Gravitational Vacuum Condensate Stars in the Effective Theory of Gravity

Abstract

The low energy effective theory of gravity comprises two elements of quantum theory joined to classical general relativity. The first is the quantum conformal anomaly, which is responsible for macroscopic correlations on light cones and a stress tensor that can strongly modify the classical geometry at black hole horizons. The second is the formulation of vacuum energy as $\Lambda_{\rm eff}\!\propto\! F^2$ in terms of an exact $4$-form abelian gauge field strength $F\!=\!dA$. When $A$ is identified with the Chern-Simons $3$-form of the Euler class, defined in terms of the spin connection, a $J\cdot A$ interaction is generated by the conformal anomaly of massless fermions. Due to the extreme blueshifting of local frequencies in the near-horizon region of a `black hole,' the lightest fermions of the Standard Model can be treated as massless there, contributing to the anomaly and providing a $3$-current source $J$ for the `Maxwell' equation $d\ast F = \ast J$. In this phase boundary region, torsion is activated, and $F$ can change rapidly. The Schwarzschild black hole horizon is thereby replaced by a surface, with a positive surface tension and $\mathbb{R}\otimes \mathbb{S}^2$ worldtube topology, separating regions of differing vacuum energy. The result is a gravitational vacuum condensate star, a cold, compact, horizonless object with a $p_{_V}\!=\! - \rho_{_V}$ zero entropy, non-singular de Sitter interior and thin quantum phase boundary layer at the Schwarzschild radius $2GM/c^2$.