Narrow Results

The purpose of this paper is to demonstrate a new method of generating exact solutions to Einstein’s equations obtained by the Hamiltonian reduction. The key element to the successful Hamiltonian reduction is finding the privileged spacetime coordinates in which physical degrees of freedom manifestly reside in the conformal two-metric, and all the other metric components are determined by the conformal two-metric. In the privileged coordinates, Einstein’s constraint equations become trivial; the Hamiltonian and momentum constraints are simply the defining equations of a nonvanishing gravitational Hamiltonian and momentum densities in terms of conformal two-metric and its conjugate momentum, respectively. Thus, given any conformal two-metric, which is a constraint-free data, one can construct the whole four-dimensional spacetime by integrating the first-order superpotential equations. As the first examples of using Hamiltonian reduction in solving Einstein’s equations, we found two exact solutions to Einstein’s equations in the privileged coordinates. Suitable coordinate transformations from the privileged to the standard coordinates show that they are just the Einstein–Rosen wave and the Schwarzschild solution. The local gravitational Hamiltonian and momentum densities of these spacetimes are also presented in the privileged coordinates.

A system consisting of a point particle coupled to gravity is investigated. The set of constraints is derived. It was found that a suitable superposition of those constraints is the generator of the infinitesimal transformations of the time coordinate $t\equiv x^0$ and serves as the Hamiltonian which gives the correct equations of motion. Besides that, the system satisfies the mass shell constraint, $p^{\mu }{p}_{\mu }-m^2=0$, which is the generator of the worldline reparametrizations, where the momenta $p_{\mu }$, $\mu =0,1,2,3$, generate infinitesimal changes of the particle’s position $X^{\mu }$ in spacetime. Consequently, the Hamiltonian contains $p_0$, which upon quantization becomes the operator $-i\partial /\partial T$, occurring on the right-hand side of the Wheeler–DeWitt equation. Here, the role of time has the particle coordinate $X^0\equiv T$, which is a distinct concept than the spacetime coordinate $x^0\equiv t$. It is also shown how the ordering ambiguities can be avoided if a quadratic form of the momenta is cast into the form that instead of the metric contains the basis vectors.

The cosmological implications of an evolutionary quantum gravity are analyzed in the context of a generic inhomogeneous model. The Schrödinger problem is formulated and solved in the presence of a scalar field, an ultrarelativistic matter and a perfect gas regarded as the dust-clock. Considering the actual phenomenology, it is shown how the evolutionary approach overlaps the Wheeler-DeWitt one.

The problem of time is an unsolved issue of canonical General Relativity. A possible solution is the Brown-Kuchař mechanism which couples matter to the gravitational field and recovers a physical, i.e. non vanishing, observable Hamiltonian functional by manipulating the set of constraints. Two cases are analyzed. A generalized scalar fluid model provides an evolutionary picture, but only in a singular case. The Schutz' model provides an interesting singularity free result: the entropy per baryon enters the definition of the physical Hamiltonian. Moreover in the co-moving frame one is able to identify the time variable τ with the logarithm of entropy.

We present a novel solution to the low entropy and arrow of time puzzles of the initial state of the universe. Our approach derives from the physics of a specific generalization of Matrix theory put forth in earlier work as the basis for a quantum theory of gravity. The particular dynamical state space of this theory, the infinite-dimensional analogue of the Fubini–Study metric over a complex nonlinear Grassmannian, has recently been studied by Michor and Mumford. The geodesic distance between any two points on this space is zero. Here we show that this mathematical result translates to a description of a hot, zero entropy state and an arrow of time after the Big Bang. This is modeled as a far from equilibrium, large fluctuation driven, "freezing by heating" metastable ordered phase transition of a nonlinear dissipative dynamical system.

In this paper, we will first derive the Wheeler–DeWitt equation for the generalized geometry which occurs in M-theory. Then we will observe that M2-branes act as probes for this generalized geometry, and as M2-branes have an extended structure, their extended structure will limits the resolution to which this generalized geometry can be defined. We will demonstrate that this will deform the Wheeler–DeWitt equation for the generalized geometry. We analyze such a deformed Wheeler–DeWitt equation in the minisuperspace approximation, and observe that this deformation can be used as a solution to the problem of time. This is because this deformation gives rise to time crystals in our universe due to the spontaneous breaking of time reparametrization invariance.

An approach to the quantization of gravity in the presence of matter is examined which starts from the classical Einstein–Hilbert action and matter approximated by “point” particles minimally coupled to the metric. Upon quantization, the Hamilton constraint assumes the form of the Schrödinger equation: it contains the usual Wheeler–DeWitt term and the term with the time derivative of the wave function. In addition, the wave function also satisfies the Klein–Gordon equation, which arises as the quantum counterpart of the constraint among particles’ momenta. Comparison of the novel approach with the usual one in which matter is represented by scalar fields is performed, and shown that those approaches do not exclude, but complement each other. In final discussion it is pointed out that the classical matter could consist of superparticles or spinning particles, described by the commuting and anticommuting Grassmann coordinates, in which case spinor fields would occur after quantization.

In quantum gravity, one seeks to combine quantum mechanics and general relativity. In attempting to do so, one comes across the "problem of time" impasse: the notion of time is conceptually different in each of these theories. In this paper, I consider the timeless records approach to resolving this. Records are localized, information-containing subconfigurations of a single instant. Records theory is the study of these and of how science (or history) is to be abstracted from correlations between them. I critically evaluate motivations for this approach that have previously appeared in the literature. I provide a ground-level structure for records theory and discuss what kind of further tools are needed, illustrated with some toy models: ordinary mechanics, relational particle dynamics, detector models and inhomogeneous perturbations about homogeneous cosmology.

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.

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.

I apply the Hamiltonian reduction procedure to spacetimes of 4 dimensions without isometries in the (2+2) formalism and find privileged spacetime coordinates in which the physical Hamiltonian is solely expressed in terms of the conformal two metric and its conjugate momentum. Physical time is the area element of the spatial cross-section of null hypersurfaces, and the physical radial coordinate is defined by equipotential surfaces on a given spacelike hypersurface of constant physical time. The physical Hamiltonian is local and positive in the privileged coordinates. I present the complete set of Hamilton’s equations and find that they coincide with the Einstein’s equations written in the privileged coordinates. This shows that the Hamiltonian reduction is self-consistent and respects general covariance.

I apply the prescription of Hamiltonian reduction first proposed by Arnowitt-Misner-Deser to general spacetimes of 4 dimensions in the (2+2) formalism and find privileged spacetime coordinates in which the physical Hamiltonian is expressed in terms of the conformal two metric and its conjugate momentum. Physical time is the area element of the spatial cross section of null hypersurfaces. I present the complete set of Einstein’s equations in the privileged spacetime coordinates in a constraint-free form.

The following sections are included:

• Introduction

• The Smallest RPM Model of Quantum Cosmology

• Conclusion

• References