Redshifts and Killing Vectors

Courses in introductory special and general relativity have increasingly become part of the curriculum for upper-level undergraduate physics majors and master's degree candidates. One of the topics rarely discussed is symmetry, particularly in the theory of general relativity. The principal tool for its study is the Killing vector. We provide an elementary introduction to the concept of a Killing vector field, its properties, and as an example of its utility apply these ideas to the rigorous determination of gravitational and cosmological redshifts.Comment: 16 Latex pages, 6 postscript figures, submitted to Am. J. Phy

Evanescent Black Holes

A renormalizable theory of quantum gravity coupled to a dilaton and conformal matter in two space-time dimensions is analyzed. The theory is shown to be exactly solvable classically. Included among the exact classical solutions are configurations describing the formation of a black hole by collapsing matter. The problem of Hawking radiation and backreaction of the metric is analyzed to leading order in a $1/N$ expansion, where $N$ is the number of matter fields. The results suggest that the collapsing matter radiates away all of its energy before an event horizon has a chance to form, and black holes thereby disappear from the quantum mechanical spectrum. It is argued that the matter asymptotically approaches a zero-energy ``bound state'' which can carry global quantum numbers and that a unitary $S$-matrix including such states should exist.Comment: 14 page

Thermal divergences on the event horizons of two-dimensional black holes

The expectation value of the stress-energy tensor \langleT_{\mu\nu}\rangle of a free conformally invariant scalar field is computed in a general static two-dimensional black hole spacetime when the field is in either a zero temperature vacuum state or a thermal state at a nonzero temperature. It is found that for every static two-dimensional black hole the stress-energy diverges strongly on the event horizon unless the field is in a state at the natural black hole temperature which is defined by the surface gravity of the event horizon. This implies that both extreme and nonextreme two-dimensional black holes can only be in equilibrium with radiation at the natural black hole temperature.Comment: 13 pages, REVTe

A rotating black ring in five dimensions

The vacuum Einstein equations in five dimensions are shown to admit a solution describing an asymptotically flat spacetime regular on and outside an event horizon of topology S^1 x S^2. It describes a rotating ``black ring''. This is the first example of an asymptotically flat vacuum solution with an event horizon of non-spherical topology. There is a range of values for the mass and angular momentum for which there exist two black ring solutions as well as a black hole solution. Therefore the uniqueness theorems valid in four dimensions do not have simple higher dimensional generalizations. It is suggested that increasing the spin of a five dimensional black hole beyond a critical value results in a transition to a black ring, which can have an arbitrarily large angular momentum for a given mass.Comment: 4 pages, 3 figures; v2: minor improvement

Semiclassical Stability of the Extreme Reissner-Nordstrom Black Hole

The stress-energy tensor of a free quantized scalar field is calculated in the extreme Reissner-Nordstr\"{o}m black hole spacetime in the zero temperature vacuum state. The stress-energy appears to be regular on the event horizon, contrary to the suggestion provided by two-dimensional calculations. An analytic calculation on the event horizon for a thermal state shows that if the temperature is nonzero then the stress-energy diverges strongly there.Comment: 10 pages, REVTeX, 4 figures in separate uuencoded compressed fil

Calibrated Sub-Bundles in Non-Compact Manifolds of Special Holonomy

This paper is a continuation of math.DG/0408005. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S^n by looking at the conormal bundle of appropriate submanifolds of S^n. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey-Lawson for submanifolds in R^n in their pioneering paper. We also construct calibrated submanifolds in complete metrics with special holonomy G_2 and Spin(7) discovered by Bryant and Salamon on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy.Comment: 20 pages; for Revised Version: Minor cosmetic changes, some paragraphs rewritten for improved clarit

Cosmological Consequences of Spontaneous Lepton Number Violation in SO(10)SO(10) Grand Unification, EFI-93-07

Cosmological constraints on grand unified theories with spontaneous lepton number violation are analysed. We concentrate on $SO(10)$, the simplest of the models possessing this property. It has been noted previously that the consistency of these models with the observed baryon asymmetry generically implies strict upper bounds on the light neutrino masses. In this paper, we analyze the situation in detail. We find that minimal models of fermion masses face difficulties, but that it is possible for these models to generate an adequate baryon asymmetry via non-equilibrium lepton number violating processes when the right-handed neutrino masses are near their maximum possible values. This condition uniquely picks out the minimal gauge symmetry breaking scheme. A non-minimal model is also analyzed, with somewhat different conclusions due to the nature of the imposed symmetries.Comment: uses harvmac.tex, epsf.tex, and tables.tex; 4 figures submitted as tar-compressed-uuencoded postscript files; beautiful compressed postscript version available by anonymous ftp from rainbow.uchicago.edu:/pt-preprints/efi-93-07.ps.