Orthonormal Frame and SO(3) Kaluza-Klein Dyon

TL;DR

This paper presents an alternative method using orthonormal frames and Cartan's structure equations to derive an SO(3) Kaluza-Klein dyon solution, confirming previous results obtained via Einstein's equations.

Contribution

It introduces a new approach employing orthonormal frames and Cartan's structure equations to analyze the Kaluza-Klein dyon, validating earlier findings with a different mathematical framework.

Findings

Both methods yield identical solutions.

The orthonormal frame approach is effective for this problem.

Results confirm the consistency of the Kaluza-Klein dyon solution.

Abstract

In previous paper, we present an SO(3) Wu-Yang-like Kaluza-Klein dyon so- lution satisfies the Einstein equation in the seven-dimensional spacetimes. In this note, we will show an alternative approach using an orthonormal frame, the Cartan's structure equations, and calculating the affine spin connection one-form,curvature tensor and Ricci tensor. The results from these two different methods are coincident.

Equations

$$d{\bar{s}}^2$$

$$=$$

$$d{\bar{s}}^2=$$

$$L_a\equiv -i{\zeta }_a^m{\partial }_m,$$

$$[L_a,L_b]=i{ϵ}_{abc}{L}_c,$$

$${\zeta }_a^m{\partial }_m{\zeta }_b^n-{\zeta }_b^m{\partial }_m{\zeta }_a^n=-{ϵ}_{abc}{\zeta }_c^n.$$

$$B_{\mu }^m={\zeta }_a^m{A}_{\mu }^a,$$

$${\tilde{\mathcal{F}}}_{\mu \nu }^m$$

$$=$$

$$F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+{ϵ}_{abc}{A}_{\mu }^b{A}_{\nu }^c.$$

$${\zeta }_a^5$$

$${\zeta }_a^6$$

$${\zeta }_a^7$$

$$A_0^a$$

$$A_2^a$$

$$F_{01}^a=\frac{1}{r^2}{\hat{r}}^a,$$

$$F_{23}^a=-\sin \theta {\hat{r}}^a.$$

$$B_{\mu }^5$$

$$B_0^6$$

$$B_2^6$$

$$B_3^6$$

$${\tilde{\mathcal{F}}}_{01}^6$$

$${\tilde{\mathcal{F}}}_{23}^6$$

$${\bar{g}}_{AB}={\eta }_{\bar{a}\bar{b}}{e}_A^{\bar{a}}{e}_B^{\bar{b}},\bar{a},\bar{b}=0,1,2,3,5,6,7,$$

$${\eta }_{\bar{a}\bar{b}}=diag(-1,1,1,1,1,1,1,)$$

$$E_{\bar{a}}^A={\eta }_{\bar{a}\bar{b}}{\bar{g}}^{AB}{e}_B^{\bar{b}}$$

$$E_{\bar{a}}^A{e}_A^{\bar{b}}={\delta }_{\bar{a}}^{\bar{b}},{\eta }^{\bar{a}\bar{b}}{E}_{\bar{a}}^A{E}_{\bar{b}}^B={\bar{g}}^{AB}.$$

$$e^{\bar{a}}=e_A^{\bar{a}}d{\bar{x}}^A.$$

$$e^0=e^{\Psi }dt,e^1=e^{\Lambda }dr,e^2=rd\theta ,e^3=rsin\theta d\varphi ,$$

$$e^5=dR,e^6=R(d\Theta +B_{\mu }^{6}d{x}^{\mu }),e^7=Rsin\Theta (d\Phi +B_{\mu }^{7}d{x}^{\mu }).$$

$$d{e}^{\bar{a}}+{\omega }_{\bar{b}}^{\bar{a}}\wedge e^{\bar{b}}=0$$

$${\omega }_{\bar{a}\bar{b}}=-{\omega }_{\bar{b}\bar{a}}.$$

$$R_{\bar{b}}^{\bar{a}}=d{\omega }_{\bar{b}}^{\bar{a}}+{\omega }_{\bar{c}}^{\bar{a}}\wedge {\omega }_{\bar{b}}^{\bar{c}}={\bar{R}}_{\bar{b}\bar{c}\bar{d}}^{\bar{a}}{e}^{\bar{c}}\wedge e^{\bar{d}}.$$

$${\bar{R}}_{\bar{a}\bar{b}\bar{c}\bar{d}}=-{\bar{R}}_{\bar{b}\bar{a}\bar{c}\bar{d}}=-{\bar{R}}_{\bar{a}\bar{b}\bar{d}\bar{c}}={\bar{R}}_{\bar{c}\bar{d}\bar{a}\bar{b}}.$$

$${\bar{R}}_{\bar{a}\bar{b}\bar{c}\bar{d}}+{\bar{R}}_{\bar{a}\bar{c}\bar{d}\bar{b}}+{\bar{R}}_{\bar{a}\bar{d}\bar{b}\bar{c}}=0.$$

Full text

PACS numbers: 04.50.Cd

1 Introduction

In previous paper,[1] we have shown that a new SO(3) Wu-Yang-like [2] seven-dimensional Kaluza-Klein(KK) dyon solution satisfies the Einstein equation, by calculating the Christoffel symbols and the Ricci tensor. In this note, we will show an alternative approach [3] using an orthonormal frame, the Cartan’s structure equations and calculating the affine spin connection one-form, curvature tensor and Ricci tensor.

Now consider the Kaluza-Klein theory [4] in a $(4+N)$-manifold $M$ with a metric $\bar{g}_{AB}(x)$ on $M$ in local coordinates $\bar{x}^{A}$. The line element is

[TABLE]

where $x$ parametrizes four-dimensional spacetimes, $y$ parametrizes extra dimensions. We use $A,B,C...$ indices to represent the total spacetimes; $\mu,\nu,\rho...$ to represent the four-dimensional spacetimes; $m,n,l...$ to represent the extra dimensions. $g_{\mu\nu}$ is only a function of $x$, and $\gamma_{mn}$ is only a function of $y$.

The ans$\ddot{a}$tz [1] of the Kaluza-Klein dyonic metric admitting $SO(3)$ Killing vectors is

[TABLE]

$r,\theta,\phi$ are three coordinates of the ordinary three-dimensional spherical coordinate system, $(r,\theta,\phi)=(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})=(x^{1},x^{2},x^{3})$. $R,\Theta,\Phi$ are another three coordinates of the spherical coordinate system in the extra dimensions, $(R,\Theta,\Phi)=(\bar{x}^{5},\bar{x}^{6},\bar{x}^{7})=(y^{5},y^{6},y^{7})$. $\Psi$ and $\Lambda$ are two functions of $r$. We have $g_{00}=-e^{2\Psi}$, $g_{11}=e^{2\Lambda}$, $g_{22}=r^{2}$, $g_{33}=r^{2}\sin^{2}\theta$, $\gamma_{55}=1$, $\gamma_{66}=R^{2}$, $\gamma_{77}=R^{2}\sin^{2}\Theta$.

$B_{\mu}^{m}$ cannot be identified as the Yang-Mills field. To extract the true Yang-Mills field, one has to introduce the Killing vectors

[TABLE]

which generate an $SO(3)$ algebra,

[TABLE]

Inserting $L_{a}$ of (4) into equation (5), one gets the Killing’s equation

[TABLE]

With these Killing vectors, one can define

[TABLE]

where $A_{\mu}^{a}$ is the true Yang-Mills field and $\zeta_{a}^{m}$ is only a function of $y$. Defining

[TABLE]

where $F_{\mu\nu}^{a}$ is the true field strength tensor of the Yang-Mills field,

[TABLE]

It can be checked that the components of $\zeta_{a}^{m}$,

[TABLE]

satisfy the Killing equation (6). The gauge-field components of the Wu-Yang-like KK dyon are

[TABLE]

$A_{1}^{a}$,$A_{2}^{a}$,$A_{3}^{a}$ are just the spherical coordinate representation of the Wu-Yang monopole field in the ordinary gauge theory of four-dimensional spacetimes. The electric field of the KK dyon is

[TABLE]

while the magnetic field is

[TABLE]

The fields $B_{\mu}^{m}$ in (7) can be rewritten as

[TABLE]

The nonzero components of $\widetilde{\mathcal{F}}_{\mu\nu}^{m}$ are

[TABLE]

2 Orthonormal Frame

We now decompose the metric into vielbeins as

[TABLE]

where the $\eta_{\bar{a}\bar{b}}$ is a seven-dimensional flat Minkowski space metric,

[TABLE]

The inverse of $e^{\bar{a}}_{\;A}$ is defined by

[TABLE]

which obeys

[TABLE]

$e^{\bar{a}}_{\;A}$ is the matrix which tranforms the coordinate basis $d\bar{x}^{A}$ of the cotangent space $T^{*}_{x}(M)$ to an orthonormal basis of the same space $T^{*}_{x}(M)$,

[TABLE]

The vielbein basis of the SO(3) KK dyonic metric can be written as

[TABLE]

[TABLE]

The affine spin connection one-form $\omega^{\bar{a}}_{\;\;\bar{b}}$ are introduced by

[TABLE]

and the metricity condition

[TABLE]

The cuvature 2-form is defined as

[TABLE]

Equations (31) and (33) are called Cartan’s structure equations. The components of the curvature tensor have the relations,

[TABLE]

and satisfy the Bianchi identity,

[TABLE]

The components of the affine spin connection one-form $\omega^{\bar{a}}_{\;\;\bar{b}}$ of the $SO(3)$ KK dyon can be written more explicitely as

[TABLE]

The nonzero components of the curvature tensor are

[TABLE]

[TABLE]

We have tacitly omitted nearly one half of the nonzero components of the curvature tensor, because of the relations in (34) , for shorthand.

3 Einstein Equation

Contracting the curvature tensor, one will get the Ricci tensor

[TABLE]

Substituting the components of the dyonic metric,

[TABLE]

where $r_{s}$ is the Schwarzschild radius, then we can obtain almost $\bar{R}_{\bar{a}\bar{b}}$ are zero except

[TABLE]

Then one has the Ricci scalar curvature,

[TABLE]

From the fields $\widetilde{\mathcal{F}}_{\mu\nu}^{m}$ in (22) and (23), the components of the Ricci tensor, $(84)\sim(87)$, can be recast into the form,

[TABLE]

or

[TABLE]

Since the identity, $\gamma_{mn}\widetilde{\mathcal{F}}_{\mu\nu}^{m}\widetilde{\mathcal{F}}^{\mu\nu n}=0$, holds, the right-hand side of the equation (92) can be identified as $8\pi$ times the stress-energy tensor of the Yang-Mills field,

[TABLE]

Then the Einstein equation, $\bar{R}_{AB}-{1\over 2}\bar{g}_{AB}\bar{R}=8\pi\bar{T}_{AB}$, is satisfied, where some components of $\bar{T}_{AB}$ are zero, $\bar{T}_{\mu m}=0$ and $\bar{T}_{mn}=0.$

The $SO(3)$ KK dyonic metric satisfies the Einstein equation in the seven-dimensional. The results from two different methods are coincident. .