The cosmological term as a source of mass

In the spherically symmetric case, the dominant energy condition, together with the requirement of regularity at the centre, asymptotic flatness and finiteness of the ADM mass, defines the family of asymptotically flat globally regular solutions to the Einstein equations which includes the class of metrics asymptotically de Sitter as r → 0. The source term corresponds to an r-dependent cosmological term Λ_μν invariant under boosts in the radial direction and evolving from the de Sitter vacuum Λg_μν in the origin to the Minkowski vacuum at infinity. The ADM mass is related to a cosmological term by m = (2G)⁻¹∫₀^∞Λ_t^tr² dr, with the de Sitter vacuum replacing a central singularity at the scale of symmetry restoration. Spacetime symmetry changes smoothly from the de Sitter group near the centre to the Lorentz group at infinity through radial boosts in between. In the range of masses m ≥ m_crit, de Sitter–Schwarzschild geometry describes a vacuum nonsingular black hole (ΛBH), and for m < m_crit, it describes a G-lump—a vacuum self-gravitating particle-like structure without horizons. The quantum energy spectrum of the G-lump is shifted down by the binding energy and zero-point vacuum mode is fixed at the value corresponding (up to the coefficient) to the Hawking temperature from the de Sitter horizon.