Soap bubbles in outer space: Interaction of a domain wall with a black hole

We discuss the generalized Plateau problem in the (3+1)-dimensional Schwarzschild background. This represents the physical situation, which could for instance have appeared in the early universe, where a cosmic membrane (thin domain wall) is located near a black hole. Considering stationary axially symmetric membranes, three different membrane topologies are possible depending on the boundary conditions at infinity: 2+1 Minkowski topology, 2+1 wormhole topology, and 2+1 black hole topology. Interestingly, we find that the different membrane topologies are connected via phase transitions of the form first discussed by Choptuik in investigations of scalar field collapse. More precisely, we find a first order phase transition (finite mass gap) between wormhole topology and black hole topology, the intermediate membrane being an unstable wormhole collapsing to a black hole. Moreover, we find a second order phase transition (no mass gap) between Minkowski topology and black hole topology, the intermediate membrane being a naked singularity. For the membranes of black hole topology, we find a mass scaling relation analogous to that originally found by Choptuik. However, in our case the parameter $p$ is replaced by a 2-vector $\overrightarrow{p}$ parametrizing the solutions. We find that $\mathrm{mass}\propto |\overrightarrow{p}-p_{*}{|}^{\gamma }$ where $\gamma \approx \mathrm{0.66.}$ We also find a periodic wiggle in the scaling relation. Our results show that black hole formation as a critical phenomenon is far more general than expected.