Collapse of Kaluza-Klein bubbles

TL;DR

The paper analyzes Brill-Horowitz Kaluza-Klein bubble initial data and shows that all negative-energy bubbles start out expanding from the moment of time symmetry, while positive-energy bubbles can start out either expanding or contracting. This makes collapse to a naked singularity unlikely for the negative-energy bubbles.

Key findings

• All negative-energy bubble solutions in the Brill-Horowitz family begin expanding away from the moment of time symmetry; positive-energy bubbles can start expanding or contracting depending on whether \(r_+\) is greater than or less than \(2m\), respectively.
• The total ADM energy is \(m/2\) in units with the reduced 4-dimensional gravitational coupling constant \(4G=1\); the zero-energy Witten bubble expands exponentially and is known in closed form as an analytic continuation of the five-dimensional Schwarzschild solution.
• Because negative-energy bubbles initially expand, it is unlikely that they collapse and produce a naked singularity.
• The vacuum bubble data come from the four-dimensional Euclidean Reissner-Nordström solution; the electrovac data include a magnetic field component \(F_{r\sigma}=-2k/r^3\) and \(U=1-\frac{2m}{r}+\frac{b}{r^2}-\frac{4k^2}{3r^4}\).

Methods

• Uses the three-parameter Brill-Horowitz initial data at a moment of time symmetry for five-dimensional Kaluza-Klein bubbles with spatial topology \(S^2\times R^2\), metric \(ds^2=U\,d\sigma^2+U^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\varphi^2)\), with \(U=1-\frac{2m}{r}+\frac{b}{r^2}-\frac{4k^2}{3r^4}\); the bubble is at the largest root \(U(r_+)=0\), and regularity requires period \(P=4\pi/U'(r_+)\).
• Since the time-dependent solutions are not known in closed form, the paper determines whether the bubble starts collapsing or expanding by the sign of the second proper-time derivative of the bubble area \(A=4\pi g_{\theta\theta}(r_+,t)\) at \(t=0\).
• It uses the geodesic deviation equation for the radial geodesic congruence at the bubble surface, obtaining \(g_{\theta\theta,tt}=-2R_{\theta t\theta t}=2(R_{\theta\theta}-{}^{(4)}R_{\theta\theta})=2(T_{\theta\theta}-\frac13 T g_{\theta\theta}-{}^{(4)}R_{\theta\theta})\), then evaluates this with the vacuum and electrovac data.

Limitations

• The paper states that the analysis provides no insight into “why” the negative-energy bubbles start out expanding, and it is unknown whether the result generalizes beyond the particular Brill-Horowitz initial data considered.
• It does not determine whether a bubble that starts expanding or contracting will continue to do so for all time; it could turn around or bounce. At a later moment of time symmetry, the data need not belong to the Brill-Horowitz family, so the sign of \({}^{(4)}R_{\theta\theta}\) at the bubble is not fixed by the known constraints.
• The time evolution of these data is not known in closed form, so only the initial tendency is established. The authors also note they have not thought carefully about black hole entropy in Kaluza-Klein theory.