Energy spectrum of a quantum spacetime with boundary

In this paper, I revisit the microcanonical partition function, or density of states (DOS), of general relativity. By using the minisuperspace path integral approximation, I directly calculate the S² × Disc topology sector of the DOS of a (quantum) spacetime with an S² × R Lorentzian boundary from the microcanonical path integral, in contrast with previous works in which DOSs are derived by inverse Laplace transformation from various canonical partition functions. Although I found there always exists only one saddle point for any given boundary data, it does not always dominate the possible integration contours. There is another contribution to the path integral other than the saddle point. One of the obtained DOSs has behavior similar to that of the previous DOSs; that is, it exhibits exponential Bekenstein-Hawking entropy for the limited energy range (1 - √ 2/3) <GE/Rb < (1 + √ 2/3), where energy E is defined by the Brown-York quasi-local energy momentum tensor and Rb is the radius of the boundary S2. In that range, the DOS is dominated by the saddle point. However, for sufficiently high energy, where the saddle point no longer dominates, the DOS approaches a positive constant, different from the previous ones, which approach zero.