Cosmic time and reduced phase space of general relativity

Abstract

In an ever-expanding spatially closed universe, the fractional change of the volume is the preeminent intrinsic time interval to describe evolution in General Relativity. The expansion of the universe serves as a subsidiary condition which transforms Einstein's theory from a first class to a second class constrained system when the physical degrees of freedom (d.o.f.) are identified with transverse traceless excitations. The super-Hamiltonian constraint is solved by eliminating the trace of the momentum in terms of the other variables, and spatial diffeomorphism symmetry is tackled explicitly by imposing transversality. The theorems of Maskawa-Nishijima appositely relate the reduced phase space to the physical variables in canonical functional integral and Dirac's criterion for second class constraints to non-vanishing Faddeev-Popov determinants in the phase space measures. A reduced physical Hamiltonian for intrinsic time evolution of the two physical d.o.f. emerges. Freed from the first class Dirac algebra, deformation of the Hamiltonian constraint is permitted, and natural extension of the Hamiltonian while maintaining spatial diffeomorphism invariance leads to a theory with Cotton-York term as the ultra-violet completion of Einstein's theory.

Intrinsic Time in Geometrodynamics of Closed Manifolds

“…Earlier to construct a scalar, the Minkowskian metric as a background one was used for asymptotically flat spaces 19 . The intrinsic time interval δD as a scalar field was implemented in the symplectic 1-form 34,35 . For splitting one degree of freedom, the average of the trace of the momentum density was used as the York time in the shape dynamics 36 .…”

On the Physical Implications of Hawking's Spacetime Foam

“…This formula for the stationary point is a bit different from the result of Hawking since we use a slightly different action. Furthermore, Hawking also appears to make the substitution V = f 2 / only in the amplitude and not in the entire Equation (19) which leads to different signs for the expressions within in the square root. The different signs in front of the square roots from Equation (21) come from the 2 contribution in the amplitude.…”

“…Then, there are works like or the recent proposal, which suggest the cosmological constant could have something to do with boundary terms of instantons and the problem of time, but these papers fails to provide a precise value of the constant. In another attempt, Barrow and Shaw threat the cosmological constant as a variable field in quantum gravity.…”

Intrinsic time gravity, heat kernel regularization, and emergence of Einstein’s theory

“…Quantum geometrodynamics with first-order Schrödinger evolution in intrinsic time, i δΨ δT = H Phys. Ψ, has been advocated in a series of articles [1][2][3][4][5][6][7]. In this work, we elucidate the Hamiltonian of the theory which contains a Cotton-York term; in addition, heat kernel regularization will be employed to demonstrate the remarkable emergence of Einstein's Ricci scalar potential.…”

“…A full canonical analysis of the resultant two physical d.o.f. and the reduced phase space, together with a discussion of the system of second class Dirac constraints, and the associated Faddeev-Popov determinants and functional integral measures has been carried out [6]. Spatial diffeomorphism invariance is maintained provided the Hamiltonian density,…”