(category:Th=Theoretical,ODE=Solve ODE,PDE=Solve PDE;Ash=Ashtekar formulation,BS= Boson Star,Cos=Cosmology,GW=Gravitational Waves,NR=Numerical Relativity,NS=Neutron Star,PN=Post Newtonian approx.,ST=Scalar Tensor theories )

title | conf. | gr-qc # |

Re-formulating the Einstein equations for stable numerical simulations | JGRG12, Tokyo, 2002 | |

Re-formulating the Einstein equations for stable numerical simulations | TH-2002, Paris, 2002 | |

Adjusted ADM systems and their expected stability properties | JGRG11, Tokyo, 2002 | |

Will hyperbolic formulation help numerical relativity? | JGRG10, Osaka, 2000 | [gr-qc/0103031] |

Hyperbolic formulations and numerical relativity | MG9, Rome, 2000 | |

Dynamical evolution of boson stars | NAP98, Tokyo, 1998 | |

Newtonian and post-Newtonian binary neutron star mergers | MG8, Jerusalem, 1997 | [gr-qc/9710073] |

Lorentzian dynamics in Ashtekar gravity | MG8, Jerusalem, 1997 | [gr-qc/9710074] |

Can we detect Brans-Dicke scalar gravitational waves in Gravitational Collapse? | Texas Symp, Chicago, 1996 | |

Inflation in a planar universe | MG7, San Francisco, 1994 | |

Cosmic no hair conjecture in a planar universe | Yamada Conf., Tokyo, 1993 | |

Gravitational waves in expanding universes with cosmological constant | Waseda Conf., Tokyo, 1992 |

T. Ebisuzaki, H. Katori, J. Makino, A. Noda, H. Shinkai, and T. Tamagawa

submitted to Int. J. Mod. Phys. D. [pdf]

*Abstract:*

The new technique of measuring frequency by optical lattice clocks now approaches to the relative precision of $(\Delta f/f)=O(10^{-18})$.
We propose to place such precise clocks in space and to use Doppler tracking method for detecting low-frequency gravitational wave below 1 Hz.
Our idea is to locate three satellites at one A.U. distance (say at L1, L4 \& L5 of the Sun-Earth orbit), and apply the Doppler tracking method by communicating ``the time" each other.
Applying the current available technologies, we obtain the sensitivity for gravitational wave with three or four-order improvement ($h_{\rm n}\sim 10^{-17}$ or $10^{-18}$ level in $10^{-5}$Hz -- $1$ Hz) than that of Cassini satellite in 2001.
This sensitivity enables us to observe black-hole mergers of their mass greater than $10^5 M_\odot$ in the cosmological scale.
Based on the hierarchical growth model of black-holes in galaxies, we estimate the event rate of detection will be 20-50 a year.
We nickname "INO" (Interplanetary Network of Optical Lattice Clocks) for this system, named after Tadataka Ino (1745--1818), a Japanese astronomer, cartographer, and geodesist.

KAGRA collaboration

Prog. Theor. Exp. Phys. (2018) 013F01 (23 pages) [pdf] [arXiv:1712.00148] ; Refers to Cited by

*Abstract:*

Major construction and initial-phase operation of a second-generation gravitational-wave detector KAGRA has been completed. The entire 3-km detector is installed underground in a mine in order to be isolated from background seismic vibrations on the surface. This allows us to achieve a good sensitivity at low frequencies and high stability of the detector. Bare-bones equipment for the interferometer operation has been installed and the first test run was accomplished in March and April of 2016 with a rather simple configuration. The initial configuration of KAGRA is named * iKAGRA*. In this paper, we summarize the construction of KAGRA, including the study of the advantages and challenges of building an underground detector and the operation of the iKAGRA interferometer together with the geophysics interferometer that has been constructed in the same tunnel.

Hisa-aki Shinkai, and Takashi Torii

Physical Review D96 (2017) 044009 (14 pages) [pdf] ;

[arXiv:1706.02070] ; Refers to Cited by

*Abstract:*

We numerically investigated how the non-linear dynamics depends on the dimensionality and on the higher-order curvature corrections in the form of Gauss-Bonnet (GB) terms. We especially monitored the processes of appearances of a singularity (or black-hole) in two models: (i) perturbed wormhole throat in spherically symmetric space-time, and (ii) colliding scalar pulses in plane symmetric space-time.
We used a dual-null formulation for evolving the field equations, which enables us to locate the trapping horizons directly, and also enables us to follow close to the large curvature region due to its causal integrating scheme.
We observed that the fate of a perturbed wormhole is either a black-hole or an expanding throat depending on the total energy of the structure, and its threshold depends on the coupling constant of the GB terms (alpha_{GB}).
We also observed that a collision of large scalar pulses will produce large curvature region, of which magnitude also depends on alpha_{GB}. For both models, the normal corrections (alpha_{GB}>0) work for avoiding the appearance of singularity, although it is inevitable. We also found that in the critical situation for forming a black-hole, the existence of the trapped region in the Einstein-GB gravity does not directly indicate a formation of a black-hole.

Hisa-aki Shinkai, Nobuyuki Kanda, and Toshikazu Ebisuzaki

Astrophysical Journal 835 (2017) 276 [pdf] ;

[arXiv:1610.09505]; Refers to Cited by

*Abstract:*

Based on a dynamical formation model of a supermassive black hole (SMBH), we estimate the expected observational profile of gravitational wave at ground-based detectors, such as KAGRA or advanced LIGO/VIRGO. Noting that the second generation of detectors have enough sensitivity from 10 Hz and up (especially with KAGRA owing to its location at less seismic noise), we are able to detect the ring-down gravitational wave of a BH with the mass M < 2 10^3 Msun. This enables us to check the sequence of BH mergers to SMBHs via intermediate-mass BHs. We estimate the number density of galaxies from the halo formation model and estimate the number of BH mergers from the giant molecular cloud model assuming hierarchical growth of merged cores. At the designed KAGRA (and/or advanced LIGO/VIRGO), we find that the BH merger of its total mass M sim 60 Msun is at the peak of the expected mass distribution. With its signal-to-noise ratio rho=10 (30), we estimate the event rate R sim 200 (20) per year in the most optimistic case, and we also find that BH mergers in the range M < 150 Msun are R>1 per year for rho=10. Thus, if we observe a BH with more than 100 Msun in future gravitational-wave observations, our model naturally explains its source.

Takashi Torii and Hisa-aki Shinkai

Physical Review D 88, 064027 (2013) (6 pages); [pdf]

[arXiv:1309.2058]; Refers to Cited by

*Abstract:*

We derive the simplest traversable wormhole solutions in $n$-dimensional general relativity,
assuming static and spherically symmetric spacetime with
ghost scalar field.
This is the generalization of
the Ellis solution (or the so-called Morris-Thorne's traversable wormhole)
into higher-dimension.
We also study their stability using linear
perturbation analysis.
We obtain the master equation for perturbed gauge-invariant variable,
and search their eigenvalues.
Our analysis shows that all higher-dimensional wormholes have an unstable mode
against the perturbations with which the throat radius is changed.
The instability is consistent with the earlier numerical analysis in
four-dimensional solution.

Takuya Tsuchiya, Gen Yoneda and Hisa-aki Shinkai

Physical Review D 85, 044018 (2012) (12 pages); [pdf]

[arXiv:1109.5782]; Refers to Cited by

*Abstract:*

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 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 2--10 times longer than those of the standard Baumgarte-Shapiro-Shibata-Nakamura evolutions.

Hisa-aki Shinkai and Yuta Yamada

Int. J. Mod. Phys. Conf. Ser. 7 (2012) 148, as a proceedings of "2011 Shanghai Asia-Pacific School and Workshop on Gravitation (Shanghai, Feb. 2011)"; [pdf]

*Abstract:*

We introduce our numerical studies of gravitational collapses
in five-dimensional (5D) space-time, with a purpose of studying
the cosmic censorship hypothesis and the hoop conjecture.
The first model is the collapse of spindle matter which was
performed by Shapiro and Teukolsky (1991) who announced
an appearance of a naked singularity in 4D.
Comparing with 4D cases, we found that 5D collapses proceed more rapidly, the final
configurations tend to be spherical, and apparent horizon (AH) forms in wider parameter ranges.
We also observed positive evidence for formation of a
naked singularity in highly spindle cases as well.
The second model is the formation of black-ring in 5D.
Our code does not include angular momentum, but the model would be helpful for
basic understandings.
We constructed an initial data sequence with ring-shaped matter,
and observed the topology of AHs, if formed.
We found a critical ring radius for ring-shaped AH, and it suggests
a dynamical transition of AH topology from ring-shaped to spherical.
We demonstrate such an example in time evolution.

Yuta Yamada and Hisa-aki Shinkai

Physical Review D 83, 064006 (2011) (5 pages); [pdf]

[arXiv:1102.2090]; Refers to Cited by

*Abstract:*

We numerically investigate the gravitational collapse of collisionless particles
in spheroidal configurations both in four and five-dimensional (5D) space-time.
We repeat the simulation performed by Shapiro and Teukolsky (1991) that announced
an appearance of a naked singularity, and also find that the similar results
in 5D version.
That is, in a collapse of a highly prolate spindle, the Kretschmann invariant blows up
outside the matter and no apparent horizon forms.
We also find that the collapses in 5D proceed rapidly than in 4D, and
the critical prolateness for
appearance of apparent horizon in 5D is loosened compared to 4D cases.
We also show how collapses differ with spatial symmetries comparing 5D
evolutions in single-axisymmetry, SO(3), and those in double-axisymmetry,
U(1) x U(1).

Takuya Tsuchiya, Gen Yoneda and Hisa-aki Shinkai

Physical Review D 83, 064032 (2011) (10 pages); [pdf]

[arXiv:1103.1202]; Refers to Cited by

*Abstract:*

With a purpose of constructing a robust evolution system against numerical instability for integrating the Einstein equations, we propose a new formulation by adjusting the ADM evolution
equations with constraints. We apply an adjusting method proposed by Fiske (2004) which uses
the norm of the constraints, $C^2$. One of the advantages of this method is that the e ective signature of adjusted terms (Lagrange multipliers) for constraint-damping evolution is pre-determined.
We demonstrate this fact by showing the eigenvalues of constraint propagation equations. We also
perform numerical tests of this adjusted evolution system using polarized Gowdy-wave propagation,
which show robust evolutions against the violation of the constraints than that of the standard
ADM formulation.

Yuta Yamada and Hisa-aki Shinkai

Class. Quant. Gravity 27 (2010) 045012; [pdf]

[arXiv:0907.2570] ; Refers to Cited by

*Abstract:*

We numerically investigated the sequences of initial data of thin spindle and thin ring in five-dimensional space-time in the context of the cosmic censorship conjecture. We modeled the matter in non-rotating homogeneous spheroidal or toroidal configurations under the momentarily static assumption, solved the Hamiltonian constraint equation, and searched the apparent horizon. We found both $S^3$ (black hole) and $S^2 times S^1$ (black ring) horizons ("black objects"), only when the matter configuration is not sharp. By monitoring the location of the maximum Kretchmann invariant, an appearance of `naked singularity' or `naked ring' under the special situations is suggested. We also discuss the validity of the "hyper-hoop" conjecture using minimum "area" around the object, and show that the appearance of the ring horizon does not match with this hoop.

Takashi Torii and Hisa-aki Shinkai

Physical Review D 78, 084037 (2008) (13 pages); [pdf]

[arXiv:0810.1790] ; Refers to Cited by

*Abstract:*

Towards the investigation of the full dynamics in higher-dimensional and/or
stringy gravitational model, we present
the basic equations of the Einstein-Gauss-Bonnet gravity theory.
We show $(N+1)$-dimensional version of the ADM decomposition
including Gauss-Bonnet terms, which shall be the standard approach
to treat the space-time as a Cauchy problem.
Due to the quasi-linear property of the Gauss-Bonnet gravity, we find that
the evolution equations can be in a treatable form in numerics.
We also show the conformally-transformed constraint equations
for constructing an initial data. We discuss how the constraints can be
simplified by tuning the powers of conformal factors.
Our equations can be used both for timelike and spacelike foliations.

Hisa-aki Shinkai

Journal of Korean Physical Society, Invited Paper (15 pages)

[arXiv:0805.0068] [pdf] ; Refers to Cited by

Invited Lecture at APCTP Winter School on Black Hole Astrophysics, Daejeon and Pohang, Korea, January 24-29, 2008

*Abstract:*

We review recent efforts to re-formulate the Einstein equations for fully relativistic numerical simulations. In order to complete a long-term and accurate simulations of binary compact objects, people seek a robust set of equations against the violation of constraints. Many trials have revealed that mathematically equivalent sets of evolution equations show different numerical stability in free evolution schemes. In this article, we overview the efforts of the community, categorizing them into three directions: (1) modifications of the standard Arnowitt-Deser-Misner equations initiated by the Kyoto group (the so-called Baumgarte-Shapiro-Shibata-Nakamura equations), (2) rewriting the evolution equations in a hyperbolic form, and (3) construction of an "asymptotically constrained" system. We then introduce our series of works that tries to explain these evolution behaviors in a unified way using eigenvalue analysis of the constraint propagation equations. The modifications of (or adjustments to) the evolution equations change the character of constraint propagation, and several particular adjustments using constraints are expected to damp the constraint-violating modes. We show several set of adjusted ADM/BSSN equations, together with their numerical demonstrations.

Kenta Kiuchi and Hisa-aki Shinkai

Physical Review D 77, 044010 (2008) (11 pages); [pdf]

[arXiv:0711.3575], Refers to Cited by

*Abstract:*

We present our numerical comparisons between the BSSN formulation which
is widely used in numerical relativity today
and its adjusted versions using constraints.
We performed three testbeds; gauge-wave, linear wave, and
Gowdy-wave tests, which were proposed by Mexico workshop on the
formulation problem of the Einstein equation.
We tried three kinds of adjustments which were previously proposed
from the analysis of the constraint propagation equations, and investigated
how they improve the accuracy and stability of evolutions.
We observed that the signature of the proposed Lagrange multipliers are
always right, and the adjustments improve the convergence
and stability of the simulations. When the original BSSN system
already shows satisfactory good evolutions (e.g. linear wave test),
the adjusted versions also coincide with those evolutions;
while in some cases (e.g. gauge-wave or Gowdy-wave tests)
the adjusted version makes 10 times
longer stable simulations than the original system.
Our demonstrations imply a potential to construct a robust evolution
system against constraint violations, for more stable and accurate
simulations even in the highly dynamical situations.

TAMA Collaboration

Physical Review D 74, 122002 (2006) (8 pages)

[gr-qc/0610064]; Refers to Cited by

*Abstract:*

We analyze the data of TAMA300 detector to search for gravitational waves
from inspiraling compact star binaries with masses of the component stars in
the range 1-3Msolar. In this analysis, 2705 hours of data, taken during the
years 2000-2004, are used for the event search. We combine the results of
different observation runs, and obtained a single upper limit on the rate of
the coalescence of compact binaries in our Galaxy of 20 per year at a 90%
confidence level. In this upper limit, the effect of various systematic errors
such like the uncertainty of the background estimation and the calibration of
the detector's sensitivity are included.

LIGO Scientific Collaboration, TAMA Collaboration

Physical Review D 73, 102002 (2006) (10 pages)

[gr-qc/0512078]; Refers to Cited by

*Abstract:*

We search for coincident gravitational wave signals from inspiralling neutron star binaries using LIGO and TAMA300 data taken during early 2003. Using a simple trigger exchange method, we perform an inter-collaboration coincidence search during times when TAMA300 and only one of the LIGO sites were operational. This data set is complementary to that used in the LIGO S2 search. The observation time of the search is 648 hours. We find no evidence of any gravitational wave signals. We place an observational upper limit on the rate of binary neutron star coalescence with component masses between 1 and 3 M_sun of 49 per year per Milky Way equivalent galaxy at a 90% confidence level.

LIGO Scientific Collaboration, TAMA Collaboration

Physical Review D72 (2005) 122004 (16 pages)

[gr-qc/0507081]; Refers to Cited by

*Abstract:*

We report on the first joint search for gravitational waves by the TAMA and LIGO collaborations. We looked for millisecond-duration unmodelled gravitational-wave bursts in 473 hr of coincident data collected during early 2003. No candidate signals were found. We set an upper limit of 0.12 events per day on the rate of detectable gravitational-wave bursts, at 90% confidence level. From simulations, we estimate that our detector network was sensitive to bursts with root-sum-square strain amplitude above approximately 1-3x10^{-19} Hz^{-1/2} in the frequency band 700-2000 Hz. We describe the details of this collaborative search, with particular emphasis on its advantages and disadvantages compared to searches by LIGO and TAMA separately using the same data. Benefits include a lower background and longer observation time, at some cost in sensitivity and bandwidth. We also demonstrate techniques for performing coincidence searches with a heterogeneous network of detectors with different noise spectra and orientations. These techniques include using coordinated signal injections to estimate the network sensitivity, and tuning the analysis to maximize the sensitivity and the livetime, subject to constraints on the background.

the TAMA collaboration

Masaki Ando, Koji Arai, Youichi Aso, Peter Beyersdorf, Kazuhiro Hayama, Yukiyoshi Iida, Nobuyuki Kanda, Seiji Kawamura, Kazuhiro Kondo, Norikatsu Mio, Shinji Miyoki, Shigenori Moriwaki, Shigeo Nagano, Kenji Numata, Shuichi Sato, Kentaro Somiya, Hideyuki Tagoshi, Hirotaka Takahashi, Ryutaro Takahashi, Daisuke Tatsumi, Yoshiki Tsunesada, Zong-Hong Zhu, Tomomi Akutsu, Tomotada Akutsu, Akito Araya, Hideki Asada, Mark A. Barton, Youhei Fujiki, Masa-Katsu Fujimoto, Ryuichi Fujita, Mitsuhiro Fukushima, Toshifumi Futamase, Yusaku Hamuro, Tomiyoshi Haruyama, Hideaki Hayakawa, Gerhard Heinzel, Gen'ichi Horikoshi, Hideo Iguchi, Kunihito Ioka, Hideki Ishitsuka, Norihiko Kamikubota, Takaharu Kaneyama, Yoshikazu Karasawa, Kunihiko Kasahara, Taketoshi Kasai, Mayu Katsuki, Keita Kawabe, Mari Kawamura, Nobuki Kawashima, Fumiko Kawazoe, Yasufumi Kojima, Keiko Kokeyama, Yoshihide Kozai, Hideaki Kudoh, Kazuaki Kuroda, Takashi Kuwabara, Namio Matsuda, Kazuyuki Miura, Osamu Miyakawa, Shoken Miyama, Hiromi Mizusawa, Mitsuru Musha, Yoshitaka Nagayama, Ken'ichi Nakagawa, Takashi Nakamura, Hiroyuki Nakano, Ken-ichi Nakao, Yuhiko Nishi, Yujiro Ogawa, Masatake Ohashi, Naoko Ohishi, Akira Okutomi, Ken-ichi Oohara, Shigemi Otsuka, Yoshio Saito, Shihori Sakata, Misao Sasaki, Kouichi Sato, Nobuaki Sato, Youhei Sato, Hidetsugu Seki, Aya Sekido, Naoki Seto, Masaru Shibata, Hisaaki Shinkai, Takakazu Shintomi, Kenji Soida, Toshikazu Suzuki, Akiteru Takamori, Shuzo Takemoto, Kohei Takeno, Takahiro Tanaka, Keisuke Taniguchi, Shinsuke Taniguchi, Toru Tanji, C. T. Taylor, Souichi Telada, Kuniharu Tochikubo, Masao Tokunari, Takayuki Tomaru, Kimio Tsubono, Nobuhiro Tsuda, Takashi Uchiyama, Akitoshi Ueda, Ken-ichi Ueda, Fumihiko Usui, Koichi Waseda, Yuko Watanabe, Hiromi Yakura, Akira Yamamoto, Kazuhiro Yamamoto, Toshitaka Yamazaki, Yuriko Yanagi, Tatsuo Yoda, Jun'ichi Yokoyama, and Tatsuru Yoshida

Physical Review D 71, 082002 (2005)

[gr-qc/0411027]; Refers to Cited by

*Abstract:*

We present data-analysis schemes and results of observations with the TAMA300 gravitational wave detector, targeting burst signals from stellar-core collapse events. In analyses for burst gravitational waves, the detection and fake-reduction schemes are different from well-investigated ones for a chirp wave analysis, because precise waveform templates are not available. We used an excess -power filter for the extraction of gravitational wave candidates, and developed two methods for the reduction of fake events caused by nonstationary noises of the detector. These analysis schemes were applied to real data from the TAMA300 interferometric gravitational wave detector. As a result, fake events were reduced by a factor of about 1000 in the best cases. In addition, in order to interpret the event candidates from an astronomical viewpoint, we performed a Monte-Carlo simulation with an assumed Galactic event distribution model and with burst waveforms obtained from numerical simulations of stellar-core collapses. We set an upper limit of 2.2 x 10^3 events/sec on the burst gravitational wave event rate in our Galaxy with a confidence level of 90%. This work shows prospects on the search for burst gravitational waves, by establishing an analysis scheme for the observation data from an interferometric gravitational wave.

Tatsushi Matsubayashi, Hisa-aki Shinkai and Toshikazu Ebisuzaki (RIKEN)

Astrophysical J. 614 (October 20, 2004) 864-868

[pdf]

*Abstract:*

The discovery of an intermediate-mass black hole (IMBH) supports a runaway path of supermassive black hole
(SMBH) formation in galactic nuclei. No concrete model to explain all the steps of this bottom-up scenario for
SMBHs is yet known, but here we propose to use gravitational radiation to probe the merging history of IMBHs.
Collisions of black holes of mass $10^3 -10^6 M_\odot$ will produce gravitational radiation of $10^{-1}$ to
$10^2$ Hz in their final
merging phase. We assume that a thousand $10^3 M_\odot$ IMBHs form a $10^6 M_\odot$
black hole in each galaxy via two
different merging histories - hierarchical growth and monopolistic growth - using a theoretical model of quasar
formation having a peak at $z \approx 2.5$. We find that there would be 22 - 67 IMBH merging events per year in the
universe and that the event numbers of the two models apparently differ in the frequency of gravitational radiation.
Most of the bursts by these events will be detectable by currently proposed space gravitational wave antennas,
such as LISA or DECIGO. We conclude that the statistics of the signals would provide both a galaxy distribution
and a formation model of SMBHs.

Hisa-aki Shinkai (RIKEN) and Gen Yoneda (Waseda U.)

Gen. Rel. Grav. 36 (2004) 1931-1937

[pdf]

[gr-qc/0402101]; Refers to Cited by

*Abstract:*

Higher dimensional space-time models provide us an alternative
interpretation of nature, and give us different dynamical aspects
than the traditional four-dimensional space-time models.
Motivated by such recent interests, especially for
future numerical research of higher-dimensional space-time,
we study the dimensional dependence of constraint propagation
behavior.
The $N+1$ Arnowitt-Deser-Misner evolution
equation has matter terms which depend on $N$,
but the constraints and constraint propagation equations remain the
same. This indicates that there would be problems with accuracy and
stability when we directly apply the $N+1$ ADM formulation to numerical
simulations as we
have experienced in four-dimensional cases. However, we also conclude
that previous efforts in re-formulating the Einstein equations
can be applied if they are based on constraint propagation analysis.

Mexico Numerical Relativity Workshop 2002 Participants

M. Alcubierre, G. Allen, C. Bona, D. Fiske, T. Goodale, F.S. Guzman, I. Hawke, S. Hawley, S. Husa, M. Koppitz, C. Lechner, D. Pollney, D. Rideout, M. Salgado, E. Schnetter, E. Seidel, H. Shinkai, D. Shoemaker, B. Szilagyi, R. Takahashi, and J. Winicour

Class. Quant. Gravity 21 (2004) 589-613

[pdf]

[gr-qc/0305023]; Refers to Cited by

*Abstract:*

In recent years, many different numerical evolution schemes for Einstein's
equations have been proposed to address stability and accuracy problems that
have plagued the numerical relativity community for decades. Some of these
approaches have been tested on different spacetimes, and conclusions have been
drawn based on these tests. However, differences in results originate from many
sources, including not only formulations of the equations, but also gauges,
boundary conditions, numerical methods, and so on. We propose to build up a
suite of standardized testbeds for comparing approaches to the numerical
evolution of Einstein's equations that are designed to both probe their
strengths and weaknesses and to separate out different effects, and their
causes, seen in the results. We discuss general design principles of suitable
testbeds, and we present an initial round of simple tests with periodic
boundary conditions. This is a pivotal first step toward building a suite of
testbeds to serve the numerical relativists and researchers from related
fields who wish to assess the capabilities of numerical relativity
codes. We present some examples of how these tests can be quite effective in
revealing various limitations of different approaches, and illustrating their
differences. The tests are presently limited to vacuum spacetimes, can be run on
modest computational resources, and can be used with many different approaches
used in the relativity community.

Gen Yoneda (Waseda U.) and Hisa-aki Shinkai (RIKEN)

Class. Quantum Gravity 20 (2003) L31-36 (Letter)

[pdf]

[gr-qc/0209106]; Refers to Cited by

*Abstract:*

In order to obtain stable and accurate general relativistic simulations,
re-formulations of the Einstein equations are necessary. In a series of our
works, we have proposed to use eigenvalue analysis of constraint propagation
equations for evaluating violation behavior of constraints. In this article, we
classify asymptotical behaviors of constraint-violation into three
(asymptotically constrained, asymptotically bounded, and diverge), and give
their necessary and sufficient conditions. We find that degeneracy of
eigenvalues sometimes leads constraint evolution to diverge (even if its
real-part is not positive), and conclude that it is quite useful to check the
diagonalizability of constraint propagation matrix. The discussion is general
and can be applied to any numerical treatments of constrained dynamics.

Hisa-aki Shinkai (RIKEN) and Sean A. Hayward (Ewha Womans Univ., Korea)

Physical Review D 66, 044005 (2002)

[pdf]

[gr-qc/0205041]; Refers to Cited by

Introduced in New Scientist Journal (UK)

*Abstract:*

We study numerically the stability of Morris amp; Thorne's first traversible
wormhole, shown previously by Ellis to be a solution for a massless ghost
Klein-Gordon field. Our code uses a dual-null formulation for spherically
symmetric space-time integration, and the numerical range covers both universes
connected by the wormhole. We observe that the wormhole is unstable against
Gaussian pulses in either exotic or normal massless Klein-Gordon fields. The
wormhole throat suffers a bifurcation of horizons and either explodes to form
an inflationary universe or collapses to a black hole, if the total input
energy is respectively negative or positive. As the perturbations become small
in total energy, there is evidence for critical solutions with a certain
black-hole mass or Hubble constant. The collapse time is related to the initial
energy with an apparently universal critical exponent. For normal matter, such
as a traveller traversing the wormhole, collapse to a black hole always
results. However, carefully balanced additional ghost radiation can maintain
the wormhole for a limited time. The black-hole formation from a traversible
wormhole confirms the recently proposed duality between them. The inflationary
case provides a mechanism for inflating, to macroscopic size, a Planck-sized
wormhole formed in space-time foam.

Gen Yoneda (Waseda U.) and Hisa-aki Shinkai (RIKEN)

Physical Review D66, 124003 (2002)

[pdf] [ps]

[gr-qc/0204002]; Refers to Cited by

*Abstract:*

Several numerical relativity groups are using a modified ADM formulation
for their simulations, which was developed by Nakamura et al
(and widely cited as Baumgarte-Shapiro-Shibata-Nakamura system).
This so-called BSSN formulation is shown to be more stable
than the standard ADM formulation in many cases,
and there have been many attempts to explain why
this re-formulation has such an advantage.
We try to explain the background mechanism of the BSSN equations
by using eigenvalue analysis of constraint propagation equations.
This analysis has been applied and has succeeded in explaining
other systems in our series of works.
We derive the full set of the constraint propagation equations,
and study it in the flat background space-time.
We carefully examine how the replacements
and adjustments in the equations change the propagation
structure of the constraints, i.e.
whether violation of constraints (if it exists) will decay or propagate away.
We conclude that the better stability of
the BSSN system is obtained by their adjustments in the equations,
and that the combination of the adjustments is in a good balance, i.e.
a lack of their adjustments might fail to obtain the present stability.
We further propose other adjustments to
the equations, which may offer more stable features than the current BSSN
equations.

Hisa-aki Shinkai (RIKEN) and Gen Yoneda (Waseda U.)

Class. Quantum Grav. 19 (2002) 1027-1049

[pdf] [ps]

[gr-qc/0110008]; Refers to Cited by

*Abstract:*

In order to find a way to have a better formulation for numerical
evolution of the Einstein equations,
we study the propagation equations of the
constraints based on the Arnowitt-Deser-Misner formulation.
By adjusting constraint terms in the evolution equations,
we try to construct an ``asymptotically constrained system"
which is expected to be robust against violation of the constraints,
and to enable
a long-term stable and accurate numerical simulation.
We first provide useful expressions for analyzing constraint propagation in
general spacetime, then apply it to Schwarzschild spacetime.
We search when and where the negative real or non-zero imaginary
eigenvalues of the homogenized
constraint propagation matrix appear, and how they depend on the choice
of coordinate system and adjustments.
Our analysis includes the proposal of
Detweiler (1987), which is still the best one according to our conjecture
but has a growing mode of error near the horizon.
Some examples are snapshots
of a maximally sliced Schwarzschild black hole.
The predictions here may help the community to make further improvements.

Gen Yoneda (Waseda U.) and Hisa-aki Shinkai (PennState, CGPG)

Physical Review D 63 (2001) 124019 (9 pages)

[pdf] [ps]

[gr-qc/0103032]; Refers to Cited by

*Abstract:*

The current important issue in numerical relativity is to determine
which formulation of the Einstein equations provides us with stable and
accurate simulations.
Based on our previous work on "asymptotically constrained" systems,
we here present constraint propagation equations
and their eigenvalues for the Arnowitt-Deser-Misner
(ADM) evolution equations with additional
constraint terms (adjusted terms) on the right hand side.
We conjecture that the system is robust against violation
of constraints if
the amplification factors (eigenvalues of Fourier-component
of the constraint propagation equations)
are negative or pure-imaginary.
We show such a system can be obtained by choosing multipliers of
adjusted terms.
Our discussion covers Detweiler's proposal (1987)
and Frittelli's analysis (1997),
and we also mention the so-called conformal-traceless ADM systems.

Hisa-aki Shinkai and Sean A. Hayward (PennState, CGPG)

Physical Review D 64 (2001) 044002 (8 pages)

[pdf] [ps]

[gr-qc/0008075]; Refers to Cited by

*Abstract:*

We numerically implement a quasi-spherical approximation scheme
for computing gravitational waveforms for coalescing black holes,
testing it against angular momentum by applying it to Kerr black holes.
As error measures,
we take the conformal strain and specific energy
due to spurious gravitational radiation.
The strain is found to be monotonic rather than wavelike.
The specific energy is found to be at least an order of magnitude smaller than
the 1\% level expected from typical black-hole collisions,
for angular momentum up to about 60\% to 70\% of the maximum,
for an initial surface as close as $r=3m$.

Andrew Chamblin (MIT), Harvey S. Reall (Cambridge), Hisa-aki Shinkai (PennState, CGPG) and Tetsuya Shiromizu (AEI/Tokyo)

Physical Review D63 (2001) 064015 (11 pages)

[ps] [pdf]

[hep-th/0008177]; Refers to Cited by

*Abstract:*

We study charged brane-world black holes in the model of Randall and
Sundrum in which our universe is viewed as a domain wall in asymptotically
anti-de Sitter space. Such black holes can carry two types of "charge",
one arising from the bulk Weyl tensor and one from a gauge field
trapped on the wall. We use a combination of analytical and numerical
techniques
to study how these black holes behave in the bulk. It has been shown that
a Reissner-Nordstrom geometry is induced on the wall when only Weyl charge
is present. However, we show that such solutions exhibit pathological
features in the bulk. For more general charged black holes, our
results suggest that the extent of the horizon in the fifth
dimension is usually less than for an uncharged black hole
that has the same mass or horizon radius on the wall.

Gen Yoneda (Waseda U.) and Hisa-aki Shinkai (PennState, CGPG)

Class. Quantum Grav. 18 (2001) 441 - 462

[pdf]

[gr-qc/0007034]; Refers to Cited by

*Abstract:*

We study asymptotically constrained systems for numerical integration
of the Einstein equation, which is intended to be robust against
perturbative errors for the free evolution of the initial data.
We, first, examine the previously proposed "$\lambda$-system",
which introduces artificial flows to constrained surfaces based on
the symmetric hyperbolic formulation. We show that this system works
as expected for the wave propagation problem in the Maxwell system and
and in general relativity using Ashtekar's connection formulation.
We, second, propose a new mechanism to control the stability,
which we named "adjusted system". This is simply obtained by adding
constraint terms in the dynamical equations and adjusting its
multipliers. We explain why a particular choice of multiplier
reduce the numerical errors by non-positiveness (or non-zero) of
the eigenvalues of the adjusted constraint propagation equations.
This "adjusted system" is also tested in the Maxwell system and in the
Ashtekar's system. This mechanism affects more than the system's
symmetric hyperbolicity.

Hisa-aki Shinkai (PennState, CGPG) and Gen Yoneda (Waseda U.)

Class. Quantum Grav. 17 (2000) 4799 - 4822

[pdf]

[gr-qc/0005003]; Refers to Cited by

*Abstract:*

In order to perform stable long-time numerical integration of the Einstein
equation, several hyperbolic systems have been proposed.
We here present our numerical comparisons
between weakly hyperbolic, strongly hyperbolic,
and symmetric hyperbolic systems based on Ashtekar's connection variables.
The primary advantage for using this connection formulation in this experiment
is that we can keep using the same dynamical variables for all levels of
hyperbolicity.
Our numerical code demonstrates gravitational wave propagation in
plane symmetric spacetimes, and we compare "the stability" by monitoring
the violation of the constraints.
By comparing the results obtained from the weakly hyperbolic system, we
observe the strongly and symmetric hyperbolic system show better stability
properties, but not so much difference between the latter two.
Rather, we find that the symmetric hyperbolic system is not always
the best for controlling stability.
Similar conclusions are obtained also in the Maxwell system.

This study is the premier for presenting full numerical simulations
using Ashtekar's variables. We also describe our procedures in detail.

Hisa-aki Shinkai (PennState, CGPG) and Tetsuya Shiromizu (Cambridge/Tokyo)

Physical Review D 62 (2000) 024010 (8 pages)

[ps]

[hep-th/0003066]; Refers to Cited by

*Abstract:*

We numerically study classical time evolutions of Kaluza-Klein
bubble space-time which has negative energy after a decay of vacuum.
As the zero energy Witten's bubble space-time,
where the bubble expands infinitely,
the subsequent evolutions of Brill and Horowitz's momentarily static
initial data show that the bubble will expand in terms of the
circumference radius. At first glance, this result may support
Corley and Jacobson's conjecture that the
bubble will expand forever as well as the Witten's bubble.
The irregular signatures, however, can be seen in the behavior
of the lapse
function in the maximal slicing gauge
and the divergence of the Kretchman invariant.
Since there is no appearance of the apparent horizon, we
suspect an appearance of a naked singularity as the final fate of
this space-time.

Hisa-aki Shinkai (PennState, CGPG) and Gen Yoneda (Waseda U.)

Physical Review D 60 (1999) 101502 (Rapid Communication, 5 pages)

[ps]

[gr-qc/9906062]; Refers to Cited by

*Abstract:*

We present a set of dynamical equations based on Ashtekar's extension
of the Einstein equation. The system forces the space-time to evolve to
the manifold that satisfies the constraint equations or the reality conditions
or both as the attractor against perturbative errors. This is an application
of the idea by Brodbeck, Frittelli, Huebner and Reula who constructed an
asymptotically stable (i.e., constrained) system for the Einstein equation,
adding dissipative forces in the extended space. The obtained systems may
be useful for future numerical studies using Ashtekar's variables.

Gen Yoneda (Waseda U.) and Hisa-aki Shinkai (PennState, CGPG)

International J. Modern Physics D 9 (2000) 13. [pdf local]

[tex] [ps]

[gr-qc/9901053]; Refers to Cited by

*Abstract:*

Hyperbolic formulations of the equations of motion are essential technique
for proving the well-posedness of the Cauchy problem of a system, and are
also helpful for implementing stable long time evolution in numerical applications.
We, here, present three kinds of hyperbolic systems in the Ashtekar formulation
of general relativity for Lorentzian vacuum spacetime. We exhibit several
(I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric
hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's
original equations form a weakly hyperbolic system. We discuss how gauge
conditions and reality conditions are constrained during each step toward
constructing a symmetric hyperbolic system.

Physical Review D60 (1999) 067504 (4 pages)

Hisa-aki Shinkai (Washington U., St. Louis)

[ps text] [fig1.ps] [fig2.ps] [fig3.ps]

[gr-qc/9807008]; Refers to Cited by

*Abstract:*

As a preliminary step towards simulating binary neutron star coalescing
problem, we test a post-Newtonian approach by constructing a single neutron
star model. We expand the Tolman-Oppenheimer-Volkov equation of hydrostatic
equilibrium by the power of $c^{-2}$, where $c$ is the speed of light,
and truncate at the various order. We solve the system using the polytropic
equation of state with index $\Gamma=5/3, 2$ and 3, and show how this approximation
converges together with mass-radius relations. Next, we solve the Hamiltonian
constraint equation with these density profiles as trial functions, and
examine the differences in the final metric. We conclude the second `post-Newtonian'
approximation is close enough to describe general relativistic single star.
The result of this report will be useful for further binary studies.

Gen Yoneda (Waseda U.) and Hisa-aki Shinkai (Washington U., St. Louis)

Physical Review Letters 82 (1999) 263-266

[ps]

[gr-qc/9803077]; Refers to Cited by

*Abstract:*

We present a first-order symmetric hyperbolic system in the Ashtekar
formulation of general relativity for vacuum spacetime. We add terms from
constraint equations to the evolution equations with appropriate combinations,
which is the same technique used by Iriondo et al ( Phys. Rev. Lett. 79,
4732 (1997) ). However our system is different from theirs in the points
that we primarily use Hermiticity of a characteristic matrix of the system
to characterize our system symmetric, discuss the consistency of this system
with reality condition, and show the characteristic speeds of the system.

Jay Balakrishna and Hisa-aki Shinkai (Washington U., St. Louis)

Physical Review D58 (1998) 044016 (13 pages)

[ps] [ps, 2 column]

[gr-qc/9712065]; Refers to Cited by

*Abstract:*

We study the dynamics of a self-gravitating scalar field solitonic
object (boson star) in the Jordan-Brans-Dicke (BD) theory of gravity. We
show dynamical processes of this system such as (i) black hole formation
of perturbed equilibrium configuration on an unstable branch; (ii) migration
of perturbed equilibrium configuration from the unstable branch to stable
branch; (iii) transition from excited state to a ground state. We find
that the dynamical behavior of boson stars in BD theory is quite similar
to that in general relativity (GR), with comparable scalar wave emission.
We also demonstrate the formation of a stable boson star from a Gaussian
scalar field packet with flat gravitational scalar field initial data.
This suggests that boson stars can be formed in the BD theory in much the
same way as in GR.

G.L. Comer (St. Louis U.) and Hisa-aki Shinkai (Washington U., St. Louis)

Class. Quantum Grav. 15 (1998) 669

[ps]

[gr-qc/9708071]; Refers to Cited by

*Abstract:*

Boson stars in zero-, one-, and two-node equilibrium states are modeled
numerically within the framework of Scalar-Tensor Gravity. The complex
scalar field is taken to be both massive and self-interacting. Configurations
are formed in the case of a linear gravitational scalar coupling (the Brans-Dicke
case) and a quadratic coupling which has been used previously in a cosmological
context. The coupling parameters and asymptotic value for the gravitational
scalar field are chosen so that the known observational constraints on
Scalar-Tensor Gravity are satisfied. It is found that the constraints are
so restrictive that the field equations of General Relativity and Scalar-Tensor
gravity yield virtually identical solutions. We then use catastrophe theory
to determine the dynamically stable configurations. It is found that the
maximum mass allowed for a stable state in Scalar-Tensor gravity in the
present cosmological era is essentially unchanged from that of General
Relativity. We also construct boson star configurations appropriate to
earlier cosmological eras and find that the maximum mass for stable states
is smaller than that predicted by General Relativity, and the more so for
earlier eras. However, our results also show that if the cosmological era
is early enough then only states with positive binding energy can be constructed.

Gen Yoneda (Waseda U.), Hisa-aki Shinkai (Washington U., St. Louis) and Akika Nakamichi (Waseda U.)

Physical Review D56 (1997) 2086-2093

[gr-qc/9704081]; Refers to Cited by

*Abstract:*

We examine one of the advantages of Ashtekar's formulation of general
relativity: a tractability of degenerate points from the point of view
of following the dynamics of classical spacetime. Assuming that all dynamical
variables are finite, we conclude that an essential trick for such a continuous
evolution is in complexifying variables. In order to restrict the complex
region locally, we propose some `reality recovering' conditions on spacetime.
Using a degenerate solution derived by pull-back technique, and integrating
the dynamical equations numerically, we show that this idea works in an
actual dynamical problem. We also discuss some features of these applications.

Motoyuki Saijo (Waseda U.), Hisa-aki Shinkai (Washington U., St. Louis) and Kei-ichi Maeda (Waseda U.)

Physical Review D56 (1997) 785-797

[gr-qc/9701001]; Refers to Cited by

*Abstract:*

Analyzing test particles falling into a Kerr black hole, we study gravitational
waves in Brans-Dicke theory of gravity. First we consider a test particle
plunging with a constant azimuthal angle into a rotating black hole and
calculate the waveform and emitted energy of both scalar and tensor modes
of gravitational radiation. We find that the waveform as well as the energy
of the scalar gravitational waves weakly depends on the rotation parameter
of black hole $a$ and on the azimuthal angle. Secondly, using a model of
a non-spherical dust shell of test particles falling into a Kerr black
hole, we study when the scalar modes dominate. When a black hole is rotating,
the tensor modes do not vanish even for a "spherically symmetric" shell,
instead a slightly oblate shell minimizes their energy but with non-zero
finite value, which depends on Kerr parameter $a$. As a result, we find
that the scalar modes dominate only for highly spherical collapse, but
they never exceed the tensor modes unless the Brans-Dicke parameter $\omega_{BD}
\lsim 750 $ for $a/M=0.99$ or unless $\omega_{BD} \lsim 20,000 $ for $a/M=0.5$,
where $M$ is mass of black hole. We conclude that the scalar gravitational
waves with $\omega_{BD} \lsim$ several thousands do not dominate except
for very limited situations (observation from the face-on direction of
a test particle falling into a Schwarzschild black hole or highly spherical
dust shell collapse into a Kerr black hole). Therefore observation of polarization
is also required when we determine the theory of gravity by the observation
of gravitational waves.

Classical and Quantum Gravity, 13, 783-790 (1996)

[gr-qc/9602026]; Refers to Cited by

*Abstract:*

We show how to treat the constraints and reality conditions in the
*SO(3)*-ADM (Ashtekar) formulation of general relativity, for the
case of a vacuum spacetime with a cosmological constant. We clarify the
difference between the reality conditions on the metric and on the triad.
Assuming the triad reality condition, we find a new variable, allowing
us to solve the gauge constraint equations and the reality conditions simultaneously.

**Dynamics of topological defects and inflation**

Nobuyuki Sakai, Hisa-aki Shinkai, Takashi Tachizawa and Kei-ichi Maeda (Waseda U.)

Physical Review D53 (1996) 655-661

[gr-qc/9506068];
Refers to
Cited by

*Abstract:*

We study the dynamics of topological defects in the context of "topological
inflation" proposed by Vilenkin and Linde independently. Analysing the
time evolution of planar domain walls and of global monopoles, we find
that the defects undergo inflationary expansion if $\eta\stackrel{>}{\sim}0.33m_{Pl}$,
where $\eta$ is the vacuum expectation value of the Higgs field and $m_{Pl}$
is the Planck mass. This result confirms the estimates by Vilenkin and
Linde. The critical value of $\eta$ is independent of the coupling constant
$\lambda$ and the initial size of the defect. Even for defects with an
initial size much greater than the horizon scale, inflation does not occur
at all if $\eta$ is smaller than the critical value. We also examine the
effect of gauge fields for static monopole solutions and find that the
spacetime with a gauge monopole has an attractive nature, contrary to the
spacetime with a global monopole. It suggests that gauge fields affect
the onset of inflation.

**A `3+1' method for finding principal null
directions**

Laurens Gunnarsen, Hisa-aki Shinkai and Kei-ichi Maeda (Waseda U.)

Classical and Quantum Gravity, 12, 133-140 (1995)

[gr-qc/9406003];
Refers to
Cited by

*Abstract:*

We present a new method for finding principal null directions (PNDs).
Because our method assumes as input the intrinsic metric and extrinsic
curvature of a spacelike hypersurface, we expect it will be useful to numerical
relativists. We illustrate our method by finding the PNDs of the Kastor-Traschen
spacetimes, which contain arbitrarily many $Q=M$ black holes in a de Sitter
back-ground.

**Generality of inflation in a planar universe**

Hisa-aki Shinkai and Kei-ichi Maeda (Waseda U.)

Physical Review D49 (1994) 6367-6378

[gr-qc/9402022];
Refers to
Cited by

*Abstract:*

We study a generality of an inflationary scenario by integrating the
Einstein equations numerically in a plane-symmetric spacetime. We consider
the inhomogeneous spacetimes due to (i) localized gravitational waves with
a positive cosmological constant $\Lambda$, and (ii) an inhomogeneous inflaton
field $\Phi$ with a potential $\frac12 m^2 \Phi^2$. For the case (i), we
find that any initial inhomogeneities are smoothed out even if waves collide,
so that we conclude that inhomogeneity due to gravitational waves do not
prevent the onset of inflation. As for the case (ii), if the mean value
of the inflaton field is initially as large as the condition in an isotropic
and homogeneous inflationary model (i.e., the mean value is larger than
several times Planck mass), the field is soon homogenized and the universe
always evolves into de Sitter spacetime. These support the cosmic no hair
conjecture in a planar universe. We also discuss the effects of an additional
massless scalar field, which is introduced to set initial data in usual
analysis.

**Can gravitational waves prevent inflation?**

Hisa-aki Shinkai and Kei-ichi Maeda (Waseda U.)

Physical Review D48 (1993) 3910-3913

[gr-qc/9305014];
Refers to
Cited by

Abstract:

To investigate the cosmic no hair conjecture, we analyze numerically
1-dimensional plane symmetrical inhomogeneities due to gravitational waves
in vacuum spacetimes with a positive cosmological constant. Assuming periodic
gravitational pulse waves initially, we study the time evolution of those
waves and the nature of their collisions. As measures of inhomogeneity
on each hypersurface, we use the 3-dimensional Riemann invariant ${\cal
I}\equiv ~^{(3)\!}R_{ijkl}~^{(3)\!}R^{ijkl}$ and the electric and magnetic
parts of the Weyl tensor. We find a temporal growth of the curvature in
the waves's collision region, but the overall expansion of the universe
later overcomes this effect. No singularity appears and the result is a
"no hair" de Sitter spacetime. The waves we study have amplitudes between
$0.020\Lambda \leq {\cal I}^{1/2} \leq 125.0\Lambda$ and widths between
$0.080l_H \leq l \leq 2.5l_H$, where $l_H=(\Lambda/3)^{-1/2}$, the horizon
scale of de Sitter spacetime. This supports the cosmic no hair conjecture.

**Bistability in an Ising model with non-Hamiltonian
dynamics**

J.R.Heringa (Delft Tech.
U., NL), H. Shinkai (Waseda U.), H.W.J.Bloete, A.Hoogland and R.K.P.Zia (Delft Tech.
U., NL)

Physical Review B45 (1992) 5707-5709

*Abstract:*

We investigate the phenomenon of magnetization bistability in a two-
dimensional Ising model with a non-Hamiltonian Glauber dynamics by means
of Monte Carlo simulations. This effect has previously been observed in
the Toom model, which supports two stable phases with different magnetizations,
even in the presence of a nonzero field. We find that such bistability
is also present in an Ising model in which the transition probabilities
are expressed in terms of Boltzmann factors depending only on the nearest-neighbor
spins and the associated bond strengths. The strength on each bond assumes
different values with respect to the spins at either of its ends, introducing
an asymmetry like that of the Toom model.

The 12th Japan GRG, Tokyo Univ., November 2002 [JGRG-12 version, pdf]

TH-2002, Paris, July 2002 [TH-2002 version, pdf]

*Abstract:*

We review recent efforts to re-formulate the Einstein equations
for fully relativistic numerical simulations.
The so-called numerical
relativity (computational simulations in general relativity) is
a promising research field matching with ongoing
astrophysical observations such as gravitational wave astronomy.
Many trials for
longterm stable and accurate simulations of binary compact objects
have revealed that mathematically
equivalent sets of evolution equations show different numerical
stability in free evolution schemes.
In this article, we first review
the efforts of the community,
categorizing them into the following three directions:
(1) modifications of the standard Arnowitt-Deser-Misner equations
initiated by the Kyoto group,
(2) rewriting of the evolution equations in hyperbolic form, and
(3) construction of an ``asymptotically constrained" system.
We next introduce our idea
for explaining these evolution behaviors in a unified way using
eigenvalue analysis of the
constraint
propagation equations. The modifications of (or adjustments to) the evolution
equations change the character of constraint propagation, and
several particular
adjustments using constraints are expected to diminish the
constraint-violating modes.
We propose several new adjusted evolution equations, and include
some numerical demonstrations. We conclude by discussing some directions
for future research.

with Gen Yoneda (Waseda University)

The 11th Japan GRG, Waseda Univ., January 2002

[pdf]

*Abstract:*

In order to find a way to have a better formulation for numerical
evolution of the Einstein equations,
we study the propagation equations of the
constraints based on the Arnowitt-Deser-Misner formulation.
By adjusting constraint terms in the evolution equations,
we try to construct an ``asymptotically constrained system"
which is expected to be robust against violation of the constraints,
and to enable
a long-term stable and accurate numerical simulation.
We first provide useful expressions for analyzing constraint propagation in a
general spacetime, then apply it to Schwarzschild spacetime.
We search when and where the negative real or non-zero imaginary
eigenvalues of the homogenized
constraint propagation matrix appear, and how they depend on the choice
of coordinate system and adjustments.
The predictions here may help the community to make further improvements.

with Gen Yoneda (Waseda University)

The 10th Japan GRG, Osaka, September 11-14 2000 (to be published)

[ps]

*Abstract:*

In order to perform accurate and stable long-term numerical integration
of the Einstein equations, several hyperbolic systems have been proposed.
We here report our numerical comparisons
between weakly hyperbolic, strongly hyperbolic,
and symmetric hyperbolic systems based on Ashtekar's connection variables.
The primary advantage for using this connection formulation
is that we can keep using the same dynamical variables for all levels of
hyperbolicity.
Our numerical code demonstrates gravitational wave propagation in
plane symmetric spacetimes, and we compare the
accuracy of the simulation by monitoring
the violation of the constraints.
By comparing with results obtained from the weakly hyperbolic system, we
observe the strongly and symmetric hyperbolic system show better
numerical performance (yield less constraint violation),
but not so much difference between the latter two.

We also study asymptotically constrained systems for
numerical integration of the Einstein equations, which are intended to be
robust against perturbative
errors for the free evolution of the initial data.
First, we examine the previously
proposed "$\lambda$-system", which introduces artificial flows to
constraint surfaces based on the symmetric hyperbolic formulation.
We show that this system works as expected for the wave propagation
problem in the Maxwell system and
in general relativity using Ashtekar's connection formulation.
Second, we propose a new mechanism to control the stability,
which we call
the "adjusted system". This is simply obtained by adding constraint terms
in the dynamical equations and adjusting its multipliers. We explain
why a particular choice of multiplier reduces the numerical errors
from non-positive or pure-imaginary eigenvalues
of the adjusted constraint propagation equations.
This "adjusted system" is also tested in the Maxwell system and in the
Ashtekar's system. This mechanism affects more than the system's
symmetric hyperbolicity.

with Gen Yoneda (Waseda University)

The 9th Marcel Grossmann meeting, Rome, July 2000

Proceedings, p 1717-1718 (eds by V.G. Gurzadyan, R.T. Jantzen, R. Ruffini, World Scientific, 2003)

[ps]

*Abstract:*

We present our numerical comparisons
between weakly hyperbolic, strongly hyperbolic,
and symmetric hyperbolic systems based on Ashtekar's connection variables.
Our numerical code demonstrates gravitational wave propagation in
plane symmetric spacetime, and we compare the stability and/or accuracy
by monitoring the violation of the constraints.
We also study alternative approaches to obtain stable evolutions,
that can be called "asymptotically constrained systems".
We demonstrate "$\lambda$-system" and "adjusted-system"
for the Ashtekar system, and show they work as desired.
We propose a new mechanism to control the stability by
evaluating the eigenvalues of constraint propagation equations,
and
this mechanism affects more than the system's
symmetric hyperbolicity.

with Gen Yoneda (Waseda University)

The 8th Marcel Grossmann meeting, Jerusalem, June 1997

Proceedings, p774-776 (ed. Tsvi Piran, World Scientific, 1999)

[ps]

*Abstract:*

We examine the advantages of the $SO(3)$-ADM (Ashtekar) formulation
of general relativity, from the point of following the dynamics of the
Lorentzian spacetime in direction of applying this into numerical relativity.
We describe our strategy how to treat new constraints and reality conditions,
together with a proposal of new variables. We show an example of passing
a degenerate point in flat spacetime numerically by posing `reality recovering'
conditions on spacetime. We also discuss some available advantages in numerical
relativity.

with Wai-Mo Suen, F. Douglas Swesty, Malcolm Tobias, Edward Y. M. Wang and Clifford M. Will (Washington University, St. Louis)

The 8th Marcel Grossmann meeting, Jerusalem, June 1997

Proceedings, p771-773 (ed. Tsvi Piran, World Scientific, 1999)

[ps]

*Abstract:*

We present two of our efforts directed toward the numerical analysis
of neutron star mergers, which are the most plausible sources for gravitational
wave detectors that should begin operating in the near future. First we
present Newtonian 3D simulations including radiation reaction (2.5PN) effects.
We discuss the gravitational wave signals and luminosity from the merger
with/without radiation reaction effects. Second we present the matching
problem between post-Newtonian formulations and general relativity in numerical
treatments. We prepare a spherical, static neutron star in a post-Newtonian
matched spacetime, and find that discontinuities at the matching surface
become smoothed out during fully relativistic evolution if we use a proper
slicing condition.

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