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

Abstract

The framework of a theory of gravity from the quantum to the classical regime is presented. The paradigm shift from full spacetime covariance to spatial diffeomorphism invariance, together with clean decomposition of the canonical structure, yield transparent physical dynamics and a resolution of the problem of time. The deep divide between quantum mechanics and conventional canonical formulations of quantum gravity is overcome with a Schrödinger equation for quantum geometrodynamics that describes evolution in intrinsic time. Unitary time development with gaugeinvariant temporal ordering is also viable. All Kuchar observables become physical; and classical spacetime, with direct correlation between its proper times and intrinsic time intervals, emerges from constructive interference. The framework not only yields a physical Hamiltonian for Einstein's theory, but also prompts natural extensions and improvements towards a well behaved quantum theory of gravity. It is a consistent canonical scheme to discuss Horava-Lifshitz theories with intrinsic time evolution, and of the many possible alternatives that respect 3-covariance (rather than the more restrictive 4-covariance of Einstein's theory), Horava's "detailed balance" form of the Hamiltonian constraint is essentially pinned down by this framework. Issues in quantum gravity that depend on radiative corrections and the rigorous definition and regularization of the Hamiltonian operator are not addressed in this work.

Intrinsic time in Wheeler–DeWitt conformal superspace

“…As the saying goes [39,40], the superspace is the arena of Geometrodynamics. Denote by (3) G * the space of corresponding canonically conjugated densities of momenta. According to the scaled Dirac's mapping (53), we have the functional mapping of WDW phase superspace of metrics γ ij and corresponding densities of their momenta π ij to WDW conformal superspace of metricsγ ij , densities of momentaπ ij ; the local intrinsic time D, and the Hamiltonian density π D :…”

“…After the procedure of deparametrization is implemented the volume of the Universe occurs its global intrinsic time. In modern papers [3][4][5][6][7][8][9][10] one can see applications of local intrinsic time interval in Geometrodynamics.…”

Intrinsic time gravity and the Lichnerowicz–York equation

“…This is a rewriting of the Hamiltonian constraint, but it is no longer to be viewed as a constraint; rather, it is the evolution equation for π. This is the fundamental equation for intrinsic time gravity; and it can be interpreted respectively as the Hamilton-Jacobi equation and Schrödinger equation in the semi-classical and quantum regimes [2]. We choose the positive root forH when we take the square-root.…”

“…A major difficulty with the canonical quantization of gravity program is this freedom to choose N [2]. Each choice of N gives a different slicing of spacetime, and reflects the 4-covariance of the Einstein equations.…”

“…The great advantage of Misner's choice is that a local reduced Hamiltonian appears. Recently, two of us [2] presented a theory of gravity passing from classical to quantum regimes with a paradigm shift from 4-covariance to 3-covariance. Its framework revealed the primacy of a local reduced Hamiltonian and intrinsic time proportional to ln g 1/3 .…”