Are Black Holes Leaky?

Abstract

We continue a study by Adler and Ramazanoǧlu (AR) of black holes as modified by a scale invariant dark energy action. For the spherically symmetric Schwarzschild-like case, (AR) found that there is no event horizon; hence spacetime is not divided by the black hole into causally disconnected regions. We review the formalism for locating trapped surfaces and apparent horizons, and show that the modified black hole has no trapped surfaces. Thus one expects that it is “leaky”, and that there will be a “black hole wind” of particles streaming out from the location of the nominal horizon. This will have astrophysical consequences, for example, the “wind” may feed and stabilize star formation in the vicinity of the black hole. We initiate a study of the stationary axially-symmetric rotating black hole as modified by a scale invariant dark energy action, i.e, one that is Kerr-like. To set up the axial case, we note that the conserving completion of the dark energy stress energy tensor can be calculated algebraically by solving a 2×2 matrix equation. Using Mathematica we calculate and simplify the modified system of Einstein equations for the axial case in quasi-isotropic coordinates, and give appropriate boundary conditions. Solution of these equations is rendered tricky by a residual general coordinate invariance of quasi-isotropic coordinates. This leads to the interesting mathematical question of representing a positive planar function as the gradient squared of a harmonic function, discussed in Appendices.