Vacuum around a spherical thin shell looks like a black hole outside as the shell nears its gravitational radius
This paper examines how quantum fields behave in the simple setting of a spherical, static thin shell of matter that has an empty (Minkowski) interior. The authors study the Boulware vacuum, the natural vacuum state for static, horizonless configurations that matches ordinary empty space far away but becomes singular at a black-hole horizon. They compute two central quantum quantities for massless scalar fields: the vacuum polarization (a measure of how the field’s average value is affected by the geometry) and the renormalized stress-energy tensor (the field’s energy and momentum after removing standard infinities). Their focus is how these quantities change as the shell is pushed closer and closer to the size at which a black hole would form (the black hole or gravitational radius).
To get these results the team used a mode-sum renormalization method called the extended-coordinate prescription. Technically, they performed calculations on a Wick-rotated or “Euclidean” version of the spacetime, which simplifies mode sums. For the Boulware vacuum the Euclidean time is not periodic, so the frequency spectrum is continuous rather than discrete. Near the shell surface they derived analytic leading-order behaviour with two independent methods: a high-frequency or WKB (Wentzel–Kramers–Brillouin) expansion of the field modes, and a weak-field approximation. Away from those regions they produced numerical results covering many shell compactnesses and different field couplings.
Close to the shell surface, the authors recovered the expected singular behaviour of both the vacuum polarization and the stress-energy tensor. At the center of the shell they found non-local, Casimir-like contributions — effects similar to the Casimir force that arise from the global arrangement of space rather than local curvature. Importantly, these central, non-local contributions remain finite even as the shell approaches the black hole limit. Inside the shell the usual local, renormalization-dependent terms vanish for massless fields, leaving only these non-local effects.