We study cosmological tensor perturbations on a quantized background within the hybrid quantization approach. In particular, we consider a flat, homogeneous and isotropic spacetime and small tensor inhomogeneities on it. We truncate the action to second order in the perturbations. The dynamics is ruled by a homogeneous scalar constraint. We carry out a canonical transformation in the system where the Hamiltonian for the tensor perturbations takes a canonical form. The new tensor modes now admit a standard Fock quantization with a unitary dynamics. We then combine this representation with a generic quantum scheme for the homogeneous sector. We adopt a Born-Oppenheimer ansatz for the solutions to the constraint operator, previously employed to study the dynamics of scalar inhomogeneities. We analyze the approximations that allow us to recover, on the one hand, a Schrödinger equation similar to the one emerging in the dressed metric approach and, on the other hand, the ones necessary for the effective evolution equations of these primordial tensor modes within the hybrid approach to be valid. Finally, we consider loop quantum cosmology as an example where these quantization techniques can be applied and compare with other approaches.
We study the collapse in spherical symmetry of a massless scalar field minimally coupled to gravity using the semiclassical equations that are expected from loop quantum gravity. We find critical behavior of the mass as a function of the parameters of the initial data similar to that found by Choptuik in classical general relativity for a large set of initial data and values of the polymerization parameter. Contrary to wide expectations for quantum gravity, our semiclassical field equations have an exact scale invariance, as do the classical field equations. As one would then expect, we numerically find that the phase transition is second order, again as in the classical case.Choptuik [1] studied numerically the collapse of a massless, minimally coupled, scalar field coupled to general relativity. For a one parameter family of initial data he noted that there exists a critical value of the parameter. Below it, the scalar field disperses to infinity. Above it, a black hole forms through a second order phase transition. The dependence of the final mass of the black hole on the parameter of the initial data has a universal form M BH ∼ (p − p * ) β where p * is the critical value and β ∼ 0.37 is a universal exponent, independent of the choice of parameter and initial data, provided p * is non-vanishing. This critical behavior and universal scaling has been observed for several other systems (see [2] for a review). While it seemed likely that the transition would be second order as there was no natural length scale in the problem, before these numerical studies the order of the phase transition was unsettled [3]. This opens the question of how things could change in a quantum treatment of the collapse. Quantum gravity has a natural length scale, the Planck length. Indeed, previous studies of polymerized dynamics of metric general relativity seemed to suggest that the transition becomes first order [4]. Even today, a complete quantum treatment of the problem is not available.Here we study the critical collapse of massless scalar fields, minimally coupled to the semi-classical equations that stem from loop quantum gravity with spherical symmetry [5]. In it, the classical variables for gravity are given by a the spherical remnants of the triads in the radial and transverse directions E x and E ϕ and their canonically conjugate momenta K x and K ϕ . The metric of space-time can be written as,
We propose a new polymerization scheme for scalar fields coupled to gravity. It has the advantage of being a (non-bijective) canonical transformation of the fields, and therefore ensures the covariance of the theory. We study it in detail in spherically symmetric situations and compare to other approaches.
The objective of this study was the evaluation of feasibility of producing particleboard for general use using cotton gin waste generated in Argentina and urea formaldehyde resin. The chemical composition and size distribution of particles of the ginning residue as well as mechanical and physical properties of the particleboards obtained were investigated. Density and flexural strength of particleboards produced with varying levels of urea-formaldehyde resin between 8.3 and 19.3% (solid to solid ratio) were evaluated. The effect of incorporating jute reinforcement on the mechanical properties of these boards was also analyzed. Particle boards with densities between 530 and 700 kg/m3 and variable flexural strength between 0.30 and 5.85 MPa were obtained, allowing the minimum levels required for low-density boards to be reached.
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