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abstract: 'We investigate the behavior of small perturbations around the Kaluza-Klein monopole in the five dimensional space-time. We find that the even parity gravitational wave does not propagate in the five dimensional space-time with Kaluza-Klein monopole provided that the gravitational wave is constant in the fifth direction. We conclude that a gravitational wave and a U(1) magnetic monopole do not coexist in five dimensional Kaluza-Klein spacetime.'
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=-20mm =-20mm

**CAN A GRAVITATIONAL WAVE AND A MAGNETIC**

**MONOPOLE COEXIST?**

.5cm

Re: MPLA/358/97

OSAMU ABE[^1] and OSAMU TABATA[^2]

*Physics Laboratory, Asahikawa Campus*

*Hokkaido University of Education, 9 Hokumoncho*

*Asahikawa 070, Japan*

Dirac [@dirac] proposed the theory of the magnetic monopole in 1931. Dirac’s conjecture of monopole is attractive because it can explain the quantization of electric charge of elementary particles. In 1974, ’t Hooft [@thooft] and Polyakov [@polyakov] independently found that a certain class of non-abelian gauge theories not only allow but also demand the existence of superheavy magnetic monopoles. Then most of all grand unified theories demand ’t Hooft-Polyakov monopoles with mass comparable to the unification mass which is typically $10^{15-17}{\rm GeV}$.

Thus, we may expect that superheavy magnetic monopoles could have been generated just after the big bang, the birth of our universe and that there must be monopoles in the cosmic rays. Many researches have been being made for finding magnetic monopoles [@giacomelli], but no good candidate has been found so far except for the observation which was made in 1982 [@cabrera].

The simpler grand unified theories requires too many magnetic monopoles, and contradict above experimental results. Guth [@guth] gave a nice mechanism, which is called as the inflationary universe scenario, that might suppress ’t Hooft-Polyakov monopole production.

In the present paper, we would like to discuss a new mechanism that denies the existence of Dirac’s U(1) magnetic monopoles in connection with a gravitational wave. Taylor [@taylor] has shown an indirect verification of the existence of gravitational radiation. Though there is no direct evidence, the existence of the gravitational wave is beyond doubt.

Our discussion is based on the Kaluza-Klein theory [@k-k]. Originally, Kaluza and Klein suggested the possibility that gravitation and electromagnetism could be unified in a theory in the framework of five-dimensional space-time. DeWitt [@dewitt] and others [@others] presented its generalization to non-abelian gauge group in 4+d dimensional space-time. Gross and Perry [@gross] and Sorkin [@sorkin] found the existence of the regular, static and topologically stable magnetic monopoles solution of the five dimensional Einstein equation. The existence of the magnetic monopole strongly indicates that the fundamental interactions are unified. Sundaresan and Tanaka [@tanaka] discussed the stability of the of the K-K monopole against the even parity perturbation with $l =m=0$ in the positive definite metric, i.e. Euclidean space-time. In Ref. [@abe1], one of present authors discussed the existence of gravitational wave in indefinite metric without the magnetic monopole. One of present authors studied [@abe2] the odd parity perturbations around the magnetic monopole solution, and concluded that there is no parity odd gravitational wave.

In this paper, we will investigate the propagation of the parity even perturbations around the K-K monopole solution in the indefinite metric, i.e. Minkowskian space-time.

We start with a theory of gravitation described by the Einstein-Hilbert action in five dimensions S=[-116G\_5]{}d\^5xR\_5, \[einstein\] where $G_5$ is the five dimensional gravitational constant, $g_5$ is the determinant of five dimensional metric $g_{AB}$ and $R_5$ denotes the five dimensional curvature scalar of the space-time.

The magnetic monopole solution is the solution of the Einstein equation in the empty space-time R\_[AB]{} = 0. We take the metric $g_{AB}$ as [@gross] ds\^2=g\_[AB]{}dx\^Adx\^B, x\^A=(t,r,,,x\^5), where g\_[AB]{}=() \[metric\] Here, V\^[-1]{}(r) = 1 + [4Mr]{}, s = . The gauge field, $A_\mu$, is that of monopole at the origin A\_(x) = (A\_t, A\_r, A\_, A\_=A) =(0,0,0,4M()), where () =, and $A_\mu$ satisfies =[****]{} = [4M[**r**]{}r\^3]{}. In order to avoid the so-called NUT singularity, $M$ should satisfy [@gross] M = [2e]{}, where $G$ is the four dimensional gravitational constant and $e$ denotes the unit electric charge. Then the magnetic charge $g$ of the monopole is given by [@gross] g = [4M]{} =[12e]{}. Thus the monopole has one unit of Dirac charge.

Now, we consider small perturbations around the monopole solution. Then the total metric $g^T_{AB}$ is given by g\^T\_[AB]{} = g\_[AB]{} + h\_[AB]{}. According to the approach of Regge and Wheeler [@regge], we can divide small perturbation into even and odd parity: & &h\_[AB]{}\^[even]{}(t,r,,, x\^5)= & & () & & Y\_[qlm]{}(,)\^[-it]{}\^[in\^5x\^5/R]{} \[even\] and & &h\_[AB]{}\^[odd]{}(t,r,,,x\^5)=& & () & & Y\_[qlm]{}(,)\^[-it]{}\^[in\^5x\^5/R]{}. \[odd\] Here, $R$ is the radius of the circle in the fifth dimension and we have used \_[AB]{} = [g\_[AB]{}r\^2]{}, \^3 -g\^[3A]{}e\_[012A5]{} = -[1s]{}, and \^2 -g\^[2A]{}e\_[013A5]{} = s, where $e_{ABCDE}$’s are totally antisymmetric Levi-Chivita symbols in five dimensions. In Eqs. (\[even\]) and (\[odd\]), $Y_{qlm}$ denotes the monopole harmonics [@wu] which is introduced so as to avoid the singularity at $\theta =\pi$ and which satisfies following eigen value equations [@dray]. \^2Y\_[qlm]{} &&{-[1s]{}\_s\_ -[1s\^2]{}\_\^2-[2q()L\_zs\^2]{} }Y\_[qlm]{} =l (l +1)Y\_[qlm]{}, L\_zY\_[qlm]{}&&{ -i\_-q()-q} Y\_[qlm]{}=mY\_[qlm]{}. The monopole harmonics $Y_{qlm}$ coincides with the usual spherical harmonics $Y_{l m}$ when $q=0$. The number $q$ is defined by $q=eg$ and which equals to ${1\over 2}$ in the case of K-K monopole.

We will consider small perturbations with $n^5=0$ that is we consider only perturbations which are constant in the fifth direction. We impose the gauge conditions or transversality conditions on the small perturbations as follows \_Bh\^[AB]{} = 0, where $h^{AB}=-g^{AC}g^{BD}h_{CD}$. We derive equations for small perturbation $h$ as follows. The Ricci tensor will be $R_{AB}$ if it is calculated from $g_{AB}$ and $R_{AB}+\delta R_{AB}$ if it is calculated from $g_{AB}+h_{AB}$. We can get second order equations on the perturbation $h$ if we impose the condition $\delta R_{AB}=0$. This condition implies that the perturbed space is also empty space. The explicit form of the equations are R\_[AB]{}=[[[R\_A]{}\^C]{}\_B]{}[\^D]{}h\_[CD]{}+[2]{}h\_[AB]{} = 0.

Now, we will investigate whether the even parity perturbations can propagate in the five dimensional space-time with K-K monopole provided that the perturbations are constant in the fifth direction. First, we consider $(0,0)$ component of $\delta R$. We have 0&=&2R\_[00]{}&=&-\^[it]{}V Y\_[qlm]{} ( H\_0”+[2r]{}H\_0’-[l (l +1)r\^2]{}H\_0 +[\^2V]{}H\_0-[2qms\^2r\^2]{}H\_0). \[R00\] We can separate $\alpha$ dependent part from $\alpha$ independent parts using the identities \_0\^d=0,\_0\^d =-2. Thus, we can conclude $H_0\equiv 0$ from Eq. (\[R00\]).

Next, we consider $(0,5)$ component of the Ricci tensor 0&=&2R\_[05]{}=& &-V\^[-it]{} { Y\_[qlm]{}& &-[2qms\^2r\^2]{} Y\_[qlm]{}H\_[0p]{} +[(V-1)\^2sr\^2]{} \_Y\_[qlm]{}H\_[0p]{}}. \[R05\] If the functions $Y_{qlm}$, $Y_{qlm}\alpha/s^2$, and $\pd_\theta Y_{qlm}/s$ are linearly independent, i.e. the Wronskian is not vanishing, each coefficient must vanish. We can prove that above functions are linearly independent. Thus we can conclude that $H_{0p}\equiv 0$ from Eq. (\[R05\]).

Further, we consider $(1,5)$ component. That is 0&=&2R\_[15]{}=&-&V\^[-it]{}{Y\_[qlm]{}&+&[2qms\^2 r\^2]{}Y\_[qlm]{}-[4MV(V-1)sr\^3]{}\_Y\_[qlm]{}H\_[1p]{}&+&[4M(V+1)s\^2r\^3]{}\_Y\_[qlm]{}H\_[2p]{} } Analogous argument for Eq. (\[R05\]) is applicable even in this case. That is $\pd_\varphi Y_{qlm}$ is linearly independent in addition to three functions those are appeared in Eq. (\[R05\]). Thus we have $H_{1p}\equiv 0$, $H_{2p}\equiv 0$ and then $H_5\equiv 0$.

The transversality condition for 0-th component becomes 0=-\_Ah\^[A0]{}=[V\^[-it]{}r\^2]{}{ Y\_[qlm]{}+[2qms\^2]{}Y\_[qlm]{}H\_3}. From which we obtain $H_3\equiv 0$ and $H_1 = {\rm const.}/r^2$.

We can solve $\delta R_{02}=0$ for $H_1$. 0=2R\_[02]{}=-[V(V+1)\^[-it]{} r]{}\_ Y\_[qlm]{}H\_1, and have $H_1\equiv 0$.

Next, we consider the traceless conditon. Because $H_0=0$, $H_5=0$, and $H_{2p}=0$, the traceless condition becomes 0&=&[Tr]{}h\^[even]{}=&-&[\^[-it]{}s\^2 r\^3 ]{}{ s\^2rY\_[qlm]{}-16M\^2rV\^2\^2 Y\_[qlm]{}K&-&2rVqmY\_[qlm]{}L +4V\^2(V-1)Ms\_Y\_[qlm]{}L }. \[trh\] The same argument as is used for Eq. (\[R05\]) is applicable in this case. Thus, we obtain $L\equiv 0$, $K\equiv 0$, and $H_2\equiv 0$.

Finally, we consider the transversality condition for 1-st component. 0=\_Ah\^[A1]{}=[V\^2\^[-it]{}r\^2]{}Y\_[qlm]{}H\_4. Thus, we have $H_4\equiv 0$. We have proven that the small perturbations which are appeared in Eq. (\[even\]) are all vanishing.

In this paper, we have studied the properties of the small perturbations around the five dimensional Kaluza-Klein metric with magnetic monopole. We found that there is no even parity perturbation provided that the perturbation is independent of the fifth coordinate. Combining our result with the result in Ref. [@abe2], we can conclude that a gravitational wave can not coexist with Kaluza-Klein magnetic monopole in the five dimensional space-time. If we believe the existence of the gravitational wave, our result implies the absence of cllasical magnetic monopole. [**Acknowledgements**]{} .5cm One of present authors(O.A.) acknowledges useful discussions with Prof. K. Tanaka and Prof. M. K. Sundaresan. He also would like to thank the Ohio State University, Department of Physics for their hospitality, where this work was started.

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[^1]: e-mail address: osamu@asa.hokkyodai.ac.jp

[^2]: e-mail address: x3920@atson.asa.hokkyodai.ac.jp
