Do a gravitational wave and a magnetic monopole coexist?

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. Dirac [1] 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 [2] and Polyakov [3] 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 10GeV. 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 [4], but no good candidate has been found so far except for the observation which was made in 1982 [5]. The simpler grand unified theories requires too many magnetic monopoles, and 1 e-mail address: [email protected] 2 e-mail address: [email protected] Preprint submitted to Elsevier Preprint July 28, 1997 contradict above experimental results. Guth [6] gave a nice mechanism, which is called as the inflationary universe scenario, that might suppress ’t HooftPolyakov 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 [7] 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 [8]. 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 [9] and others [10] presented its generalization to non-abelian gauge group in 4+d dimensional space-time. Gross and Perry [11] and Sorkin [12] 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 [13] 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. [14], one of present authors discussed the existence of gravitational wave in indefinite metric without the magnetic monopole. One of present authors studied [15] 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 = −1 16πG5 ∫ dx √ |g5|R5, (1) where G5 is the five dimensional gravitational constant, g5 is the determinant of five dimensional metric gAB and R5 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 RAB = 0. (2) 2 We take the metric gAB as [11] ds = gABdx dx, x = (t, r, θ, φ, x), (3) where