One of the possible formation mechanisms for SMBHs involves the gravitational collapse of supermassive stars (SMSs). The timescale for this formation channel is short enough to account for the presence of SMBHs at redshifts [129]. Supermassive stars may contract directly out of the primordial gas, if radiation and/or magnetic field pressure prevent fragmentation [101, 64, 100, 157, 33, 1]. Alternatively, they may build up from fragments of stellar collisions in clusters [211, 21]. Supermassive stars are radiation dominated, isentropic and convective [221, 270, 157]. Thus, they are well represented by an polytrope. If the star’s mass exceeds , nuclear burning and electron/positron annihilation are not important.

After formation, an SMS will evolve through a phase of quasistationary cooling and contraction. If the SMS is rotating when it forms, conservation of angular momentum requires that it spins up as it contracts. There are two possible evolutionary regimes for a cooling SMS. The path taken by an SMS depends on the strength of its viscosity and magnetic fields and on the nature of its angular momentum distribution.

In the first regime, viscosity or magnetic fields are strong enough to enforce uniform rotation throughout the star as it contracts. Baumgarte and Shapiro [16] have studied the evolution of a uniformly rotating SMS up to the onset of relativistic instability. They demonstrated that a uniformly rotating, cooling SMS will eventually spin up to its mass shedding limit. The mass shedding limit is encountered when matter at the star’s equator rotates with the Keplerian velocity. The limit can be represented as . In this case, . The star will then evolve along a mass shedding sequence, losing both mass and angular momentum. It will eventually contract to the onset of relativistic instability [123, 50, 51, 221, 129].

Baumgarte and Shapiro used both a second-order, post-Newtonian approximation and a fully general relativistic numerical code to determine that the onset of relativistic instability occurs at a ratio of , where is the star’s radius and in the remainder of this section. Note that a second-order, post-Newtonian approximation was needed because rotation stabilizes the destabilizing role of nonlinear gravity at the first post-Newtonian level. If the mass of the star exceeds , the star will then collapse and possibly form a SMBH. If the star is less massive, nuclear reactions may lead to explosion instead of collapse.

The major result of Baumgarte and Shapiro’s work is that the universal values of the following ratios exist for the critical configuration at the onset of relativistic instability: , , and . These ratios are completely independent of the mass of the star or its prior evolution. Because uniformly rotating SMSs will begin to collapse from a universal configuration, the subsequent collapse and the resulting gravitational waveform will be unique.

In the opposite evolutionary regime, neither viscosity nor magnetic fields are strong enough to enforce uniform rotation throughout the cooling SMS as it contracts. In this case, it has been shown that the angular momentum distribution is conserved on cylinders during contraction [27]. Because viscosity and magnetic fields are weak, there is no means of redistributing angular momentum in the star. So, even if the star starts out rotating uniformly, it cannot remain so.

The star will then rotate differentially as it cools and contracts. In this case, the subsequent evolution depends on the star’s initial angular momentum distribution, which is largely unknown. One possible outcome is that the star will spin up to mass-shedding (at a different value of than a uniformly rotating star) and then follow an evolutionary path that may be similar to that described by Baumgarte and Shapiro [16]. The alternative outcome is that the star will encounter the dynamical bar instability prior to reaching the mass-shedding limit. New and Shapiro [185, 186] have demonstrated that a bar-mode phase is likely to be encountered by differentially rotating SMSs with a wide range of initial angular momentum distributions. This mode will transport mass and angular momentum outward and thus may hasten the onset of collapse.

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