טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentEhud Eilon
SubjectInvestigation of the Gravitational Shock Wave Inside
Charged Black Holes
DepartmentDepartment of Physics
Supervisor Full Professor Ori Amos
Full Thesis textFull thesis text - English Version


Abstract

This thesis describes our numerical investigation of the inner structure of perturbed, charged black holes. In the first part of the research, we dealt with numerical problems inherent to coordinate choice. Double-null coordinates are highly useful but become problematic in long-time simulations: Along the event horizon, the truncation error runs out of control for a sufficiently long interval of ve, the outgoing Eddington null coordinate. This problem could destroy the numerics both inside and outside the black hole at late times (i.e. large ve). We explored the origin of this problem and solved it with an adaptive gauge choice for the ingoing null coordinate u. We then generalized the solution in order to handle an analogous problem that occurs at the inner horizon. This generalized gauge, the maximal-σ gauge, allows long-v double-null numerical simulation across both the event horizon and the (outgoing) inner horizon, up to the vicinity of the spacelike r=0 singularity. We have used this adaptive gauge solution along the entire research.

In the second part of the research, we have studied the interior of a four-dimensional, asymptotically flat, spherically symmetric charged black hole perturbed by a scalar field Φ. Previous study by Marolf and Ori indicated that late infalling observers will encounter an effective shock wave as they approach the left portion of the inner horizon. This shock manifests itself as a sudden change in the values of various fields, within a tremendously short interval of proper time τ of the infalling observers. We confirmed this prediction numerically for test and self-gravitating scalar field perturbations. In both cases we demonstrated the effective shock in the scalar field by exploring Φ(τ) along a family of infalling timelike geodesics. In the self-gravitating case we also demonstrated the shock in the area coordinate r by exploring r(τ) . We confirmed the theoretical prediction concerning the shock sharpening rate, which is exponential in the time of infall into the black hole. We also employed a family of null (rather than timelike) ingoing geodesics to probe the shock in r.

In the third part of the research, we investigated the gravitational shock (the shock in r) in more complex (and more realistic) steady accretion scenarios. We considered three different scenarios: a) A charged black hole accreting a single (ingoing) null fluid; b) a charged black hole perturbed by two null fluids, ingoing and outgoing; c) a charged black hole perturbed by an ingoing null fluid and a self-gravitating scalar field. While we did not observe any evidence for a gravitational shock in the first case, we detected the shock in the other two, using ingoing timelike and null geodesics. The shock width Δτ decreases rapidly with a fairly good match to a new, generalized exponential law, Δτ ∼e -∫ κ (V)dV, where V is a specific timing parameter for ingoing timelike geodesics, and κ(V) is a generalized (Reissner-Nordström like) surface gravity of the charged black hole at the inner horizon. We also uncovered strong evidence for the existence of a spacelike r=0 singularity in the case of a charged black hole perturbed by two null fluids.