Rotating core collapse

4.3 Rotating core collapse

4.3.1 Collapse to a rotating black hole

Black hole formation in relativistic core collapse was first studied in axisymmetry by Nakamura [231 , 232 ], using the (2+1)+1 formalism [216 ]. The outcome of the simulation depends on the rotational parameter

A rotating black hole is formed only if $q < 1$ , indicating that cosmic censorship holds. Stark and Piran [289

, 243 ] use the 3+1 formalism and the radial gauge of Bardeen–Piran [25 ] to study black hole formation and gravitational wave emission in axisymmetry. In this gauge, two metric functions used in determining $g𝜃𝜃$ and $gϕϕ$ can be chosen such that at large radii they tend directly to $h+$ and $h×$ (the even and odd transverse traceless amplitudes of the gravitational waves, with $1∕r$ fall-off at large radii; note that $h +$ defined in [289 ] has the opposite sign as that commonly used, e.g., in [305 ]). In this way, the gravitational waveform is obtained at large radii directly in the numerical evolution. It is also easy to compute the gravitational energy emitted, as a simple integral over a sphere far from the source: $∫ ΔE ∼ r2 dt(h2+,r + h2×,r)$ . Using polar slicing, black hole formation appears as a region of exponentially small lapse, when $q < 𝒪 (1)$ . The initial data consists of a nonrotating, pressure deficient TOV solution, to which angular momentum is added by hand. The obtained waveform is nearly independent of the details of the collapse: It consists of a broad initial peak (since the star adjusts its initial spherical shape to a flattened shape, more consistent with the prescribed angular momentum), the main emission (during the formation of the black hole), and an oscillatory tail, corresponding to oscillations of the formed black hole spacetime. The energy of the emitted gravitational waves during the axisymmetric core collapse is found not to exceed $− 4 2 7 × 10 M ⊙c$ (to which the broad initial peak has a negligible contribution). The emitted energy scales as $q4$ , while the energy in the even mode exceeds that in the odd mode by at least an order of magnitude.

More recently, Shibata [271 ] carried out axisymmetric simulations of rotating stellar collapse in full general relativity, using a Cartesian grid, in which axisymmetry is imposed by suitable boundary conditions. The details of the formalism (numerical evolution scheme and gauge) are given in [270 ]. It is found that rapid rotation can prevent prompt black hole formation. When $q = 𝒪(1)$ , a prompt collapse to a black hole is prevented even for a rest mass that is 70 – 80% larger than the maximum allowed mass of spherical stars, and this depends weakly on the rotational profile of the initial configuration. The final configuration is supported against collapse by the induced differential rotation. In these axisymmetric simulations, shock formation for $q < 0.5$ does not result in a significant heating of the core; shocks are formed at a spheroidal shell around the high density core. In contrast, when the initial configuration is rapidly rotating ( $q = 𝒪 (1 )$ ), shocks are formed in a highly nonspherical manner near high density regions, and the resultant shock heating contributes in preventing prompt collapse to a black hole. A qualitative analysis in [271 ] suggests that a disk can form around a black hole during core collapse, provided the progenitor is nearly rigidly rotating and $q = 𝒪 (1)$ for a stiff progenitor EOS. On the other hand, $q ≪ 1$ still allows for a disk formation if the progenitor EOS is soft. At present, it is not clear how much the above conclusions depend on the restriction to axisymmetry or on other assumptions – 3-dimensional simulations of the core collapse of such initially axisymmetric configurations have still to be performed.

A new numerical code for axisymmetric gravitational collapse in the (2+1)+1 formalism is presented in [63 ].

4.3.2 Formation of rotating neutron stars

First attempts to study the formation of rotating neutron stars in axisymmetric collapse were initiated by Evans [96 , 97 ]. Recently, Dimmelmeier, Font and Müller [90 , 89 ] have successfully obtained detailed simulations of neutron star formation in rotating collapse. In the numerical scheme, HRSC methods are employed for the hydrodynamical evolution, while for the spacetime evolution the conformal flatness approximation [324 ] is used. Surprisingly, the gravitational waves obtained during the neutron star formation in rotating core collapse are weaker in general relativity than in Newtonian simulations. The reason for this result is that relativistic rotating cores bounce at larger central densities than in the Newtonian limit (for the same initial conditions). The gravitational waves are computed from the time derivatives of the quadrupole moment, which involves the volume integration of $ρr4$ . As the density profile of the formed neutron star is more centrally condensed than in the Newtonian case, the corresponding gravitational waves turn out to be weaker. Details of the numerical methods and of the gravitational wave extraction used in the above studies can be found in [91 , 92 ].

New, fully relativistic axisymmetric simulations with coupled hydrodynamical and spacetime evolution in the light-cone approach, have been obtained by Siebel et al. [280 , 281 ]. One of the advantages of the light-cone approach is that gravitational waves can be extracted accurately at null infinity, without spurious contamination by boundary conditions. The code by Siebel et al. combines the light-cone approach for the spacetime evolution with HRSC methods for the hydrodynamical evolution. In [281 ] it is found that gravitational waves are extracted more accurately using the Bondi news function than by a quadrupole formula on the null cone.

A new 2D code for axisymmetric core collapse, also using HRSC methods, has recently been introduced in [272 ].