Static Bodies Research Articles

The Hoop Conjecture on black holes

We study the minimum sizes of some static bodies made of charged perfect fluid.

Geometric constraints on nonsingular, momentarily static, axisymmetric systems in general relativity

This paper attempts to examine the relationship between system size and gravitational collapse for the case of axial symmetry. The approach here is to construct noncollapsing systems, with momentarily static matter interiors and static vacuum exteriors, and to find limitations on the validity of the construction. Specifically, the exteriors are static, axisymmetric, asymptotically flat, vacuum geometries, described by Weyl solutions of the Einstein field equations. These solutions have singular sources (naked singularities, except for the Schwarzschild solution); here, regions of the Weyl solutions containing the singularities are replaced by momentarily static material bodies. These are described by axisymmetric solutions of Brill's time-symmetric initial-value equation with non-negative energy density, joining smoothly to the Weyl geometries at the bodies' boundaries. The consistency requirements of such a construction limit the choice of surfaces in the exterior geometry suitable as matter/vacuum boundaries; general constraints on the boundary location and geometry are derived here. For the explicit examples of the GAMMA metric and the Bach-Weyl ring metric as exteriors, these constraints forbid the boundary surface to be arbitrarily near the Weyl singularity. The ''hoop conjecture'' demands, roughly, that the largest circumference of the boundary surface of such a noncollapsing system always exceed a limit ofmore » the order of the system's mass. The specific examples studied here are all consistent with the hoop conjecture, but they show that the boundary constraints derived in this paper are not in general related to boundary surface size and thus that these constraints do not embody the hoop conjecture.« less

Physical interpretation of N.U.T. metric

AbstractA metric is constructed, which represents the gravitational field of two spherical static bodies of masses +b and −b, together with a massless source of angular momentum between them. When the separation of the two masses is allowed to tend to infinity the N.U.T. metric can be obtained from the limiting process. This supports the view that the N.U.T. solution represents a static Schwarzschild body together with a rotating semi-infinite spike of pure angular momentum.

Similarities, differences, and some genetic problems of the wyoming and colorado plateau types of uranium deposits in sandstone

Uranium deposits of the Wyoming roll type and the Colorado Plateau peneconcordant type are the principal domestic resources of uranium. Both types occur in lenticular sandstone beds of continental origin, have a similar suite of elements and minerals, are associated with mildly altered rock, and are thought to have formed by reduction precipitation from ground waters before significant regional deformation. They differ in several respects. The Wyoming deposits are in unlithified sandstone that is highly arkosic, whereas the Colorado Plateau deposits are in lithified sandstone that is only slightly to moderately arkosic. Uranium is the only ore metal in the Wyoming deposits, whereas vanadium or copper is more abundant than uranium in some of the Plateau deposits. The Wyoming deposits are elongate crescent-shaped bodies that extend vertically through, or partly through, a sandstone unit and which are scattered, like widely spaced beads on a string, along miles-long interfaces between oxidized (altered) and unoxidized sandstone, whereas the Plateau deposits are thin tabular layers that are nearly concordant to bedding and which occur as discrete bodies, like raisins in raisin bread, enveloped in rock altered by reduction. The Wyoming ore rolls and interfaces were dynamic, having been pushed downdip by downward-moving oxygen-bearing water that passed through the interfaces and deposited the ore minerals on the reducing side, whereas the Plateau deposits seemingly formed as static bodies, localized by intensive reducing patches in a mildly reducing environment. These differences focus attention on genetic problems relating to the Eh of the ore-bearing and altering solutions, the shape and localization of deposits, and the source of the uranium.

A CRYSTALLOGRAPHIC STUDY OF VERTEBRATE OTOLITHS

1. The crystallographic properties of otoliths from 58 vertebrate species were investigated by means of polarization microscopy and x-ray diffraction.2. The otoliths occur as statoconia, microstatoliths, and statoliths. Usually only one kind is present in the labyrinth but in some vertebrates a combination of two or all three types may be found.3. While statoliths and microstatoliths always are polycrystalline, statoconia may either be polycrystalline or single crystals.4. Five different minerals, viz., apatite, calcite, aragonite, vaterite and calcium carbonate monohydrate, compose the endogenous otoliths in the vertebrate labycarbonate monohydrate, compose the endogenous otoliths in the vertebrate labyrinth. Some elasmobranchs have in addition exogenous statoconia consisting of sea-sand.5. The distribution of statoliths and/or statoconia, their texture and their composition within the vertebrate series, show remarkable consistencies; within each class the same kind of static bodies is usually present.6. Some crystallographic and biological aspects are discussed and it is suggested that the findings may be of some value for the study of the phylogeny of vertebrates and also may aid towards a better understanding of the function of the otolith organ.