Journal articles: 'Spacetime initial data'

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Eichmair, Michael, Gregory J. Galloway, and Abraão Mendes. "Initial Data Rigidity Results." Communications in Mathematical Physics 386, no. 1 (February 27, 2021): 253–68. http://dx.doi.org/10.1007/s00220-021-04033-x.

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AbstractWe prove several rigidity results related to the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature.

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Bernstein, David, David Hobill, Edward Seidel, and Larry Smarr. "Initial data for the black hole plus Brill wave spacetime." Physical Review D 50, no. 6 (September 15, 1994): 3760–82. http://dx.doi.org/10.1103/physrevd.50.3760.

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Cole, Michael J., and Juan A. Valiente Kroon. "A Geometric Invariant Characterising Initial Data for the Kerr–Newman Spacetime." Annales Henri Poincaré 18, no. 11 (August 7, 2017): 3651–93. http://dx.doi.org/10.1007/s00023-017-0606-x.

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ØKSENDAL, BERNT, FRANK PROSKE, and MIKAEL SIGNAHL. "THE CAUCHY PROBLEM FOR THE WAVE EQUATION WITH LÉVY NOISE INITIAL DATA." Infinite Dimensional Analysis, Quantum Probability and Related Topics 09, no. 02 (June 2006): 249–70. http://dx.doi.org/10.1142/s0219025706002330.

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In this paper we study the Cauchy problem for the wave equation with spacetime Lévy noise initial data in the Kondratiev space of stochastic distributions. We prove that this problem has a strong and unique C2-solution, which takes an explicit form. Our approach is based on the use of the Hermite transform.

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Wang, Yaohua, Naqing Xie, and Xiao Zhang. "The positive energy theorem for asymptotically anti-de Sitter spacetimes." Communications in Contemporary Mathematics 17, no. 04 (June 22, 2015): 1550015. http://dx.doi.org/10.1142/s0219199715500157.

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We establish the inequality for Henneaux–Teitelboim's total energy–momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary t-slice in anti-de Sitter spacetime. In particular, when t = 0, it generalizes Chruściel–Maerten–Tod's inequality in the center of AdS mass coordinates. We also show that the determinant of energy–momentum endomorphism Q is the geometric invariant of asymptotically anti-de Sitter spacetimes.

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Ku, Cheng-Yu, Chih-Yu Liu, Jing-En Xiao, Wei-Po Huang, and Yan Su. "A spacetime collocation Trefftz method for solving the inverse heat conduction problem." Advances in Mechanical Engineering 11, no. 7 (July 2019): 168781401986127. http://dx.doi.org/10.1177/1687814019861271.

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In this article, a novel spacetime collocation Trefftz method for solving the inverse heat conduction problem is presented. This pioneering work is based on the spacetime collocation Trefftz method; the method operates by collocating the boundary points in the spacetime coordinate system. In the spacetime domain, the initial and boundary conditions are both regarded as boundary conditions on the spacetime domain boundary. We may therefore rewrite an initial value problem (such as a heat conduction problem) as a boundary value problem. Hence, the spacetime collocation Trefftz method is adopted to solve the inverse heat conduction problem by approximating numerical solutions using Trefftz base functions satisfying the governing equation. The validity of the proposed method is established for a number of test problems. We compared the accuracy of the proposed method with that of the Trefftz method based on exponential basis functions. Results demonstrate that the proposed method obtains highly accurate numerical solutions and that the boundary data on the inaccessible boundary can be recovered even if the accessible data are specified at only one-fourth of the overall spacetime boundary.

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Duggal, Krishan L., and Ramesh Sharma. "Conformal killing vector fields on spacetime solutions of Einstein's equations and initial data." Nonlinear Analysis: Theory, Methods & Applications 63, no. 5-7 (November 2005): e447-e454. http://dx.doi.org/10.1016/j.na.2004.09.034.

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CHAKRABORTY, SUBENOY, and SANJUKTA CHAKRABORTY. "DYNAMICAL SYMMETRY IN GRAVITATIONAL COLLAPSE WITH GENERAL INITIAL AREA RADIUS." Modern Physics Letters A 21, no. 18 (June 14, 2006): 1467–79. http://dx.doi.org/10.1142/s0217732306019773.

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In this work, gravitational collapse has been studied for quasi-spherical spacetime with dust or anisotropic pressure as the matter content. A linear transformation on the initial data set and of the area radius shows the invariance of the physical parameters as well as the final fate of collapse, considering an arbitrary function of r as the initial area radius.

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Deppe, Nils, and Andrew R. Frey. "Classes of stable initial data for massless and massive scalars in Anti-de Sitter spacetime." Journal of High Energy Physics 2015, no. 12 (December 2015): 1–31. http://dx.doi.org/10.1007/jhep12(2015)004.

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JAIN, PANKAJ. "A FLAT SPACETIME MODEL OF THE UNIVERSE." Modern Physics Letters A 27, no. 36 (November 11, 2012): 1250201. http://dx.doi.org/10.1142/s021773231250201x.

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We propose a model of the Universe based on Minkowski flat spacetime metric. In this model the spacetime does not evolve. Instead the matter evolves such that all the mass parameters increase with time. We construct a model based on unimodular gravity to show how this can be accomplished within the framework of flat spacetime. We show that the model predicts the Hubble law if the masses increase with time. Furthermore, we show that it fits the high z supernova data in a manner almost identical to the standard Big Bang model. Furthermore, we show that at early times the Universe is dominated by radiative energy density. The phenomenon of recombination also arises in our model and hence predicts the existence of CMBR. However, a major difference with respect to the standard Big Bang is that there is no initial singularity and the radiative temperature and energy density do not evolve in our model. Furthermore, we argue that the basic motivation for inflation is absent in our model.

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Seidel, Edward, and Wai-Mo Suen. "NUMERICAL RELATIVITY." International Journal of Modern Physics C 05, no. 02 (April 1994): 181–87. http://dx.doi.org/10.1142/s012918319400012x.

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The present status of numerical relativity is reviewed. There are five closely interconnected aspects of numerical relativity: (1) Formulation. The general covariant Einstein equations are reformulated in a way suitable for numerical study by separating the 4-dimensional spacetime into a 3-dimensional space evolving in time. (2) Techniques. A set of tools is developed for determining gauge choices, setting boundary and initial conditions, handling spacetime singularities, etc. As required by the special physical and mathematical properties of general relativity, such techniques are indispensable for the numerical evolutions of spacetime. (3) Coding. The optimal use of parallel processing is crucial for many problems in numerical relativity, due to the intrinsic complexity of the theory. (4) Visualization. Numerical relativity is about the evolutions of 3-dimensional geometric structures. There are special demands on visualization. (5) Interpretation and Understanding. The integration of numerical data in relativity into a consistent physical picture is complicated by gauge and coordinate degrees of freedoms and other difficulties. We give a brief overview of the progress made in these areas.

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Rein, Gerhard. "Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 4 (May 1996): 739–62. http://dx.doi.org/10.1017/s0305004100074569.

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AbstractThe Vlasov-Einstein system describes a self-gravitating, collisionless gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with spherical, plane, or hyperbolic symmetry and prove that for small initial data solutions exist up to a spacetime singularity which is a curvature and a crushing singularity. An important tool in the analysis is a local existence result with a continuation criterion saying that solutions can be extended as long as the momenta in the support of the phase-space distribution of the matter remain bounded.

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Beyer, Florian, Jörg Frauendiener, and Jörg Hennig. "Explorations of the infinite regions of spacetime." International Journal of Modern Physics D 29, no. 10 (July 2020): 2030007. http://dx.doi.org/10.1142/s0218271820300074.

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An important concept in Physics is the notion of an isolated system. It is used in many different areas to describe the properties of a physical system which has been isolated from its environment. The interaction with the “outside” is then usually reduced to a scattering process, in which incoming radiation interacts with the system, which in turn emits outgoing radiation. In Einstein’s theory of gravitation, isolated systems are modeled as asymptotically flat spacetimes. They provide the appropriate paradigm for the study of gravitational waves and their interaction with a material system or even only with themselves. In view of the emerging era of gravitational wave astronomy, in which gravitational wave signals from many different astrophysical sources are detected and interpreted, it is necessary to have a foundation for the theoretical and numerical treatments of these signals. Furthermore, from a purely mathematical point of view, it is important to have a full understanding of the implications of imposing the condition of asymptotic flatness onto solutions of the Einstein equations. While it is known that there exists a large class of asymptotically flat solutions of Einstein’s equations, it is not known what the necessary and sufficient conditions at infinity are that have to be imposed on initial data so that they evolve into regular asymptotically flat spacetimes. The crux lies in the region near spacelike infinity [Formula: see text] where incoming and outgoing radiation “meet”. In this paper, we review the current knowledge and some of the analytical and numerical work that has gone into the attempt to understand the structure of asymptotically flat spacetimes near spacelike and null-infinity.

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Noutchegueme, Norbert, and David Dongo. "Global existence of solutions for the Einstein–Boltzmann system in a Bianchi type I spacetime for arbitrarily large initial data." Classical and Quantum Gravity 23, no. 9 (April 4, 2006): 2979–3003. http://dx.doi.org/10.1088/0264-9381/23/9/013.

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Han, Qing, and Marcus Khuri. "The Conformal Flow of Metrics and the General Penrose Inequality." Advances in Mathematical Physics 2018 (June 3, 2018): 1–9. http://dx.doi.org/10.1155/2018/7390148.

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The conformal flow of metrics has been used to successfully establish a special case of the Penrose inequality, which yields a lower bound for the total mass of a spacetime in terms of horizon area. Here we show how to adapt the conformal flow of metrics, so that it may be applied to the Penrose inequality for general initial data sets of the Einstein equations. The Penrose conjecture without the assumption of time symmetry is then reduced to solving a system of PDE with desirable properties.

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ANDRÉASSON, HÅKAN, and GERHARD REIN. "The asymptotic behaviour in Schwarzschild time of Vlasov matter in spherically symmetric gravitational collapse." Mathematical Proceedings of the Cambridge Philosophical Society 149, no. 1 (January 5, 2010): 173–88. http://dx.doi.org/10.1017/s0305004109990454.

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AbstractGiven a static Schwarzschild spacetime of ADM mass M, it is well known that no ingoing causal geodesic starting in the outer domain r > 2M will cross the event horizon r = 2M in finite Schwarzschild time. We show that in gravitational collapse of Vlasov matter this behaviour can be very different. We construct initial data for which a black hole forms and all matter crosses the event horizon as Schwarzschild time goes to infinity, and show that this is a necessary condition for geodesic completeness of the event horizon. In addition to a careful analysis of the asymptotic behaviour of the matter characteristics our proof requires a new argument for global existence of solutions to the spherically symmetric Einstein–Vlasov system in an outer domain, since our initial data have non-compact support in the radial momentum variable and previous methods break down.

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Grangé, Pierre, and Ernst Werner. "Light-front and conformal field theories in two dimensions." Modern Physics Letters A 33, no. 22 (July 19, 2018): 1850119. http://dx.doi.org/10.1142/s0217732318501195.

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Light-front (LF) quantization of massless fields in two spacetime dimensions is a long-standing and much debated problem. Even though the classical wave-equation is well-documented for almost two centuries, either as problems with initial values in spacetime variables or with initial data on characteristics in light-cone variables, the way to a consistent quantization in both types of frames is still a puzzle in many respects. This is in contrast to the most successful Conformal Field Theoretic (CFT) approach in terms of complex variables [Formula: see text], [Formula: see text] pioneered by Belavin, Polyakov and Zamolodchikov in the ’80s. It is shown here that the 2D-massless canonical quantization in both reference frames is completely consistent provided that quantum fields are treated as Operator-Valued Distributions (OPVD) with Partition of Unity (PU) test functions. We recall first that classical fields have to be considered as distributions (e.g. generalized functions in the Russian literature). Then, a necessary condition on the PU test function follows from the required matching of the classical solutions of the massless differential equations in both types of reference frame. Next we use a mathematical formulation for OPVD, developed in the recent past. Specifically, smooth [Formula: see text] fields are introduced through the convolution operation in the distributional context. Due to the specific behavior of the Fourier-transform of the initial test function, this convolution transform has a well-defined integral in the dual space, whatever the initial choice of the reference frame. The relation to the conformal fields method follows immediately from the transition to Euclidean time and leads directly to explicit calculations of a few correlation functions of the scalar field and its energy–momentum tensor. The LF derivation of the Virasoro algebra is then obtained from the [Formula: see text] and [Formula: see text] expansions of the canonical fields as distributional Laplace-transform in these variables. Finally, the popular and problematic Discretized Light Cone Quantization (DLCQ) method is scrutinized with respect to its zero mode and ultraviolet content as encompassed in the continuum OPVD formulation.

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Saha, Abhisek, and Soma Sanyal. "Temperature fluctuations and Tsallis statistics in relativistic heavy ion collisions." Modern Physics Letters A 36, no. 22 (July 19, 2021): 2150152. http://dx.doi.org/10.1142/s0217732321501522.

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In this paper, we study temperature fluctuations in the initial stages of the relativistic heavy ion collision using a multiphase transport model. We consider the plasma in the initial stages after collision before it has a chance to equilibrate. We have considered [Formula: see text] collision with a center-of-mass energy of 200 GeV. We use the nonextensive Tsallis statistics to find the entropic index in the partonic stages of the relativistic heavy ion collisions. We find that the temperature and the entropic index have a linear relationship during the partonic stages of the heavy ion collision. This has already been observed in the hadronic phase. A detailed analysis of the dependence of the entropic index on the system shows that for increasing spacetime rapidity, the entropic index of the partonic system increases. The entropic index also depends on the beam collision energy. The calculation of the entropic index from the experimental data fitting of the transverse momenta deals with the hadronic phase. However, our study shows that the behavior of the entropic index in the initial nonequilibrium stage of the collision is very similar to the behavior of the entropic index in the hadronic stage.

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CARLINI, A., and I. D. NOVIKOV. "TIME MACHINES AND THE PRINCIPLE OF SELF-CONSISTENCY AS A CONSEQUENCE OF THE PRINCIPLE OF STATIONARY ACTION (II): THE CAUCHY PROBLEM FOR A SELF-INTERACTING RELATIVISTIC PARTICLE." International Journal of Modern Physics D 05, no. 05 (October 1996): 445–79. http://dx.doi.org/10.1142/s021827189600028x.

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We consider the action principle to derive the classical, relativistic motion of a selfinteracting particle in a 4D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. In particular, we study the case of a pointlike particle subject to a “hard-sphere” self-interaction potential and which can traverse the wormhole an arbitrary number of times, and show that the only possible trajectories for which the classical action is stationary are those which are globally self-consistent. Generically, the multiplicity of these trajectories (defined as the number of self-consistent solutions to the equations of motion beginning with given Cauchy data) is finite, and it becomes infinite if certain constraints on the same initial data are satisfied. This confirms the previous conclusions (for a nonrelativistic model) by Echeverria, Klinkhammer and Thorne that the Cauchy initial value problem in the presence of a wormhole “time machine” is classically “ill-posed” (far too many solutions). Our results further extend the recent claim by Novikov et al. that the “principle of self-consistency” is a natural consequence of the “principle of minimal action.”

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Benisty, David, Eduardo I. Guendelman, Emil Nissimov, and Svetlana Pacheva. "Dynamically Generated Inflationary ΛCDM." Symmetry 12, no. 3 (March 20, 2020): 481. http://dx.doi.org/10.3390/sym12030481.

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Our primary objective is to construct a plausible, unified model of inflation, dark energy and dark matter from a fundamental Lagrangian action first principle, wherein all fundamental ingredients are systematically dynamically generated starting from a very simple model of modified gravity interacting with a single scalar field employing the formalism of non-Riemannian spacetime volume-elements. The non-Riemannian volume element in the initial scalar field action leads to a hidden, nonlinear Noether symmetry which produces an energy-momentum tensor identified as the sum of a dynamically generated cosmological constant and dust-like dark matter. The non-Riemannian volume-element in the initial Einstein–Hilbert action upon passage to the physical Einstein-frame creates, dynamically, a second scalar field with a non-trivial inflationary potential and with an additional interaction with the dynamically generated dark matter. The resulting Einstein-frame action describes a fully dynamically generated inflationary model coupled to dark matter. Numerical results for observables such as the scalar power spectral index and the tensor-to-scalar ratio conform to the latest 2018 PLANCK data.

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VILLAIN, L., and S. BONAZZOLA. "NUMERICAL TIME EVOLUTION OF INERTIAL MODES IN SLOWLY ROTATING NEUTRON STARS." International Journal of Modern Physics A 17, no. 20 (August 10, 2002): 2780. http://dx.doi.org/10.1142/s0217751x02012077.

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A numerical and linear study of modes driven toward instability by PN radiation-reaction force (current quadrupole) in slowly rotating neutron stars was presented. It consists in a time evolution using spectral methods in spherical coordinates for spatial operators. Whatever the noisy initial data, there exists a hydrodynamic instability. Yet, depending on the background, the symmetric properties of the mode may change. Thus, in a rigid rotating Newtonian star, the expected and purely axial r-mode1 is growing. But, when the Newtonian background is assumed to be differentially rotating or when the evolution is done in the framework of general relativity with Cowling approximation (frozen spacetime), a coupling between axial and polar modes appears. Then, the mode driven to instability no longer belongs to one of these subclasses. Preliminary results were presented, showing the common features and main differences between Newtonian and relativistic cases2.

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Dempsey, David, and Sam R. Dolan. "Waves and null congruences in a draining bathtub." International Journal of Modern Physics D 25, no. 09 (August 2016): 1641004. http://dx.doi.org/10.1142/s0218271816410042.

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We study wave propagation in a draining bathtub: a black hole analogue in fluid mechanics whose perturbations are governed by a Klein–Gordon equation on an effective Lorentzian geometry. Like the Kerr spacetime, the draining bathtub geometry possesses an (effective) horizon, an ergosphere and null circular orbits. We propose here that a ‘pulse’ disturbance may be used to map out the light-cone of the effective geometry. First, we apply the eikonal approximation to elucidate the link between wavefronts, null geodesic congruences and the Raychaudhuri equation. Next, we solve the wave equation numerically in the time domain using the method of lines. Starting with Gaussian initial data, we demonstrate that a pulse will propagate along a null congruence and thus trace out the light-cone of the effective geometry. Our new results reveal features, such as wavefront intersections, frame-dragging, winding and interference effects, that are closely associated with the presence of null circular orbits and the ergosphere.

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Cao, Zhoujian, Pei Fu, Li-Wei Ji, and Yinhua Xia. "Application of local discontinuous Galerkin method to Einstein equations." International Journal of Modern Physics D 28, no. 01 (January 2019): 1950014. http://dx.doi.org/10.1142/s0218271819500147.

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Finite difference and pseudo-spectral methods have been widely used in the numerical relativity to solve the Einstein equations. As the third major category method to solve partial differential equations, finite element method is less frequently used in numerical relativity. In this paper, we design a finite element algorithm to solve the evolution part of the Einstein equations. This paper is the second one of a systematic investigation of applying adaptive finite element method to the Einstein equations, especially aiming for binary compact objects simulations. The first paper of this series has been contributed to the constrained part of the Einstein equations for initial data. Since applying finite element method to the Einstein equations is a big project, we mainly propose the theoretical framework of a finite element algorithm together with local discontinuous Galerkin method for the Einstein equations in the current work. In addition, we have tested our algorithm based on the spherical symmetric spacetime evolution. In order to simplify our numerical tests, we have reduced the problem to a one-dimensional space problem by taking the advantage of the spherical symmetry. Our reduced equation system is a new formalism for spherical symmetric spacetime simulation. Based on our test results, we find that our finite element method can capture the shock formation which is introduced by numerical error. In contrast, such shock is smoothed out by numerical dissipation within the finite difference method. We suspect this is partly the reason that the accuracy of finite element method is higher than the finite difference method. At the same time, different kinds of formulation parameters setting are also discussed.

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Dain, Sergio. "Initial data for stationary spacetimes near spacelike infinity." Classical and Quantum Gravity 18, no. 20 (October 4, 2001): 4329–38. http://dx.doi.org/10.1088/0264-9381/18/20/312.

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ALCOFORADO, M. A., H. P. DE OLIVEIRA, and E. L. RODRIGUES. "INITIAL DATA FOR NUMERICAL RELATIVITY: THE USE OF SPECTRAL METHODS." International Journal of Modern Physics: Conference Series 03 (January 2011): 417–27. http://dx.doi.org/10.1142/s2010194511000936.

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The determination of physical initial data is an important task in numerical relativity. In this direction we have applied the Galerkin and collocation methods to solve the Hamiltonian constraints resulting from the Cauchy formulation in the cases of spacetimes containing black holes as described by Ref. 1. We have shown that a considerable improvement in the accuracy is obtained if the basis functions are chosen such that the boundary conditions are satisfied. We have also introduced a new approach to solve numerically the constraint equations which consists in transforming them into parabolic equations after introducing fictitious diffusion terms. As a consequence, the application of Galerkin or collocation methods produces a dynamical system whose stationary solution corresponds to the initial data.

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Stoica, Ovidiu-Cristinel. "Spacetimes with singularities." Analele Universitatii "Ovidius" Constanta - Seria Matematica 20, no. 2 (June 1, 2012): 213–38. http://dx.doi.org/10.2478/v10309-012-0050-3.

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Abstract We report on some advances made in the problem of singularities in general relativity.First is introduced the singular semi-Riemannian geometry for metrics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like the Riemann curvature are usually prohibited by the fact that the metric is not invertible. The things become even worse at the points where the signature changes. We show that we can still do many of these operations, in a different framework which we propose. This allows the writing of an equivalent form of Einstein's equation, which works for degenerate metric too.Once we make the singularities manageable from mathematical viewpoint, we can extend analytically the black hole solutions and then choose from the maximal extensions globally hyperbolic regions. Then we find space-like foliations for these regions, with the implication that the initial data can be preserved in reasonable situations. We propose qualitative models of non-primordial and/or evaporating black holes.We supplement the material with a brief note reporting on progress made since this talk was given, which shows that we can analytically extend the Schwarzschild and Reissner-Nordström metrics at and beyond the singularities, and the singularities can be made degenerate and handled with the mathematical apparatus we developed.

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Wang, Yaohua, and Xiao Zhang. "Positive energy theorem for asymptotically anti-de Sitter spacetimes with distributional curvature." International Journal of Mathematics 30, no. 13 (December 2019): 1940003. http://dx.doi.org/10.1142/s0129167x19400032.

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Wessels, Ewald. "Exact analytic characteristic initial data for axisymmetric, non-rotating, vacuum spacetimes, with an application to the binary black hole problem." Classical and Quantum Gravity 15, no. 8 (August 1, 1998): 2509–22. http://dx.doi.org/10.1088/0264-9381/15/8/025.

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Gajic, Dejan, and Claude Warnick. "Quasinormal Modes in Extremal Reissner–Nordström Spacetimes." Communications in Mathematical Physics 385, no. 3 (June 27, 2021): 1395–498. http://dx.doi.org/10.1007/s00220-021-04137-4.

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AbstractWe present a new framework for characterizing quasinormal modes (QNMs) or resonant states for the wave equation on asymptotically flat spacetimes, applied to the setting of extremal Reissner–Nordström black holes. We show that QNMs can be interpreted as honest eigenfunctions of generators of time translations acting on Hilbert spaces of initial data, corresponding to a suitable time slicing. The main difficulty that is present in the asymptotically flat setting, but is absent in the previously studied asymptotically de Sitter or anti de Sitter sub-extremal black hole spacetimes, is that $$L^2$$ L 2 -based Sobolev spaces are not suitable Hilbert space choices. Instead, we consider Hilbert spaces of functions that are additionally Gevrey regular at infinity and at the event horizon. We introduce $$L^2$$ L 2 -based Gevrey estimates for the wave equation that are intimately connected to the existence of conserved quantities along null infinity and the event horizon. We relate this new framework to the traditional interpretation of quasinormal frequencies as poles of the meromorphic continuation of a resolvent operator and obtain new quantitative results in this setting.

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Alho, Artur, Grigorios Fournodavlos, and Anne T. Franzen. "The wave equation near flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang singularities." Journal of Hyperbolic Differential Equations 16, no. 02 (June 2019): 379–400. http://dx.doi.org/10.1142/s0219891619500140.

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We consider the wave equation, [Formula: see text], in fixed flat Friedmann–Lemaître–Robertson–Walker and Kasner spacetimes with topology [Formula: see text]. We obtain generic blow up results for solutions to the wave equation toward the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to the solutions that blow up in an open set of the Big Bang hypersurface [Formula: see text]. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate [Formula: see text]-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the [Formula: see text] norms of the solutions blow up toward the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates, respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

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Costa, João L., José Natário, and Pedro F. C. Oliveira. "Decay of solutions of the wave equation in expanding cosmological spacetimes." Journal of Hyperbolic Differential Equations 16, no. 01 (March 2019): 35–58. http://dx.doi.org/10.1142/s0219891619500024.

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We study the decay of solutions of the wave equation in some expanding cosmological spacetimes, namely flat Friedmann–Lemaître–Robertson–Walker (FLRW) models and the cosmological region of the Reissner–Nordström–de Sitter (RNdS) solution. By introducing a partial energy and using an iteration scheme, we find that, for initial data with finite higher order energies, the decay rate of the time derivative is faster than previously existing estimates. For models undergoing accelerated expansion, our decay rate appears to be (almost) sharp.

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Kroon, Juan Antonio Valiente. "Time asymmetric spacetimes near null and spatial infinity: II. Expansions of developments of initial data sets with non-smooth conformal metrics." Classical and Quantum Gravity 22, no. 9 (April 8, 2005): 1683–707. http://dx.doi.org/10.1088/0264-9381/22/9/015.

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Oliver, Jesús. "A vector field method for non-trapping, radiating spacetimes." Journal of Hyperbolic Differential Equations 13, no. 04 (December 2016): 735–90. http://dx.doi.org/10.1142/s021989161650020x.

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We study the global decay properties of solutions to the linear wave equation in 1+3 dimensions on time-dependent, weakly asymptotically flat spacetimes. Assuming non-trapping of null geodesics and a local energy decay estimate, we prove that sufficiently regular solutions to this equation have bounded conformal energy. As an application we also show a conformal energy estimate with vector fields applied to the solution as well as a global [Formula: see text] decay bound in terms of a weighted norm on initial data. For solutions to the wave equation in these dynamical backgrounds, our results reduce the problem of establishing the classical pointwise decay rate [Formula: see text] in the interior and [Formula: see text] along outgoing null cones to simply proving that local energy decay holds.

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RINGSTRÖM, HANS. "CURVATURE BLOW UP ON A DENSE SUBSET OF THE SINGULARITY IN T3-GOWDY." Journal of Hyperbolic Differential Equations 02, no. 02 (June 2005): 547–64. http://dx.doi.org/10.1142/s021989160500052x.

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This paper is concerned with the Einstein vacuum equations under the additional assumption of T3-Gowdy symmetry. We prove that there is a generic set of initial data such that the corresponding solutions exhibit curvature blow up on a dense subset of the singularity. By generic, we mean a countable intersection of open sets (i.e. a Gδ set) which is also dense. Furthermore, the set of initial data is given the C∞ topology. This result was presented at a conference in Miami 2004. Recently, we have obtained a stronger result, but the argument to prove it is different and much longer. Therefore, we here wish to present the original argument. Finally, combining the results presented here with a paper by Chruściel and Lake, one obtains strong cosmic censorship for T3-Gowdy spacetimes.

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ARANHA, R. F., I. DAMIÃO SOARES, H. P. OLIVEIRA, and E. V. TONINI. "ENERGY AND MOMENTUM LOSS BY GRAVITATIONAL RADIATION EMISSION IN THE COLLISION OF TWO SCHWARZSCHILD BLACK HOLES." International Journal of Modern Physics A 24, no. 08n09 (April 10, 2009): 1583–87. http://dx.doi.org/10.1142/s0217751x09045042.

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We analyze the head-on collision of two boosted Schwarzschild black holes and the mass-energy loss of the system by gravitational wave emission, in the realm of Robinson-Trautman (RT) spacetimes. The characteristic initial data for the problem are constructed, and evolved by RT equation integrated by numerical codes based on the Galerkin method. The emission of gravitational waves is typical bremsstrahlung at early times and the final configuration is that of a boosted black hole with larger (Bondi) rest mass and smaller velocity parameter. The efficiency Δ of the process of energy extraction by gravitational radiation is evaluated and satisfies a nonextensive distribution with entropic index q ≃ 1/2. The final momentum of the remnant black hole has a maximum which depends on the ratio of the masses of the initial black holes.

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ANDRÉASSON, HÅKAN, and GERHARD REIN. "FORMATION OF TRAPPED SURFACES FOR THE SPHERICALLY SYMMETRIC EINSTEIN–VLASOV SYSTEM." Journal of Hyperbolic Differential Equations 07, no. 04 (December 2010): 707–31. http://dx.doi.org/10.1142/s0219891610002268.

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We consider the spherically symmetric, asymptotically flat, non-vacuum Einstein equations, using as matter model a collisionless gas as described by the Vlasov equation. We find explicit conditions on the initial data which guarantee the formation of a trapped surface in the evolution which in particular implies that weak cosmic censorship holds for these data. We also analyze the evolution of solutions after a trapped surface has formed and we show that the event horizon is future complete. Furthermore we find that the apparent horizon and the event horizon do not coincide. This behavior is analogous to what is found in certain Vaidya spacetimes. The analysis is carried out in Eddington–Finkelstein coordinates.

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BLOHMANN, CHRISTIAN, MARCO CEZAR BARBOSA FERNANDES, and ALAN WEINSTEIN. "GROUPOID SYMMETRY AND CONSTRAINTS IN GENERAL RELATIVITY." Communications in Contemporary Mathematics 15, no. 01 (January 22, 2013): 1250061. http://dx.doi.org/10.1142/s0219199712500617.

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When the vacuum Einstein equations are cast in the form of Hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold [Formula: see text] of Riemannian metrics on a Cauchy hypersurface Σ. As in every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection. In an appendix, we develop some aspects of diffeology, the basic framework for our treatment of function spaces.

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Angelopoulos, Yannis, Stefanos Aretakis, and Dejan Gajic. "Logarithmic corrections in the asymptotic expansion for the radiation field along null infinity." Journal of Hyperbolic Differential Equations 16, no. 01 (March 2019): 1–34. http://dx.doi.org/10.1142/s0219891619500012.

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We obtain the second-order late-time asymptotics for the radiation field of solutions to the wave equation on spherically symmetric and asymptotically flat backgrounds including the Schwarzschild and sub-extremal Reissner–Nordström families of black hole spacetimes. These terms appear as logarithmic corrections to the leading-order asymptotic terms which were rigorously derived in our previous work. Such corrections have been heuristically and numerically derived in the physics literature in the case of a non-vanishing Newman–Penrose constant. In this case, our results provide a rigorous confirmation of the existence of these corrections. On the other hand, the precise logarithmic corrections for spherically symmetric compactly supported initial data (and hence, with a vanishing Newman–Penrose constant) explicitly obtained here appear to be new.

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Góźdź, Andrzej, Włodzimierz Piechocki, Grzegorz Plewa, and Tomasz Trześniewski. "Hunting for Gravitational Quantum Spikes." Universe 7, no. 3 (February 28, 2021): 49. http://dx.doi.org/10.3390/universe7030049.

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We present the result of our examination of quantum structures called quantum spikes. The classical spikes that are known in gravitational systems, occur in the evolution of the inhomogeneous spacetimes. A different kind of spikes, which we name strange spikes, can be seen in the dynamics of the homogeneous sector of the Belinski–Khalatnikov–Lifshitz scenario. They can be made visible if the so-called inhomogeneous initial data are used. The question to be explored is whether the strange spikes may survive quantization. The answer is in the affirmative. However, this is rather a subtle effect that needs further examination using sophisticated analytical and numerical tools. The spikes seem to be of fundamental importance, both at classical and quantum levels, as they may serve as seeds of real structures in the universe.

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Gómez-Lobo, Alfonso García-Parrado, and Juan A. Valiente Kroon. "Initial data sets for the Schwarzschild spacetime." Physical Review D 75, no. 2 (January 23, 2007). http://dx.doi.org/10.1103/physrevd.75.024027.

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41

Alaee, Aghil, and Hari K. Kunduri. "Mass functional for initial data in4+1-dimensional spacetime." Physical Review D 90, no. 12 (December 29, 2014). http://dx.doi.org/10.1103/physrevd.90.124078.

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Almaraz, Sérgio, Levi Lopes de Lima, and Luciano Mari. "Spacetime Positive Mass Theorems for Initial Data Sets with Non-Compact Boundary." International Mathematics Research Notices, September 7, 2020. http://dx.doi.org/10.1093/imrn/rnaa226.

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Abstract In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positive mass inequalities under the assumption that the underlying manifold is spin. In the asymptotically flat case, we also prove a rigidity statement when the energy-momentum vector is light-like. Our treatment aims to underline both the common features and the differences between the asymptotically Euclidean and hyperbolic settings, especially regarding the boundary DECs.

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Cederbaum, Carla, and Anna Sakovich. "On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity." Calculus of Variations and Partial Differential Equations 60, no. 6 (August 27, 2021). http://dx.doi.org/10.1007/s00526-021-02060-z.

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AbstractWe propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a equivariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau (Invent Math 124:281–311, 1996). We show that a unique STCMC-foliation exists near infinity of any asymptotically Euclidean initial data set with non-vanishing energy which allows for the definition of a new notion of total center of mass for isolated systems. This STCMC-center of mass transforms equivariantly under the asymptotic Poincaré group of the ambient spacetime and in particular evolves under the Einstein evolution equations like a point particle in Special Relativity. The new definition also remedies subtle deficiencies in the CMC-approach to defining the total center of mass suggested by Huisken and Yau (Invent Math 124:281–311, 1996) which were described by Cederbaum and Nerz (Ann Henri Poincaré 16:1609–1631, 2015).

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Hilditch, David, Juan A. Valiente Kroon, and Peng Zhao. "Improved existence for the characteristic initial value problem with the conformal Einstein field equations." General Relativity and Gravitation 52, no. 9 (September 2020). http://dx.doi.org/10.1007/s10714-020-02734-7.

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Abstract We adapt Luk’s analysis of the characteristic initial value problem in general relativity to the asymptotic characteristic problem for the conformal Einstein field equations to demonstrate the local existence of solutions in a neighbourhood of the set on which the data are given. In particular, we obtain existence of solutions along a narrow rectangle along null infinity which, in turn, corresponds to an infinite domain in the asymptotic region of the physical spacetime. This result generalises work by Kánnár on the local existence of solutions to the characteristic initial value problem by means of Rendall’s reduction strategy. In analysing the conformal Einstein equations we make use of the Newman–Penrose formalism and a gauge due to J. Stewart.

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Lienert, Matthias, and Markus Nöth. "Existence of relativistic dynamics for two directly interacting Dirac particles in 1 + 3 dimensions." Reviews in Mathematical Physics, May 6, 2021, 2150023. http://dx.doi.org/10.1142/s0129055x21500239.

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Here we prove the existence and uniqueness of solutions of a class of integral equations describing two Dirac particles in 1+3 dimensions with direct interactions. This class of integral equations arises naturally as a relativistic generalization of the integral version of the two-particle Schrödinger equation. Crucial use of a multi-time wave function [Formula: see text] with [Formula: see text] is made. A central feature is the time delay of the interaction. Our main result is an existence and uniqueness theorem for a Minkowski half-space, meaning that the Minkowski spacetime is cut off before [Formula: see text]. We furthermore show that the solutions are determined by Cauchy data at the initial time; however, no Cauchy problem is admissible at other times. A second result is to extend the first one to particular FLRW spacetimes with a Big Bang singularity, using the conformal invariance of the Dirac equation in the massless case. This shows that the cutoff at [Formula: see text] can arise naturally and be fully compatible with relativity. We thus obtain a class of interacting, manifestly covariant and rigorous models in [Formula: see text] dimensions.

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Burikham, Piyabut, Supakchai Ponglertsakul, and Taum Wuthicharn. "Quasi-normal modes of near-extremal black holes in generalized spherically symmetric spacetime and strong cosmic censorship conjecture." European Physical Journal C 80, no. 10 (October 2020). http://dx.doi.org/10.1140/epjc/s10052-020-08528-0.

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AbstractA number of near-extremal conditions are utilized to simplify the equation of motion of the neutral scalar perturbations in generalized spherically symmetric black hole background into a differential equation with the Pöschl–Teller potential. An analytic formula for quasinormal frequencies is obtained. The analytic formula is then used to investigate strong cosmic censorship conjectures (SCC) of the generalized black hole spacetime for the smooth initial data. The Christodoulou version of the SCC is found to be violated for certain regions of the black hole parameter space including the black holes in general relativity while the $$C^{1}$$ C 1 version of the SCC is always valid.

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47

Bigorgne, Léo. "Asymptotic Properties of the Solutions to the Vlasov–Maxwell System in the Exterior of a Light Cone." International Mathematics Research Notices, July 27, 2020. http://dx.doi.org/10.1093/imrn/rnaa062.

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Abstract This paper is concerned with the asymptotic behavior of small data solutions to the three-dimensional Vlasov–Maxwell system in the exterior of a light cone. The plasma does not have to be neutral, and no compact support assumptions are required on the data. In particular, the initial decay in the velocity variable of the particle density is optimal and we only require an $L^2$ bound on the electromagnetic field with no additional weight. We use vector field methods to derive improved decay estimates in null directions for the electromagnetic field, the particle density, and their derivatives. In contrast with [ 2], where we studied the behavior of the solutions in the whole spacetime, the initial data have less decay and we do not need to modify the commutation vector fields of the relativistic transport operator. To control the solutions under these assumptions, we crucially use the strong decay satisfied by the particle density in the exterior of the light cone, null properties of the Vlasov equation, and certain hierarchies in the energy norms.

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Enciso, Alberto, and Daniel Peralta-Salas. "Approximation Theorems for the Schrödinger Equation and Quantum Vortex Reconnection." Communications in Mathematical Physics, July 28, 2021. http://dx.doi.org/10.1007/s00220-021-04177-w.

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AbstractWe prove the existence of smooth solutions to the Gross–Pitaevskii equation on $$\mathbb {R}^3$$ R 3 that feature arbitrarily complex quantum vortex reconnections. We can track the evolution of the vortices during the whole process. This permits to describe the reconnection events in detail and verify that this scenario exhibits the properties observed in experiments and numerics, such as the $$t^{1/2}$$ t 1 / 2 and change of parity laws. We are mostly interested in solutions tending to 1 at infinity, which have finite Ginzburg–Landau energy and physically correspond to the presence of a background chemical potential, but we also consider the cases of Schwartz initial data and of the Gross–Pitaevskii equation on the torus. In the proof, the Gross–Pitaevskii equation operates in a nearly linear regime, so the result applies to a wide range of nonlinear Schrödinger equations. Indeed, an essential ingredient in the proofs is the development of novel global approximation theorems for the Schrödinger equation on $$\mathbb {R}^n$$ R n . Specifically, we prove a qualitative approximation result that applies for solutions defined on very general spacetime sets and also a quantitative result for solutions on product sets in spacetime $$D\times \mathbb {R}$$ D × R . This hinges on frequency-dependent estimates for the Helmholtz–Yukawa equation that are of independent interest.

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Rogatko, Marek. "Mass formulae and staticity condition for dark matter charged black holes." European Physical Journal C 80, no. 9 (September 2020). http://dx.doi.org/10.1140/epjc/s10052-020-08432-7.

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AbstractThe Arnowitt–Deser–Misner formalism is used to derive variations of mass, angular momentum and canonical energy for Einstein–Maxwell dark matter gravity in which the auxiliary gauge field coupled via kinetic mixing term to the ordinary Maxwell one, which mimics properties of hidden sector. Inspection of the initial data for the manifold with an interior boundary, having topology of $$S^2$$ S 2 , enables us to find the generalised first law of black hole thermodynamics in the aforementioned theory. It has been revealed that the stationary black hole solution being subject to the condition of encompassing a bifurcate Killing horizon with a bifurcation sphere, which is non-rotating, must be static and has vanishing magnetic Maxwell and dark matter sector fields, on static slices of the spacetime under consideration.

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Heller, Michal P., Alexandre Serantes, Michał Spaliński, Viktor Svensson, and Benjamin Withers. "Transseries for causal diffusive systems." Journal of High Energy Physics 2021, no. 4 (April 2021). http://dx.doi.org/10.1007/jhep04(2021)192.

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Abstract The large proper-time behaviour of expanding boost-invariant fluids has provided many crucial insights into quark-gluon plasma dynamics. Here we formulate and explore the late-time behaviour of nonequilibrium dynamics at the level of linearized perturbations of equilibrium, but without any special symmetry assumptions. We introduce a useful quantitative approximation scheme in which hydrodynamic modes appear as perturbative contributions while transients are nonperturbative. In this way, solutions are naturally organized into transseries as they are in the case of boost-invariant flows. We focus our attention on the ubiquitous telegrapher’s equation, the simplest example of a causal theory with a hydrodynamic sector. In position space we uncover novel transient contributions as well as Stokes phenomena which change the structure of the transseries based on the spacetime region or the choice of initial data.

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Journal articles on the topic 'Spacetime initial data'

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