arXiv subjects

Carsten Gundlach

Publications and source records attributed to Carsten Gundlach.

At least 19 recordsLinked to original sources

arXiv2026

Critical collapse of vacuum spacetimes: Nakamura wave initial data

We report on numerical simulations of critical phenomena in the collapse of axisymmetric vacuum gravitational waves, adopting families of initial data that, to the best of our knowledge, have not been used in this context before. Like Teukolsky waves, the data are based on linear wave solutions to the Einstein equations. We follow Nakamura's construction and encode the wave content in the extrinsic curvature rather than the spatial curvature, which leads to several simplifications when the data are "dressed up" so that they satisfy the nonlinear constraint equations. We are able to fine-tune these data to the onset of black hole formation slightly better than in our previous simulations, allowing us to observe and examine one more echo in the approximately self-similar threshold solution. Our findings are consistent with earlier studies: while we find threshold solutions that are approximately discretely self-similar, the self-similarity is not exact, and we find no evidence for a unique critical solution. We discuss common features between the different threshold solutions, including the appearance of alternating maxima in the direction of the poles and the equator.

gr-qc↗

arXiv2025

Simulations of gravitational collapse in null coordinates IV: evolving through the event horizon, with an application to the spherical charged scalar field

We consider line elements of the form $-2G\,du\,(dx+B\,du) + R^2(...)$, where $(...)$ does not contain $dx$. Surfaces of constant $u$ are then null surfaces, and their affinely parameterised generators have tangent vector $G^{-1}\partial_x$. Considering $u$ as the time coordinate, we can evolve either $R$ or $G$, with the other one found by solving the Raychaudhuri equation along the null generators, or we can evolve both. This choice of {\em formulation} is independent from the remaining {\em gauge} choice $x\to x'(u,x,...)$ in the line element above, which is fixed incrementally by the choice of $B$. For example, we can evolve $G$, in order to be able to evolve through an event horizon, and use $B$ to adapt the coordinates to type-II critical collapse. As a demonstration of these ideas, we consider a charged scalar field in spherical symmetry. We consider two settings: a domain where the outgoing null cones emanate from a regular centre $R=0$, and a domain where they emanate from an ingoing-null boundary. In both settings, we demonstrate convergence with resolution, within each formulation and between the three formulations. As testbeds, we compute the critical exponents and periodic fine-structures of the black hole charge and mass scaling laws in a one-parameter family of charged regular initial data, and examples of perturbed extremal Reissner-Nordstr\"om solutions.

gr-qc↗

arXiv2025

Critical Phenomena in Gravitational Collapse

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. Major additions since the 2010 version of this review are numerical simulations beyond spherical symmetry, in particular of vacuum critical collapse, and new sections on mathematical results in PDE blowup (as a toy model for singularity formation) and on naked singularity formation in GR.

gr-qc↗

arXiv2025

Laboratory Three-dimensional X-ray Micro-beam Laue Diffraction

The development of three-dimensional (3D) non-destructive X-ray characterization techniques in home laboratories is essential for enabling many more researchers to perform 3D characterization daily, overcoming the limitations imposed by competitive and scarce access to synchrotron facilities. Recent efforts have focused on techniques such as laboratory diffraction contrast tomography (LabDCT), which allows 3D characterization of recrystallized grains with sizes larger than 15-20 $μ$m, offering a boundary resolution of approximately 5$μ$m using commercial X-ray computed tomography (CT) systems. To enhance the capabilities of laboratory instruments, we have developed a new laboratory-based 3D X-ray micro-beam diffraction (Lab-3D$μ$XRD) technique. Lab-3D$μ$XRD combines the use of a focused polychromatic beam with a scanning-tomographic data acquisition routine to enable depth-resolved crystallographic orientation characterization. This work presents the first realization of Lab-3D$μ$XRD, including hardware development through the integration of a newly developed Pt-coated twin paraboloidal capillary X-ray focusing optics into a conventional X-ray $μ$CT system, as well as the development of data acquisition and processing software. The results are validated through comparisons with LabDCT and synchrotron phase contrast tomography. The findings clearly demonstrate the feasibility of Lab-3D$μ$XRD, particularly in detecting smaller grains and providing intragranular information. Finally, we discuss future directions for developing Lab-3D$μ$XRD into a versatile tool for studying materials with smaller grain sizes and high defect densities, including the potential of combining it with LabDCT and $μ$CT for multiscale and multimodal microstructural characterization.

cond-mat.mtrl-sci↗

arXiv2024

MozzaVID: Mozzarella Volumetric Image Dataset

Influenced by the complexity of volumetric imaging, there is a shortage of established datasets useful for benchmarking volumetric deep-learning models. As a consequence, new and existing models are not easily comparable, limiting the development of architectures optimized specifically for volumetric data. To counteract this trend, we introduce MozzaVID -- a large, clean, and versatile volumetric classification dataset. Our dataset contains X-ray computed tomography (CT) images of mozzarella microstructure and enables the classification of 25 cheese types and 149 cheese samples. We provide data in three different resolutions, resulting in three dataset instances containing from 591 to 37,824 images. While targeted for developing general-purpose volumetric algorithms, the dataset also facilitates investigating the properties of mozzarella microstructure. The complex and disordered nature of food structures brings a unique challenge, where a choice of appropriate imaging method, scale, and sample size is not trivial. With this dataset, we aim to address these complexities, contributing to more robust structural analysis models and a deeper understanding of food structure. The dataset can be explored through: https://papieta.github.io/MozzaVID/

cs.CV↗

arXiv2024

BugNIST -- a Large Volumetric Dataset for Object Detection under Domain Shift

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.

cs.CV↗

arXiv2024

The two-body problem in 2+1 spacetime dimensions with negative cosmological constant: two point particles

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.

gr-qc↗

arXiv2024

Simulations of gravitational collapse in null coordinates: I. Formulation and weak-field tests in generalised Bondi gauges

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 $(u,x,y,φ)$, where the retarded time $u$ labels outgoing null cones emerging from a regular central worldline, the angles $(θ,φ)$ 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 $Δu$ for an explicit numerical scheme is limited by the radial resolution $Δx$, rather than $Δx(Δθ)^2$. We present convergence tests in the weak-field regime, where we have exact solutions to the linearised scalar and gravitational-wave equations.

gr-qc↗

arXiv2024

Simulations of gravitational collapse in null coordinates: II. Critical collapse of an axisymmetric scalar field

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.

gr-qc↗

arXiv2024

Simulations of gravitational collapse in null coordinates: III. Hyperbolicity

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.

gr-qc↗

arXiv2023

Critical phenomena in the collapse of gravitational waves

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.

gr-qc↗

arXiv2023

Critical phenomena in the collapse of quadrupolar and hexadecapolar gravitational waves

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.

gr-qc↗

arXiv2023

A Hybrid Approach to Full-Scale Reconstruction of Renal Arterial Network

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.

cs.CE↗

arXiv2022

Critical phenomena in gravitational collapse with competing scalar field and gravitational waves, in 4+1 dimensions

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 Bizón, 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.

gr-qc↗

arXiv2022

Critical collapse of an axisymmetric ultrarelativistic fluid in $2+1$ dimensions

We carry out numerical simulations of the gravitational collapse of a rotating perfect fluid with the ultrarelativistic equation of state $P=κρ$, in axisymmetry in $2+1$ spacetime dimensions with $Λ<0$. We show that for $κ\lesssim 0.42$, the critical phenomena are type I and the critical solution is stationary. The picture for $κ\gtrsim 0.43$ 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.

gr-qc↗

arXiv2021

Fully constrained, high-resolution shock-capturing, formulation of the Einstein-fluid equations in $2+1$ dimensions

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.

gr-qc↗

arXiv2021

Critical collapse of a spherically symmetric ultrarelativistic fluid in $2+1$ dimensions

We carry out numerical simulations of the gravitational collapse of a perfect fluid with the ultrarelativistic equation of state $P=κρ$, in spherical symmetry in $2+1$ spacetime dimensions with $Λ<0$. 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 $κ\gtrsim 0.43$, and type I critical phenomena (lifetime scales as logarithm of distance from the threshold) for $κ\lesssim 0.42$. 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.

gr-qc↗

arXiv2020

Rigidly rotating perfect fluid stars in $2+1$ dimensions

Cataldo has found all rigidly rotating self-gravitating perfect fluid solutions in 2+1 dimensions with a negative cosmological constant $Λ$, 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 $p(ρ)$. 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 $\tilde J:=\sqrt{-Λ}\,J$, 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 $|\tilde J|<M$ (whose extent depends on the equation of state), there are precisely two solutions for each $(\tilde J,M)$. Hence exterior solutions exist in all three classes of BTZ solution (black hole, point particle and overspinning), but not all possible values of $(\tilde J,M)$ can be realised as stars. Regardless of the values of $\tilde J$ and $M$, the causal structure of all stars for all equations of state is that of anti-de Sitter space, without horizons or closed timelike curves.

gr-qc↗