Spherically Symmetric Gravitational Collapse of Perfect Fluids

Formulating a perfect fluid filled spherically symmetric metric utilizing the 3+1 formalism for general relativity, we show that the metric coefficients are completely determined by the mass-energy distribution, and its time rate of change on an initial spacelike hypersurface. Rather than specifying Schwarzschild coordinates for the exterior of the collapsing region, we let the interior dictate the form of the solution in the exterior, and thus both regions are found to be written in one coordinate patch. This not only alleviates the need for complicated matching schemes at the interface, but also finds a new coordinate system for the Schwarzschild spacetime expressed in generalized Painleve-Gullstrand coordinates.Comment: 3 pages, To appear in the proceedings of the eleventh Marcel Grossmann meeting on general relativity (MGXI), 23-29 July, 2006, Berli

1+1+2 Electromagnetic perturbations on general LRS space-times: Regge-Wheeler and Bardeen-Press equations

We use the, covariant and gauge-invariant, 1+1+2 formalism developed by Clarkson and Barrett, and develop new techniques, to decouple electromagnetic (EM) perturbations on arbitrary locally rotationally symmetric (LRS) space-times. Ultimately, we derive 3 decoupled complex equations governing 3 complex scalars. One of these is a new Regge-Wheeler (RW) equation generalized for LRS space-times, whereas the remaining two are new generalizations of the Bardeen-Press (BP) equations. This is achieved by first using linear algebra techniques to rewrite the first-order Maxwell equations in a new complex 1+1+2 form which is conducive to decoupling. This new complex system immediately yields the generalized RW equation, and furthermore, we also derive a decoupled equation governing a newly defined complex EM 2-vector. Subsequently, a further decomposition of the 1+1+2 formalism into a 1+1+1+1 formalism is developed, allowing us to decompose the complex EM 2-vector, and its governing equations, into spin-weighted scalars, giving rise to the generalized BP equations

Spherically Symmetric Gravitational Collapse of General Fluids

We express Einstein's field equations for a spherically symmetric ball of general fluid such that they are conducive to an initial value problem. We show how the equations reduce to the Vaidya spacetime in a non-null coordinate frame, simply by designating specific equations of state. Furthermore, this reduces to the Schwarzschild spacetime when all matter variables vanish. We then describe the formulation of an initial value problem, whereby a general fluid ball with vacuum exterior is established on an initial spacelike slice. As the system evolves, the fluid ball collapses and emanates null radiation such that a region of Vaidya spacetime develops. Therefore, on any subsequent spacelike slice there exists three regions; general fluid, Vaidya and Schwarzschild, all expressed in a single coordinate patch with two free-boundaries determined by the equations. This implies complicated matching schemes are not required at the interfaces between the regions, instead, one simply requires the matter variables tend to the appropriate equations of state. We also show the reduction of the system of equations to the static cases, and show staticity necessarily implies zero ``heat flux''. Furthermore, the static equations include a generalization of the Tolman-Oppenheimer-Volkoff equations for hydrostatic equilibrium to include anisotropic stresses in general coordinates.Comment: 11 pages, 3 figures, submitted to Phys. Rev.

Gravitational collapse of spherically symmetric plasmas in Einstein-Maxwell spacetimes

We utilize a recent formulation of a spherically symmetric spacetime endowed with a general decomposition of the energy momentum tensor [Phys. Rev. D, 75, 024031 (2007)] to derive equations governing spherically symmetric distributions of electromagnetic matter. We show the system reduces to the Reissner-Nordstrom spacetime in general, spherically symmetric coordinates in the vacuum limit. Furthermore, we show reduction to the charged Vaidya spacetime in non-null coordinates when certain equations of states are chosen. A model of gravitational collapse is discussed whereby a charged fluid resides within a boundary of finite radial extent on the initial hypersurface, and is allowed to radiate charged particles. Our formalism allows for the discussion of all regions in this model without the need for complicated matching schemes at the interfaces between successive regions. As further examples we consider the collapse of a thin shell of charged matter onto a Reissner-Nordstrom black hole. Finally, we reduce the entire system of equations to the static case such that we have the equations for hydrostatic equilibrium of a charged fluid.Comment: Accepted for publication in Phys. Rev.

Two-component Bose-Einstein Condensates with Large Number of Vortices

We consider the condensate wavefunction of a rapidly rotating two-component Bose gas with an equal number of particles in each component. If the interactions between like and unlike species are very similar (as occurs for two hyperfine states of $^{87}$Rb or $^{23}$Na) we find that the two components contain identical rectangular vortex lattices, where the unit cell has an aspect ratio of $\sqrt{3}$, and one lattice is displaced to the center of the unit cell of the other. Our results are based on an exact evaluation of the vortex lattice energy in the large angular momentum (or quantum Hall) regime.Comment: 4 pages, 2 figures, RevTe

1+1+2 Electromagnetic perturbations on non-vacuum LRS class II space-times: Decoupling scalar and 2-vector harmonic amplitudes

We use the covariant and gauge-invariant 1+1+2 formalism of Clarkson and Barrett \cite{Clarkson2003} to analyze electromagnetic (EM) perturbations on non-vacuum {\it locally rotationally symmetric} (LRS) class II space-times. Ultimately, we show how to derive six real decoupled equations governing the total of six EM scalar and 2-vector harmonic amplitudes. Four of these are new, and result from expanding the complex EM 2-vector which we defined in \cite{Burston2007} in terms of EM 2-vector harmonic amplitudes. We are then able to show that there are four precise combinations of the amplitudes that decouple, two of these are polar perturbations whereas the remaining two are axial. The remaining two decoupled equations are the generalized Regge-Wheeler equations which were developed previously in \cite{Betschart2004}, and these govern the two EM scalar harmonic amplitudes. However, our analysis generalizes this by including a full description and classification of energy-momentum sources, such as charges and currents.Comment: 9 page

Local Spin-Gauge Symmetry of the Bose-Einstein Condensates in Atomic Gases

The Bose-Einstein condensates of alkali atomic gases are spinor fields with local ``spin-gauge" symmetry. This symmetry is manifested by a superfluid velocity ${\bf u}_{s}$ (or gauge field) generated by the Berry phase of the spin field. In ``static" traps, ${\bf u}_{s}$ splits the degeneracy of the harmonic energy levels, breaks the inversion symmetry of the vortex nucleation frequency ${\bf \Omega}_{c1}$, and can lead to {\em vortex ground states}. The inversion symmetry of ${\bf \Omega}_{c1}$, however, is not broken in ``dynamic" traps. Rotations of the atom cloud can be generated by adiabatic effects without physically rotating the entire trap.Comment: Typos in the previous version corrected, thanks to the careful reading of Daniel L. Cox. 13 pages + 2 Figures in uuencode + gzip for