# Introduction

## Introduction

The remnants of the first generation of massive stars formed at redshift $z \gtrsim 15$, Population III (Pop III) stars, might be a population of black holes (BHs) which could grow to become supermassive black holes (SMBHs). SMBHs are believed to be the engines that power quasars and AGNs. A stellar-mass BH seed born at high redshift and radiating at the Eddington limit with $10\%$ radiative efficiency may have difficulty reaching supermassive size by $z\sim 7$ when the earliest quasars are observed. In fact, BHs may not grow at the Eddington limit during all earlier times. Photoionization, heating, and radiation pressure modify the accretion flow and reduce it on average to $\sim 32\%$ of the Eddington-limited rate.

An alternative scenario to building SMBHs is the collapse of supermassive stars (SMSs), which could provide the seeds for SMBHs. Stellar evolution calculations have recently shown that SMSs with a stellar radius of $R\simeq 10^4 R_\odot$ can be formed if rapid mass accretion ($\dot{M}\gtrsim 0.1M_\odot/\text{yr}$) takes place onto a star. Because of this larger stellar radius, the temperature never reaches the point where photoionization is activated. Also, mergers of SMBHs can help grow the seeds, as well as accretion.

For sufficiently massive stars, $M \gtrsim 10^6 M_\odot$, on the other hand, the accretion can build up enough entropy so that the radiation pressure can slow down and halt the accretion onto SMSs. Radiative cooling accompanied by mass loss then will induce quasistatic contraction to spin up the star to rotate at the mass-shedding limit on a secular timescale. Contraction will bring the star to the onset of a relatvistic dynamical instability, at which point it collapses to a spinning BH. Viscosity and/or magnetic turbulence will drive the rotation to be uniform. The resulting ratios $M_{BH}/M$ and $a/M$ turn out to be independent of the initial seed mass. Notice that the equation of state (EoS) for these SMSs is dominated by thermal radiation pressure and can be described by a simple $\Gamma = 4/3$ law.

Our previous axisymetric calculations of uniformly rotating SMSs spinning at the mass-shedding limit found that about $90-95\%$ of the initial stellar mass forms a spinning BH with spin parameter $a_{BH}/M_{H} \sim 0.7-0.75$ surrounded by a hot accretion disk. An axisymmetric GRMHD calculation we performed in which the stellar interior of a uniformly rotating SMS spinning at the mass-shedding limit was initially seeded with a poloidal magnetic field showed that the final configuration of the SMS consists of a central BH surrounded by a collimated magnetic field and a hot accretion disk.

In this work, we generalize the above study by performing GRMHD simulations of SMBHs in full 3D. We adopt the BSSN formulation of Einstein's equations in a puncture gauge for the gravitational field and a HRSC scheme for the MHD. A vector potential is introduced to solve Maxwell's induction equation in a new Lorentz gauge. In addition to the dipole $\textit{interior}$ magnetic model previously used, we also evolve a model in which the SMS is initially endowed with a dipole-like magnetic field which extends from the stellar $\textit{interior}$ into the $\textit{exterior}$. In all of our magnetized cases, we set the magnetic to rotational energy ratio to $\mathcal{M}/T = 0.1$. Since $T/|W|=0.009$ at mass-shedding, with $W$ the gravitational potential energy, the magnetic field turns out to be a small perturbation to the initial star configurations. We find that the final mass and spin parameters of the BH are close to those reported earlier. At about $t\sim 570M\approx 2600(M/10^6M_\odot)$s after the BH formation, the magnetized configurations launch a strongly magnetized, collimated, and mildly relativistic outflow-- an incipient jet. Based on the Poynting luminosity magnitude and scaling, the angular velocity of the force-free magnetic field, and other factors, we conclude that the Blandford-Znajek process is operating in our systems to drive the jet.

Magnetized SMSs therefore could be viable engines for long gamma-ray bursts (LGRBs). Recent observations of LGRBs at redshift $z>5$ by the $\textit{FERMI}$ and $\textit{SWIFT}$ satellites suggest that some of these bursts may originate from the collapse of massive stars (see e.g. GRB 140304A, GRB 090423).

The collapse of SMSs is also a source of gravitational waves (GWs) with frequencies in the frequency band of space laser interferometric detectors like LISA. The gravitational signal produced during the collapse of a SMS of $M=10^6 \text{M}_{\odot}$ at redshift $z=3$ has an amplitude of $5\times 10^{-21}$ at the frequency $5\text{mHz}$. Detections of these signals can test the SMS-collapse scenario as the seed of SMBHs.

Simulations were performed on the Blue Waters supercomputer at UIUC. The Illinois GRMHD code, which implements the BSSN formulation of GR with moving box adaptive mesh refinement, was used for all simulations.