Nonlinear Perturbations of the Kaluza-Klein Monopole

We consider the nonlinear stability of the Kaluza-Klein monopole viewed as the static solution of the five-dimensional vacuum Einstein equations. Using both numerical and analytical methods, we give evidence that the Kaluza-Klein monopole is asymptotically stable within the cohomogeneity-two biaxial Bianchi type-IX ansatz recently introduced by Bizoń, Chmaj, and Schmidt [Phys. Rev. Lett. 95, 071102 (2005)]. We also show that for sufficiently large perturbations the Kaluza-Klein monopole loses stability and collapses to a Kaluza-Klein black hole. The relevance of our results for the stability of Bogomol’nyi-Prasad-Sommerfield states in M or string theory is briefly discussed.

Figure 1

Asymptotic stability of the Kaluza-Klein monopole. For initial data (13) with a small amplitude ($p=0.1$, $R=3$, $s=1$), we plot a series of snapshots of the function $B(t,r)$ (where $t$ is central proper time). The dashed line shows the unperturbed KK monopole. During the evolution, the excess energy of the perturbation is clearly seen to be radiated away to infinity and the solution returns to equilibrium.

Figure 2

The convergence to the Kaluza-Klein monopole. For the same initial data as in Fig. 1, we plot the time series $\ln |P(t,r_0)|$ at $r_0=2$. The asymptotic power-law tail is seen at late times.

Figure 3

Instability of the Kaluza-Klein monopole for large perturbations. For initial data (13) with a large amplitude ($p=0.3$, $R=3$, $s=1$), we plot a series of snapshots of the function $A(t,r)$ (where $t$ is central proper time). During the evolution, $A(t,r)$ drops to zero at $r=r_H\approx 1.68$, which signals the formation of an apparent horizon there. Outside the horizon, the solution relaxes to the Kaluza-Klein black hole (15) (shown by the dashed line) with $M=0.502$.

Figure 4

Critical behavior. For supercritical solutions, we plot the horizon radius vs the distance to the threshold on the log-log scale. The fit of the power law $\ln (r_H)=(\gamma /2)\ln (p-p^{*})+\mathrm{const}$ yields $\gamma \approx 0.33$, which agrees (up to numerical errors) with the critical exponent obtained in Ref. 3. The echoing period $\Delta \approx 0.47$ (read off from the period of wiggles around the linear fit above and computed independently from the spatial shift between the echoes) is also the same.