Baumgarte-Shapiro-Shibata-Nakamura Research Articles

Gauge preheating with full general relativity

We study gauge preheating following pseudoscalar-driven inflation in full general relativity. We implement the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) scheme to solve the full nonlinear evolution of the metric alongside the dynamics of the pseudoscalar and gauge fields. The dynamics of the background and emission of gravitational waves are broadly consistent with simulations in a Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. We find large, localized overdensities in the BSSN simulations of order δ = δρ/ρ ∼ 30, and the dimensionless power spectrum of δ peaks above unity. These overdense regions are seeded on length scales only slightly smaller than the horizon, and have a compactness C ∼ 0.1. The scale of peak compactness is shorter than the Jeans length, which implies that pressure of the matter fields plays an important role in the evolution of these objects.

Numerical solutions for the f(R)-Klein–Gordon system

We construct a numerical relativity code based on the Baumgarte–Shapiro–Shibata–Nakamura (BSSN) formulation for the gravitational quadratic f(R) Starobinsky model. By removing the assumption that the determinant of the conformal 3-metric is unity, we first generalize the BSSN formulation for general f(R) gravity theories in the metric formalism to accommodate arbitrary coordinates for the first time. We then describe the implementation of this formalism to the paradigmatic Starobinsky model. We apply the implementation to three scenarios: the Schwarzschild black hole solution, flat space with non-trivial gauge dynamics, and a massless Klein–Gordon (KG) scalar field. In each case, long-term stability and second-order convergence is demonstrated. The case of the massless KG scalar field is used to exercise the additional terms and variables resulting from the f(R) contributions. For this model, we show for the first time that additional damped oscillations arise in the subcritical regime as the system approaches a stable configuration.

SPHINCS_BSSN: a general relativistic smooth particle hydrodynamics code for dynamical spacetimes

We present a new methodology for simulating self-gravitating general-relativistic fluids. In our approach the fluid is modelled by means of Lagrangian particles in the framework of a general-relativistic (GR) smoothed particle hydrodynamics (SPH) formulation, while the spacetime is evolved on a mesh according to the Baumgarte–Shapiro–Shibata–Nakamura (BSSN) formulation that is also frequently used in Eulerian GR-hydrodynamics. To the best of our knowledge this is the first Lagrangian fully general relativistic hydrodynamics code (all previous SPH approaches used approximations to GR-gravity). A core ingredient of our particle–mesh approach is the coupling between the gas (represented by particles) and the spacetime (represented by a mesh) for which we have developed a set of sophisticated interpolation tools that are inspired by other particle–mesh approaches, in particular by vortex-particle methods. One advantage of splitting the methodology between matter and spacetime is that it gives us more freedom in choosing the resolution, so that—if the spacetime is smooth enough—we obtain good results already with a moderate number of grid cells and can focus the computational effort on the simulation of the matter. Further advantages of our approach are the ease with which ejecta can be tracked and the fact that the neutron star surface remains well-behaved and does not need any particular treatment. In the hydrodynamics part of the code we use a number of techniques that are new to SPH, such as reconstruction, slope limiting and steering dissipation by monitoring entropy conservation. We describe here in detail the employed numerical methods and demonstrate the code performance in a number of benchmark problems ranging from shock tube tests, over Cowling approximations to the fully dynamical evolution of neutron stars in self-consistently evolved spacetimes.

A spectral method algorithm for numerical simulations of gravitational fields

A numerical study of the Einstein field equations, based on the 3 + 1 foliation of the spacetime, is presented. A pseudo-spectral technique has been employed for simulations in vacuum, within two different formalisms, namely the Arnowitt–Deser–Misner (ADM) and the conformal Baumgarte–Shapiro–Shibata–Nakamura (BSSN) approach. The numerical code is based on the Fourier decomposition, accompanied by different filtering techniques. The role of the dealiasing, as well as the influence of the filter type, has been investigated. The algorithms have been stabilized via a novel procedure that controls self-consistently the regularity of the solutions. The accuracy of the model has been validated through standard testbeds, revealing that the filtered pseudo-spectral technique is among the most accurate approaches. Finally, the procedure has been stressed via black hole dynamics and a new strategy, based on hyperviscous dissipation that suppresses spurious boundary problems, has been proposed. The model represents a valid tool of investigation, particularly suitable for the inspection of small scale nonlinear phenomena in gravitational dynamics.

Departures from the Friedmann-Lemaitre-Robertston-Walker Cosmological Model in an Inhomogeneous Universe: A Numerical Examination.

While the use of numerical general relativity for modeling astrophysical phenomena and compact objects is commonplace, the application to cosmological scenarios is only just beginning. Here, we examine the expansion of a spacetime using the Baumgarte-Shapiro-Shibata-Nakamura formalism of numerical relativity in synchronous gauge. This work represents the first numerical cosmological study that is fully relativistic, nonlinear, and without symmetry. The universe that emerges exhibits an average Friedmann-Lemaître-Robertson-Walker (FLRW) behavior; however, this universe also exhibits locally inhomogeneous expansion beyond that expected in linear perturbation theory around a FLRW background.

Integration of inhomogeneous cosmological spacetimes in the BSSN formalism

We present cosmological-scale numerical simulations of an evolving universe in full general relativity (GR) and introduce a new numerical tool, {\sc CosmoGRaPH}, which employs the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism on a 3-dimensional grid. Using {\sc CosmoGRaPH}, we calculate the effect of an inhomogeneous matter distribution on the evolution of a spacetime. We also present the results of a set of standard stability tests to demonstrate the robustness of our simulations.

Black hole critical behavior with the generalized BSSN formulation

The development of hyperbolic formulations of Einstein's equations has revolutionized our ability to perform long-time, stable, accurate numerical simulations of strong field gravitational phenomena. However, hyperbolic methods have seen relatively little application in one area of interest, type II critical collapse, where the challenges for a numerical code are particularly severe. Using the critical collapse of a massless scalar field in spherical symmetry as a test case, we study a generalization of the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation due to Brown that is suited for use with curvilinear coordinates. We adopt standard dynamical gauge choices, including 1+log slicing and a shift that is either zero or evolved by a Gamma-driver condition. With both choices of shift we are able to evolve sufficiently close to the black hole threshold to (1) unambiguously identify the discrete self-similarity of the critical solution (2) determine an echoing exponent consistent with previous calculations and (3) measure a mass scaling exponent, also in accord with prior computations. Our results can be viewed as an encouraging first step towards the use of hyperbolic formulations in more generic type II scenarios, including the as yet unresolved problem of critical collapse of axisymmetric gravitational waves.

Improved fast-rotating black hole evolution simulations with modified Baumgarte-Shapiro-Shibata-Nakamura formulation

Different formulations of Einstein's equations used in numerical relativity can affect not only the stability but also the accuracy of numerical simulations. In the original Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation, the loss of the angular momentum, $J$, is non-negligible in highly spinning single black hole evolutions. This loss also appears, usually right after the merger, in highly spinning binary black hole simulations, The loss of $J$ may be attributed to some unclear numerical dissipation. Reducing unphysical dissipation is expected to result in more stable and accurate evolutions. In the previous work [H.-J. Yo et al., Phys. Rev. D 86, 064027 (2012).] we proposed several modifications which are able to prevent black hole evolutions from the unphysical dissipation, and the resulting simulations are more stable than in the traditional BSSN formulation. Specifically, these three modifications (M1, M2, and M3) enhance the effects of stability, hyperbolicity, and dissipation of the formulation. We experiment further in this work with these modifications, and demonstrate that these modifications improve the accuracy and also effectively suppress the loss of $J$, particularly in the black hole simulations with an initially large ratio of $J$ and a square of the ADM mass.

Gravitational collapse of charged scalar fields

In order to study the gravitational collapse of charged matter we analyze the simple model of an self-gravitating massless scalar field coupled to the electromagnetic field in spherical symmetry. The evolution equations for the Maxwell-Klein-Gordon sector are derived in the 3+1 formalism, and coupled to gravity by means of the stress-energy tensor of these fields. To solve consistently the full system we employ a generalized Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of General Relativity that is adapted to spherical symmetry. We consider two sets of initial data that represent a time symmetric spherical thick shell of charged scalar field, and differ by the fact that one set has zero global electrical charge while the other has non-zero global charge. For compact enough initial shells we find that the configuration doesn't disperse and approaches a final state corresponding to a sub-extremal Reissner-N\"ordstrom black hole with $|Q|<M$. By increasing the fundamental charge of the scalar field $q$ we find that the final black hole tends to become more and more neutral. Our results support the cosmic censorship conjecture for the case of charged matter.

Accuracy issues for numerical waveforms

We study the convergence properties of our implementation of the moving punctures approach at very high resolutions for an equal-mass, nonspinning, black-hole binary. We find convergence of the Hamiltonian constraint on the horizons and the ${L}_{2}$ norm of the Hamiltonian constraint in the bulk for sixth- and eighth-order finite difference implementations. The momentum constraint is more sensitive, and its ${L}_{2}$ norm shows clear convergence for a system with consistent sixth-order finite differencing, while the momentum and Baumgarte-Shapiro-Shibata-Nakamura (BSSN) constraints on the horizons show convergence for both sixth- and eighth-order systems. We analyze the gravitational waveform error from the late inspiral, merger, and ringdown. We find that using several lower-order techniques for increasing the speed of numerical relativity simulations actually lead to apparently nonconvergent errors. Even when using standard high-accuracy techniques, rather than seeing clean convergence, where the waveform phase is a monotonic function of grid resolution, we find that the phase tends to oscillate with resolution, possibly due to stochastic errors induced by grid refinement boundaries. Our results seem to indicate that one can obtain gravitational waveform phases to within 0.05 rad. (and possibly as small as 0.015 rad), while the amplitude error can be reduced to 0.1%. We then compare with the waveforms obtained using the CCZ4 formalism. We find that the CCZ4 waveforms have larger truncation errors for a given resolution, but the Richardson extrapolation phase of the CCZ4 and BSSN waveforms agrees to within 0.01 rad, even during the ringdown.

Modifications for numerical stability of black hole evolution

We experiment with several new modifications for the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of the Einstein field equations and demonstrate how these modifications affect the stability of numerical black hole evolution calculations. With these modifications, we evolve both single non-rotating excised Kerr-Schild black holes and punctured binary black holes, and obtain accurate and stable simulations.

Constraint propagation ofC2-adjusted formulation. II. Another recipe for robust Baumgarte-Shapiro-Shibata-Nakamura evolution system

In order to obtain an evolution system which is robust against the violation of constraints, we present a new set of evolution systems based on the so-called Baumgarte-Shapiro-Shibata-Nakamura (BSSN) equations.The idea is to add functional derivatives of the norm of constraints, $C^2$, to the evolution equations, which was proposed by Fiske (2004) and was applied to the ADM formulation in our previous study. We derive the constraint propagation equations, discuss the behavior of constraint damping, and present the results of numerical tests using the gauge-wave and polarized Gowdy wave spacetimes. The construction of the $C^2$-adjusted system is straightforward. However, in BSSN, there are two kinetic constraints and three algebraic constraints; thus, the definition of $C^2$ is a matter of concern. By analyzing constraint propagation equations, we conclude that $C^2$ should include all the constraints, which is also confirmed numerically. By tuning the parameters, the lifetime of the simulations can be increased as 2-10 times as longer than those of the standard BSSN evolutions.

Binary black hole mergers in gaseous disks: Simulations in general relativity

Simultaneous gravitational and electromagnetic wave observations of merging black hole binaries (BHBHs) can provide unique opportunities to study gravitation physics, accretion, and cosmology. Here we perform fully general-relativistic, hydrodynamic simulations of equal-mass, nonspinning BHBHs coalescing in a circumbinary disk. We evolve the metric using the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of Einstein's field equations with standard moving puncture gauge conditions. We handle the hydrodynamics via a high-resolution shock-capturing scheme. These initial simulations are exploratory in nature and simplified accordingly. We track the inspiral starting from a binary separation of $10M$, where $M$ is the total binary mass. We take the disks to have an inner radius at ${R}_{\mathrm{in}}\ensuremath{\approx}15M$ to account for the hollow created by the binary torques. Our disks extend to $R\ensuremath{\approx}65M$ and have an initial scale height of $H/R\ensuremath{\approx}0.03--0.11$. The gas is governed by a $\ensuremath{\Gamma}$-law equation of state, with $\ensuremath{\Gamma}$ equal to $5/3$, $4/3$, and 1.1. Disks are allowed to relax in the ``early inspiral'' epoch to provide quasistationary realistic initial data. We then evolve the spacetime metric and matter during the ``late inspiral and merger'' epochs. The later simulations are designed to track BHBH inspiral following disk-binary decoupling, through merger and ringdown, terminating before viscosity has time to fill the hollow about the black hole remnant. We compute the gas flow and accretion rate and estimate the electromagnetic luminosity due to bremsstrahlung and synchrotron emission as a perturbation for optically thin disks. The synchrotron component of the luminosity peaks in the infrared band and should be detectable by WFIRST and possibly the LSST for a ${10}^{8}{M}_{\ensuremath{\bigodot}}$ binary embedded in a disk with a density $n\ensuremath{\sim}{10}^{12}\text{ }\text{ }{\mathrm{cm}}^{\ensuremath{-}3}$ at $z=1$, beginning with a maximum value of $L\ensuremath{\sim}{10}^{46}{n}_{12}^{2}{M}_{8}^{3}\text{ }\text{ }\mathrm{erg}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$ at decoupling, and decreasing steadily over a time scale of $\ensuremath{\sim}100{M}_{8}$ hours to a value of $L\ensuremath{\sim}{10}^{45}{n}_{12}^{2}{M}_{8}^{3}\text{ }\text{ }\mathrm{erg}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$ at merger.

Formulations of the 3+1 evolution equations in curvilinear coordinates

Following Brown, in this paper we give an overview of how to modify standard hyperbolic formulations of the 3+1 evolution equations of General Relativity in such a way that all auxiliary quantities are true tensors, thus allowing for these formulations to be used with curvilinear sets of coordinates such as spherical or cylindrical coordinates. After considering the general case for both the Nagy-Ortiz-Reula (NOR) and the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulations, we specialize to the case of spherical symmetry and also discuss the issue of regularity at the origin. Finally, we show some numerical examples of the modified BSSN formulation at work in spherical symmetry.

Action principle for numerical-relativity evolution systems

A Lagrangian density is provided that allows the recovery of the Z4 evolution system from an action principle. The resulting system is then strongly hyperbolic when supplemented by gauge conditions like ``$1+\mathrm{log}$'' or ``freezing shift,'' suitable for numerical evolution. The physical constraint ${Z}_{\ensuremath{\mu}}=0$ can be imposed just on the initial data. The corresponding canonical equations are also provided. This opens the door to analogous results for other numerical-relativity formalisms, like BSSN (Baumgarte-Shapiro-Shibata-Nakamura), that can be derived from Z4 by a symmetry-breaking procedure. The harmonic formulation can be easily recovered by a slight modification of the procedure. This provides a mechanism for deriving both the field evolution equations and the gauge conditions from the action principle, with a view on using symplectic integrators for a constraint-preserving numerical evolution. The gauge sources corresponding to the ``puncture gauge'' conditions are identified in this context.

Discontinuous Galerkin method for the spherically reduced Baumgarte-Shapiro-Shibata-Nakamura system with second-order operators

We present a high-order accurate discontinuous Galerkin method for evolving the spherically-reduced Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system expressed in terms of second-order spatial operators. Our multi-domain method achieves global spectral accuracy and long-time stability on short computational domains. We discuss in detail both our scheme for the BSSN system and its implementation. After a theoretical and computational verification of the proposed scheme, we conclude with a brief discussion of issues likely to arise when one considers the full BSSN system.

Relativistic magnetohydrodynamics in dynamical spacetimes: A new adaptive mesh refinement implementation

We have written and tested a new general relativistic magnetohydrodynamics (GRMHD) code, capable of evolving MHD fluids in dynamical spacetimes with adaptive-mesh refinement (AMR). Our code solves the Einstein-Maxwell-MHD system of coupled equations in full 3+1 dimensions, evolving the metric via the Baumgarte-Shapiro Shibata-Nakamura (BSSN) formalism and the MHD and magnetic induction equations via a conservative, high-resolution shock-capturing scheme. The induction equations are recast as an evolution equation for the magnetic vector potential, which exists on a grid that is staggered with respect to the hydrodynamic and metric variables. The divergenceless constraint div(B)=0 is enforced by the curl of the vector potential. Our MHD scheme is fully compatible with AMR, so that fluids at AMR refinement boundaries maintain div(B)=0. In simulations with uniform grid spacing, our MHD scheme is numerically equivalent to a commonly used, staggered-mesh constrained-transport scheme. We present code validation test results, both in Minkowski and curved spacetimes. They include magnetized shocks, nonlinear Alfv\'en waves, cylindrical explosions, cylindrical rotating disks, magnetized Bondi tests, and the collapse of a magnetized rotating star. Some of the more stringent tests involve black holes. We find good agreement between analytic and numerical solutions in these tests, and achieve convergence at the expected order.

Covariant formulations of Baumgarte, Shapiro, Shibata, and Nakamura and the standard gauge

The Baumgarte-Shapiro-Shibata-Nakamura (BSSN) and standard gauge equations are written in covariant form with respect to spatial coordinate transformations. The BSSN variables are defined as tensors with no density weights. This allows us to evolve a given set of initial data using two different coordinate systems and to relate the results using the familiar tensor transformation rules. Two variants of the covariant equations are considered. These differ from one another in the way that the determinant of the conformal metric is evolved.

BSSN in spherical symmetry

The BSSN (Baumgarte–Shapiro–Shibata–Nakamura) formulation of the Einstein evolution equations is written in spherical symmetry. These equations can be used to address a number of technical and conceptual issues in numerical relativity in the context of a single Schwarzschild black hole. One of the benefits of spherical symmetry is that the numerical grid points can be tracked on a Kruskal–Szekeres diagram. Boundary conditions suitable for puncture evolution of a Schwarzschild black hole are presented. Several results are shown for puncture evolution using a fourth-order finite difference implementation of the equations.

Nada: A new code for studying self-gravitating tori around black holes

We present a new two-dimensional numerical code called $\mathtt{N}\mathtt{a}\mathtt{d}\mathtt{a}$ designed to solve the full Einstein equations coupled to the general relativistic hydrodynamics equations. The code is mainly intended for studies of self-gravitating accretion disks (or tori) around black holes, although it is also suitable for regular spacetimes. Concerning technical aspects the Einstein equations are formulated and solved in the code using a formulation of the standard $3+1$ Arnowitt-Deser-Misner canonical formalism system, the so-called Baumgarte-Shapiro Shibata-Nakamura approach. A key feature of the code is that derivative terms in the spacetime evolution equations are computed using a fourth-order centered finite difference approximation in conjunction with the Cartoon method to impose the axisymmetry condition under Cartesian coordinates (the choice in $\mathtt{N}\mathtt{a}\mathtt{d}\mathtt{a}$), and the puncture/moving puncture approach to carry out black hole evolutions. Correspondingly, the general relativistic hydrodynamics equations are written in flux-conservative form and solved with high-resolution, shock-capturing schemes. We perform and discuss a number of tests to assess the accuracy and expected convergence of the code, namely, (single) black hole evolutions, shock tubes, and evolutions of both spherical and rotating relativistic stars in equilibrium, the gravitational collapse of a spherical relativistic star leading to the formation of a black hole. In addition, paving the way for specific applications of the code, we also present results from fully general relativistic numerical simulations of a system formed by a black hole surrounded by a self-gravitating torus in equilibrium.