Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

Nano-urchin: The formation and structure of high-density spherical clusters of vanadium oxide nanotubes

We report the observation of urchin-like nanostructures consisting of high-density spherical nanotube radial arrays of vanadium oxide nanocomposite, successfully synthesized by a simple chemical route using an ethanolic solution of vanadium tri-isopropoxide and alkyl amine hexadecylamine for 7 days at 180oC. The results show that the growth process of the NanoUrchin occurs in stages, starting with a radial self-organized arrangement of lamina followed by the rolling of the lamina into nanotubes. The longest nanotubes are measured to be several micrometers in length with diameters of ~120 nm and hollow centers typically measured to be ~75 nm. The NanoUrchin have an estimated density of nanotubes of ~40 sr-1. The tube walls comprise layers of vanadium oxide with the organic surfactant intercalated between atomic layers. The interlayer distance is measured to be 2.9 ± 0.1 nm and electron diffraction identified the vanadate phase in the VOx nanocomposite as orthorhombic V2O5. These nanostructures may be used as three-dimensional composite materials and as supports for other materials.