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We numerically construct the spectrum of the Laplacian on Page's inhomogeneous Einstein metric on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ by reducing the problem to a (singular) Sturm-Liouville problem in one dimension. We perform a perturbative analysis based upon a closely related, exactly solvable problem that strongly supports our results. We also study the spectrum of the Lichnerowicz Laplacian on symmetric traceless transverse two-tensors. The method relies on both the isometries of the Page metric and pseudospectral methods to numerically solve the resulting ODEs.

During a binary black hole merger, multiple intermediary marginally outer trapped tubes connect the initial pair of apparent horizons with the final (single) apparent horizon. The marginally outer trapped surfaces (MOTSs) that foliate these tubes can have complicated geometries as well as non-spherical topologies. In particular, toroidal MOTSs form inside both of the original black holes during the early stages of a head-on merger that starts from time-symmetric initial data [1]. We show that toroidal MOTSs also form in the maximal analytic extension of the Schwarzschild spacetime as Kruskal time advances from the $T=0$ moment of time symmetry. As for the merger simulations, they cross the Einstein-Rosen bridge and are tightly sandwiched between the apparent horizons in the two asymptotic regions at early times. This strongly suggests that their formation is a consequence of the initial conditions rather than merger physics. Finally, we consider MOTSs of spherical topology in the Kruskal-Szekeres slicing and study their properties. All of these are contained within the apparent horizon but some do not enclose the wormhole., Comment: 14 pages, 16 figures