4.1 Numerical evolution of equilibrium models

4.1.1 Stable equilibrium

Figure 13: Time evolution of the rotational velocity profile for a stationary, rapidly rotating relativistic star (in the Cowling approximation), using the 3rd order PPM scheme and a 116³ grid. The initial rotational profile is preserved to a high degree of accuracy, even after 20 rotational periods. (Figure 1 of Stergioulas and Font [293].)

The long-term stable evolution of rotating relativistic stars in 3D simulations has become possible through the use of High-Resolution Shock-Capturing (HRSC) methods (see [103] for a review). Stergioulas and Font [293] evolve rotating relativistic stars near the mass-shedding limit for dozens of rotational periods (evolving only the equations of hydrodynamics) (see Figure 13), while accurately preserving the rotational profile, using the 3rd order PPM method [65]. This method was shown to be superior to other, commonly used methods, in 2D evolutions of rotating relativistic stars [106].

Fully coupled hydrodynamical and spacetime evolutions in 3D have been obtained by Shibata [269] and by Font et al. [105]. In [269], the evolution of approximate (conformally flat) initial data is presented for about two rotational periods, and in [105] the simulations extend to several full rotational periods (see Movie 14), using numerically exact initial data and a monotonized central difference (MC) slope limiter [315]. The MC slope limiter is somewhat less accurate in preserving the rotational profile of equilibrium stars than the 3rd order PPM method, but, on the other hand, it is easier to implement in a numerical code.

New evolutions of uniformly and differentially rotating stars in 3D, using different gauges and coordinate systems, are presented in [93], while new 2D evolutions are presented in [272].

4.1.2 Instability to collapse

Shibata, Baumgarte, and Shapiro [274] study the stability of supramassive neutron stars rotating at the mass-shedding limit, for a polytropic EOS. Their 3D simulations in full general relativity show that stars on the mass-shedding sequence, with central energy density somewhat larger than that of the maximum mass model, are dynamically unstable to collapse. Thus, the dynamical instability of rotating neutron stars to axisymmetric perturbations is close to the corresponding secular instability. The initial data for these simulations are approximate, conformally flat axisymmetric solutions, but their properties are not very different from exact axisymmetric solutions even near the mass-shedding limit [73]. It should be noted that the approximate minimal distortion (AMD) shift condition does not prove useful in the numerical evolution, once a horizon forms. Instead, modified shift conditions are used in [274]. In the above simulations, no massive disk around the black hole is formed, as the equatorial radius of the initial model is inside the radius which becomes the ISCO of the final black hole. This could change if a different EOS is chosen.

4.1.3 Dynamical bar-mode instability

Shibata, Baumgarte, and Shapiro [273] study the dynamical bar-mode instability in differentially rotating neutron stars, in fully relativistic 3D simulations. They find that stars become unstable when rotating faster than a critical value of . This is only somewhat smaller than the Newtonian value of . Models with rotation only somewhat above critical become differentially rotating ellipsoids, while models with much larger than critical also form spiral arms, leading to mass ejection (see Figure 15, and Movies 16 and 17). In any case, the differentially rotating ellipsoids formed during the bar-mode instability have , indicating that they will be secularly unstable to bar-mode formation (driven by gravitational radiation or viscosity). The decrease of the critical value of for dynamical bar formation due to relativistic effects has been confirmed by post-Newtonian simulations [258].

Figure 15: Density contours and velocity flow for a neutron star model that has developed spiral arms, due to the dynamical bar-mode instability. The computation was done in full General Relativity. (Figure 4 of Shibata, Baumgarte, and Shapiro [273]; used with permission).

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Figure 16: mpg-Movie (1655 KB) Simulation of the development of the dynamical bar-mode instability in a rapidly rotating relativistic star. Spiral arms form within a few rotational periods. The different colors correspond to different values of the density, while the computation was done in full general relativity. Movie produced at the University of Illinois by T.W. Baumgarte, S.L. Shapiro, and M. Shibata, with the assistance of the Illinois Undergraduate Research Team [31]; used with permission.