Evolution of the maximal hypersurface in a D-dimensional dynamical spacetime

Abstract

In this article we set up a variational problem to arrive at the equation of a maximal hypersurface inside a spherically symmetric evolving trapped region. In the first part of the article, we present the Lagrangian and the corresponding Euler Lagrange equations that maximize the interior volume of a trapped region that is formed dynamically due to infalling matter in D-dimensions, with and without the cosmological constant. We explore the properties of special points in these maximal hypersurfaces at which the Kodama vector becomes tangential to the hypersurface. These points which we call steady state points, are shown to play a crucial role in approximating the maximal interior volume of a black hole. We derive a formula to locate these points on the maximal hypersurface in terms of coordinate invariants like area radius, principle values of energy momentum tensor, Misner Sharp mass and cosmological constant. Based on this formula, we estimate the location of these steady state points in various scenarios: (a) the case of static BTZ black holes in 2+1 dimensions and for the Schwarzschild, Schwarzschild-deSitter and Schwarzschild-Anti-deSitter black holes in D-dimensions. We plot the location of the steady state points in relation to the event horizon and cosmological horizon in a static D-dimensional scenario, (b) cosmological case: we prove that these steady state points do not exist for homogeneous evolving dust for the zero and negative cosmological constant but exist in the presence of positive cosmological constant when the scale factor is greater than a critical value.