The Thermodynamics of Transient Trapped Surfaces in the Geon Collapse

It is shown that transient trapped surfaces form in a class of emerging globally hyperbolic spacetimes, within punctured Planck-scale neighbourhoods of the geon supported on intersecting singular supports whose intersection forms a characteristic core in a non-strongly causal setting. These neighbourhoods shrink towards the intersecting singular support in the distributional geometry. In particular, the trapped surfaces occur near the characteristic limit corresponding to the unstable equilibrium of the self-gravitating geon. They act as an effective classical barrier for descriptions formulated purely within smooth differential geometry. The area of these trapped-surface configurations, computed on Planck-referenced neighbourhoods, is shown to tend to zero both in the asymptotically flat limit of the emerging spacetime and in the geon limit. Thus, transient trapped surfaces evaporate in the sense that their area vanishes as classical and asymptotically flat spacetime emerges within the quantum foam framework. A state-counting generating function for the transient trapped surfaces is constructed from the coherent-state density operator. This generating function maps microscopic occupation-number sectors to macroscopic data and thereby allows a definition of Boltzmann entropy (not to be confused with the von Neumann entropy, which is zero for any pure coherent state). Since the coherent state is constructed to implement the correspondence principle, expectation values of the relevant quantised observables reproduce their classical values. In particular, the expectation value of the bosonic occupation-number operator serves as a microstate-counting variable in the coherent sector. The generating function takes the form of an exponential of this expectation value, leading to an entropy–area relation consistent with the Hawking–Bekenstein scaling.