Black holes in de Sitter space: Masses, energies, and entropy bounds

In this paper we consider spacetimes in vacuum general relativity—possibly coupled to a scalar field—with a positive cosmological constant $\Lambda .$ We employ the isolated horizons (IH) formalism where the boundary conditions imposed are that of two horizons, one of black hole type and the other, serving as outer boundary, a cosmological horizon. As particular cases, we consider the Schwarzschild–de Sitter spacetime, in both $2+1$ and $3+1$ dimensions. Within the IH formalism, it is useful to define two different notions of energy for the cosmological horizon, namely, the “mass” and the “energy.” Empty de Sitter space provides a striking example of such a distinction: its horizon energy is zero but the horizon mass takes a finite value given by $\pi /\left(2\mathrm{}\sqrt{\Lambda }\right).\mathrm{}$ For both horizons we study their thermodynamic properties, compare our results with those of Euclidean Hamiltonian methods and construct some generalized Bekenstein entropy bounds. We discuss these new entropy bounds and compare them with some recently proposed entropy bounds in the cosmological setting.