Arbitrarily negative energy for small Kaluza-Klein bubbles

TL;DR

The paper constructs explicit time-symmetric Kaluza-Klein initial data for bubbles of nothing whose ADM energy is unbounded from below even when the asymptotic circle size and the minimal \(S^2\) bubble size are fixed. This suggests the standard Kaluza-Klein vacuum is more unstable than previously thought, while the analogous asymptotically AdS initial data does not show arbitrarily negative energy.

Key findings

• For a family of Kaluza-Klein bubble initial data with fixed bubble size \(\rho_0\) and fixed asymptotic circle radius \(R\), the ADM energy becomes arbitrarily negative as a parameter \(p\) grows: \(E = -p^3/(2^9 R^2) + O(p^2)\) (Eq. 27).
• The result holds for very small bubbles, indicating that the standard Kaluza-Klein vacuum is more unstable than previously thought.
• The construction satisfies the Brill-Pfister bound \(E > -2B\); the paper reports that both \(E\) and \(B\) are \(O(p^3)\) and checks \(E > -2B\) numerically.
• In the asymptotically AdS analog, the energy does not become arbitrarily negative; for parameters \(R=\ell=G_5=d=1\), \(\rho_0=1/4\), \(C_0=-7/24\), the numerical plot shows the energy remains bounded, consistent with AdS/CFT.

Methods

• The paper studies time-symmetric initial data with \(SO(3)\times U(1)\) symmetry, metric \(ds^2 = f_1(\rho)d\rho^2 + \rho^2 d\Omega^2 + f_2(\rho)d\phi^2\), with \(f_1=(\alpha\beta)^{-1}\), \(f_2=\alpha\), and solves the Hamiltonian constraint \((4)R=0\) for \(\beta\) given \(\alpha\).
• It derives a general ADM energy expression (Eq. 13) and chooses an explicit \(\alpha(\rho)=\frac{\rho-\rho_0}{\rho-\rho_0+d}\exp(-p/\rho)\) with \(d>0\), \(p\ge 0\), then analyzes the large-\(p\) limit analytically and uses numerical integration for \(I_2\).
• For the AdS case, it uses analogous time-symmetric data with negative cosmological constant, derives the energy expression (Eq. 40), and evaluates the integrals numerically for representative parameters.

Limitations

• The paper states that it does not have the corresponding instantons for the new small zero-energy solutions, that it remains an open question whether any time-symmetric Kaluza-Klein bubble with negative energy always starts out expanding, and that it has no proof that the AdS energy is bounded from below—only trials of other scenarios and functions.
• The paper notes that in the main Kaluza-Klein example the curvature at the bubble grows exponentially in \(|E|^{1/3}\), so strong quantum gravity corrections may impose a lower bound once curvature exceeds the Planck scale; Appendix B gives another family with only polynomial curvature growth.
• Evident from the text: the AdS conclusion relies on numerical integration and specific parameter choices rather than an analytic proof, and the main unbounded-energy family has rapidly growing curvature, limiting its direct classical reliability.