraft (see Ref. [9]). We develop a theory finding spherically symmetric solutions (5-dimensional electromagnetic case) with particle-like properties and generalized plane gravito-electromagnetic waves (see Refs [3], [10]). A dielectric model of a confinement of charge (see Ref. [10]) has been posed and developed. Afterwards such a model has been extended to a non-Abelian dielectric model of color confinement (see Ref. [11]). The Nonsymmetric Kaluza–Klein Theory with spontaneous symmetry breaking and Higgs’ mechanism has been developed. In particular, a bosonic part of GSW (Glashow–Salam–Weinberg) has been derived with correct Weinberg angle and masses of $W^{\pm},Z^{0}$ and Higgs’ particle agreed with experiment (see Ref. [11]). A problem of hierarchy of symmetry breaking in the Nonsymmetric Kaluza–Klein Theory has been derived (see Ref. [12]). Cosmological models in the Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory with quintessence and quintessential inflation have been obtained ([13]). All of the above investigations are in line of the so-called programme of geometrization and unification of fundamental physical interactions (see Refs [14], [15]).

The paper has been organized as follows. In the second section we write down all important formulas in the Nonsymmetric Kaluza–Klein Theory (5-dimensional case), which we should use in Sections 3 and 4. Section 3 is devoted to axially-symmetric stationary fields in (5-dimensional) Nonsymmetric Kaluza–Klein Theory. In Section 4 we consider cylindrical gravito-electromagnetic waves in the Nonsymmetric Kaluza–Klein Theory. We use results calculated by our programmes written in Mathematica. We write down field equations for the mentioned cases. In the fifth section we consider generalized Kerner–Wong–Kopczyński equation in GSW model as a possibility to travel in higher dimensions.

In Conclusions and Prospects we give some prospects for further investigations. The programmes have been quoted in Appendix A and Appendix B. In Appendix A for axially-symmetric and stationary case. In Appendix B for cylindrical gravito-electromagnetic waves.

Let us remind to the reader that someone considers cylindrical waves in Einstein’s unified field theory (see Refs [16], [17]) and in Bonnor’s and Schrödinger’s unified field theory (see Ref. [18]). In NGT (Nonsymmetric Gravitation Theory) the problem of gravitational radiation from axisymmetric sources has been taken into account (see Refs [19], [20], [21]).

We use such system of units that $G_{N}=c=1$.

2 The Nonsymmetric Kaluza–Klein Theory

Let us give some details of the Nonsymmetric Kaluza–Klein Theory. The all full theory has been described in Refs [1], [2], [3], [4], [5], [6]. In this paper we do not repeat our considerations from previous papers. The Nonsymmetric Kaluza–Klein Theory unifies gravity and electromagnetism. Gravity is described by NGT (Nonsymmetric Gravitational Theory, see Ref. [8]). In this way in the theory we have nonsymmetric metric

[TABLE]

and two connections ${\overline{W}{}^{\alpha}}_{\!\beta}$ and ${\overline{\omega}{}^{\alpha}}_{\!\beta}$ such that

[TABLE]

where $\overline{W}$ is a one form on $E$ (a space-time),

[TABLE]

where $\overline{\theta}{}^{\mu}$ is a frame on $E$

[TABLE]

For connection ${\overline{\omega}{}^{\alpha}}_{\!\beta}$ we suppose the following conditions

[TABLE]

where $\overline{D}$ is an exterior covariant derivative with respect to a connection ${\overline{\omega}{}^{\alpha}}_{\!\beta}$ and ${\overline{Q}{}^{\alpha}}_{\!\beta\gamma}(\overline{\varGamma})$ its torsion.

[TABLE]

According to our general scheme we metrize nonsymmetrically an electromagnetic fibre bundle $\underline{P}$ over a space-time $E$. The metrization is of course biinvariant with respect to an action of a group $\text{U}(1)$.

We introduce two types of affine connections on $\underline{P}$ (a metrized electromagnetic fibre bundle) and calculate a scalar curvature for those affine connections. One gets

[TABLE]

where $H^{\alpha\mu}$ is a tensor on $E$ such that

[TABLE]

We have as usual

[TABLE]

From Palatini variational principle with respect to ${\overline{W}{}^{\alpha}}_{\!\beta}$, $g_{\alpha\beta}$ and $A_{\mu}$ one gets

[TABLE]

or

[TABLE]

We have

[TABLE]

Equations (2.12)–(2.15) can be rewritten in the following form:

[TABLE]

where $\overline{R}_{\alpha\beta}(\overline{\varGamma})$ is a Moffat–Ricci tensor for the connection

[TABLE]

For an energy-momentum tensor of an electromagnetic field we have three equivalent forms.

[TABLE]

It is easy to see that

[TABLE]

All forms of an energy-momentum tensor for an electromagnetic field are equivalent modulo equation (2.10) and also modulo the conditions:

[TABLE]

derived from the condition (2.10). The condition (2.10) can be solved (see Ref. [10]). Moreover, from a theoretical point of view it is more convenient to work with the tensor $H_{\mu\nu}$.

Equation (2.10) can be solved (as we mentioned before) and one gets

[TABLE]

In this way a lagrangian for an electromagnetic field in the Nonsymmetric Kaluza–Klein Theory

[TABLE]

can be rewritten in the form

[TABLE]

or

[TABLE]

where

[TABLE]

Using (2.31) we can reconsider an energy-momentum tensor for an electromagnetic field. One gets

[TABLE]

where

[TABLE]

is an energy-momentum tensor of the electromagnetic field in NGT,

[TABLE]

and

[TABLE]

The second part of Maxwell equations in the Nonsymmetric Kaluza–Klein Theory (i.e. Eq. (2.15) or Eq. (2.16)) can be rewritten in the following form:

[TABLE]

where ${J{}^{\alpha}}_{\!t}$ is a topological current and ${J{}^{\alpha}}_{\!p}$ is a polarization current

[TABLE]

$M^{\alpha\mu}$ is a polarization tensor induced by a nonsymmetric part of the metric

[TABLE]

From Eq. (2.31) one gets

[TABLE]

This tensor has an interpretation as a tensor of torsion in the fifth dimension

[TABLE]

The topological current

[TABLE]

is conserved by its definition. Its current density is equal to

[TABLE]

and one gets

[TABLE]

The first part of Maxwell equation is of course a Bianchi identity (due to this a fourpotential exists)

[TABLE]

where

[TABLE]

$\alpha$ is an electromagnetic connection defined on an electromagnetic bundle $\underline{P}$, $e$ is a local section of a bundle $\underline{P}$ ($e:E\supset U\to P$). We have as usual

[TABLE]

and

[TABLE]

$\widetilde{g}^{(\alpha\beta)}$ is an inverse tensor of $g_{(\alpha\beta)}$.

According to Refs [22], [23] a condition (2.22)–(2.23) or (2.5) can be exactly solved. One gets

[TABLE]

where ${\widetilde{\overline{\varGamma}}{}{}^{\lambda}}_{\!{\mu\nu}}$ is a Levi-Civita connection induced by $g_{(\alpha\beta)}$ on $E$ and

[TABLE]

is a torsion of an affine connection ${\overline{\varGamma}{}^{\lambda}}_{\!{\mu\nu}}$.

[TABLE]

where $\widetilde{\overline{\nabla}}{}$ is a covariant derivative with respect to a Levi-Civita connection ${\widetilde{\overline{\varGamma}}{}{}^{\alpha}}_{\!\beta\gamma}$ defined on $E$ and induced by a tensor $g_{(\alpha\beta)}$. In particular ${\widetilde{\overline{\varGamma}}{}{}^{\alpha}}_{\!\beta\gamma}$ can be a Christoffel symbol built for $g_{(\alpha\beta)}$.

One can rewrite the remaining Einstein equations and the second pair of Maxwell equations in the following way.

[TABLE]

In this way we get field equation in a GR (General Relativity) shape.

$\widetilde{\overline{R}}_{\beta\gamma}$ is a Ricci tensor for a Levi-Civita connection defined on $E$, induced by $g_{(\alpha\beta)}$.

Moreover, in future applications we use full field equation (2.20)–(2.24) and with a known solution for $H_{\nu\mu}$ (Eq. (2.31)) and a solution (2.54) for a connection ${\overline{\varGamma}{}^{\lambda}}_{\!{\mu\nu}}$ (solution of conditions (2.22)–(2.23) or (2.5)). We will work in holonomical or in unholonomical frames.

In the case of a holonomic frame we have the following points to introduce or calculate symbolically:

$g_{\mu\nu}$ 2. 2)

$g^{\mu\nu}$, $g^{[{\mu\nu}]}$, {\mathrel{\lower 5.0pt\hbox{{\scriptstyle\sim}}\hskip-5.0ptg}}^{[{\mu\nu}]}=\sqrt{-g}\,g^{[{\mu\nu}]}, \sqrt{-g}\,g^{\mu\nu}={\mathrel{\lower 5.0pt\hbox{{\scriptstyle\sim}}\hskip-5.0ptg}}^{\mu\nu} 3. 3)

$g=\det g_{\mu\nu}$ 4. 4)

$g_{({\mu\nu})}$, $g_{[{\mu\nu}]}$ 5. 5)

$\widetilde{g}=\det g_{({\mu\nu})}$, $\widetilde{g}{}^{(\alpha\beta)}$ 6. 6)

$F_{\mu\nu}$ 7. 7)

$H_{\mu\nu}=F_{\mu\nu}-\widetilde{g}{}^{(\tau\alpha)}F_{\alpha\nu}g_{[\mu\tau]}+\widetilde{g}{}^{(\tau\alpha)}F_{\alpha\mu}g_{[\nu\tau]}$ 8. 8)

$H^{\mu\nu}=g^{\beta\nu}g^{\gamma\nu}H_{\beta\gamma}$, {\mathrel{\lower 4.0pt\hbox{{\scriptstyle\sim}}\hskip-7.5ptH}}^{\mu\nu}=\sqrt{-g}H^{\mu\nu} 9. 9)

8\pi\overset{\rm em}{T}_{\!\alpha\beta}=2\mathopen{\hbox{\left[\vbox to11.5pt{}\right.\n@space}}g_{\gamma\beta}g^{\tau\mu}g^{\varepsilon\gamma}H_{\mu\alpha}H_{\tau\varepsilon}-2g^{[{\mu\nu}]}F_{\mu\nu}F_{\alpha\beta}-\frac{1}{4}\,g_{\alpha\beta}\mathopen{\hbox{\left(\vbox to9.5pt{}\right.\n@space}}H^{\mu\nu}F_{\mu\nu}-2(g^{[{\mu\nu}]}F_{\mu\nu})^{2}\mathclose{\hbox{\left.\vbox to9.5pt{}\right)\n@space}}\mathclose{\hbox{\left.\vbox to11.5pt{}\right]\n@space}} 10. 10)

{\widetilde{\overline{\varGamma}}{}{}^{\alpha}}_{\!\beta\gamma}=\frac{1}{2}\,\widetilde{g}{}^{(\alpha\delta)}\mathopen{\hbox{\left(\vbox to11.5pt{}\right.\n@space}}\frac{\partial g_{(\delta\beta)}}{\partial x^{\gamma}}+\frac{\partial g_{(\delta\gamma)}}{\partial x^{\beta}}-\frac{\partial g_{(\beta\gamma)}}{\partial x^{\delta}}\mathclose{\hbox{\left.\vbox to11.5pt{}\right)\n@space}} 11. 11)

$\widetilde{\overline{\nabla}}{}_{\alpha}g_{[\beta\gamma]}=\partial_{\alpha}g_{[\beta\gamma]}-\widetilde{\overline{\varGamma}}{}{}^{\delta}_{\beta\alpha}g_{[\delta\gamma]}-\widetilde{\overline{\varGamma}}{}{}^{\delta}_{\beta\alpha}g_{[\beta\delta]}$ 12. 12)

$K_{\alpha\beta\gamma}=-\widetilde{\overline{\nabla}}{}_{\alpha}g_{[\beta\gamma]}-\widetilde{\overline{\nabla}}{}_{\beta}g_{[\gamma\alpha]}+\widetilde{\overline{\nabla}}{}_{\gamma}g_{[\alpha\beta]}$ 13. 13)

${K{}_{\gamma\mu}}^{\!\nu}=K_{\gamma\mu\alpha}\widetilde{g}{}^{(\alpha\nu)}$ 14. 14)

${{g{}_{[\rho}}^{\!\beta}}{}_{\!]}=\widetilde{g}{}^{(\alpha\beta)}g_{[\rho\alpha]}$ 15. 15)

{\overline{Q}{}^{\nu}}_{\!\gamma\mu}=\frac{1}{2}\,\mathopen{\hbox{\left(\vbox to9.5pt{}\right.\n@space}}{K{}_{\gamma\mu}}^{\!\nu}-2{{g{}_{\mathopen{\hbox{\dm[}}[\mu}}^{\!\alpha}}{}_{\!]}K_{\gamma\mathclose{\hbox{\dm]}}\alpha\beta}\widetilde{g}{}^{(\nu\rho)}\widetilde{g}^{(\beta\delta)}g_{[\rho\delta]}\mathclose{\hbox{\left.\vbox to9.5pt{}\right)\n@space}} 16. 16)

{\varDelta{}^{\nu}}_{\!\gamma\mu}=\widetilde{g}{}^{(\nu\delta)}\mathopen{\hbox{\left{\vbox to11.5pt{}\right.\n@space}}{K{}_{\delta(\gamma}}^{\!\alpha}g_{[\mu)\alpha]}+{{g{}_{[\rho}}^{\!\beta}}{}_{\!]}\mathopen{\hbox{\left[\vbox to9.5pt{}\right.\n@space}}{{g{}_{([\mu}}^{\!\rho}}{}_{\!]}K_{\gamma)\alpha\beta}{{g{}_{[\delta}}^{\!\alpha}}{}_{\!]}-K_{\delta\alpha\beta}{{g{}_{([\gamma}}^{\!\alpha}}{}_{\!]}{{g{}_{[\mu)}}^{\!\rho}}{}_{\!]}\mathclose{\hbox{\left.\vbox to9.5pt{}\right]\n@space}}\mathclose{\hbox{\left.\vbox to11.5pt{}\right}\n@space}} 17. 17)

${\overline{\varGamma}{}^{\nu}}_{\!\gamma\mu}={\widetilde{\overline{\varGamma}}{}^{\nu}}_{\!\gamma\mu}+{\overline{Q}{}^{\nu}}_{\!\gamma\mu}+{\varDelta{}^{\nu}}_{\!\gamma\mu}$ 18. 18)

${\overline{R}{}^{\alpha}}_{\!\beta\gamma\delta}=2{\overline{\varGamma}{}^{\alpha}}_{\!\beta{[\delta,\gamma]}}+2{\overline{\varGamma}{}^{\alpha}}_{\!\rho[\gamma}{\overline{\varGamma}{}^{\rho}}_{\!|\beta|\delta]}$ 19. 19)

$\overline{R}_{\mu\nu}={\overline{R}{}^{\alpha}}_{\!{\mu\nu}\alpha}+\frac{1}{2}\,{\overline{R}{}^{\alpha}}_{\!\alpha{\mu\nu}}$ (a Moffat–Ricci tensor) 20. 20)

$\overline{R}_{({\mu\nu})}$ 21. 21)

$\overline{R}_{[{\mu\nu}]}$ 22. 22)

$8\pi\overset{\rm em}{T}_{\!(\alpha\beta)}$ 23. 23)

$8\pi\overset{\rm em}{T}_{\![\alpha\beta]}$ 24. 24)

$\overline{R}_{(\alpha\beta)}=8\pi\overset{\rm em}{T}_{\!(\alpha\beta)}$ 25. 25)

$\overline{R}_{\mathopen{\hbox{\dm[}}[{\mu\nu}],\gamma\mathclose{\hbox{\dm]}}}=8\pi\overset{\rm em}{T}_{\!\mathopen{\hbox{\dm[}}[{\mu\nu}],\gamma\mathclose{\hbox{\dm]}}}$ 26. 26)

2{\mathrel{\lower 5.0pt\hbox{{\scriptstyle\sim}}\hskip-5.0ptg}}^{[\alpha\beta]}\partial_{\beta}(g^{[{\mu\nu}]}F_{\mu\nu}) 27. 27)

\partial_{\mu}({\mathrel{\lower 4.0pt\hbox{{\scriptstyle\sim}}\hskip-7.5ptH}}^{\alpha\mu}) 28. 28)

\partial_{\mu}({\mathrel{\lower 4.0pt\hbox{{\scriptstyle\sim}}\hskip-7.5ptH}}^{\alpha\mu})=2{\mathrel{\lower 5.0pt\hbox{{\scriptstyle\sim}}\hskip-5.0ptg}}^{[\alpha\beta]}\partial_{\beta}(g^{[{\mu\nu}]}F_{\mu\nu}) 29. 29)

{{\mathrel{\lower 5.0pt\hbox{{\scriptstyle\sim}}\hskip-5.0ptg}}{}^{[{\mu\nu}]}}_{\!,\nu}=0.

In the case of unholonomic frames we have some differences concerning points 10) and 18). In this case ,ν means an action of a vector field dual to the unholonomic frame. Section 3 gives us an example of this situation.

Let us introduce external (material) sources in the Nonsymmetric Kaluza–Klein Theory as in Refs [1], [3]. Let us notice that one introduces material (external) sources in the Kaluza–Klein Theory (Kaluza–Klein Theory with torsion) in Ref. [7] (see also Refs [24], [25], [26]).

In order to do this we write down a material sources lagrangian in the following way:

[TABLE]

One gets

[TABLE]

where

[TABLE]

$T^{\mu\nu}$ is an energy momentum tensor of material sources, $j^{\mu}$ an electric current and $A_{\mu}$ a four-potential.

Usually we suppose a conservation of an electric charge

[TABLE]

One can consider also $T^{\mu\nu}$ as a sum of a hydrodynamical tensor and an interaction term

[TABLE]

where $u^{\mu}$ is the four-velocity of a fluid, $p$ its pressure, $h^{\nu}$ a four-vector of an enthalpy. In this way in Eqs (2.12), (2.20), (2.21) we should pass from $\overset{\rm em}{T}_{\!{\mu\nu}}$ to $\overset{\rm\rm eff}{T}_{\!{\mu\nu}}$, where

[TABLE]

Eq. (2.24) should be rewritten as

[TABLE]

In some cases we can consider only

[TABLE]

neglecting the hydrodynamic part. An electric current can be written also as

[TABLE]

where $q$ is an electric charge and $\rho$ the density of a fluid.

External sources in Kaluza–Klein Theory (any type—nonsymmetric, symmetric, with torsion etc.) are in some sense against a unification approach, where everything has been geometrized. Moreover, those external sources can be obtained from more unified theory (also Kaluza–Klein or Jordan–Thiry type) where 5-dimensional theory (an electromagnetic theory unified with gravity) is only a part of a full theory which is non-Abelian Kaluza–Klein theory with Higgs’ fields and spontaneous symmetry breaking. This theory geometrized all physical interactions (a bosonic part of the physical world) and only as external sources remain fermion which are coupled to the multidimensional geometry via a minimal coupling scheme. It means they are coupled by new type of gauge derivatives. All fermions are unified in one multidimensional spinor. All physical interactions are described by only one affine connection defined on the Kaluza–Klein (Jordan–Thiry) manifold. In this setting a cosmological constant which we consider in the 5-dimensional case is also a part of external sources which can be derived from more extended theory.

Let us remind to the reader what we mean by a geometrization in the Einstein programme. In this programme all physical quantities concerning fundamental physical interaction should get geometrical interpretation. They should be geometrical quantities as connections, curvatures, metric tensors, torsions, nonmetricity tensors etc. The best way to achieve it is to construct a lagrangian of the theory as a scalar curvature of some affine connection defined on a manydimensional manifold. In this way classical equations of fields obtained from variational principle are vacuum equations. On the right-hand side we have only zero, on the left-hand side geometrical quantities only. If this lagrangian contains all physical interactions unified in this one affine connection, we get simultaneously a unification of those interactions. Moreover, 5-dimensional theory which describes only a unification of a gravity and an electrodynamics, even it is given by a lagrangian which is a scalar curvature of an affine connection is not a full story. Thus external sources are not breaking the idea of a full geometrization, even they are not of geometrical nature and in the field equations they are on the right hand side of field equations.

They can be derived from more unified theory (in higher dimensions, more than five dimensions). 5-dimensional theory in our geometrization and unification is pure geometrical because $\overset{\rm em}{T}_{\!{\mu\nu}}$ (an energy-momentum tensor) is derived from geometry and the lagrangian of the theory is a scalar curvature. We have the same in more advanced theory i.e. a bosonic part of GSW-model (see Ref. [11]).

3 Axially-symmetric and stationary fields

in Nonsymmetric Kaluza–Klein Theory

Let us consider axially-symmetric and stationary fields in the Nonsymmetric Kaluza–Klein Theory. First of all we write down a symmetric part of the metric (see Refs [27]–[36])

[TABLE]

where $\rho$ and $z$, $\varphi$ are usual cylindrical coordinates. A signature is here ${}+{+}+{-}$. We introduce a new nonholonomical frame

[TABLE]

(functions are functions of $\rho$ and $z$ only). In this new frame a metric (3.1) has a form

[TABLE]

Let us introduce new functions $r_{1}$ and $r_{2}$ such that

[TABLE]

and $r_{1}^{2}r_{2}^{2}=\rho^{2}$. Simultaneously we change for a convenience the signature to ${}-{-}-{+}$. One gets

[TABLE]

In this way a symmetric part of a metric has the following shape

[TABLE]

is an inverse tensor for $g_{({\mu\nu})}$.

Now we rename variables such that $1-x$, $2-y$ and of course $r_{1}^{2}r_{2}^{2}=y^{2}$. One gets

[TABLE]

The frame considered by us is nonholonomical and we get

[TABLE]

We can recapitulate it in a one formula

[TABLE]

The remaining coefficients are zero.

Now we calculate a Levi-Civita connection coefficients induced by (3.10) in our nonholonomic frames from the formulas

[TABLE]

One gets

[TABLE]

Let us introduce a skew-symmetric part of the metric in the mentioned nonholonomic frame

[TABLE]

In this way our nonsymmetric metric (axially-symmetric and stationary) takes a form

[TABLE]

$b$ is a function of $z$ and $\rho$ (or $r$). In our contemporary notation it is a function of $x$ and $y$ only.

If we define $g_{[{\mu\nu}]}=K_{\mu\nu}$ we can define a very important object

[TABLE]

and we get

[TABLE]

The remaining $K_{\omega{\mu\nu}}$ equal zero.

$K_{\omega{\mu\nu}}$ are important in order to construct a connection ${\overline{\varGamma}{}^{\mu}}_{\!\alpha\beta}$ compatible with a full nonsymmetric metric $g_{\mu\nu}$.

In order to calculate a curvature tensor for the connection ${\widetilde{\overline{\varGamma}}{}{}^{\mu}}_{\!\alpha\beta}$ (we should remember that we are working in an unholonomic frame) we write down

[TABLE]

One gets

[TABLE]

For

[TABLE]

one gets

[TABLE]

,γ means an action of a vector field dual to $\overline{\theta}{}^{\gamma}$. We suppose the following shape of $F_{\mu\nu}$ tensor

[TABLE]

We have the following relation:

[TABLE]

In Appendix A we calculate using symbolic manipulations in Mathematica all important quantities in our theory.

Let us write down field equations for axially-symmetric, stationary field configurations in the Nonsymmetric Kaluza–Klein Theory. All independent equations are written down below. We use of course our programme to calculate $\overline{R}_{(\alpha\beta)}$, $\overline{R}_{[\alpha\beta]}$, $\overset{\rm em}{T}_{\!(\alpha\beta)}$, $\overset{\rm em}{T}_{\![\alpha\beta]}$, $W^{\mu\nu}$. We get some zeros $\overset{\rm em}{T}_{\!(13)}=\overset{\rm em}{T}_{\!(34)}=\overset{\rm em}{T}_{\![13]}=\overset{\rm em}{T}_{\![23]}=\overset{\rm em}{T}_{\![34]}=W^{13}=W^{23}=W^{34}=0$.

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

Eq. (3.49) $(\overline{R}_{[34]}={\rm const})$ is coming from the equations $\frac{\partial}{\partial\rho}(\overline{R}_{[34]}-8\pi\overset{\rm em}{T}_{\![34]})=\frac{\partial}{\partial z}(\overline{R}_{[43]}-8\pi\overset{\rm em}{T}_{\![34]})=0$ and $\overset{\rm em}{T}_{\![34]}=0$.

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

Let us consider the second pair of Maxwell equations in our theory

[TABLE]

{\mathrel{\lower 4.0pt\hbox{{\scriptstyle\sim}}\hskip-7.5ptW}}^{\mu\nu}=\sqrt{-g}\mathopen{\hbox{\left(\vbox to9.5pt{}\right.\n@space}}H^{\mu\nu}-2g^{[{\mu\nu}]}(g^{[\alpha\beta]}F_{\alpha\beta})\mathclose{\hbox{\left.\vbox to9.5pt{}\right)\n@space}}. One gets

[TABLE]

From (3.53) one gets

[TABLE]

and eventually

[TABLE]

where $g(z)$, $h(z)$ and $C(z)$ are arbitrary functions of $z$.

We can introduce potentials $A_{1}=A_{1}(\rho,z)$ and $A_{2}=A_{2}(\rho,z)$ for Eqs (3.50) and (3.51) such that

[TABLE]

In particular one gets

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The above equations can be solved exactly or numerically. In order to solve them exactly we should develop new methods of exact solution similar to the inverse scattering method, Bäcklund transformation or Hirota method. First of all we should transform the system of the equations to some kind Ernst-like equation (see Refs [27]–[36]). This will be a subject of a future development.

4 Cylindrical gravito-electromagnetic waves in the Nonsymmetric Kaluza–Klein Theory

Let us consider the following nonsymmetric metric tensor (cylindrical gravito-electromagnetic wave)

[TABLE]

in cylindrical coordinates $r,\theta,z$.

$l,n,s$ are functions of $r$ and $t$. The strength of an electromagnetic field takes a form

[TABLE]

$b$ is a function of $r$ and $t$.

One gets

[TABLE]

(see Ref. [37]),

[TABLE]

From the equation

[TABLE]

one gets that

[TABLE]

(see Ref. [37]).

One gets

[TABLE]

It is easy to see that

[TABLE]

The first pair of Maxwell equations is satisfied.

[TABLE]

Let us calculate

[TABLE]

From the second pair of Maxwell equations in the Nonsymmetric Kaluza–Klein Theory one gets

[TABLE]

For $\nu=1$

[TABLE]

$\nu=3$

[TABLE]

$\nu=4$

[TABLE]

From Eq. (4.37) and Eq. (4.39) one gets

[TABLE]

where $C=\rm const$.

Eq. (3.27) gives us an existence of a potential $A=A(r,t)$ such that

[TABLE]

The function $A(r,t)$ satisfies the equation

[TABLE]

Moreover $b(r,t)$ satisfies the equation

[TABLE]

Equations (4.40), (4.41), (4.42) and (4.43) give us an additional interaction between gravitational and electromagnetic fields which is absent in GR theory.

All functions are functions of $r,t$ in the following way:

[TABLE]

Let us write the remaining field equations

[TABLE]

Using the programme from Appendix B one gets field equations (4.45)–(4.46).

There are some zeros:

[TABLE]

In the remaining cases we get

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

or

[TABLE]

We can introduce a potential $B=B(r,t)$ such that

[TABLE]

In particular one gets

[TABLE]

[TABLE]

In future papers we will try to solve these equations to get cylindrical gravito-electromagnetic waves, possibly exactly or numerically.

Let us consider material sources for a cylindrically symmetric gravito-electromagnetic waves in the Nonsymmetric Kaluza–Klein Theory. In cylindrical coordinates we write an electric current

[TABLE]

$j_{r},j_{\theta},j_{z}$ and $j_{t}$ are functions of $r$ and $t$ only.

Let us consider a conservation law for an electric charge

[TABLE]

Let us suppose that $j_{\theta}=0$. Eventually one gets

[TABLE]

where

[TABLE]

We introduce a potential $d(r,t)$ such that

[TABLE]

One gets

[TABLE]

where $d=d(r,t)$ is a potential. From Eq. (2.69) one gets

[TABLE]

One also gets

[TABLE]

where $\widetilde{A}(r)$, $\widetilde{B}(t)$ are arbitrary functions of one variable and eventually

[TABLE]

Thus an electric current has a shape

[TABLE]

An interacting lagrangian in this case has the following shape

[TABLE]

and

[TABLE]

If we suppose that $j_{z}=0$

[TABLE]

The simplest example of an interaction of electromagnetic and gravitational (skewon-gravitational) waves in the Nonsymmetric Kaluza–Klein Theory gives us a possibility to generate gravito-electromagnetic wave by an external source. The following equations give us in some sense a theory of such a generator. In this case our equations (the second pair of Maxwell equations) look as follows:

[TABLE]

where as usual

[TABLE]

Moreover a more interesting case can be derived if we suppose that $\widetilde{A}(r)=\widetilde{B}(t)=0$ and $j_{z}(r,t)\neq 0$ in such a way that

[TABLE]

In this case one gets

[TABLE]

and

[TABLE]

One can even design an experiment to check whether we are right or not in gravito-electromagnetic waves generation problem. Moreover we should pass in all equations (4.49)–(4.57) from $\overset{\rm em}{T}_{\!{\mu\nu}}$ to

[TABLE]

where $g_{\mu\nu}$ is given by Eq. (4.5).

One gets:

[TABLE]

Let us consider an external electromagnetic (it can be in reality e.g. a laser field) field such that

[TABLE]

and let $j_{z}(r,t)$ be a source of this field

[TABLE]

Let

[TABLE]

One gets in the place of Eq. (4.92)

[TABLE]

Eqs (4.90) and (4.91) do not change.

Let us consider an initial value problem for Eqs (4.90)–(4.91) and Eq. (4.98). In this way we should consider an initial value problem for a metric tensor $g_{\mu\nu}$ given by Eq. (4.5). The best way to do this is to take $g_{tt}$ in cylindrical coordinates from spherically symmetric solution in NGT

[TABLE]

($m$ is a mass and $\widetilde{l}$ is an additional parameter) and to equal it to $g_{tt}|_{t=0}$ of the metric (4.5), i.e.

[TABLE]

Supposing $|z|\ll r$ one gets

[TABLE]

In this case we do not add any energy-momentum tensor to $\overset{\rm em}{T}_{\!\alpha\beta}$ and $\overset{\rm\rm eff}{T}_{\!\alpha\beta}=\overset{\rm em}{T}_{\!\alpha\beta}$.

We suppose that external electromagnetic field does not change a geometry of space-time. In this way Eqs (4.48)–(4.55) are considered as consistency conditions for equations of a generator of gravito-electromagnetic (electromagnetic-gravitational) waves.

In some applications one can suppose

[TABLE]

5 A generalized Kerner–Wong–Kopczyński equation as a possibility to travel in higher dimensions

Let us consider generalized Kerner–Wong–Kopczyński equation in the Nonsymmetric Kaluza–Klein Theory for GSW (Glashow–Salam–Weinberg) model, i.e. Eqs (C.49)–(C.51) and Eqs (C.53)–(C.58) from Ref. [11]. Let us quote those equations here.

[TABLE]

[TABLE]

All the details concerning the Nonsymmetric Kaluza–Klein Theory (in general) with and without spontaneous symmetry breaking can be found in Ref. [11]. Some applications for GSW model of the theory can be found also here. Let us make some simplification of the Kerner–Wong–Kopczyński equation in a particular application in GSW model.

Let $F_{\beta\gamma}=0$ ($A_{\mu}=0$), and $Z^{0u}_{\beta\gamma}=0$ ($Z^{0u}_{\mu}=0$). Simultaneously let some gauge charges be equal to zero, $q=\widetilde{Q}{}^{iu}=\widetilde{Q}{}^{0u}=0$ (i.e. an electric charge and weak interaction charges) and also ${Q{}^{u}}_{\!\gamma}={Q{}^{u}}_{\!-\gamma}=q_{-\alpha}=0$ (i.e. some charges connected to Higgs’ field). Let us remind to the reader that Higgs’ field in our approach is also a gauge field, but over additional dimensions, in particular for GSW model over fifth and sixth dimensions. In this case additional dimensions are dimensions of $S^{2}$ sphere ($\psi$ and $\varphi$, $\psi\in\langle-\frac{\pi}{2},\frac{\pi}{2}\rangle$, $\varphi\in\langle 0,2\pi)$).

One gets the following set of equations:

[TABLE]

In the equations from [11] we have

[TABLE]

where $\{{\alpha\atop\beta\gamma}\}$ are Christoffel’s symbols formed for $g_{(\alpha\beta)}$, ${W{}^{\pm u}}_{\!\gamma}$ are charged weak interactions gauge fields in the gauge $U$, $H(x)$ is a Higgs’ field after a spontaneous symmetry breaking, $v$ is a vacuum expectation value of a Higgs’ field, $r$ is the radius of the sphere $S^{2}$. Let us notice the following fact: a charge $q_{\alpha}$ is a reason to connect the motion on space-time $E$ to higher dimension on $S^{2}$, $m_{0}$ is a mass of a test particle. Let us remind to the reader the following fact: Eqs (5.1) are equations of a test particle equipped with many charges in our theory.

After some simplifications given here we get equations in six dimensions on $E\times S^{2}$ (not four as on $E$). Additional dimensions, fifth and sixth, are dimensions on $S^{2}$. Moreover, the radius $r$ of $S^{2}$ is very small ($r\approx 2.39\cdot 10^{-18}$ m).

The set of those equations gives us a possibility of travelling in higher dimensions due to the existence of Higgs’ field and a charge $q_{\alpha}$ connected to it. This is in some sense a Lorentz force term involving Higgs’ field. For a Higgs’ field is a gauge field over fifth and sixth dimensions, the force involves higher dimensions. Such a situation can give a possibility to go to distant points in space crosspassing in short (shortcut), travelling in time, or to go out from compact, 3-dimensional objects without crossing boundaries. Such solutions of Eqs (5.2)–(5.5) are under considerations. One easily gets from Eq. (5.5)

[TABLE]

If $C_{\alpha}=0$, an effect of higher dimensions disappears.

Let us consider a more general situation where not only $q_{\alpha}\neq 0$, but also $q_{-\alpha}\neq 0$. In this case from Eqs (5.1) one gets supposing as before $Z_{\mu}^{0u}=Z_{\beta\gamma}^{0u}=F_{\beta\gamma}=A_{\mu}=q={Q{}^{u}}_{\!\gamma}={Q{}^{u}}_{\!-\gamma}=0$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and Eqs (5.10)–(5.11) for $q_{\alpha}$. From Eq. (5.15) one easily gets

[TABLE]

where $C_{-\alpha}=q_{-\alpha}(\tau_{0})={\rm const.}$, $C_{-\alpha}>0$. If $C_{-\alpha}=C_{\alpha}=0$, an effect of higher dimensions disappears.

Let us notice that such a physics was considered in [38, 39, 40]. Here we give mathematical and physical reasons to realize such situations.

Let us consider a very special situation where $C_{\alpha}=C_{-\alpha}=0$, i.e. $q_{\alpha}=q_{-\alpha}=0$. In this case one gets the following set of equations from Eqs (5.12)–(5.14):

[TABLE]

Eq. (5.17) is a geodetic equation on $E$ with a metric tensor $g_{(\alpha\beta)}$. Eqs (5.18)–(5.19) are geodetic equations on $S^{2}$. The last can easily be solved and we get

[TABLE]

where $a$ and $\tau_{0}$ are constants.

In the case where all the charges are zero the movements on $E$ and on $S^{2}$ decouple. Those movements also decouple in the case of non-zero electric charge $q$. Moreover, if the charges $\widetilde{Q}{}^{iu},\widetilde{Q}{}^{0u},Q^{u}_{\gamma},Q^{u}_{-\gamma}$ are non-zero the coupling between movement on $E$ and on $S^{2}$ is evident. Let us notice that a value of a radius of a sphere enters the equations.

Let $g_{(\alpha\beta)}=\eta_{\alpha\beta}={\rm diag}(-1,-1,-1,+1)$ (a Minkowski space) and let $\psi$ and $\varphi$ be described by Eqs (5.20)–(5.21). Let us consider Eq. (5.12). One gets

[TABLE]

Moreover,

[TABLE]

In this way one gets

[TABLE]

Eqs (5.25) and (5.20)–(5.21) govern a motion in higher dimensions under some assumptions. We can simplify Eq. (5.25) even more taking $H(x)=0$ and $W^{-u}_{\delta}=0$. One gets

[TABLE]

If $C_{-\alpha}=0$ ($q_{-\alpha}=0$) one gets even more simple

[TABLE]

and Eqs (5.20)–(5.21).

Let $W^{+u}_{\delta}={\rm const}$. One gets

[TABLE]

These equations are easy to be solved exactly.

[TABLE]

where $\alpha^{i}$, $i=1,2,3$, and $\alpha^{4}$ are constants.

$\tau$ is here of course a proper time. A coordinate time is $t_{1}=x^{4}$.

Thus we should solve a transcendent equation

[TABLE]

getting

[TABLE]

where $f(t)$ is a solution of the transcendental equation:

[TABLE]

Taking for simplicity $\alpha^{4}=0$ one finds

[TABLE]

under the condition

[TABLE]

In this case one gets

[TABLE]

under the condition (5.39).

Eqs (5.40)–(5.42) give us a trajectory of a travel in higher dimensions on $E\times S^{2}$ in a function of a coordinate time on $E$ (a Minkowski space).

Let us identify $\widetilde{x}{}^{1}=x$, $\widetilde{x}{}^{2}=y$, $\widetilde{x}{}^{3}=z$; $(x,y,z)$—a Cartesian coordinate system in $E^{3}$. Let us introduce a Cartesian coordinate system in the 3-dimensional $\mathbb{R}^{3}$ space where our $S^{2}$ sphere is located.

[TABLE]

In this way our trajectory looks like

[TABLE]

where $r\approx 2.39\cdot 10^{-18}$ m is a scale of GSW-model,

[TABLE]

under the condition (5.39).

Let us take the following transformation for a time. Let

[TABLE]

In this way one gets

[TABLE]

Let us consider the condition (5.39) in more details

[TABLE]

where

[TABLE]

One gets

[TABLE]

If $\varepsilon>0$, $t_{1}<0$; if $\varepsilon<0$, $t_{1}>0$.

The trajectory (5.45)–(5.52) is a trajectory in six dimensions. Moreover, an additional movement in higher dimensions is on a sphere $S^{2}$ embedded in $E^{3}$. We can get more complicated trajectory if we disturb the solution (5.20)–(5.21), allowing in Eqs (5.13)–(5.14) some additional terms, i.e.

[TABLE]

and Eqs (5.10)–(5.11). In this case we expect some additional effects of impossible physics. We will consider such possibility in further papers.

Conclusions and prospects for further research

In the paper we consider axially-symmetric and stationary fields in the Nonsymmetric Kaluza–Klein Theory, i.e.  a nonsymmetric metric (symmetric metric and skewsymmetric part—skewon part) and an electromagnetic field. We derive equations of the theory using symbolic manipulation programme written in Mathematica. There are some prospects for future research:

1. 1∘

Solve these equations exactly or numerically. 2. 2∘

Extend this formalism for non-Abelian Nonsymmetric Kaluza–Klein Theory also with spontaneous symmetry breaking and Higgs’ mechanism. In this case we should consider also spherically symmetric and stationary solutions, cylindrical and spherical Yang–Mills’ waves.

We consider also in the paper a cylindrical gravito-electromagnetic wave in the Nonsymmetric Kaluza–Klein Theory. Why it is called gravito-electromagnetic wave is explained in Ref. [10] and we do not repeat our explanation in details here. There are some prospects for future research.

1. 1∘

Find exact (or numerical) solutions for field equations. 2. 2∘

Extend a formalism to spherical gravito-electromagnetic wave with derivations of exact (or numerical solutions).

For gravitational waves have been detected (three events: GW150914, LVT151012 (a lower significance), GW151226) by LIGO-Virgo interferometers (see Refs [41], [42], [43], [44]), the existence of gravito-electromagnetic waves is highly probable. We concentrate in future investigations on a possible detection of these waves.

Let us notice the following fact. One considers a notion of electromagnetic-gravitational waves earlier as a conclusion of “interference effects” in the Nonsymmetric Kaluza–Klein (Jordan–Thiry) Theory.

In Ref. [1], p. 289, point 3) one writes in the following way on a further prospect of research: To find wave-like solutions of the field equations in the Abelian and non-Abelian cases. This could, in the electromagnetic case offer a solution which could be treated as a kind of electromagnetogravitational wave (nonlinear wave) with nontrivial interactions between fields. The objective of this hope is related to points 4) and 5) (see p. 6 of Ref. [1]) in the list of “interference effects” (we recall that the displacement current in the classical Maxwell equations leads us to the nontrivial interaction between the electric and magnetic field—the raison d’être of the wave solutions for Maxwell equations. However, this is only a historical remark. By a nontrivial interaction, we mean that the flow of energy is possible from one field to another in a quasi-periodic way. One can try to use the following Ansatz for the ….

Points 4) and 5) from page 6 of Ref. [1] are as follows:

4. The source in the second pair of Maxwell (Yang–Mills) equations, i.e. a current $j_{\mu}$ $(j^{a}_{\mu})$.

5. The polarization of vacuum $M_{\mu\nu}=-\frac{1}{4\pi}(H_{\mu\nu}-F_{\mu\nu})$ $(M^{a}_{\mu\nu}=-\frac{1}{4\pi}(L^{a}_{\mu\nu}-H^{a}_{\mu\nu}))$ with an interpretation as the torsion in the 5th dimension (in higher dimensions in Yang–Mills case).

Let us remind to the reader that in Ref. [10] we consider gravito-electromagnetic waves in the Nonsymmetric Kaluza–Klein Theory. In some sense they are similar to Ansatz from Ref. [1], p. 289. They are generalized plane waves for $g_{\mu\nu}$ (nonsymmetric) and generalized electromagnetic plane waves. We use some achievements from Einstein Unified Field Theory to find solutions for gravito-electromagnetic waves (generalized plane gravito-electromagnetic waves). Let us notice that in the limit of zero skewon and electromagnetic fields we get generalized plane waves from General Relativity (see Ref. [45]). The electromagnetic field of our solution has remarkable properties to have both invariants of the electromagnetic field $S=F_{\mu\nu}F^{\mu\nu}$ and $P=F_{\mu\nu}F^{\ast{\mu\nu}}$ equal to zero. The electromagnetic field depends on arbitrary function of $(z-t)$. Skewon part of the solution describes a wave in this sense that it depends on $(z-t)$ (a Riemann invariant, see Ref. [46]). It depends also on an arbitrary function of $(z-t)$. Travelling in higher dimensions can also be considered.

Recently the LIGO and Virgo collaborations achieved a three-detector observation of gravitational waves from a binary black hole coalescence (see Ref. [47]). This is the first case of gravitational waves detection with three-detector observation.

Appendix A

In this appendix we give a programme written in Mathematica (kalinowski_1044pr1w2.nb) to calculate $\overline{R}_{(\alpha\beta)}$, $\overline{R}_{[\alpha\beta]}$, $\overset{\rm em}{T}_{\!(\alpha\beta)}$, $\overset{\rm em}{T}_{\![\alpha\beta]}\frac{}{}$, $W^{\mu\nu}$ for axially-symmetric, stationary case.

x[1] = x x[2] = y x[3] = z x[4] = t1 r2[x,y]=f2[x,y]^(1/2) e2[x,y]=D[w1[x,y],x[1]]*Exp[-n1[x,y]]*f2[x,y]^(1/2) e3[x,y]=D[w1[x,y],x[2]]*Exp[-n1[x,y]]f2[x,y]^(1/2) r1[x, y]=xf2[x,y]^(-1/2) tt = Table[0, {i, 1, 4}, {j, 1, 4}] tt[[1, 1]] = -1 tt[[2, 2]] = -1 tt[[3, 3]] =-r1[x, y]^2 tt[[4, 4]] = r2[x, y]^2 tt[[1, 4]] =b2[x,y]*f2[x,y]^(1/2) tt[[4, 1]] =-b2[x,y]f2[x,y]^(1/2) n1[x,y]=m1[x,y]-1/2Log[f2[x,y]] ff = Table[0, {i, 1, 4}, {j, 1, 4}] ff[[2, 4]] = D[a3[x, y], x[2]]*Exp[-n1[x, y]]*f2[x,y]^(1/2) ff[[4, 1]] = - D[a3[x, y], x[1]]*Exp[-n1[x, y]]*f2[x,y]^(1/2) ff[[4, 2]] = - D[a3[x, y], x[2]]*Exp[-n1[x, y]]*f2[x,y]^(1/2) ff[[1, 4]] = D[a3[x, y], x[1]]*Exp[-n1[x, y]]f2[x,y]^(1/2) MatrixForm[tt] MatrixForm[ff] h1h = Table[0, {i, 1, 4}, {j, 1, 4}, {k, 1, 4}] h1h[[1, 2, 1]] = (D[n1[x, y], x[2]] - 1/2D[f2[x, y], x[2]]*f2[x, y]^(-1))*Exp[-n1[x, y]]f2[x,y]^(1/2) h1h[[1, 1, 2]] = -(D[n1[x, y], x[2]] - 1/2D[f2[x, y], x[2]]*f2[x, y]^(-1))*Exp[-n1[x, y]]f2[x,y]^(1/2) h1h[[2, 1, 2]] = (D[n1[x, y], x[1]] - 1/2D[f2[x, y], x[1]]*f2[x, y]^(-1))*Exp[-n1[x, y]]f2[x,y]^(1/2) h1h[[2, 2, 1]] = -(D[n1[x, y], x[1]] - 1/2D[f2[x, y], x[1]]*f2[x, y]^(-1))*Exp[-n1[x, y]]f2[x,y]^(1/2) h1h[[4,1,3]]=-e2[x,y] h1h[[4,3,1]]=e2[x,y] h1h[[4,2,3]]=-e3[x,y] h1h[[4,3,2]]=e3[x,y] gam = Table[0, {i, 1, 4}, {j, 1, 4}, {k, 1, 4}] gam[[1, 2, 2]] = -(D[n1[x, y], x[1]] - 1/2D[f2[x, y], x[1]]*f2[x, y]^(-1))*Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[2, 1, 1]] = -(D[n1[x, y], x[2]] - 1/2D[f2[x, y], x[2]]*f2[x, y]^(-1))*Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[1, 2, 1]] =(D[n1[x, y], x[2]] - 1/2D[f2[x, y], x[2]]*f2[x, y]^(-1))*Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[2, 1, 2]] =(D[n1[x, y], x[1]] - 1/2D[f2[x, y], x[1]]*f2[x, y]^(-1))*Exp[-n1[x, y]]*f2[x,y]^(1/2) gam[[1, 3, 3]] =- r1[x, y]*D[r1[x, y], x[1]]*Exp[-n1[x, y]]*f2[x,y]^(1/2) gam[[2, 3, 3]] = -r1[x, y]*D[r1[x, y], x[2]]*Exp[-n1[x, y]]*f2[x,y]^(1/2) gam[[1, 4, 4]] =r2[x, y]*D[r2[x, y], x[1]]Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[2, 4, 4]] = r2[x, y]D[r2[x, y], x[2]]Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[3, 1, 3]] = 1/r1[x, y]D[r1[x, y], x[1]]Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[3, 3, 1]] = 1/r1[x, y]D[r1[x, y], x[1]]Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[4, 1, 4]] = 1/r2[x, y]D[r2[x, y], x[1]]Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[4, 4, 1]] = 1/r2[x, y]D[r2[x, y], x[1]]Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[3, 2, 3]] = 1/r1[x, y]D[r1[x, y], x[2]]Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[3, 3, 2]] = 1/r1[x, y]D[r1[x, y], x[2]]Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[4, 4, 2]] = 1/r2[x, y]D[r2[x, y], x[2]]Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[4, 2, 4]] = 1/r2[x, y]D[r2[x, y], x[2]]Exp[-n1[x, y]]f2[x,y]^(1/2) gam[[4, 1, 3]] =-1/2 e2[x, y] gam[[4, 3, 1]] =1/2e2[x, y] gam[[4, 3, 2]] =1/2e3[x, y] gam[[4, 2, 3]] =-1/2e3[x, y] gam[[3, 4, 1]] =1/2r2[x, y]^2r1[x, y]^(-2) e2[x, y] gam[[3, 1, 4]] =1/2r2[x, y]^2r1[x, y]^(-2) e2[x, y] gam[[3, 4, 2]] =1/2r2[x, y]^2r1[x, y]^(-2) e3[x, y] gam[[3, 2, 4]] =1/2r2[x, y]^2r1[x, y]^(-2) e3[x, y] gam[[1, 3, 4]] =-1/2r2[x, y]^2 e2[x, y] gam[[1, 4, 3]] =-1/2r2[x, y]^2 e2[x, y] gam[[2, 3, 4]] =-1/2r2[x, y]^2 e3[x, y] gam[[2, 4, 3]] =-1/2r2[x, y]^2 e3[x, y] tts = FullSimplify[1/2(tt + Transpose[tt])] tti = FullSimplify[Inverse[tt]] ttsi = FullSimplify[Inverse[tts]] sdet = FullSimplify[Det[tt]] sdets = FullSimplify[Det[tts]] tta = FullSimplify[1/2(tt - Transpose[tt])] ttia = FullSimplify[1/2(tti - Transpose[tti])] ccudd = h1h; gsuu = ttsi; gdd = tt; guu = tti; ffdd = ff; MatrixForm[tts] MatrixForm[ttsi] gsdd = 1/2(gdd + Transpose[gdd]); kkdd = 1/2(gdd - Transpose[gdd]); tkkuu[a_, b_] := Sum[Sum[gsuu[[l, a]]*gsuu[[m, b]]kkdd[[l, m]], {m, 1, 4}], {l, 1, 4}]; kkuu = Table[tkkuu[a, b], {a, 1, 4}, {b, 1, 4}]; aguu= 1/2(-Transpose[guu] + guu); tkkddd[a_, b_, c_] := D[kkdd[[b, c]], x[a]] - Sum[(gam[[d, b, a]]*kkdd[[d, c]] - gam[[d, c, a]]*kkdd[[b, d]]), {d, 1, 4}]; kkddd = Table[tkkddd[a, b, c], {a, 1, 4}, {b, 1, 4}, {c, 1, 4}]; takkddd[a_, b_, c_] := -kkddd[[a, b, c]] + kkddd[[c, a, b]] - kkddd[[b, c, a]]; akkddd = Table[takkddd[a, b, c], {a, 1, 4}, {b, 1, 4}, {c, 1, 4}]; takkddu[n_, w_, m_] := Sum[gsuu[[m, p]]*akkddd[[n, w, p]], {p, 1, 4}]; akkddu = Table[takkddu[n, w, m], {n, 1, 4}, {w, 1, 4}, {m, 1, 4}]; takkdud[l_, a_, w_] := Sum[gsuu[[a, p]]*akkddd[[l, p, w]], {p, 1, 4}]; akkdud = Table[takkdud[l, a, w], {l, 1, 4}, {a, 1, 4}, {w, 1, 4}]; tkkdu[m_, a_] := Sum[gsuu[[a, p]]kkdd[[m, p]], {p, 1, 4}]; kkdu = Table[tkkdu[m, a], {m, 1, 4}, {a, 1, 4}]; tgamudd[n_, w_, m_] := gam[[n, w, m]] + 1/2(akkddu[[w, m,n]] - Sum[Sum[kkdu[[m, a]]*akkddd[[w, a, b]]kkuu[[n, b]] - kkdu[[w, a]]akkddd[[m, a, b]]kkuu[[n, b]], {a, 1, 4}], {b, 1, 4}]) + Sum[Sum[gsuu[[n,l]](1/2(akkddu[[l, a, w]]kkdd[[m, a]] + akkddu[[l, m, a]]kkdd[[ w,a]])), {a, 1, 4}], {l, 1, 4}] + Sum[Sum[Sum[Sum[1/2(gsuu[[n, l]](kkdu[[c,b]] (kkdu[[m, c]]*akkddd[[w, a, b]]*kkdu[[l, a]] + kkdu[[w, c]]*akkddd[[m, a, b]]*kkdu[[l, a]] - akkddd[[l, a, b]]*kkdu[[w, a]]*kkdu[[m, c]] - akkddd[[l,a,b]]*kkdu[[m, a]]kkdu[[w, c]]))), {c, 1, 4}], {l, 1, 4}], {a, 1, 4}], {b, 1, 4}]; gamudd = Table[tgamudd[n, w, m], {n, 1, 4}, {w, 1, 4}, {m, 1, 4}]; ruddd[a_, b_, r_, s_] := (-D[gamudd[[a, b, r]], x[s]] + D[gamudd[[a, b, s]], x[r]]) Exp[-n1[x, y]]*f2[x,y]^(1/2) + Sum[(gamudd[[a, c, r]]gamudd[[c, b, s]] - gamudd[[a, c, s]]gamudd[[c, b, r]]), {c, 1, 4}] + Sum[gamudd[[a, b, c]]ccudd[[c, r, s]], {c, 1, 4}]; tabruddd = Table[ruddd[a, b, r, s], {a, 1, 4}, {b, 1, 4}, {r, 1, 4}, {s, 1, 4}]; ricdd[b_, r_] := Sum[tabruddd[[a, b, r, a]] + 1/2tabruddd[[a, a, b, r]], {a, 1, 4}]; tabricdd = Table[ricdd[b, r], {b, 1, 4}, {r, 1, 4}] ; sricdd =Simplify[ 1/2(tabricdd + Transpose[tabricdd])]; aricdd =Simplify[ 1/2(tabricdd - Transpose[tabricdd])]; hhdd[w_, m_] := ffdd[[w, m]] + Sum[Sum[-gsuu[[t, a]]*ffdd[[a, w]]*kkdd[[m, t]] + gsuu[[t, a]]*ffdd[[a, m]]*kkdd[[w, t]], {a, 1, 4}], {t, 1, 4}]; tabhhdd = Table[hhdd[w, m], {w, 1, 4}, {m, 1, 4}]; hhguu[m_, a_] := Sum[Sum[guu[[b, m]]*guu[[c, a]]*tabhhdd[[b, c]], {b, 1, 4}], {c, 1, 4}]; tabhhguu = Table[hhguu[m, a], {m, 1, 4}, {a, 1, 4}] aadd[a_, b_] := Sum[Sum[gdd[[c, b]]tabhhdd[[m, a]]tabhhguu[[m, c]], {m, 1, 4}], {c, 1, 4}]; tabaadd = Table[aadd[a, b], {a, 1, 4}, {b, 1, 4}]; bbdd[a_, b_] := -2Sum[Sum[aguu[[m, c]]ffdd[[m, c]]ffdd[[a, b]], {m, 1, 4}], {c, 1, 4}]; tabbbdd = Table[bbdd[a, b], {a, 1, 4}, {b, 1, 4}]; ccdd[a_, b_] := -1/4gdd[[a, b]]Sum[Sum[tabhhguu[[m, n]]ffdd[[m, n]] - 2(aguu[[m, n]]ffdd[[m, n]])^2, {n, 1, 4}], {m, 1, 4}]; tabccdd = Table[ccdd[a, b], {a, 1, 4}, {b, 1, 4}]; tedd = tabaadd + tabbbdd + tabccdd; tesdd =Simplify[ 1/2(tedd + Transpose[tedd])]; teadd =Simplify[1/2(tedd - Transpose[tedd])]; wwuu[a_, m_] := (-sdet)^(1/2)(tabhhguu[[a, m]] - 2aguu[[a, m]]Sum[Sum[aguu[[n, b]]ffdd[[n, b]], {n, 1, 4}], {b, 1, 4}]); wwu[a_] := Sum[D[wwuu[a, m], x[m]], {m, 1, 4}]; ee=Table[0,{a,1,4},{b,1,4}] ee = sricdd ; ee1=2tesdd; eea=Table[0,{a,1,4},{b,1,4}] eea = aricdd ; eea1=2teadd; vv[a_] := Sum[D[(-sdet)^(1/2)*aguu[[a, b]], x[b]], {b, 1, 4}]; ww = FullSimplify[Table[wwu[a], {a, 1, 4}]] vs = FullSimplify[Table[vv[a], {a, 1, 4}]]

Let us notice the following fact. In our programme to calculate symbolically we use several notions. Moreover, in the text we change a notation for convenience. In this way one gets

[TABLE]

Appendix B

In this appendix we give a programme (kalinowski_24mmkw.nb) written in Mathematica to calculate symbolically all quantities from Section 2 for cylindrical gravito-electromagnetic wave.

x[1] = r1 x[2] = th x[3] = z1 x[4] = t1 n1 = n2[x[1], (x[1] - x[4])] a1 = D[b2[x[1], (x[1] - x[4])], x[1]] s1 = s2[x[1] - x[4]](x[1])^(-1) l1 = l2[x[1], (x[1] - x[4])] b1 = -D[b2[x[1], (x[1] - x[4])], x[4]] tt = Table[0, {i, 1, 4}, {j, 1, 4}] tt[[1, 1]] = -Exp[2(n1 - l1)] tt[[1, 3]] = s1Exp[2l1] tt[[2, 2]] = -(x[1])^2Exp[-2l1] tt[[3, 1]] = -s1Exp[2l1] tt[[3, 3]] = -Exp[2l1] tt[[3, 4]] = s1Exp[2l1] tt[[4, 4]] = Exp[2(n1 - l1)] tt[[4, 3]] = -s1Exp[2l1] ff = Table[0, {i, 1, 4}, {j, 1, 4}] ff[[1, 3]] = a1 ff[[3, 1]] = -a1 ff[[3, 4]] = b1 ff[[4, 3]] = -b1 MatrixForm[tt] MatrixForm[ff] sdet = FullSimplify[Det[tt]] gdd = tt; ffdd = ff; tti = FullSimplify[Inverse[tt]] guu = tti; Table[0, {i, 1, 4}, {j, 1, 4}] tts = FullSimplify[1/2*(tt + Transpose[tt])] MatrixForm[tts] sdets = FullSimplify[Det[tts]] ttsi = FullSimplify[Inverse[tts]] gsuu = ttsi; MatrixForm[ttsi] gsdd =1/2*(gdd + Transpose[gdd]); kkdd =1/2*(gdd - Transpose[gdd]); tkkuu[a_, b_] := Sum[Sum[gsuu[[l, a]]gsuu[[m, b]]kkdd[[l, m]], {m, 1, 4}], {l, 1, 4}]; kkuu = Table[tkkuu[a, b], {a, 1, 4}, {b, 1, 4}]; aguu = 1/2(-Transpose[guu] + guu); tchrudd[c_, b_, m_] := Sum[1/2gsuu[[a, c]]*(D[gsdd[[a, b]], x[m]] + D[gsdd[[a, m]], x[b]] - D[gsdd[[b, m]], x[a]]), {a, 1, 4}]; chrudd = Table[tchrudd[c, b, m], {c, 1, 4}, {b, 1, 4}, {m, 1, 4}]; tkkddd[a_, b_, c_] := D[kkdd[[b, c]], x[a]] - Sum[(chrudd[[d, b, a]]*kkdd[[d, c]] + chrudd[[d, c, a]]*kkdd[[b, d]]), {d, 1, 4}]; kkddd = Table[tkkddd[a, b, c], {a, 1, 4}, {b, 1, 4}, {c, 1, 4}]; takkddd[a_, b_, c_] := -kkddd[[a, b, c]] + kkddd[[c, a, b]] - kkddd[[b, c, a]]; akkddd = Table[takkddd[a, b, c], {a, 1, 4}, {b, 1, 4}, {c, 1, 4}]; takkddu[ w_, m_,n_] := Sum[gsuu[[n, p]]*akkddd[[ w, m,p]], {p, 1, 4}]; akkddu = Table[takkddu[n, w, m], {n, 1, 4}, {w, 1, 4}, {m, 1, 4}]; chrudd = Table[tchrudd[c, b, m], {c, 1, 4}, {b, 1, 4}, {m, 1, 4}]; tkkddd[a_, b_, c_] := D[kkdd[[b, c]], x[a]] - Sum[(chrudd[[d, b, a]]*kkdd[[d, c]] + chrudd[[d, c, a]]*kkdd[[b, d]]), {d, 1, 4}]; kkddd = Table[tkkddd[a, b, c], {a, 1, 4}, {b, 1, 4}, {c, 1, 4}]; takkddd[a_, b_, c_] := -kkddd[[a, b, c]] + kkddd[[c, a, b]] - kkddd[[b, c, a]]; akkddd = Table[takkddd[a, b, c], {a, 1, 4}, {b, 1, 4}, {c, 1, 4}]; takkddu[ w_, m_,n_] := Sum[gsuu[[n, p]]*akkddd[[ w, m,p]], {p, 1, 4}]; akkddu = Table[takkddu[n, w, m], {n, 1, 4}, {w, 1, 4}, {m, 1, 4}]; takkdud[l_, a_, w_] := Sum[gsuu[[a, p]]*akkddd[[l, p, w]], {p, 1, 4}]; akkdud = Table[takkdud[l, a, w], {l, 1, 4}, {a, 1, 4}, {w, 1, 4}]; tkkdu[m_, a_] := FullSimplify[Sum[gsuu[[a, p]]kkdd[[m, p]], {p, 1, 4}]]; kkdu = Table[tkkdu[m, a], {m, 1, 4}, {a, 1, 4}]; tgamudd[n_, w_, m_] := chrudd[[n, w, m]] + 1/2(akkddu[[w, m,n]] - Sum[Sum[kkdu[[m, a]]*akkddd[[w, a, b]]kkuu[[n, b]] - kkdu[[w, a]]akkddd[[m, a, b]]kkuu[[n, b]], {a, 1, 4}], {b, 1, 4}]) + Sum[Sum[gsuu[[n, l]](1/2(akkddu[[l, a, w]]kkdd[[m, a]] + akkddu[[l, m, a]]kkdd[[ w,a]])), {a, 1, 4}], {l, 1, 4}] + Sum[Sum[Sum[Sum[1/2(gsuu[[n, l]](kkdu[[c, b]](kkdu[[m, c]]*akkddd[[w, a, b]]*kkdu[[l, a]] + kkdu[[w, c]]*akkddd[[m, a, b]]*kkdu[[l, a]] - akkddd[[l, a, b]]*kkdu[[w, a]]kkdu[[m, c]] - akkddd[[l,a,b]] kkdu[[m, a]]kkdu[[w, c]]))), {c, 1, 4}], {l, 1, 4}], {a, 1, 4}], {b, 1, 4}]; gamudd = Table[tgamudd[n, w, m], {n, 1, 4}, {w, 1, 4}, {m, 1, 4}]; truddd[a_, b_, r_, s_] := FullSimplify[-D[gamudd[[a, b, r]], x[s]] + D[gamudd[[a, b, s]], x[r]] + Sum[(gamudd[[a, c, r]]gamudd[[c, b, s]] - gamudd[[a, c, s]]gamudd[[c, b, r]]), {c, 1, 4}]]; tabruddd= Table[truddd[a, b, r, s], {a, 1, 4}, {b, 1, 4}, {r, 1, 4}, {s, 1, 4}]; ricdd[b_, r_] := Sum[tabruddd[[a, b, r, a]] + 1/2tabruddd[[a, a, b, r]], {a, 1, 4}]; tabricdd = Table[ricdd[b, r], {b, 1, 4}, {r, 1, 4}] ; sricdd =Simplify[ 1/2(tabricdd + Transpose[tabricdd])]; aricdd =Simplify[ 1/2(tabricdd - Transpose[tabricdd])]; hhdd[w_, m_] := ffdd[[w, m]] + Sum[Sum[-gsuu[[t, a]]*ffdd[[a, w]]*kkdd[[m, t]] + gsuu[[t, a]]*ffdd[[a, m]]*kkdd[[w, t]], {a, 1, 4}], {t, 1, 4}]; tabhhdd = Table[hhdd[w, m], {w, 1, 4}, {m, 1, 4}]; hhguu[m_, a_] := Sum[Sum[guu[[b, m]]*guu[[c, a]]tabhhdd[[b, c]], {b, 1, 4}], {c, 1, 4}]; tabhhguu = Table[hhguu[m, a], {m, 1, 4}, {a, 1, 4}]; aricdd =Simplify[ 1/2(tabricdd - Transpose[tabricdd])]; hhdd[w_, m_] := ffdd[[w, m]] + Sum[Sum[-gsuu[[t, a]]*ffdd[[a, w]]*kkdd[[m, t]] + gsuu[[t, a]]*ffdd[[a, m]]*kkdd[[w, t]], {a, 1, 4}], {t, 1, 4}]; tabhhdd = Table[hhdd[w, m], {w, 1, 4}, {m, 1, 4}]; hhguu[m_, a_] := Sum[Sum[guu[[b, m]]guu[[c, a]]tabhhdd[[b, c]], {b, 1, 4}], {c, 1, 4}]; tabhhguu = Table[hhguu[m, a], {m, 1, 4}, {a, 1, 4}] aadd[a_, b_] := Sum[Sum[gdd[[c, b]]tabhhdd[[m, a]]tabhhguu[[m, c]], {m, 1, 4}], {c, 1, 4}]; tabaadd = Table[aadd[a, b], {a, 1, 4}, {b, 1, 4}]; bbdd[a_, b_] := -2 Sum[Sum[aguu[[m, c]]ffdd[[m, c]]ffdd[[a, b]], {m, 1, 4}], {c, 1, 4}]; tabbbdd = Table[bbdd[a, b], {a, 1, 4}, {b, 1, 4}]; ccdd[a_, b_] := -1/4gdd[[a, b]]Sum[Sum[tabhhguu[[m, n]]ffdd[[m, n]] - 2(aguu[[m, n]]ffdd[[m, n]])^2, {n, 1, 4}], {m, 1, 4}]; tabccdd = Table[ccdd[a, b], {a, 1, 4}, {b, 1, 4}]; tedd = tabaadd + tabbbdd + tabccdd; tesdd = Simplify[ 1/2(tedd + Transpose[tedd])]; teadd =Simplify[ 1/2(tedd - Transpose[tedd])]; wwuu[a_, m_] :=FullSimplify[(-sdet)^(1/2)(tabhhguu[[a, m]] - 2aguu[[a, m]] Sum[Sum[aguu[[n, b]]ffdd[[n, b]], {n, 1, 4}], {b, 1, 4}])]; wwu[a_] := Sum[D[wwuu[a, m], x[m]], {m, 1, 4}]; ee=Table[0,{a,1,4},{b,1,4}] ee = sricdd ; ee1=2tesdd; eea=Table[0,{a,1,4},{b,1,4}] eea = aricdd ; eea1=2teadd; vv[a_] := FullSimplify[ Sum[D[(-sdet)^(1/2)*aguu[[a, b]], x[b]], {b, 1, 4}]]; ww = Table[wwu[a], {a, 1, 4}] vs = Table[vv[a], {a, 1, 4}]

Here also we change a notation in the text for convenience:

[TABLE]

Acknowledgement

I would like to thank Professor B. Lesyng for the opportunity to carry out computations using Mathematica™ 9111Mathematica™ is the registered mark of Wolfram Co. in the Centre of Excellence BioExploratorium, Faculty of Physics, University of Warsaw, Poland.