Extrinsic Curvature and the Einstein Constraints

The Einstein initial-value equations in the extrinsic curvature (Hamiltonian) representation and conformal thin sandwich (Lagrangian) representation are brought into complete conformity by the use of a decomposition of symmetric tensors which involves a weight function. In stationary spacetimes, there is a natural choice of the weight function such that the transverse traceless part of the extrinsic curvature (or canonical momentum) vanishes.Comment: 8 pages, no figures; added new section; significant polishing of tex

Comparing Criteria for Circular Orbits in General Relativity

We study a simple analytic solution to Einstein's field equations describing a thin spherical shell consisting of collisionless particles in circular orbit. We then apply two independent criteria for the identification of circular orbits, which have recently been used in the numerical construction of binary black hole solutions, and find that both yield equivalent results. Our calculation illustrates these two criteria in a particularly transparent framework and provides further evidence that the deviations found in those numerical binary black hole solutions are not caused by the different criteria for circular orbits.Comment: 4 pages; to appear in PRD as a Brief Report; added and corrected reference

Corotating and irrotational binary black holes in quasi-circular orbits

A complete formalism for constructing initial data representing black-hole binaries in quasi-equilibrium is developed. Radiation reaction prohibits, in general, true equilibrium binary configurations. However, when the timescale for orbital decay is much longer than the orbital period, a binary can be considered to be in quasi-equilibrium. If each black hole is assumed to be in quasi-equilibrium, then a complete set of boundary conditions for all initial data variables can be developed. These boundary conditions are applied on the apparent horizon of each black hole, and in fact force a specified surface to be an apparent horizon. A global assumption of quasi-equilibrium is also used to fix some of the freely specifiable pieces of the initial data and to uniquely fix the asymptotic boundary conditions. This formalism should allow for the construction of completely general quasi-equilibrium black hole binary initial data.Comment: 13 pages, no figures, revtex4; Content changed slightly to reflect fact that regularized shift solutions do satisfy the isometry boundary condition

Gauge conditions for binary black hole puncture data based on an approximate helical Killing vector

We show that puncture data for quasicircular binary black hole orbits allow a special gauge choice that realizes some of the necessary conditions for the existence of an approximate helical Killing vector field. Introducing free parameters for the lapse at the punctures we can satisfy the condition that the Komar and ADM mass agree at spatial infinity. Several other conditions for an approximate Killing vector are then automatically satisfied, and the 3-metric evolves on a timescale smaller than the orbital timescale. The time derivative of the extrinsic curvature however remains significant. Nevertheless, quasicircular puncture data are not as far from possessing a helical Killing vector as one might have expected.Comment: 11 pages, 6 figures, 2 table

A model problem for conformal parameterizations of the Einstein constraint equations

We investigate the possibility that the conformal and conformal thin sandwich (CTS) methods can be used to parameterize the set of solutions of the vacuum Einstein constraint equations. To this end we develop a model problem obtained by taking the quotient of certain symmetric data on conformally flat tori. Specializing the model problem to a three-parameter family of conformal data we observe a number of new phenomena for the conformal and CTS methods. Within this family, we obtain a general existence theorem so long as the mean curvature does not change sign. When the mean curvature changes sign, we find that for certain data solutions exist if and only if the transverse-traceless tensor is sufficiently small. When such solutions exist, there are generically more than one. Moreover, the theory for mean curvatures changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations.Comment: 40 pages, 4 figure

Comparing initial-data sets for binary black holes

We compare the results of constructing binary black hole initial data with three different decompositions of the constraint equations of general relativity. For each decomposition we compute the initial data using a superposition of two Kerr-Schild black holes to fix the freely specifiable data. We find that these initial-data sets differ significantly, with the ADM energy varying by as much as 5% of the total mass. We find that all initial-data sets currently used for evolutions might contain unphysical gravitational radiation of the order of several percent of the total mass. This is comparable to the amount of gravitational-wave energy observed during the evolved collision. More astrophysically realistic initial data will require more careful choices of the freely specifiable data and boundary conditions for both the metric and extrinsic curvature. However, we find that the choice of extrinsic curvature affects the resulting data sets more strongly than the choice of conformal metric.Comment: 18 pages, 12 figures, accepted for publication in Phys. Rev.

Conformal-thin-sandwich initial data for a single boosted or spinning black hole puncture

Sequences of initial-data sets representing binary black holes in quasi-circular orbits have been used to calculate what may be interpreted as the innermost stable circular orbit. These sequences have been computed with two approaches. One method is based on the traditional conformal-transverse-traceless decomposition and locates quasi-circular orbits from the turning points in an effective potential. The second method uses a conformal-thin-sandwich decomposition and determines quasi-circular orbits by requiring the existence of an approximate helical Killing vector. Although the parameters defining the innermost stable circular orbit obtained from these two methods differ significantly, both approaches yield approximately the same initial data, as the separation of the binary system increases. To help understanding this agreement between data sets, we consider the case of initial data representing a single boosted or spinning black hole puncture of the Bowen-York type and show that the conformal-transverse-traceless and conformal-thin-sandwich methods yield identical data, both satisfying the conditions for the existence of an approximate Killing vector.Comment: 13 pages, 2 figure

Circular orbits of corotating binary black holes: comparison between analytical and numerical results

We compare recent numerical results, obtained within a ``helical Killing vector'' (HKV) approach, on circular orbits of corotating binary black holes to the analytical predictions made by the effective one body (EOB) method (which has been recently extended to the case of spinning bodies). On the scale of the differences between the results obtained by different numerical methods, we find good agreement between numerical data and analytical predictions for several invariant functions describing the dynamical properties of circular orbits. This agreement is robust against the post-Newtonian accuracy used for the analytical estimates, as well as under choices of resummation method for the EOB ``effective potential'', and gets better as one uses a higher post-Newtonian accuracy. These findings open the way to a significant ``merging'' of analytical and numerical methods, i.e. to matching an EOB-based analytical description of the (early and late) inspiral, up to the beginning of the plunge, to a numerical description of the plunge and merger. We illustrate also the ``flexibility'' of the EOB approach, i.e. the possibility of determining some ``best fit'' values for the analytical parameters by comparison with numerical data.Comment: Minor revisions, accepted for publication in Phys. Rev. D, 19 pages, 6 figure

An Invitation to Higher Gauge Theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincar\'e 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a 'tangent 2-group', which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an 'inner automorphism 2-group', which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an 'automorphism 2-group', which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a 'string 2-group'. We also touch upon higher structures such as the 'gravity 3-group' and the Lie 3-superalgebra that governs 11-dimensional supergravity.Comment: 60 pages, based on lectures at the 2nd School and Workshop on Quantum Gravity and Quantum Geometry at the 2009 Corfu Summer Institut

National Beef Quality Audit-2016: Transportation, mobility, and harvest-floor assessments of targeted characteristics that affect quality and value of cattle, carcasses, and by-products

The National Beef Quality Audit-2016 (NBQA-2016) was conducted to assess current transportation, mobility, and quality characteristics of U.S. fed steers and heifers. Data were collected at 17 beef processing facilities between March and November 2016. About 8,000 live cattle were evaluated for transportation and mobility, and about 25,000 carcasses were evaluated on the slaughter floor. Cattle were in transit to the slaughter facility for a mean duration of 2.7 h from a mean distance of 218.5 km using trailers with dimensions ranging from 17.84 m2 to 59.09 m2. Area allotted per animal averaged 1.13 m2 and ranged from 0.85 m2 to 2.28 m2. A total of 96.8% of cattle received a mobility score of 1 (walks easily, no apparent lameness). Identification types (35.1% had multiple) were lot visual tags (61.5%), individual tags (55.0%), electronic tags (16.9%), metal-clip tags (9.2%), bar-coded tags (0.05%), wattles (0.01%), and other (2.6%). Cattle were black-hided (57.8%), Holstein (20.4%), red-hided (10.5%), yellow-hided (4.8%), gray-hided (2.9%), brown-hided (1.3%), and white-hided (1.1%). Unbranded hides were observed on 74.3% of cattle; 18.6% had brands located on the butt, 6.3% on the side, and 1.3% on the shoulder (values exceed 100% due to multiple brands). For hide-on carcasses, 37.7% displayed no mud or manure; specific locations for mud or manure were legs (40.8%), belly (33.0%), tail region (15.5%), side (6.8%), and top-line (3.9%). Cattle without horns represented 83.3% of the sample, and cattle that did have horns measured: \u3c 2.54 cm (5.5%), 2.54 to 12.7 cm (8.3%), and \u3e 12.7 cm (2.9%). Carcasses without bruises represented 61.1% of those sampled, whereas 28.2% had 1, 8.2% had 2, 2.1% had 3, and 0.3% had 4 bruises. Of those carcasses with a bruise, the bruise was located on the loin (29.7%), round (27.8%), chuck (16.4%), rib (14.4%), and brisket/plate/flank (11.6%). Frequencies of offal condemnations were livers (30.8%), lungs (18.2%), viscera (16.3%), hearts (11.1%), heads (2.7%), and tongues (2.0%). Compared to NBQA-2011, fewer cattle were identified for traceability, fewer were black-hided, a greater number were Holstein cattle, more with no brand and no horns, fewer without bruises, more liver, lung, and viscera condemnations, and fewer heads and tongues were condemned. The NBQA remains an influential survey for the U.S. beef industry to provide benchmarks and strategic plans for continued improvement of beef quality and consistency