Abstract:
We report a new implementation for axisymmetric simulation in full general relativity. In this implementation, the Einstein equations are solved using the Nakamura-Shibata formulation with the so-called cartoon method to impose an axisymmetric boundary condition, and the general relativistic hydrodynamic equations are solved using a high-resolution shock-capturing scheme based on an approximate Riemann solver. As tests, we performed the following simulations: (i) long-term evolution of non-rotating and rapidly rotating neutron stars, (ii) long-term evolution of neutron stars of a high-amplitude damping oscillation accompanied with shock formation, (iii) collapse of unstable neutron stars to black holes, and (iv) stellar collapses to neutron stars. The tests (i)--(iii) were carried out with the $\Gamma$-law equation of state, and the test (iv) with a more realistic parametric equation of state for high-density matter. We found that this new implementation works very well: It is possible to perform the simulations for stable neutron stars for more than 10 dynamical time scales, to capture strong shocks formed at stellar core collapses, and to accurately compute the mass of black holes formed after the collapse and subsequent accretion. In conclusion, this implementation is robust enough to apply to astrophysical problems such as stellar core collapse of massive stars to a neutron star and black hole, phase transition of a neutron star to a high-density star, and accretion-induced collapse of a neutron star to a black hole. The result for the first simulation of stellar core collapse to a neutron star started from a realistic initial condition is also presented.

Abstract:
Latest general relativistic simulations for merger of binary neutron stars with realistic equations of state (EOSs) show that a hypermassive neutron star of an ellipsoidal figure is formed after the merger if the total mass is smaller than a threshold value which depends on the EOSs. The effective amplitude of quasiperiodic gravitational waves from such hypermassive neutron stars is $\sim 6$--$7 \times 10^{-21}$ at a distance of 50 Mpc, which may be large enough for detection by advanced laser interferometric gravitational wave detectors although the frequency is high $\sim 3$ kHz. We point out that the detection of such signal may lead to constraining the EOSs for neutron stars.

Abstract:
We present relativistic hydrostatic equations for obtaining irrotational binary neutron stars in quasi equilibrium states in 3+1 formalism. Equations derived here are different from those previously given by Bonazzola, Gourgoulhon, and Marck, and have a simpler and more tractable form for computation in numerical relativity. We also present hydrostatic equations for computation of equilibrium irrotational binary stars in first post-Newtonian order.

Abstract:
We study the final state of the gravitational collapse of uniformly rotating supramassive neutron stars by axisymmetric simulations in full general relativity. The rotating stars provided as the initial condition are marginally stable against quasiradial gravitational collapse and its equatorial radius rotates with the Kepler velocity (i.e., the star is at the mass-shedding limit). To model the neutron stars, we adopt the polytropic equations of state for a wide range of the polytropic index as $n=2/3$, 4/5, 1, 3/2 and 2. We follow the formation and evolution of the black holes, and show that irrespective of the value of $n (2/3\leq n \leq 2)$, the final state is a Kerr black hole and the disk mass is very small ($< 10^{-3}$ of the initial stellar mass).

Abstract:
We have carried out simulations of the coalescence between two relativistic clusters of collisionless particles using a 3D numerical relativity code. We have adopted a new spatial gauge condition obtained by slightly modifying the minimum distortion gauge condition proposed by Smarr and York and resulting in a simpler equation for the shift vector. Using this gauge condition, we have performed several simulations of the merger between two identical clusters in which we have varied the compaction, the type of internal motion in the clusters, and the magnitude of the orbital velocity. As a result of the coalescence, either a new rotating cluster or a black hole is formed. In the case in which a black hole is not formed, simulations could be carried out for a time much longer than the dynamical time scale, and the resulting gravitational waveforms were calculated fairly accurately: In these cases, the amplitude of gravitational waves emitted can be $\sim 10^{-18}(M/10^6M_{\odot})$ at a distance 4000Mpc, and $\sim 0.5%$ of the rest mass energy may be dissipated by the gravitational wave emission in the final phase of the merger. These results confirm that the new spatial gauge condition is promising in many problems at least up to the formation of black holes. In the case in which a black hole is formed, on the other hand, the gauge condition seems to be less adequate, but we suggest a strategy to improve it in this case. All of the results obtained confirm the robustness of our formulation and the ability of our code for stable evolution of strong gravitational fields of compact binaries.

Abstract:
Motivated by a recent paper by the Potsdam numerical relativity group, we have constructed a new numerical code for hydrodynamic simulation of axisymmetric systems in full general relativity. In this code, we solve the Einstein field equation using Cartesian coordinates with appropriate boundary conditions. On the other hand, the hydrodynamic equations are solved in cylindrical coordinates. Using this code, we perform simulations to study axisymmetric collapse of rotating stars, which thereby become black holes or new compact stars, in full general relativity. To investigate the effects of rotation on the criterion for prompt collapse to black holes, we first adopt a polytropic equation of state, $P=K\rho^{\Gamma}$, where $P$, $\rho$, and $K$ are the pressure, rest mass density, and polytropic constant, with $\Gamma=2$. In this case, the collapse is adiabatic (i.e., no change in entropy), and we can focus on the bare effect of rotation. As the initial conditions, we prepare rigidly and differentially rotating stars in equilibrium and then decrease the pressure to induce collapse. In this paper, we consider cases in which $q \equiv J/M_g^2 < 1$, where $J$ and $M_g$ are the angular momentum and the gravitational mass. It is found that the criterion of black hole formation is strongly dependent on the angular momentum parameter $q$. For $q < 0.5$, the criterion is not strongly sensitive to $q$; more precisely, if the rest mass is slightly larger than the maximum allowed value of spherical stars, a black hole is formed. However, for $q \alt 1$, it changes significantly: For $q \simeq 0.9$, the maximum allowed rest mass becomes $\sim 70$ - 80% larger than that for spherical stars.

Abstract:
We present our first successful numerical results of 3D general relativistic simulations in which the Einstein equation as well as the hydrodynamic equations are fully solved. This paper is especially devoted to simulations of test problems such as spherical dust collapse, stability test of perturbed spherical stars, and preservation of (approximate) equilibrium states of rapidly rotating neutron star and/or corotating binary neutron stars. These test simulations confirm that simulations of coalescing binary neutron stars are feasible in a numerical relativity code. It is illustrated that using our numerical code, simulations of these problems, in particular those of corotating binary neutron stars, can be performed stably and fairly accurately for a couple of dynamical timescales. These numerical results indicate that our formulation for solving the Einstein field equation and hydrodynamic equations are robust and make it possible to perform a realistic simulation of coalescing binary neutron stars for a long time from the innermost circular orbit up to formation of a black hole or neutron star.

Abstract:
We review the current status of general relativistic studies for the coalescence of black hole-neutron star (BH-NS) binaries. First, procedures for a solution of BH-NS binaries in quasi-equilibrium circular orbits and the numerical results, such as quasi-equilibrium sequence and mass-shedding limit, of the high-precision computation, are summarized. Then, the current status of numerical-relativity simulations for the merger of BH-NS binaries is described. We summarize our understanding for the merger and/or tidal disruption processes, the criterion for tidal disruption, the properties of the remnant formed after the tidal disruption, gravitational waveform, and gravitational-wave spectrum.

Abstract:
We derive a formalism of numerical relativity for higher-dimensional spacetimes and develop numerical codes for simulating a wide variety of five-dimensional (5D) spacetimes for the first time. First, the Baumgarte-Shapiro-Shibata-Nakamura formalism is extended for arbitrary spacetime dimensions $D \ge 4$, and then, the so-called cartoon method, which was originally proposed as a robust method for simulating axisymmetric 4D spacetimes, is described for 5D spacetimes of several types of symmetries. Implementing 5D numerical relativity codes with the cartoon methods, we perform test simulations by evolving a 5D Schwarzschild spacetime and a 5D spacetime composed of a gravitational-wave packet of small amplitude. The numerical simulations are stably performed for a sufficiently long time, as done in the 4D case, and the obtained numerical results agree well with the analytic solutions: The numerical solutions are shown to converge at the correct order. We also confirm that a longterm accurate evolution of the 5D Schwarzschild spacetime is feasible using the so-called puncture approach. In addition, we derive the Landau-Lifshitz pseudo tensor in arbitrary dimensions, and show that it gives a robust tool for computing the energy flux of gravitational waves. The formulations and methods developed in this paper provide a powerful tool for studying nonlinear dynamics of higher-dimensional gravity.

Abstract:
Numerical-relativity simulation is performed for rapidly spinning black holes (BHs) in a higher-dimensional spacetime of special symmetries for the dimensionality $6 \leq d \leq 8$. We find that higher-dimensional BHs, spinning rapidly enough, are dynamically unstable against nonaxisymmetric bar-mode deformation and spontaneously emit gravitational waves, irrespective of $d$ as in the case $d=5$ \cite{SY09}. The critical values of a nondimensional spin parameter for the onset of the instability are $q:=a/\mu^{1/(d-3)} \approx 0.74$ for $d=6$, $\approx 0.73$ for $d=7$, and $\approx 0.77$ for $d=8$ where $\mu$ and $a$ are mass and spin parameters. Black holes with a spin smaller than these critical values ($q_{\rm crit}$) appear to be dynamically stable for any perturbation. Longterm simulations for the unstable BHs are also performed for $d=6$ and 7. We find that they spin down as a result of gravitational-wave emission and subsequently settle to a stable stationary BH of a spin smaller than $q_{\rm crit}$. For more rapidly spinning unstable BHs, the timescale, for which the new state is reached, is shorter and fraction of the spin-down is larger. Our findings imply that a highly rapidly spinning BH with $q > q_{\rm crit}$ cannot be a stationary product in the particle accelerators, even if it would be formed as a consequence of a TeV-gravity hypothesis. Its implications for the phenomenology of a mini BH are discussed.