Mass function and particle creation in Schwarzschild-de Sitter spacetime

Understand

We construct a mass or energy function for the non-Nariai class Schwarzschild-de Sitter black hole spacetime in the region between the black hole and the cosmological event horizons.

• The mass function is local, positive definite, continuous and increases monotonically with the radial distance from the black hole event horizon.
• We derive the Smarr formula using this mass function, and demonstrate that the mass function reproduces the two-temperature Schwarzschild-de Sitter black hole thermodynamics, along with a term corresponding to the negative pressure due to positive cosmological constant.
• We further give a field theoretic derivation of the particle creation by both the horizons and discuss its connection with the mass function.