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 ≤ 9 8 R s = 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 = 3 R 1 − 8 9 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 = − &rgr; ¯ ?> 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 τ s = Δ κ / 8 &pgr; G ?> , where Δ κ = κ + − κ − = 2 κ + = 1 / R s ?> is the difference of equal and opposite surface gravities between the exterior and interior Schwarzschild solutions. The First Law, d M = d E V + τ s d A ?> 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.