Kuchař hypertime formalism for cylindrically symmetric spacetimes with interacting scalar fields

Abstract

The Kuchař canonical transformation for vacuum geometrodynamics in the presence of cylindrical symmetry is applied to a general non-vacuum case. The resulting constraints are highly nonlinear and non-local in the momenta conjugate to the Kuchař embedding variables. However, it is demonstrated that the constraints can be solved for these momenta and thus the dynamics of cylindrically symmetric models can be cast in a form suitable for the construction of a hypertime functional Schrödinger equation.

Hypertime formalism for spherically symmetric black holes and wormholes

“…The resulting constraints and momenta are very closely related to those found for the cylindrically symmetric system discussed in Ref. [6].…”

“…Just such a case was investigated in Ref. [6], when no extra boundary terms were needed to achieve the correct dynamics. Now we are ready to reduce spherically symmetric gravity to a Hamiltonian of the form given by Equation ( 5), and construct the corresponding P Λ functionals.…”

“…We will write the spherically symmetric metric, with charge q, in a form that will lead to an action that mirrors the one for cylindrically symmetric metrics used in Ref. [6]…”

“…We are directly motivated by Ref. [6], and therefore use basically the same hypertime variable (which is derived from those originally discovered by Kuchař for vacuum cylindrically symmetric spacetimes [17]). We define…”

“…One possible solution to the problem of time is to view the constraints given by Equation (2) as representing evolution with respect to other variables, constructed out of the gravitational and matter phase-space variables themselves, instead of with respect to the simple label t. The latter are called embedding variables [3,4,5] and they can be considered as representing the location of a given spacelike hypersurface in an embedding into a surrounding spacetime, thus leading to the term hypertime for such formalisms. In a recent paper [6], we looked at this method for the case of cylindrically symmetric spacetimes, and succeeded in constructing hypertime variables for those models. We will be extending that work in this paper to the physically more interesting case of spherically symmetric black holes with self-interacting matter fields.…”