Effects of charge on the interior volume of BTZ black holes

In this article we extend the variational technique for maximal volume estimation of a black hole developed by Christodoulou and Rovelli (CR) to the case of a charged BTZ black hole in 2+1 dimensions. The technique involves a study of the equation of motion of a hypothetical particle moving in an auxiliary manifold defined by spacetime variables. We then compare this estimation with the volume computed using maximization method and extrinsic curvature method. The charge Q of the black hole appears as a log term in the metric and hence an analytical solution for the volume does not exist. So first we compute the steady state radius and the volume for limiting case when the charge Q is very small i.e. Q << 1 and then carry out a numerical analysis to solve for the volume for more generic values of the charge. We find that the volume grows monotonically with the advance time. We further investigate the functional behaviour of the entropy of a massless scalar field living on the maximal hypersurface of a near extremal black hole. We show that this volume entropy exhibits a very different functional form compared to the horizon entropy.Comment: 11 pages, 5 figure

Quantum Gravitational Collapse and Hawking Radiation in 2+1 Dimensions

We develop the canonical theory of gravitational collapse in 2+1 dimensions with a negative cosmological constant and obtain exact solutions of the Wheeler--DeWitt equation regularized on a lattice. We employ these solutions to derive the Hawking radiation from black holes formed in all models of dust collapse. We obtain an (approximate) Planck spectrum near the horizon characterized by the Hawking temperature $T_{\mathrm H}=\hbar\sqrt{G\Lambda M}/2\pi$, where $M$ is the mass of a black hole that is presumed to form at the center of the collapsing matter cloud and $-\Lambda$ is the cosmological constant. Our solutions to the Wheeler-DeWitt equation are exact, so we are able to reliably compute the greybody factors that result from going beyond the near horizon region.Comment: 27 pages, no figure

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

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.Comment: 13 pages, 12 figure

Test of Transitivity in Quantum Field theory using Rindler spacetime

We consider a massless scalar field in Minkowski spacetime $\cal{M}$ in its vacuum state, and consider two Rindler wedges $R_1$ and $R_2$ in this space. $R_2$ is shifted to the right of $R_1$ by a distance $\Delta$. We therefore have $R_2\subset R_1 \subset \cal{M}$ with the symbol $\subset$ implying a quantum subsystem. We find the reduced state in $R_2$ using two independent ways: a) by evaluation of the reduced state from vacuum state in $\cal{M}$ which yields a thermal density matrix, b) by first evaluating the reduced state in $R_1$ from $\cal{M}$ yielding a thermal state in $R_1$, and subsequently evaluate the reduced state in $R_2$ in that order of sequence. In this article we attempt to address the question whether both these independent ways yield the same reduced state in $R_2$. To that end, we devise a method which involves cleaving the Rindler wedge $R_1$ into two domains such that they form a thermofield double. One of the domains aligns itself along the wedge $R_2$ while the other is a diamond shaped construction between the boundaries of $R_1$ and $R_2$. We conclude that both these independent methods yield two different answers, and discuss the possible implications of our result in the context of quantum states outside a non-extremal black hole formed by collapsing matter.Comment: 6 pages, 3 figure