Surface tension and negative pressure interior of a non-singular 'black hole'

The constant density interior Schwarzschild solution for a static, spherically symmetric collapsed star has a divergent pressure when its radius $R⩽\frac{9}{8}{R}_s=\frac{9}{4}GM$. We show that this divergence is integrable, and induces a non-isotropic transverse stress with a finite redshifted surface tension on a spherical surface of radius $R_0=3R\sqrt{1-\frac{8}{9}\frac{R}{R_s}}$. For $r<R_0$ the interior Schwarzschild solution exhibits negative pressure. When $R=R_s$, the surface is localized at the Schwarzschild radius itself, $R_0=R_s$, and the solution has constant negative pressure $p=-\overline{\rho }$ everywhere in the interior $r<R_s$, thereby describing a gravitational condensate star, a fully collapsed non-singular state already inherent in and predicted by classical general relativity. The redshifted surface tension of the condensate star surface is given by ${\tau }_s=\Delta \kappa /8\pi G$, where $\Delta \kappa ={\kappa }_{+}-{\kappa }_{-}=2{\kappa }_{+}=1/R_s$ is the difference of equal and opposite surface gravities between the exterior and interior Schwarzschild solutions. The First Law, $dM=d{E}_V+{\tau }_{s}dA$ is recognized as a purely mechanical classical relation at zero temperature and zero entropy, describing the volume energy and surface energy change respectively. The Schwarzschild time t of such a non-singular gravitational condensate star is a global time, fully consistent with unitary time evolution in quantum theory. A clear observational test of gravitational condensate stars with a physical surface versus black holes is the discrete surface modes of oscillation which should be detectable by their gravitational wave signatures. (paper)