Regularity of the Einstein Equations at Future Null Infinity

When Einstein's equations for an asymptotically flat, vacuum spacetime are reexpressed in terms of an appropriate conformal metric that is regular at (future) null infinity, they develop apparently singular terms in the associated conformal factor and thus appear to be ill-behaved at this (exterior) boundary. In this article however we show, through an enforcement of the Hamiltonian and momentum constraints to the needed order in a Taylor expansion, that these apparently singular terms are not only regular at the boundary but can in fact be explicitly evaluated there in terms of conformally regular geometric data. Though we employ a rather rigidly constrained and gauge fixed formulation of the field equations, we discuss the extent to which we expect our results to have a more 'universal' significance and, in particular, to be applicable, after minor modifications, to alternative formulations.Comment: 43 pages, no figures, AMS-TeX. Minor revisions, updated to agree with published versio

An axisymmetric evolution code for the Einstein equations on hyperboloidal slices

We present the first stable dynamical numerical evolutions of the Einstein equations in terms of a conformally rescaled metric on hyperboloidal hypersurfaces extending to future null infinity. Axisymmetry is imposed in order to reduce the computational cost. The formulation is based on an earlier axisymmetric evolution scheme, adapted to time slices of constant mean curvature. Ideas from a previous study by Moncrief and the author are applied in order to regularize the formally singular evolution equations at future null infinity. Long-term stable and convergent evolutions of Schwarzschild spacetime are obtained, including a gravitational perturbation. The Bondi news function is evaluated at future null infinity.Comment: 21 pages, 4 figures. Minor additions, updated to agree with journal versio

Black Hole Mergers and Unstable Circular Orbits

We describe recent numerical simulations of the merger of a class of equal mass, non-spinning, eccentric binary black hole systems in general relativity. We show that with appropriate fine-tuning of the initial conditions to a region of parameter space we denote the threshold of immediate merger, the binary enters a phase of close interaction in a near-circular orbit, stays there for an amount of time proportional to logarithmic distance from the threshold in parameter space, then either separates or merges to form a single Kerr black hole. To gain a better understanding of this phenomena we study an analogous problem in the evolution of equatorial geodesics about a central Kerr black hole. A similar threshold of capture exists for appropriate classes of initial conditions, and tuning to threshold the geodesics approach one of the unstable circular geodesics of the Kerr spacetime. Remarkably, with a natural mapping of the parameters of the geodesic to that of the equal mass system, the scaling exponent describing the whirl phase of each system turns out to be quite similar. Armed with this lone piece of evidence that an approximate correspondence might exist between near-threshold evolution of geodesics and generic binary mergers, we illustrate how this information can be used to estimate the cross section and energy emitted in the ultra relativistic black hole scattering problem. This could eventually be of use in providing estimates for the related problem of parton collisions at the Large Hadron Collider in extra dimension scenarios where black holes are produced.Comment: 16 pages, 12 figures; updated to coincide with journal versio

Axisymmetric evolution of Einstein equations and mass conservation

For axisymmetric evolution of isolated systems, we prove that there exists a gauge such that the total mass can be written as a positive definite integral on the spacelike hypersurfaces of the foliation and the integral is constant along the evolution. The conserved mass integral controls the square of the extrinsic curvature and the square of first derivatives of the intrinsic metric. We also discuss applications of this result for the global existence problem in axial symmetry.Comment: A mistake in the proof of Lemma 5.1 is corrected. This version includes the Corrigendum that appears in Class. Quantum Grav. 26 (2009) 12980

Phase-resolved magnetomotive OCT for imaging nanomolar concentrations of magnetic nanoparticles in tissues

Magnetic nanoparticles (MNPs) are increasingly important in magnetic resonance and biomedical optical imaging. We describe a method for imaging MNPs by detecting nanoscale displacements using a phaseresolved spectral-domain optical coherence tomography (OCT) system. Biological tissues and phantoms are exposed to ∼800 G magnetic fields modulated at 56 and 100 Hz to mechanically actuate embedded iron oxide MNPs (∼20 nm diameter). Sensitivity to 27 μg/g (∼2 nM) MNPs within tissue phantoms is achieved by filtering paramagnetic from diamagnetic vibrations. We demonstrate biological feasibility by imaging topically applied MNPs during their diffusion into an excised rat tumor over a 2 hour time period

Scalar Field Dark Matter: behavior around black holes

We present the numerical evolution of a massive test scalar fields around a Schwarzschild space-time. We proceed by using hyperboloidal slices that approach future null infinity, which is the boundary of scalar fields, and also demand the slices to penetrate the event horizon of the black hole. This approach allows the scalar field to be accreted by the black hole and to escape toward future null infinity. We track the evolution of the energy density of the scalar field, which determines the rate at which the scalar field is being diluted. We find polynomial decay of the energy density of the scalar field, and use it to estimate the rate of dilution of the field in time. Our findings imply that the energy density of the scalar field decreases even five orders of magnitude in time scales smaller than a year. This implies that if a supermassive black hole is the Schwarzschild solution, then scalar field dark matter would be diluted extremely fastComment: 15 pages, 21 eps figures. Appendix added, accepted for publication in JCA

Gravitational perturbations of Schwarzschild spacetime at null infinity and the hyperboloidal initial value problem

We study gravitational perturbations of Schwarzschild spacetime by solving a hyperboloidal initial value problem for the Bardeen-Press equation. Compactification along hyperboloidal surfaces in a scri-fixing gauge allows us to have access to the gravitational waveform at null infinity in a general setup. We argue that this hyperboloidal approach leads to a more accurate and efficient calculation of the radiation signal than the common approach where a timelike outer boundary is introduced. The method can be generalized to study perturbations of Kerr spacetime using the Teukolsky equation.Comment: 14 pages, 9 figure