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We work towards the general solution of the two-body problem in 2+1-dimensional general relativity with a negative cosmological constant. The BTZ solutions corresponding to black holes, point particles and overspinning particles can be considered either as objects in their own right, or as the exterior solution of compact objects with a given mass $M$ and spin $J$, such as rotating fluid stars. We compare and contrast the metric approach to the group-theoretical one of characterising the BTZ solutions as identifications of 2+1-dimensional anti-de Sitter spacetime under an isometry. We then move on to the two-body problem. In this paper, we restrict the two objects to the point particle range $|J|-1\le M<-|J|$, or their massless equivalents, obtained by an infinite boost. (Both anti-de Sitter space and massless particles have $M=-1$, $J=0$). We derive analytic expressions for the total mass $M_\text{tot}$ and spin $J_\text{tot}$ of the system in terms of the six gauge-invariant parameters of the two-particle system: the rest mass and spin of each object, and the impact parameter and energy of the orbit. Based on work of Holst and Matschull on the case of two massless, nonspinning particles, we conjecture that the black hole formation threshold is $M_\text{tot}=|J_\text{tot}|$. The threshold solutions are then extremal black holes. We determine when the global geometry is a black hole, an eternal binary system, or a closed universe.

We investigate the well-posedness of the characteristic initial-boundary value problem for the Einstein equations in Bondi-like coordinates (including Bondi, double-null and affine). We propose a definition of strong hyperbolicity of a system of partial differential equations of any order, and show that the Einstein equations in Bondi-like coordinates in their second-order form used in numerical relativity do not meet it, in agreement with results of Giannakopoulos et al for specific first-order reductions. In the principal part, frozen coefficient approximation that one uses to examine hyperbolicity, we explicitly construct the general solution to identify the solutions that obstruct strong hyperbolicity. Independently, we present a first-order symmetric hyperbolic formulation of the Einstein equations in Bondi gauge, linearised about Schwarzschild, thus completing work by Frittelli. This establishes an energy norm ($L^2$ in the metric perturbations and selected first and second derivatives), in which the initial-boundary value problem, with initial data on an outgoing null cone and boundary data on a timelike cylinder or an ingoing null cone, is well-posed, thus verifying a conjecture by Giannakopoulos et al. Unfortunately, our method does not extend to the pure initial-value problem on a null cone with regular vertex.

We present the first numerical simulations in null coordinates of the collapse of nonspherical regular initial data to a black hole. We restrict to twist-free axisymmetry, and re-investigate the critical collapse of a non-spherical massless scalar field. We find that the Choptuik solution governing scalar field critical collapse in spherical symmetry persists when fine-tuning moderately non-spherical initial data to the threshold of black hole formation. The non-sphericity evolves as an almost-linear perturbation until the end of the self-similar phase, and becomes dominant only in the final collapse to a black hole. We compare with numerical results of Choptuik et al, Baumgarte, and Marouda et al, and conclude that they have been able to evolve somewhat more non-spherical solutions. Future work with larger deviations from spherical symmetry, and in particular vacuum collapse, will require a different choice of radial coordinate that allows the null generators to reconverge locally.

We present a code for numerical simulations of the collapse of regular initial data to a black hole in null coordinates. We restrict to twist-free axisymmetry with scalar field matter. Our coordinates are $(u,x,y,\varphi)$, where the retarded time $u$ labels outgoing null cones emerging from a regular central worldline, the angles $(\theta,\varphi)$ label the null generators of each null cone, and the radial coordinate $x$ labels points along these generators. We focus on a class of generalised Bondi radial coordinates $x$ with the twin properties that $x=0$ is the central world line and that the numerical domain $(u\ge0$, $0\le x\le x_\text{max})$ is a subset of the domain of dependence of the initial data on $(u=0$, $0\le x\le x_\text{max}$). In critical collapse, an appropriate choice of these coordinates can be made to zoom in on the accumulation point of scale echos of the critical solution, without the need for explicit mesh refinement. We introduce a novel numerical scheme that in effect reduces the angular resolution at small radius, such that the time step $\Delta u$ for an explicit numerical scheme is limited by the radial resolution $\Delta x$, rather than $\Delta x(\Delta\theta)^2$. We present convergence tests in the weak-field regime, where we have exact solutions to the linearised scalar and gravitational-wave equations.

Fine-tuning generic but smooth spherically-symmetric initial data for general relativity to the threshold of dynamical black hole formation creates arbitrarily large curvatures, mediated by a universal self-similar solution that acts as an intermediate attractor. For vacuum gravitational waves, however, these critical phenomena have been elusive. We present, for the first time, excellent agreement among three independent numerical simulations of this collapse. Surprisingly, we find no universality, and observe approximate self-similarity for some families of initial data but not for others., Comment: 6 pages, 3 figures

Domain shift significantly influences the performance of deep learning algorithms, particularly for object detection within volumetric 3D images. Annotated training data is essential for deep learning-based object detection. However, annotating densely packed objects is time-consuming and costly. Instead, we suggest training models on individually scanned objects, causing a domain shift between training and detection data. To address this challenge, we introduce the BugNIST dataset, comprising 9154 micro-CT volumes of 12 bug types and 388 volumes of tightly packed bug mixtures. This dataset is characterized by having objects with the same appearance in the source and target domains, which is uncommon for other benchmark datasets for domain shift. During training, individual bug volumes labeled by class are utilized, while testing employs mixtures with center point annotations and bug type labels. Together with the dataset, we provide a baseline detection analysis, with the aim of advancing the field of 3D object detection methods., Comment: 31 pages, 12 figures, 5 tables

We report on numerical simulations of critical phenomena near the threshold of black hole formation in the collapse of axisymmetric gravitational waves in vacuum. We discuss several new features of our numerical treatment, and then compare results obtained from families of quadrupolar and hexadecapolar initial data. Specifically, we construct (nonlinear) initial data from quadrupolar and hexadecapolar, time-symmetric wavelike solutions to the linearized Einstein equations (often referred to as Teukolsky waves), and evolve these using a shock-avoiding slicing condition. While our degree of fine-tuning to the onset of black-hole formation is rather modest, we identify several features of the threshold solutions formed for the two families. Both threshold solutions appear to display an at least approximate discrete self-similarity with an accumulation event at the center, and the characteristics of the threshold solution for the quadrupolar data are consistent with those found previously by other authors. The hexadecapolar threshold solution appears to be distinct from the quadrupolar one, providing further support to the notion that there is no universal critical solution for the collapse of vacuum gravitational waves., Comment: 17 pages, 14 figures

The renal vasculature, acting as a resource distribution network, plays an important role in both the physiology and pathophysiology of the kidney. However, no imaging techniques allow an assessment of the structure and function of the renal vasculature due to limited spatial and temporal resolution. To develop realistic computer simulations of renal function, and to develop new image-based diagnostic methods based on artificial intelligence, it is necessary to have a realistic full-scale model of the renal vasculature. We propose a hybrid framework to build subject-specific models of the renal vascular network by using semi-automated segmentation of large arteries and estimation of cortex area from a micro-CT scan as a starting point, and by adopting the Global Constructive Optimization algorithm for generating smaller vessels. Our results show a statistical correspondence between the reconstructed data and existing anatomical data obtained from a rat kidney with respect to morphometric and hemodynamic parameters., Comment: 19 pages, 5 figures (excluding references and supplementary) submitted to Communications Biology

In the gravitational collapse of matter beyond spherical symmetry, gravitational waves are necessarily present. On the other hand, gravitational waves can collapse to a black hole even without matter. One might therefore wonder how the interaction and competition between the matter fields and gravitational waves affects critical phenomena at the threshold of black hole formation. As a toy model for this, we study the threshold of black-hole formation in 4+1 dimensions, where we add a massless minimally coupled scalar matter field to the gravitational wave ansatz of Biz\'on, Chmaj and Schmidt (in a nutshell, Bianchi~IX on $S^3\times\text{radius}\times\text{time}$). In order to find a stable discretisation of the equation governing the gravitational waves in 4+1 physical dimensions, which has the same principal part as the spherical wave equation in 9+1 dimensions, we first revisit the problem of critical spherical scalar field collapse in $n+2$ dimensions with large $n$. Returning to the main problem, we find numerically that weak gravitational wave perturbations of the scalar field critical solution decay, while weak scalar perturbations of the gravitational wave critical solution also decay. A dynamical systems picture then suggests the existence of a codimension-two attractor. We find numerical evidence for this attractor by evolving mixed initial data and fine-tuning both an overall amplitude and the relative strength of the two fields.

We carry out numerical simulations of the gravitational collapse of a rotating perfect fluid with the ultrarelativistic equation of state $P=\kappa\rho$, in axisymmetry in $2+1$ spacetime dimensions with $\Lambda<0$. We show that for $\kappa \lesssim 0.42$, the critical phenomena are type I and the critical solution is stationary. The picture for $\kappa \gtrsim 0.43$ is more delicate: for small angular momenta, we find type II phenomena and the critical solution is quasistationary, contracting adiabatically. The spin-to-mass ratio of the critical solution increases as it contracts, and hence so does that of the black hole created at the end as we fine-tune to the black-hole threshold. Forming extremal black holes is avoided because the contraction of the critical solution smoothly ends as extremality is approached., Comment: 14 pages, 8 figures. Typos corrected. This version has been accepted by PRD

We carry out numerical simulations of the gravitational collapse of a perfect fluid with the ultrarelativistic equation of state $P=\kappa\rho$, in spherical symmetry in $2+1$ spacetime dimensions with $\Lambda<0$. At the threshold of prompt collapse, we find type II critical phenomena (apparent horizon mass and maximum curvature scale as powers of distance from the threshold) for $\kappa\gtrsim 0.43$, and type I critical phenomena (lifetime scales as logarithm of distance from the threshold) for $\kappa\lesssim 0.42$. The type I critical solution is static, while the type II critical solution is not self-similar (as in higher dimensions) but contracting quasi-statically., Comment: 22 pages, 15 figures. Typos corrected. This version has been accepted by PRD

Four components of the axisymmetric Einstein equations in 2+1 dimensions with negative cosmological constant can be written as $\nabla_aM=\dots$ and $\nabla_aJ=\dots$, where the dots stand for stress-energy terms, and $M$ and $J$ are scalars. In vacuum, they reduce to the constant mass and angular momentum parameters of the BTZ solution of the same name. The integrability conditions for the Einstein equations give rise to two conserved stress-energy currents $\nabla_aj^a_{(M)}=0$ and $\nabla_aj^a_{(J)}=0$. The angular momentum current is just the Noether current due to axisymmetry, but the mass current is unexpected in the presence of rotation. The conserved quantity $M$ exists in all dimensions in spherical symmetry, known as the Misner-Sharp, Hawking or Kodama mass, but in 2+1 dimensions $M$ exists also in axisymmetry, even with rotation. We use $M$ and $J$ to give a fully constrained formulation of the axisymmetric Einstein equations in 2+1 dimensions, where the Einstein equations are solved by explicit integration from the center along time slices. We use the two conserved matter currents in the construction of a high-resolution shock-capturing formulation of the Einstein-perfect fluid system, in which $M$ and $J$ momentum are then exactly conserved by construction. We demonstrate convergence of the code in the test cases of generic dispersion and collapse and stable and unstable rotating stars., Comment: 23 pages, 17 figures, Typos corrected. This version has been accepted by PRD

Cataldo has found all rigidly rotating self-gravitating perfect fluid solutions in 2+1 dimensions with a negative cosmological constant $\Lambda$, for a density that is specified a priori as a function of a certain radial coordinate. We rewrite these solutions in standard polar-radial coordinates, for an arbitrary barotropic equation of state $p(\rho)$. For any given equation of state, we find the two-parameter family of solutions with a regular centre and finite total mass $M$ and angular momentum $J$ (rigidly rotating stars). For analytic equations of state, the solution is analytic except at the surface, but including at the centre. Defining the dimensionless spin $\tilde J:=\sqrt{-\Lambda}\,J$, there is precisely one solution for each $(\tilde J,M)$ in the region $|\tilde J|-1|M|$. In an adjacent compact part of the black hole region $|\tilde J|

We numerically investigate the threshold of black-hole formation in the gravitational collapse of electromagnetic waves in axisymmetry. We find approximate power-law scaling $\rho_{\rm max}\sim (\eta_*-\eta)^{-2\gamma}$ of the maximum density in the time evolution of near-subcritical data with $\gamma\simeq 0.145$, where $\eta$ is the amplitude of the initial data. We directly observe approximate discrete self-similarity in near-critical time evolutions with a log-scale echoing period of $\Delta\simeq 0.55$. The critical solution is approximately the same for two families of initial data, providing some evidence of universality. Neither the discrete self-similarity nor the universality, however, are exact. We speculate that the absence of an exactly discrete self-similarity might be caused by the interplay of electromagnetic and gravitational wave degrees of freedom, or by the presence of higher-order angular multipoles, or both, and discuss implications of our findings for the critical collapse of vacuum gravitational waves., Comment: 5 pages, 3 figures; version accepted for publication in PRL

In the gravitational collapse of matter beyond spherical symmetry, gravitational waves are necessarily present. On the other hand, gravitational waves can collapse to a black hole even without matter. One might therefore wonder whether the interaction and competition between the matter fields and gravitational waves affects critical phenomena at the threshold of black hole formation. As a toy model for this, we study type II critical collapse with two matter fields in spherical symmetry, namely a scalar field and a Yang-Mills field. On their own, both display discrete self-similarity (DSS) in type II critical collapse, and we can take either one of them as a toy model for gravitational waves. To our surprise, in numerical time evolutions we find that, for sufficiently good fine-tuning, the scalar field always dominates on sufficiently small scales. We explain our results by the conjectured existence of a "quasi-discretely self-similar" (QSS) solution shared by the two fields, equal to the known Yang-Mills critical solution at infinitely large scales and the known scalar field critical solution (the Choptuik solution) at infinitely small scales, with a gradual transition from one field to the other. This QSS solution itself has only one unstable mode, and so acts as the critical solution for any mixture of scalar field and Yang-Mills initial data., Comment: Section on the numerical method expanded

We describe imbibition in real and artificial plant seeds, using a combination of experiments and theory. In both systems, our experiments demonstrate that liquid permeates the substrate at a rate which decreases gradually over time. Tomographic imaging of soy seeds is used to confirmed this by observation of the permeating liquid using an iodine stain. To rationalize the experimental data, we propose a model based on capillary action which predicts the temporal evolution of the radius of the wet front and the seed mass. The depth of the wetting front initially evolves as $t^{1/2}$ in accord with the Lucas-Washburn law. At later times, when the sphere is almost completely filled, the front radius scales as $(1-t/t_{\text{max}})^{1/2}$ where $t_{\text{max}}$ is the time required to complete imbibition. The data obtained on both natural and artificial seeds collapse onto a single curve that agrees well with our model, suggesting that capillary phenomena contribute to moisture uptake in soy seeds.

We study critical phenomena in the collapse of rotating ultrarelativistic perfect fluids, in which the pressure $P$ is related to the total energy density $\rho$ by $P=\kappa\rho$, with $\kappa$ a constant. We generalize earlier results for radiation fluids with $\kappa=1/3$ to other values of $\kappa$, focussing on $\kappa < 1/9$. For $1/9<\kappa \lesssim 0.49$, the critical solution has only one unstable, growing mode, which is spherically symmetric. For supercritical data it controls the black hole mass, while for subcritical data it controls the maximum density. For $\kappa<1/9$, an additional axial $l=1$ mode becomes unstable. This controls either the black hole angular momentum, or the maximum angular velocity. In theory, the additional unstable $l=1$ mode changes the nature of the black hole threshold completely: at sufficiently large initial rotation rates $\Omega$ and sufficient fine-tuning of the initial data to the black hole threshold we expect to observe nontrivial universal scaling functions (familiar from critical phase transitions in thermodynamics) governing the black hole mass and angular momentum, and, with further fine-tuning, eventually a finite black hole mass almost everywhere on the threshold. In practice, however, the second unstable mode grows so slowly that we do not observe this breakdown of scaling at the level of fine-tuning we can achieve, nor systematic deviations from the leading-order power-law scalings of the black hole mass. We do see systematic effects in the black-hole angular momentum, but it is not clear yet if these are due to the predicted non-trivial scaling functions, or to nonlinear effects at sufficiently large initial angular momentum (which we do not account for in our theoretical model)., Comment: Figs. 2-5 and their discussion improved. Version accepted by PRD

We reduce the equations governing the spherically symmetric perturbations of static spherically symmetric solutions of the Einstein-Vlasov system (with either massive or massless particles) to a single stratified wave equation $-\psi_{,tt}=H\psi$, with $H$ containing second derivatives in radius, and integrals over energy and angular momentum. We identify an inner product with respect to which $H$ is symmetric, and use the Ritz method to approximate the lowest eigenvalues of $H$ numerically. For two representative background solutions with massless particles we find a single unstable mode with a growth rate consistent with the universal one found by Akbarian and Choptuik in nonlinear numerical time evolutions.

We carry out numerical simulations of the collapse of a complex rotating scalar field of the form $\Psi(t,r,\theta)=e^{im\theta}\Phi(t,r)$, giving rise to an axisymmetric metric, in 2+1 spacetime dimensions with cosmological constant $\Lambda<0$, for $m=0,1,2$, for four 1-parameter families of initial data. We look for the familiar scaling of black hole mass and maximal Ricci curvature as a power of $|p-p_*|$, where $p$ is the amplitude of our initial data and $p_*$ some threshold. We find evidence of Ricci scaling for all families, and tentative evidence of mass scaling for most families, but the case $m>0$ is very different from the case $m=0$ we have considered before: the thresholds for mass scaling and Ricci scaling are significantly different (for the same family), scaling stops well above the scale set by $\Lambda$, and the exponents depend strongly on the family. Hence, in contrast to the $m=0$ case, and to many other self-gravitating systems, there is only weak evidence for the collapse threshold being controlled by a self-similar critical solution and no evidence for it being universal., Comment: Version accepted for publication in PRD

We express the Einstein-Vlasov system in spherical symmetry in terms of a dimensionless momentum variable $z$ (radial over angular momentum). This regularises the limit of massless particles, and in that limit allows us to obtain a reduced system in independent variables $(t,r,z)$ only. Similarly, in this limit the Vlasov density function $f$ for static solutions depends on a single variable $Q$ (energy over angular momentum). This reduction allows us to show that any given static metric which has vanishing Ricci scalar, is vacuum at the centre and for $r>3M$ and obeys certain energy conditions uniquely determines a consistent $f=\bar k(Q)$ (in closed form). Vice versa, any $\bar k(Q)$ within a certain class uniquely determines a static metric (as the solution of a system of two first-order quasilinear ODEs). Hence the space of static spherically symmetric solutions of Einstein-Vlasov is locally a space of functions of one variable. For a simple 2-parameter family of functions $\bar k(Q)$, we construct the corresponding static spherically symmetric solutions, finding that their compactness is in the interval $0.7\lesssim \rm max_r(2M/r)\le 8/9$. This class of static solutions includes one that agrees with the approximately universal type-I critical solution recently found by Akbarian and Choptuik (AC) in numerical time evolutions. We speculate on what singles it out as the critical solution found by fine-tuning generic data to the collapse threshold, given that AC also found that {\em all} static solutions are one-parameter unstable and sit on the threshold of collapse., Comment: Version accepted for publication in PRD

We derive a theoretical model of mass and angular momentum scaling in type-II critical collapse with rotation. We focus on the case where the critical solution has precisely one, spherically symmetric, unstable mode. We demonstrate excellent agreement with numerical results for critical collapse of a rotating radiation fluid, which falls into this case., Comment: Version accepted for publication in PRD

We present results from the first fully relativistic simulations of the critical collapse of rotating radiation fluids. We observe critical scaling both in subcritical evolutions, in which case the fluid disperses to infinity and leaves behind flat space, and in supercritical evolutions that lead to the formation of black holes. We measure the mass and angular momentum of these black holes, and find that both show critical scaling with critical exponents that are consistent with perturbative results. The critical exponents are universal; they are not affected by angular momentum, and are independent of the direction in which the critical curve, which separates subcritical from supercritical evolutions in our two-dimensional parameter space, is crossed. In particular, these findings suggest that the angular momentum decreases more rapidly than the square of the mass, so that, as criticality is approached, the collapse leads to the formation of a non-spinning black hole. We also demonstrate excellent agreement of our numerical data with new closed-form extensions of power-law scalings that describe the mass and angular momentum of rotating black holes formed close to criticality., Comment: 5 pages, 4 figures; version accepted for publication in PRL

We carry out numerical experiments in the critical collapse of a spherically symmetric massless scalar field in 2+1 spacetime dimensions in the presence of a negative cosmological constant and compare them against a new theoretical model. We approximate the true critical solution as the $n=4$ Garfinkle solution, matched at the lightcone to a Vaidya-like solution, and corrected to leading order for the effect of $\Lambda<0$. This approximation is only $C^3$ at the lightcone and has three growing modes. We {\em conjecture} that pointwise it is a good approximation to a yet unknown true critical solution that is analytic with only one growing mode (itself approximated by the top mode of our amended Garfinkle solution). With this conjecture, we predict a Ricci-scaling exponent of $\gamma=8/7$ and a mass-scaling exponent of $\delta=16/23$, compatible with our numerical experiments., Comment: 27 pages

In numerical simulations of black hole binaries, Pretorius and Khurana [Class. Quant. Grav. {\bf 24}, S83 (2007)] have observed critical behaviour at the threshold between scattering and immediate merger. The number of orbits scales as $n\simeq -\gamma\ln|p-p_*|$ along any one-parameter family of initial data such that the threshold is at $p=p_*$. Hence they conjecture that in ultrarelavistic collisions almost all the kinetic energy can be converted into gravitational waves if the impact parameter is fine-tuned to the threshold. As a toy model for the binary, they consider the geodesic motion of a test particle in a Kerr black hole spacetime, where the unstable circular geodesics play the role of critical solutions, and calculate the critical exponent $\gamma$. Here, we incorporate radiation reaction into this model using the self-force approximation. The critical solution now evolves adiabatically along a sequence of unstable circular geodesic orbits under the effect of the self-force. We confirm that almost all the initial energy and angular momentum are radiated on the critical solution. Our calculation suggests that, even for infinite initial energy, this happens over a finite number of orbits given by $n_\infty\simeq 0.41/\eta$, where $\eta$ is the (small) mass ratio. We derive expressions for the time spent on the critical solution, number of orbits and radiated energy as functions of the initial energy and impact parameter., Comment: Version published in PRD

We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress-energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state. We discuss the Newtonian limit of our formalism and its relation to a second formalism also used in Newtonian elasticity. We validate our framework by numerically solving a set of Riemann problems in Minkowski spacetime, as well as Newtonian ones from the literature., Comment: 31 pages, 21 pdf figures

We investigate whether tidal forcing can result in sound waves steepening into shocks at the surface of a star. To model the sound waves and shocks, we consider adiabatic non-spherical perturbations of a Newtonian perfect fluid star. Because tidal forcing of sounds waves is naturally treated with linear theory, but the formation of shocks is necessarily nonlinear, we consider the perturbations in two regimes. In most of the interior, where tidal forcing dominates, we treat the perturbations as linear, while in a thin layer near the surface we treat them in full nonlinearity but in the approximation of plane symmetry, fixed gravitational field and a barotropic equation of state. Using a hodograph transformation, this nonlinear regime is also described by a linear equation. We show that the two regimes can be matched to give rise to a single mode equation which is linear but models nonlinearity in the outer layers. This can then be used to obtain an estimate for the critical mode amplitude at which a shock forms near the surface. As an application, we consider the tidal waves raised by the companion in an irrotational binary system in circular orbit. We find that shocks form at the same orbital separation where Roche lobe overflow occurs, and so shock formation is unlikely to occur., Comment: Preprint revised to reflect an erratum published in MNRAS. The error does not propagate outside Sec. 3.1, and does not affect our results

We investigate numerical methods for wave equations in $n+2$ spacetime dimensions, written in spherical coordinates, decomposed in spherical harmonics on $S^n$, and finite-differenced in the remaining coordinates $r$ and $t$. Such an approach is useful when the full physical problem has spherical symmetry, for perturbation theory about a spherical background, or in the presence of boundaries with spherical topology. The key numerical difficulty arises from lower-order $1/r$ terms at the origin $r=0$. As a toy model for this, we consider the flat space linear wave equation in the form $\dot\pi=\psi'+p\psi/r$, $\dot\psi=\pi'$, where $p=2l+n$, and $l$ is the leading spherical harmonic index. We propose a class of summation by parts (SBP) finite differencing methods that conserve a discrete energy up to boundary terms, thus guaranteeing stability and convergence in the energy norm. We explicitly construct SBP schemes that are second and fourth-order accurate at interior points and the symmetry boundary $r=0$, and first and second-order accurate at the outer boundary $r=R$., Comment: Introduction and numerical tests section expanded

We investigate the run-up of a shock wave from inside to the surface of a perfect fluid star in equilibrium and bounded by vacuum. Near the surface we approximate the fluid motion as plane-symmetric and the gravitational field as constant. We consider the "hot" equation of state $P=(\Gamma-1)\rho e$ and its "cold" (fixed entropy, barotropic) form $P=K_0\rho^\Gamma$ (the latter does not allow for shock heating). We find numerically that the evolution of generic initial data approaches universal similarity solutions sufficiently near the surface, and we construct these similarity solutions explicitly. The two equations of state show very different behaviour, because shock heating becomes the dominant effect when it is allowed. In the barotropic case, the fluid velocity behind the shock approaches a constant value, while the density behind the shock approaches a power law in space, as the shock approaches the surface. In the hot case with shock heating,the density jumps by a constant factor through the shock, while the sound speed and fluid velocity behind the shock diverge in a whiplash effect. We tabulate the similarity exponents as a function of the equation of state parameter $\Gamma$and the stratification index $n_*$., Comment: Version accepted for publication in J. Fluid Mech

We are interested in the generic behaviour of nonlinear sound waves as they approach the surface of a star, here assumed to have the polytropic equation of state $P=K\rho^\Gamma$. Restricting to spherical symmetry, and considering only the region near the surface, we generalise the methods of Carrier and Greenspan (1958) for the shallow water equations on a sloping beach to this problem. We give a semi-quantitative criterion for a shock to form near the surface during the evolution of generic initial data with support away from the surface. We show that in smooth solutions the velocity and the square of the sound speed remain regular functions of Eulerian radius at the surface., Comment: Eqs. 28 and 29 corrected to reflect erratum submitted to PRD

As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term "critical phenomena". They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically self-similar. Critical phenomena give a natural route from smooth initial data to arbitrarily large curvatures visible from infinity, and are therefore likely to be relevant for cosmic censorship, quantum gravity, astrophysics, and our general understanding of the dynamics of general relativity., Comment: 51 pages, 5 figures. Major update of Living Rev. Rel. 2 (1999) 4

The numerical relativity session at GR18 was dominated by physics results on binary black hole mergers. Several groups can now simulate these from a time when the post-Newtonian equations of motion are still applicable, through several orbits and the merger to the ringdown phase, obtaining plausible gravitational waves at infinity, and showing some evidence of convergence with resolution. The results of different groups roughly agree. This new-won confidence has been used by these groups to begin mapping out the (finite-dimensional) initial data space of the problem, with a particular focus on the effect of black hole spins, and the acceleration by gravitational wave recoil to hundreds of km/s of the final merged black hole. Other work was presented on a variety of topics, such as evolutions with matter, extreme mass ratio inspirals, and technical issues such as gauge choices., Comment: References updated, to be published in CQG

We review in generality why time-independent endstates can be reached in black hole and collapse simulations, with and without excision. We characterise the Killing states of the Bona-Masso slicing condition with time derivative along the normals to the slice ("BMn") as solutions of a mixed elliptic/hyperbolic differential equation on the slice. We show numerically that these steady states can be reached as end states from typical initial data with excision but can be reached with the puncture method only if the puncture is not numerically well resolved. During the evolution, BMn slicings often form gauge shocks. It may be that these are not seen in current 3D simulations only through lack of resolution, although we expect that they can be avoided with some care. Finally we point out that excision with BMn as currently implemented is ill-posed and therefore not expected to converge; this can be cured. In technical appendixes, we derive the equations of pure gauge systems on a fixed spacetime, and bring the BSSN/NOR equations into 3-dimensional tensor form suitable for multiple coordinate patches or spherical polar coordinates., Comment: 14 pages, LaTex, psfrag

We prove that when the equations are restricted to the principal part the standard version of the BSSN formulation of the Einstein equations is equivalent to the NOR formulation for any gauge, and that the KST formulation is equivalent to NOR for a variety of gauges. We review a family of elliptic gauge conditions and the implicit parabolic and hyperbolic drivers that can be derived from them, and show how to make them symmetry-seeking. We investigate the hyperbolicity of ADM, NOR and BSSN with implicit hyperbolic lapse and shift drivers. We show that BSSN with the coordinate drivers used in recent "moving puncture" binary black hole evolutions is ill-posed at large shifts, and suggest how to make it strongly hyperbolic for arbitrary shifts. For ADM, NOR and BSSN with elliptic and parabolic gauge conditions, which cannot be hyperbolic, we investigate a necessary condition for well-posedness of the initial-value problem.