Schwarzschild-de Sitter Spacetimes, McVittie Coordinates, and Trumpet Geometries

Trumpet geometries play an important role in numerical simulations of black hole spacetimes, which are usually performed under the assumption of asymptotic flatness. Our Universe is not asymptotically flat, however, which has motivated numerical studies of black holes in asymptotically de Sitter spacetimes. We derive analytical expressions for trumpet geometries in Schwarzschild-de Sitter spacetimes by first generalizing the static maximal trumpet slicing of the Schwarzschild spacetime to static constant mean curvature trumpet slicings of Schwarzschild-de Sitter spacetimes. We then switch to a comoving isotropic radial coordinate which results in a coordinate system analogous to McVittie coordinates. At large distances from the black hole the resulting metric asymptotes to a Friedmann-Lemaitre-Robertson-Walker metric with an exponentially-expanding scale factor. While McVittie coordinates have another asymptotically de Sitter end as the radial coordinate goes to zero, so that they generalize the notion of a "wormhole" geometry, our new coordinates approach a horizon-penetrating trumpet geometry in the same limit. Our analytical expressions clarify the role of time-dependence, boundary conditions and coordinate conditions for trumpet slices in a cosmological context, and provide a useful test for black hole simulations in asymptotically de Sitter spacetimes.Comment: 7 pages, 3 figures, added referenc

Approximate initial data for binary black holes

We construct approximate analytical solutions to the constraint equations of general relativity for binary black holes of arbitrary mass ratio in quasicircular orbit. We adopt the puncture method to solve the constraint equations in the transverse-traceless decomposition and consider perturbations of Schwarzschild black holes caused by boosts and the presence of a binary companion. A superposition of these two perturbations then yields approximate, but fully analytic binary black hole initial data that are accurate to first order in the inverse of the binary separation and the square of the black holes' momenta.Comment: 13 pages, 4 figures, added comparison to numerical calculations, accepted to PR

Trumpet Slices in Kerr Spacetimes

We introduce a new time-independent family of analytical coordinate systems for the Kerr spacetime representing rotating black holes. We also propose a (2+1)+1 formalism for the characterization of trumpet geometries. Applying this formalism to our new family of coordinate systems we identify, for the first time, analytical and stationary trumpet slices for general rotating black holes, even for charged black holes in the presence of a cosmological constant. We present results for metric functions in this slicing and analyze the geometry of the rotating trumpet surface.Comment: 5 pages, 2 figures; version published in PR

Trumpet slices of the Schwarzschild-Tangherlini spacetime

We study families of time-independent maximal and 1+log foliations of the Schwarzschild-Tangherlini spacetime, the spherically-symmetric vacuum black hole solution in D spacetime dimensions, for D >= 4. We identify special members of these families for which the spatial slices display a trumpet geometry. Using a generalization of the 1+log slicing condition that is parametrized by a constant n we recover the results of Nakao, Abe, Yoshino and Shibata in the limit of maximal slicing. We also construct a numerical code that evolves the BSSN equations for D=5 in spherical symmetry using moving-puncture coordinates, and demonstrate that these simulations settle down to the trumpet solutions.Comment: 11 pages, 6 figures, submitted to PR

Maximally Rotating Supermassive Stars at the Onset of Collapse: Effects of Gas Pressure

The "direct collapse" scenario has emerged as a promising evolutionary track for the formation of supermassive black holes early in the Universe. In an idealized version of such a scenario, a uniformly rotating supermassive star spinning at the mass-shedding (Keplerian) limit collapses gravitationally after it reaches a critical configuration. Under the assumption that the gas is dominated by radiation pressure, this critical configuration is characterized by unique values of the dimensionless parameters $J/M^2$ and $R_p/M$, where $J$ is the angular momentum, $R_p$ the polar radius, and $M$ the mass. Motivated by a previous perturbative treatment we adopt a fully nonlinear approach to evaluate the effects of gas pressure on these dimensionless parameters for a large range of masses. We find that gas pressure has a significant effect on the critical configuration even for stellar masses as large as $M \simeq 10^6 M_{\odot}$. We also calibrate two approximate treatments of the gas pressure perturbation in a comparison with the exact treatment, and find that one commonly used approximation in particular results in increasing deviations from the exact treatment as the mass decreases, and the effects of gas pressure increase. The other approximation, however, proves to be quite robust for all masses $M \gtrsim 10^4 M_{\odot}$.Comment: 13 pages, 7 figures, version accepted for publication in MNRA

A Simple Family of Analytical Trumpet Slices of the Schwarzschild Spacetime

We describe a simple family of analytical coordinate systems for the Schwarzschild spacetime. The coordinates penetrate the horizon smoothly and are spatially isotropic. Spatial slices of constant coordinate time $t$ feature a trumpet geometry with an asymptotically cylindrical end inside the horizon at a prescribed areal radius $R_0$ (with $0<R_{0}\leq M$) that serves as the free parameter for the family. The slices also have an asymptotically flat end at spatial infinity. In the limit $R_{0}=0$ the spatial slices lose their trumpet geometry and become flat -- in this limit, our coordinates reduce to Painlev\'e-Gullstrand coordinates.Comment: 7 pages, 3 figure

Analytical Tendex and Vortex Fields for Perturbative Black Hole Initial Data

Tendex and vortex fields, defined by the eigenvectors and eigenvalues of the electric and magnetic parts of the Weyl curvature tensor, form the basis of a recently developed approach to visualizing spacetime curvature. In particular, this method has been proposed as a tool for interpreting results from numerical binary black hole simulations, providing a deeper insight into the physical processes governing the merger of black holes and the emission of gravitational radiation. Here we apply this approach to approximate but analytical initial data for both single boosted and binary black holes. These perturbative data become exact in the limit of small boost or large binary separation. We hope that these calculations will provide additional insight into the properties of tendex and vortex fields, and will form a useful test for future numerical calculations.Comment: 18 pages, 8 figures, submitted to PR

Dynamical perturbations of black-hole punctures: effects of slicing conditions

While numerous numerical relativity simulations adopt a 1+log slicing condition, shock-avoiding slicing conditions form a viable and sometimes advantageous alternative. Despite both conditions satisfying similar equations, recent numerical experiments point to a qualitative difference in the behavior of the lapse in the vicinity of the black-hole puncture: for 1+log slicing, the lapse appears to decay approximately exponentially, while for shock-avoiding slices it performs approximately harmonic oscillation. Motivated by this observation, we consider dynamical coordinate transformations of the Schwarzschild spacetime to describe small perturbations of static trumpet geometries analytically. We find that the character of the resulting equations depends on the (unperturbed) mean curvature at the black-hole puncture: for 1+log slicing it is positive, predicting exponential decay in the lapse, while for shock-avoiding slices it vanishes, leading to harmonic oscillation. In addition to identifying the value of the mean curvature as the origin of these qualitative differences, our analysis provides insight into the dynamical behavior of black-hole punctures for different slicing conditions.Comment: 8 pages, 2 figure

Entanglement sharing among qudits

Consider a system consisting of n d-dimensional quantum particles (qudits), and suppose that we want to optimize the entanglement between each pair. One can ask the following basic question regarding the sharing of entanglement: what is the largest possible value Emax(n,d) of the minimum entanglement between any two particles in the system? (Here we take the entanglement of formation as our measure of entanglement.) For n=3 and d=2, that is, for a system of three qubits, the answer is known: Emax(3,2) = 0.550. In this paper we consider first a system of d qudits and show that Emax(d,d) is greater than or equal to 1. We then consider a system of three particles, with three different values of d. Our results for the three-particle case suggest that as the dimension d increases, the particles can share a greater fraction of their entanglement capacity.Comment: 4 pages; v2 contains a new result for 3 qudits with d=

Long-term Annual Aerial Surveys of Submersed Aquatic Vegetation (SAV) Support Science, Management, and Restoration

Aerial surveys of coastal habitats can uniquely inform the science and management of shallow, coastal zones, and when repeated annually, they reveal changes that are otherwise difficult to assess from ground-based surveys. This paper reviews the utility of a long-term (1984–present) annual aerial monitoring program for submersed aquatic vegetation (SAV) in Chesapeake Bay, its tidal tributaries, and nearby Atlantic coastal bays, USA. We present a series of applications that highlight the program’s importance in assessing anthropogenic impacts, gauging water quality status and trends, establishing and evaluating restoration goals, and understanding the impact of commercial fishing practices on benthic habitats. These examples demonstrate how periodically quantifying coverage of this important foundational habitat answers basic research questions locally, as well as globally, and provides essential information to resource managers. New technologies are enabling more frequent and accurate aerial surveys at greater spatial resolution and lower cost. These advances will support efforts to extend the applications described here to similar issues in other areas