Bubbles unbound: bubbles of nothing without Kaluza-Klein

TL;DR

The paper constructs analytic time-symmetric initial data for five-dimensional asymptotically flat “bubbles of nothing” that do not require an asymptotic Kaluza-Klein circle. It argues these bubbles can have arbitrary size and arbitrarily small mass, and that low-energy bubbles can initially expand, suggesting a possible new instability of higher-dimensional flat space.

Key findings

• A two-parameter family of regular asymptotically flat solutions is found, with bubbles whose sizes and mass can be chosen arbitrarily; in particular, there are bubbles of any size that are arbitrarily light.
• The ADM energy is positive definite and given by \(E = \frac{2\pi r_0^2}{G\omega}(\sqrt{1-\omega}+\sqrt{\omega}-1)\); for low energy, \(E \approx \frac{\pi r_0^2}{G\sqrt{\omega}}\).
• The initial acceleration of the y-bubble area changes from contraction to expansion at \(\omega \approx 0.803815\), corresponding to \(r'_0/r_0 \approx 0.494032\) and \(E \approx 2.65366 r_0^2\); for the x-bubble the transition is at \(\omega \approx 0.196185\), \(r'_0/r_0 \approx 2.02416\), and \(E \approx 10.8726 r_0^2\).
• For low-energy solutions, curvature is large only near the smaller bubble and small elsewhere; the maximum squared Riemann tensor is \(R^2 = \frac{24}{(1-\omega)^2 r_0^4} + O((1-\omega)^{-3/2}r_0^{-4})\), with the high-curvature region confined to a small proper volume.

Methods

• Uses a time-symmetric initial-data ansatz with diagonal \(U(1)\times U(1)\) symmetry, solving the vacuum constraint to obtain a five-dimensional metric with parameters \(r_0\) and \(\omega\).
• Analyzes asymptotics, computes ADM mass via a covariant boundary integral, and studies initial dynamics using Hamiltonian evolution equations in Gaussian coordinates.
• Computes curvature via the Gauss-Codacci relation and evaluates the square of the Riemann tensor for low-energy solutions; no numerical evolution is performed.

Limitations

• Stated: the full time evolution is not solved and must be addressed numerically; apparent horizons are technically difficult to find analytically; whether these bubbles represent a true instability remains open.
• Stated: no instanton for quantum nucleation is found, and it is not shown that the bubbles are cobordant to flat space or admit a suitable spin structure.
• Evident from the text: the expansion/contraction conclusions are based on initial data and physical argument, not on completed dynamical simulations; extensions to AdS, compactifications, and braneworlds are left as open questions.