A Time Independent Energy Estimate for Outgoing Scalar Waves in the Kerr Geometry

The Cauchy problem for the scalar wave equation in the Kerr geometry is considered, with initial data which is smooth and compactly supported outside the event horizon. A time-independent energy estimate for the outgoing wave is obtained. As an application we estimate the outgoing energy for wave-packet initial data, uniformly as the support of the initial data is shifted to infinity. The main mathematical tool is our previously derived integral representation of the wave propagator.Comment: 31 pages, LaTeX, minor changes (published version

A Rigorous Treatment of Energy Extraction from a Rotating Black Hole

The Cauchy problem is considered for the scalar wave equation in the Kerr geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting supperradiance, the wave analogue of the Penrose process, into a rigorous mathematical framework. We quantify the maximal energy gain. We also compute the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulou's result for the Penrose process. The main mathematical tool is our previously derived integral representation of the wave propagator.Comment: 19 pages, LaTeX, proof of Propositions 2.3 and 3.1 given in more detai

Quantum Oscillations Can Prevent the Big Bang Singularity in an Einstein-Dirac Cosmology

We consider a spatially homogeneous and isotropic system of Dirac particles coupled to classical gravity. The dust and radiation dominated closed Friedmann-Robertson-Walker space-times are recovered as limiting cases. We find a mechanism where quantum oscillations of the Dirac wave functions can prevent the formation of the big bang or big crunch singularity. Thus before the big crunch, the collapse of the universe is stopped by quantum effects and reversed to an expansion, so that the universe opens up entering a new era of classical behavior. Numerical examples of such space-times are given, and the dependence on various parameters is discussed. Generically, one has a collapse after a finite number of cycles. By fine-tuning the parameters we construct an example of a space-time which is time-periodic, thus running through an infinite number of contraction and expansion cycles.Comment: 8 pages, LaTeX, 4 figures, statement on energy conditions correcte

A Discussion on Dirac Field Theory, No-Go Theorems and Renormalizability

We study Dirac field equations coupled to electrodynamics with metric and torsion fields: we discuss how special spinorial solutions are incompatible with torsion; eventually these results will be used to sketch a discussion on the problem of renormalizability of point-like particles.Comment: 10 page

The Dirac Equation and the Normalization of its Solutions in a Closed Friedmann-Robertson-Walker Universe

We set up the Dirac equation in a Friedmann-Robertson-Walker geometry and separate the spatial and time variables. In the case of a closed universe, the spatial dependence is solved explicitly, giving rise to a discrete set of solutions. We compute the probability integral and analyze a space-time normalization integral. This analysis allows us to introduce the fermionic projector in a closed Friedmann-Robertson-Walker geometry and to specify its global normalization as well as its local form.Comment: 22 pages, LaTeX, sign error in equation (3.7) correcte

The repulsive nature of naked singularities from the point of view of Quantum Mechanics

We use the Dirac equation coupled to a background metric to examine what happens to quantum mechanical observables like the probability density and the radial current in the vicinity of a naked singularity of the Reissner-Nordstr\"{o}m type. We find that the wave function of the Dirac particle is regular in the point of the singularity. We show that the probability density is exactly zero at the singularity reflecting quantum-mechanically the repulsive nature of the naked singularity. Furthermore, the surface integral of the radial current over a sphere in the vicinity of the naked singularity turns out to be also zero.Comment: 11 page

Some recent progress in classical general relativity

In this short survey paper, we shall discuss certain recent results in classical gravity. Our main attention will be restricted to two topics in which we have been involved; the positive mass conjecture and its extensions to the case with horizons, including the Penrose conjecture (Part I), and the interaction of gravity with other force fields and quantum-mechanical particles (Part II). © 2000 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70615/2/JMAPAQ-41-6-3943-1.pd

Particle-Like Solutions of the Einstein-Dirac Equations

The coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state are derived. Using numerical methods, we construct an infinite number of soliton-like solutions of these equations. The stability of the solutions is analyzed. For weak coupling (i.e., small rest mass of the fermions), all the solutions are linearly stable (with respect to spherically symmetric perturbations), whereas for stronger coupling, both stable and unstable solutions exist. For the physical interpretation, we discuss how the energy of the fermions and the (ADM) mass behave as functions of the rest mass of the fermions. Although gravitation is not renormalizable, our solutions of the Einstein-Dirac equations are regular and well-behaved even for strong coupling.Comment: 31 pages, LaTeX, 21 PostScript figures, some references adde

Tires, Tracks, and Tethering: Idaho Steep Slope Harvesting

Steep slope timber harvesting often falls under scrutiny of labor, safety, and operational challenges, but is beginning to advance past these barriers through substantial technological progression. Across previous decades, large advancements of technology have occurred in ground-based timber harvesting systems, giving mechanized options to every phase of timber harvesting. These progressions have created outcomes including, but not limited to, improved worker safety and reduced risk, increased productivity and reduced harvest cost, while also increasing consistent harvest output through seasonal conditions. Timber harvesting methods on steep slopes historically involved motor-manual tree felling and labor-intensive extraction, but are now giving way to mechanization in steep slope harvesting. Tether-assist technology is now bringing the decades of progression from groundbased harvesting systems onto steep slopes. With the ability of ground-based harvesting systems to now traverse slopes steeper than previously possible, there is much to learn of their impacts and relationships with the landscape. Previously developed state and federal government policies in the Pacific Northwest (PNW), within the United States of America (USA), limit ground-based harvesting equipment on steep slopes. Different levels of regulation come into application by means of restriction for traditional ground based harvesting equipment above specified slope thresholds, in addition to extra requirements and some restrictions for tether-assist technology. This research is a case study showcasing soil impacts of traditional steep slope cable harvesting systems alongside developing tether-assist ground-based harvesting systems in similar terrain and timber conditions. Felling methods vary from motor-manual to mechanized directional felling head, while extraction methods incorporate grapple skidder, shovel, and cable logging, each exhibiting a different interaction with the site. Pre-harvest and post-harvest observations were collected of bulk density and penetrometer resistance for impact characterization and comparison. Bulk density measures work to capture differences in top-soil disturbance, while penetrometer resistance captures soil profile differences at increased depths. Sampling consisted of pre-operation and post-operation measurements taken at repeated locations on an established grid, allowing for paired testing of observations. The results from this study have shown differences in harvest system and operational area impacts, with each configuration contributing a unique distribution of soil impact to the harvest area. Through a variety of cable, tracked, and rubber tire equipment, this is to be expected due to the differing contact relationships and payload interactions with the soils in the harvest area. Machine passes and spatial distribution of machine activity was also found to be variable between harvest system configurations. These differing outcomes led to support traditional trends found in ground-based harvesting soil disturbance studies, with grapple skidding exhibiting the greatest impacts followed by shovel, and cable logging. Although trends in the data led to this comparative conclusion, significant differences were not found between either of the tether-assisted skidder or shovel systems. Further development of tethered logging system research is necessary, as trends may be similar to flat ground, yet additional forces via tether tension and extra payload may be entering new magnitudes