The Compton-Schwarzschild correspondence from extended de Broglie relations

The Compton wavelength gives the minimum radius within which the mass of a particle may be localized due to quantum effects, while the Schwarzschild radius gives the maximum radius within which the mass of a black hole may be localized due to classial gravity. In a mass-radius diagram, the two lines intersect near the Planck point $(l_P,m_P)$, where quantum gravity effects become significant. Since canonical (non-gravitational) quantum mechanics is based on the concept of wave-particle duality, encapsulated in the de Broglie relations, these relations should break down near $(l_P,m_P)$. It is unclear what physical interpretation can be given to quantum particles with energy $E \gg m_Pc^2$, since they correspond to wavelengths $\lambda \ll l_P$ or time periods $T \ll t_P$ in the standard theory. We therefore propose a correction to the standard de Broglie relations, which gives rise to a modified Schr{\" o}dinger equation and a modified expression for the Compton wavelength, which may be extended into the region $E \gg m_Pc^2$. For the proposed modification, we recover the expression for the Schwarzschild radius for $E \gg m_Pc^2$ and the usual Compton formula for $E \ll m_Pc^2$. The sign of the inequality obtained from the uncertainty principle reverses at $m \approx m_P$, so that the Compton wavelength and event horizon size may be interpreted as minimum and maximum radii, respectively. We interpret the additional terms in the modified de Broglie relations as representing the self-gravitation of the wave packet.Comment: 40 pages, 7 figures, 2 appendices. Published version, with additional minor typos corrected (v3

Bose-Einstein condensates in standing waves: The cubic nonlinear Schroedinger equation with a periodic potential

We present a new family of stationary solutions to the cubic nonlinear Schroedinger equation with a Jacobian elliptic function potential. In the limit of a sinusoidal potential our solutions model a dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided the ratio of the height of the variations of the condensate to its DC offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Numerical simulations suggest such stationary states are experimentally observable.Comment: 4 pages, 4 figure

Constraints on Cosmic Strings due to Black Holes Formed from Collapsed Cosmic String Loops

The cosmological features of primordial black holes formed from collapsed cosmic string loops are studied. Observational restrictions on a population of primordial black holes are used to restrict $f$, the fraction of cosmic string loops which collapse to form black holes, and $\mu$, the cosmic string mass-per-unit-length. Using a realistic model of cosmic strings, we find the strongest restriction on the parameters $f$ and $\mu$ is due to the energy density in $100 MeV$ photons radiated by the black holes. We also find that inert black hole remnants cannot serve as the dark matter. If earlier, crude estimates of $f$ are reliable, our results severely restrict $\mu$, and therefore limit the viability of the cosmic string large-scale structure scenario.Comment: (Plain Tex, uses tables.tex -- wrapped lines corrected), 11 pages, FERMILAB-Pub-93/137-

Sub-Planckian black holes and the Generalized Uncertainty Principle

The Black Hole Uncertainty Principle correspondence suggests that there could exist black holes with mass beneath the Planck scale but radius of order the Compton scale rather than Schwarzschild scale. We present a modified, self-dual Schwarzschild-like metric that reproduces desirable aspects of a variety of disparate models in the sub-Planckian limit, while remaining Schwarzschild in the large mass limit. The self-dual nature of this solution under $M \leftrightarrow M^{-1}$ naturally implies a Generalized Uncertainty Principle with the linear form $\Delta x \sim \frac{1}{\Delta p} + \Delta p$. We also demonstrate a natural dimensional reduction feature, in that the gravitational radius and thermodynamics of sub-Planckian objects resemble that of $(1+1)$-D gravity. The temperature of sub-Planckian black holes scales as $M$ rather than $M^{-1}$ but the evaporation of those smaller than $10^{-36}$g is suppressed by the cosmic background radiation. This suggests that relics of this mass could provide the dark matter.Comment: 12 pages, 9 figures, version published in J. High En. Phy

Superfield T-duality rules

A geometric treatment of T-duality as an operation which acts on differential forms in superspace allows us to derive the complete set of T-duality transformation rules which relate the superfield potentials of D=10 type IIA supergravity with those of type IIB supergravity including Ramond-Ramond superfield potentials and fermionic supervielbeins. We show that these rules are consistent with the superspace supergravity constraints.Comment: 24 pages, latex, no figures. V2 misprints corrected. V3. One reference ([30]) and a comment on it ('Notice added') on p. 19 adde

Persistent black holes in bouncing cosmologies

In this paper we explore the idea that black holes can persist in a universe that collapses to a big crunch and then bounces into a new phase of expansion. We use a scalar field to model the matter content of such a universe {near the time} of the bounce, and look for solutions that represent a network of black holes within a dynamical cosmology. We find exact solutions to Einstein's constraint equations that provide the geometry of space at the minimum of expansion and that can be used as initial data for the evolution of hyperspherical cosmologies. These solutions illustrate that there exist models in which multiple distinct black holes can persist through a bounce, and allow for concrete computations of quantities such as the black hole filling factor. We then consider solutions in flat cosmologies, as well as in higher-dimensional spaces (with up to nine spatial dimensions). We derive conditions for the black holes to remain distinct (i.e. avoid merging) and hence persist into the new expansion phase. Some potentially interesting consequences of these models are also discussed.Comment: 37 pages, 16 figure