Adaptive mesh and geodesically sliced Schwarzschild spacetime in 3+1 dimensions

We present first results obtained with a 3+1 dimensional adaptive mesh code in numerical general relativity. The adaptive mesh is used in conjunction with a standard ADM code for the evolution of a dynamically sliced Schwarzschild spacetime (geodesic slicing). We argue that adaptive mesh is particularly natural in the context of general relativity, where apart from adaptive mesh refinement for numerical efficiency one may want to use the built in flexibility to do numerical relativity on coordinate patches.Comment: 21 pages, LaTeX, 7 figures included with eps

Binary black hole initial data from matched asymptotic expansions

We present an approximate metric for a binary black hole spacetime to construct initial data for numerical relativity. This metric is obtained by asymptotically matching a post-Newtonian metric for a binary system to a perturbed Schwarzschild metric for each hole. In the inner zone near each hole, the metric is given by the Schwarzschild solution plus a quadrupolar perturbation corresponding to an external tidal gravitational field. In the near zone, well outside each black hole but less than a reduced wavelength from the center of mass of the binary, the metric is given by a post-Newtonian expansion including the lowest-order deviations from flat spacetime. When the near zone overlaps each inner zone in a buffer zone, the post-Newtonian and perturbed Schwarzschild metrics can be asymptotically matched to each other. By demanding matching (over a 4-volume in the buffer zone) rather than patching (choosing a particular 2-surface in the buffer zone), we guarantee that the errors are small in all zones. The resulting piecewise metric is made formally $C^\infty$ with smooth transition functions so as to obtain the finite extrinsic curvature of a 3-slice. In addition to the metric and extrinsic curvature, we present explicit results for the lapse and the shift, which can be used as initial data for numerical simulations. This initial data is not accurate all the way to the asymptotically flat ends inside each hole, and therefore must be used with evolution codes which employ black hole excision rather than puncture methods. This paper lays the foundations of a method that can be sraightforwardly iterated to obtain initial data to higher perturbative order.Comment: 24 pages, 15 figures. Replaced with published version. Major editing of text, no major change to the physic

Beyond the Bowen-York extrinsic curvature for spinning black holes

It is well-known that Bowen-York initial data contain spurious radiation. Although this ``junk'' radiation has been seen to be small for non-spinning black-hole binaries in circular orbit, its magnitude increases when the black holes are given spin. It is possible to reduce the spurious radiation by applying the puncture approach to multiple Kerr black holes, as we demonstrate for examples of head-on collisions of equal-mass black-hole binaries.Comment: 10 pages, 2 figures, submitted to special "New Frontiers in Numerical Relativity" issue of Classical and Quantum Gravit

Quasi-equilibrium binary black hole sequences for puncture data derived from helical Killing vector conditions

We construct a sequence of binary black hole puncture data derived under the assumptions (i) that the ADM mass of each puncture as measured in the asymptotically flat space at the puncture stays constant along the sequence, and (ii) that the orbits along the sequence are quasi-circular in the sense that several necessary conditions for the existence of a helical Killing vector are satisfied. These conditions are equality of ADM and Komar mass at infinity and equality of the ADM and a rescaled Komar mass at each puncture. In this paper we explicitly give results for the case of an equal mass black hole binary without spin, but our approach can also be applied in the general case. We find that up to numerical accuracy the apparent horizon mass also remains constant along the sequence and that the prediction for the innermost stable circular orbit is similar to what has been found with the effective potential method.Comment: 6 pages, 3 figures, 1 tabl

Plunge waveforms from inspiralling binary black holes

We study the coalescence of non-spinning binary black holes from near the innermost stable circular orbit down to the final single rotating black hole. We use a technique that combines the full numerical approach to solve Einstein equations, applied in the truly non-linear regime, and linearized perturbation theory around the final distorted single black hole at later times. We compute the plunge waveforms which present a non negligible signal lasting for $t\sim 100M$ showing early non-linear ringing, and we obtain estimates for the total gravitational energy and angular momentum radiated.Comment: Corrected typos in the radiated ang momentum and frequenc

Numerical evolution of multiple black holes with accurate initial data

We present numerical evolutions of three equal-mass black holes using the moving puncture approach. We calculate puncture initial data for three black holes solving the constraint equations by means of a high-order multigrid elliptic solver. Using these initial data, we show the results for three black hole evolutions with sixth-order waveform convergence. We compare results obtained with the BAM and AMSS-NCKU codes with previous results. The approximate analytic solution to the Hamiltonian constraint used in previous simulations of three black holes leads to different dynamics and waveforms. We present some numerical experiments showing the evolution of four black holes and the resulting gravitational waveform.Comment: Published in PR

Chern-Simons functional and the no-boundary proposal in Bianchi IX quantum cosmology

The Chern-Simons functional $S_{\rm CS}$ is an exact solution to the Ashtekar-Hamilton-Jacobi equation of general relativity with a nonzero cosmological constant. In this paper we consider $S_{\rm CS}$ in Bianchi type IX cosmology with $S^3$ spatial surfaces. We show that among the classical solutions generated by~$S_{\rm CS}$, there is a two-parameter family of Euclidean spacetimes that have a regular NUT-type closing. When two of the three scale factors are equal, these spacetimes reduce to a one-parameter family within the Euclidean Taub-NUT-de~Sitter metrics. For a nonzero cosmological constant, $\exp(iS_{\rm CS})$ therefore provides a semiclassical estimate to the Bianchi~IX no-boundary wave function in Ashtekar's variables.Comment: 9 pages, REVTeX v3.0. (One reference added.

Numerical black hole initial data with low eccentricity based on post-Newtonian orbital parameters

Black hole binaries on non-eccentric orbits form an important subclass of gravitational wave sources, but it is a non-trivial issue to construct numerical initial data with minimal initial eccentricity for numerical simulations. We compute post-Newtonian orbital parameters for quasi-spherical orbits using the method of Buonanno, Chen and Damour (2006) and examine the resulting eccentricity in numerical simulations. Four different methods are studied resulting from the choice of Taylor-expanded or effective-one-body Hamiltonians, and from two choices for the energy flux. The eccentricity increases for unequal masses and for spinning black holes, but remains smaller than that obtained from previous post-Newtonian approaches. The effective-one-body Hamiltonian offers advantages for decreasing initial separation as expected, but in the context of this study also performs significantly better than the Taylor-expanded Hamiltonian for binaries with spin.Comment: 20 pages, 11 figures, pdflatex; V2 with minor changes according to published versio

The 3D Grazing Collision of Two Black Holes

We present results for two colliding black holes (BHs), with angular momentum, spin, and unequal mass. For the first time gravitational waveforms are computed for a grazing collision from a full 3D numerical evolution. The collision can be followed through the merger to form a single BH, and through part of the ringdown period of the final BH. The apparent horizon is tracked and studied, and physical parameters, such as the mass of the final BH, are computed. The total energy radiated in gravitational waves is shown to be consistent with the total mass of the spacetime and the final BH mass. The implication of these simulations for gravitational wave astronomy is discussed.Comment: 4 pages, 7 figures, revte

Os isotopic composition and Os and Re distribution in the active mound of the TAG hydrothermal system, Mid-Atlantic Ridge

Drilling during ODP Leg 158 took place on the active mound of the TAG hydrothermal field on the Mid-Atlantic Ridge. The dominant mineral precipitating from the hydrothermal fluid is pyrite. Its Re and Os concentration and the Os isotopic composition provide constraints on the nature of the hydrothermal fluid circulating in the TAG mound. The 187Os/186Os ratios of massive pyrite samples vary from 4.9 to 8.9. The highest ratios have been observed in the upper part of the sulfide mountain (<20 mbsf) and the lowest in the stockwork zone at ~80 mbsf. This range of Os isotopic compositions is likely the result of mixing of seawater with hydrothermal fluid. The Os concentrations are very low, ranging from 0.04 to 4.2 ppt, and the massive pyrite zone at the top of the mound is enriched in Os relative to the interior of the hydrothermal system. A hyperbolic relationship between Os isotopic composition and Os concentration reflects the systematic addition of seawater-derived Os to the hydrothermal Os component at stratigraphically shallower levels. From this relationship it is estimated that pyrite precipitating from the hydrothermal fluid contains 0.02 to 0.04 ppt Os provided the 187Os/186Os value of the fluid ranges from about 1.3 to 4.7. Because of the great mobility of Os in the high-temperature hydrothermal system, it is assumed that its partition coefficient between pyrite and hydrothermal fluid is <1. This implies that the hydrothermal fluid contains more than 0.02 ppt Os. The occurrence of anhydrite-rich lithologies at ~30–40 mbsf corroborates that seawater is penetrating the hydrothermal system and contaminating the hydrothermal fluid circulating in the upper part of the mound. This partly explains why the Os of sulfides that precipitated above this level has a strong seawater-like isotopic signature. In addition, the massive pyrite zone of the upper part of the TAG mound formed by accumulation of sulfides derived from chimneys and the fall-out material of the hydrothermal plume above the TAG field. Both sulfide components formed during mixing of seawater and hydrothermal fluid and their Os should also have a distinct seawater component. These processes, especially the entrainment of seawater, appear to control the distribution of Os and Re within the hydrothermal system. The Os enrichment in the upper part of the mound can be explained if the element is co-precipitated with sulfides or adsorbed on mineral surface during the accumulation of sulfides on the TAG mound. As a significant amount of Os can be dissolved in the hydrothermal fluid, remobilization of Os within the hydrothermal system could lead to further Os enrichment at the top of the mound but to very low Os concentrations in the stockwork zone. The Re concentrations indicate a distribution opposite to that of Os: the highest concentrations of about 60 ppb have been observed more than 15 mbsf, but the concentrations decrease from 50 to 2 ppb in samples from the top of the sulfide mound (<15 mbsf). The behavior of Re appears to be controlled by the redox conditions in the TAG hydrothermal system, which, in turn, could be determined by the relative proportions of oxidized seawater and reduced hydrothermal fluid. Deep within the mound, where the hydrothermal fluid component dominates, Re is rather immobile and becomes concentrated. In the upper part of the sulfide mound where larger quantities of seawater mix with the fluid, however, the redox potential should be more oxidizing, and Re would be more soluble and released to the ocean