On Super-Translation transition between Quasi-local Black holes

Abstract

We re-explore the symmetries of a weakly isolated horizon (WIH) from the perspective of freedom in the choice of intrinsic data. The supertranslations are realized as additional symmetries. Further, it is shown that all smooth vector fields tangent to the cross-sections are Hamiltonian. We show that joining two WIHs which differ in these Hamiltonians and boundary data, under the action of a supertranslation, necessarily require the inclusion of an intermediate dynamical phase, possibly with the inclusion of a stress energy tensor. This phase of the boundary is non-expanding but not a WIH and invariably leads to a violation of the dominant energy condition.The assumptions made allow us to reconstruct the (classically) pathological stress energy tensor also.

Dynamical horizons and Super-Translation transitions of the horizon

“…where the covariant derivative compatible with the metric on the cross section, q, has been denoted by D. On contracting both sides by n − κ dF we get the desired result found in [20].…”

“…Following the definition of an isolated horizon (IH) [22][23][24][25][26][27] the two geometric quantities that transform under a supertranslation are the pull back of the rotation one form viz. [20]. Let us denote an IH by ∆.…”

“…This essentially fixes a gauge freedom. A generalisation of the supertranslation charges given in [20] to the dynamical phase may be written as,…”

“…But in doing so one induces an extra diffeomorphism on the cross sections (due to modification of the shift vector). Since diffeomorphisms on the cross section are not symmetries in IH phase [20], the final black holes formed in each case are indeed different. Symmetry requirements can however put severe constraints on the shift vector.…”

“…The second being the fact that unlike asymptotic null infinity the zeroth order structure at the horizon is not universal and in fact becomes dynamic during the evolution process thus posing as a hurdle in the way. A simplistic approach to studying the process of inducing a supertranslation to quasi-local black holes was taken in [20]. Here we will go a step forward and study general dynamical evolutions and demonstrate how they may be induced for spherically symmetric dynamical horizon.…”