Full Space-time Covariance Research Articles

Covariant information theory and emergent gravity

Informational dependence between statistical or quantum subsystems can be described with Fisher information matrix or Fubini-Study metric obtained from variations/shifts of the sample/configuration space coordinates. Using these (noncovariant) objects as macroscopic constraints, we consider statistical ensembles over the space of classical probability distributions (i.e. in statistical space) or quantum wave functions (i.e. in Hilbert space). The ensembles are covariantized using dual field theories with either complex scalar field (identified with complex wave functions) or real scalar field (identified with square roots of probabilities). We construct space–time ensembles for which an approximate Schrodinger dynamics is satisfied by the dual field (which we call infoton due to its informational origin) and argue that a full space–time covariance on the field theory side is dual to local computations on the information theory side. We define a fully covariant information-computation tensor and show that it must satisfy certain conservation equations. Then we switch to a thermodynamic description of the quantum/statistical systems and argue that the (inverse of) space–time metric tensor is a conjugate thermodynamic variable to the ensemble-averaged information-computation tensor. In (local) equilibrium, the entropy production vanishes, and the metric is not dynamical, but away from the equilibrium the entropy production gives rise to an emergent dynamics of the metric. This dynamics can be described approximately by expanding the entropy production into products of generalized forces (derivatives of metric) and conjugate fluxes. Near equilibrium, these fluxes are given by an Onsager tensor contracted with generalized forces and on the grounds of time-reversal symmetry, the Onsager tensor is expected to be symmetric. We show that a particularly simple and highly symmetric form of the Onsager tensor gives rise to the Einstein–Hilbert term. This proves that general relativity is equivalent to a theory of nonequilibrium (thermo)dynamics of the metric, but the theory is expected to break down far away from equilibrium where the symmetries of the Onsager tensor are to be broken.

Quantum geometrodynamics with intrinsic time development

Quantum geometrodynamics with intrinsic time development is presented. Paradigm shift from full spacetime covariance to spatial diffeomorphism invariance yields a nonvanishing Hamiltonian, a resolution of the ‘problem of time’ and gauge-invariant temporal ordering in an ever expanding universe. Einstein’s general relativity is a particular realization of a wider class of theories; and the framework prompts natural extensions and improvements with the consequent dominance of Cotton–York potential at early times when the universe was small.

Multi-event universal kriging (MEUK)

Multi-event universal kriging (MEUK) is a method of interpolation that creates a series of maps, each corresponding to a specific sampling “event”, which exhibit spatial relationships that persist over time. MEUK is computed using minimum-variance unbiased linear prediction from data obtained via a sequence of events. MEUK assumes multi-event data can be described by a sum of (a) spatial trends that vary over time, (b) spatial trends that are invariant over time, and (c) spatially- and temporally-stationary correlation among the residuals from the combination of these trends. The fundamental advance made by MEUK versus traditional universal kriging (UK) lies with the generalized least squares (GLS) model and the multi-event capability it facilitates, rather than in the geostatistics, although it is shown how use of MEUK can greatly reduce predictive variances versus UK. For expediency, MEUK assumes a spatial covariance that does not change over time – although it does not have to – which is an advantage over space-time methods that employ a full space-time covariance function. MEUK can be implemented with large multi-event datasets, as demonstrated by application to a large water level dataset. Often, MEUK enables the stable solution of multiple events for similar computational effort as for a single event. MEUK provides an efficient basis for developing “wheel-and-axle” monitoring strategies [32] that combines frequently sampled locations used monitor changes over time with many more locations sampled periodically to provide synoptic depictions. MEUK can aid in the identification of the core monitoring locations, allowing for reduced sampling frequency elsewhere. Although MEUK can incorporate longitudinal variograms as in other space-time methods, doing so reduces the computational advantages of MEUK.

General Relativity without paradigm of space-time covariance, and resolution of the problem of time

The framework of a theory of gravity from the quantum to the classical regime is presented. The paradigm shift from full space-time 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 Schrodinger equation for quantum geometrodynamics that describes evolution in intrinsic time. Unitary time development with gauge-invariant temporal ordering is also viable. All Kuchar observables become physical; and classical space-time, 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 thanthemorerestrictive4-covariance ofEinstein’stheory), 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.