Intrinsic time gravity and the Lichnerowicz–York equation

Abstract

We investigate the effect on the Hamiltonian structure of general relativity of choosing an intrinsic time to fix the time slicing. 3-covariance with momentum constraint is maintained, but the Hamiltonian constraint is replaced by a dynamical equation for the trace of the momentum. This reveals a very simple structure with a local reduced Hamiltonian. The theory is easily generalized; in particular, the square of the Cotton-York tensor density can be added as an extra part of the potential while at the same time maintaining the classic 2 + 2 degrees of freedom. Initial data construction is simple in the extended theory; we get a generalized Lichnerowicz-York equation with nice existence and uniqueness properties. Adding standard matter fields is quite straightforward.

General Relativity Without Paradigm of Space-Time Covariance: Sensible Quantum Gravity and Resolution of the “Problem of Time”

“…and the role of intrinsic time in general relativity and its extensions (a related discussion on the initial data formulation can be found in Ref. [28]) can be obtained from several complementary approaches: the master constraint formulation which recovers the correct physical content from the usual starting point of canonical general relativity; the Schrödinger equation (4) (or its superspace version (14)) as the fundamental equation for quantum geometrodynamics; the generalized Baierlein-Sharp-Wheeler action (18); and also, perhaps most important to a causal quantum theory, the evolution operator U (h, h 0 ) with gauge-invariant temporal ordering.…”

Intrinsic Time Quantum Gravity

“…We had shown in [1,2] that the symplectic potential π ij δq ij = π ij δq ij +πδ ln q 1 3 , can be cleanly separated into the conjugate pair, (ln q 1 3 ,π), consisting of (one-third of) the logarithm of the determinant of the spatial metric and the trace of the momentum, from (q ij ,π ij ), the unimodular part of the spatial metric with traceless conjugate momentum that allows a deparametrization of the theory wherein ln q 1 3 plays the role of the intrinsic time variable for β 2 = l − 1 3 > 0 with l being the deformation parameter in the DeWitt supermetric; G ijkl = 1 2 (q ik q jl + q il q jk ) − lq ij q kl . This decomposition and identification of the intrinsic time variable point to a paradigm shift in the symmetries of Gravitation/space-time.…”

Symplectic Structure of Intrinsic Time Gravity

“…Starting from this canonically conjugate pair, let us define as fundamental classical variables the following barred quantitiesq ij , a unimodular metric with detq ij = 1, and a traceless momentum variable π ij via the relations [7,8] …”

“…The basic idea, as introduced in [7] and [8], is the concept of a new phase space for gravity which breaks the paradigm of four-dimensional spacetime covariance, shifting the emphasis to three dimensional spatial diffeomorphism invariance combined with a physical Hamiltonian which generates evolution with respect to intrinsic time. Through the constructive interference of wavefronts, classical spacetime emerges from the formalism, with direct correlation between intrinsic time intervals and proper time intervals of spacetime.…”