We derive the gravitational waveform from the collapse of a rapidly rotating supermassive star (SMS) core leading directly to a seed of a supermassive black hole (SMBH) in axisymmetric numerical-relativity simulations. We find that the peak strain amplitude of gravitational waves emitted during the black-hole formation is ≈ 5 × 10 −21 at the frequency f ≈ 5 mHz for an event at the cosmological redshift z = 3, if the collapsing SMS core is in the hydrogen-burning phase. Such gravitational waves will be detectable by space laser interferometric detectors like eLISA with signal-to-noise ratio ≈ 10, if the sensitivity is as high as LISA for f = 1-10 mHz. The detection of the gravitational-wave signal will provide a potential opportunity for testing the direct-collapse scenario for the formation of a seed of SMBHs.

We revisit secular stability against quasi-radial collapse for rigidly rotating supermassive stars (SMSs) in general relativity. We suppose that the SMSs are in a nuclear-burning phase and can be modeled by polytropic equations of state with the polytropic index n p slightly smaller than 3. The stability is determined in terms of the turning-point method. We find a fitting formula of the stability condition for the plausible range of n p (2.95 < ∼ n p < ∼ 3) for SMSs. This condition reconfirms that, while non-rotating SMSs with mass ∼ 10 5 M ⊙ -10 6 M ⊙ may undergo a general-relativistically induced quasi-radial collapse, rigidly rotating SMSs with a ratio of rotational to gravitational potential energy (β) of ∼ 10 −2 are likely to be stable against collapse unless they are able to accrete ∼ 5 times more mass during the (relatively brief) hydrogen-burning phase of their evolution. We discuss implications of our results.

Supermassive stars (SMSs) of mass 10 5 M are candidates for seeds of supermassive black holes found in the center of many massive galaxies. We simulate the gravitational collapse of a rigidly rotating SMS core including nuclear burning effects in axisymmetric numerical-relativity simulation. We find that for realistic initial conditions, the nuclear burning does not play an important role. After the collapse, a torus surrounding a rotating black hole is formed and a fraction of the torus material is ejected. We quantitatively study the relation between the properties of these objects and rotation. We find that if a SMS core is sufficiently rapidly rotating, the torus and outflow mass have approximately 6% and 1% of the initial mass, respectively. The typical average velocity and the total kinetic energy of the outflow are 0.2 c and 10 54−56 erg where c is the speed of light. Finally, we briefly discuss the possibility for observing the torus and outflow.

We found that the numerical code used in our paper [1] has some errors in the part of calculating the effect of nuclear burning. Then, we performed simulations for models A1-A4 again with the modified code and found that the errors made little change in the main results and conclusions of our paper.However, we found that this error changes the conclusion of Appendix A. In Appendix A, we compared the results of two simulations with the same initial conditions but with different formalisms to incorporate the effects of nuclear burning. One is the formalism described aswhere T μν , ρ, u ν , c, and _ q are the energy-momentum tensor, rest-mass density, 4-velocity of the fluid, speed of light, and energy generation rate of the nuclear burning, respectively. Here, we considered CNO cycle and triple-alpha reactions (model T1). The other is that described in Sec. II D (model T2). Then, we found that when we employed an initial condition with large metallicity (Z CNO ≳ 10 −3 ) the results of the two simulations did not agree with each other. We suspected that this difference would be due to the fact that Eq. (1.1) violates the energy-momentum conservation and this violation would be accumulated and finally induce the unnatural results for model T1.However, we simulated these models again by using the modified code and found that the numerical results of these simulations agree well each other. Figure 1 here shows the modified version of Fig. 23. A black hole is formed for all the models, and Q=jE ini j agrees well in models T1 and T2. Here, Q and E ini are the released rest-mass energy due to nuclear burning and the initial total energy, respectively. Thus, we withdraw the above argument and conclude that the formulation described in Eq. (1.1) does not lead to such unnatural results. We apologize if a reader has any inconvenience from our incorrect result.[1] H. Uchida, M. Shibata, T. Yoshida, Y. Sekiguchi, and H. Umeda, Phys. Rev. D 96, 083016 (2017). -4 -3 -2 -1 0 10000 15000 20000 25000 30000 35000 log 10 ini (Q/ E ) t[sec] T1 T2 T3FIG. 1. Time evolution of the ratio of the released rest-mass energy due to nuclear burning, Q, to the initial total energy, E ini , with the modified numerical code. Q and E ini are defined in Eqs. (A3) and (A4), respectively. The dotted, dashed, and dashed-dotted curves denote for models T1, T2, and T3, respectively. For models T1 and T2, a black hole is formed at t ≈ 30; 000 s (filled triangle and circle), and for model T3, a black hole is formed at t ≈ 31; 500 s (filled square).

We explore the formation process of a black hole (BH) through the pair-instability collapse of a rotating Population III very massive star in axisymmetric numerical relativity. As the initial condition, we employ a progenitor star which is obtained by evolving a rapidly rotating zero-age main sequence (ZAMS) star with mass 320M until it reaches a pair instability region. We find that for such rapidly rotating model, a fraction of the mass, ∼ 10M , forms a torus surrounding the remnant BH of mass ∼ 130M and an outflow is driven by a hydrodynamical effect. We also perform simulations, artificially reducing the initial angular velocity of the progenitor star, and find that only a small or no torus is formed and no outflow is driven. We discuss the possible evolution scenario of the remnant torus for the rapidly rotating model by considering the viscous and recombination effects and show that if the energy of ∼ 10 52 erg is injected from the torus to the envelope, the luminosity and timescale of the explosion could be of the orders of 10 43 erg/s and yrs, respectively. We also point out the possibility for observing gravitational waves associated with the BH formation for the rapidly rotating model by ground-based gravitational-wave detectors.