Ultrarelativistic Black Hole Formation

We study the head-on collision of fluid particles well within the kinetic energy dominated regime ($\gamma =8$ to 12) by numerically solving the Einstein-hydrodynamic equations. We find that the threshold for black hole formation is lower (by a factor of a few) than simple hoop conjecture estimates, and, moreover, near this threshold two distinct apparent horizons first form postcollision and then merge. We argue that this can be understood in terms of a gravitational focusing effect. The gravitational radiation reaches luminosities of $0.014c^5/G$, carrying $16\pm 2\%$ of the total energy.

Figure 1

Snapshots of rest mass density on a logarithmic scale from ${10}^{-2}$ to ${10}^2$ times the initial maximum density, for simulations with $\gamma =8$ (top) and $\gamma =10$ (bottom) at times (left to right) $t=0$, the initial time; $t=300M_{*}$, shortly after collision; $t=375M_{*}$, after the appearance of the smaller AHs in the $\gamma =10$ case; $t=424M_{*}$, after the appearance of the third, encompassing AH (white outline surrounding the two black regions) in the $\gamma =10$ case; and $t=700M_{*}$. The black regions are best-fit ellipses to the AHs.

Figure 2

Apparent horizon properties for the $\gamma =10$ case. The AHs labeled 1 and 2 are the two identical ones (when mirrored about the collision plane) that appear first and are later encompassed by AH 3. We show (top to bottom): the irreducible masses, the proper distance $d_p$ between (and exterior to) AHs 1 and 2 measured along the collision axis, and the ratio of the equatorial to polar circumferences.

Figure 3

Total and spin-weight 2 spherical harmonic decomposition of power in GWs from the $\gamma =10$ case (with $\gamma =8$ in the inset). The small feature at $(t-r)<0$ is an artifact of the initial data and is not included in the estimate of energy.

Figure 4

Focusing of a set of geodesics in a boosted star spacetime with $\gamma =10$ and $2M_{*}/R_{*}=1/40$. Shown are the $x$ coordinate relative to the center of the boosted star which is at $x_{*}=vt$ (top) and the $y$ coordinate (perpendicular to the boost axis) of the geodesics as a function of coordinate time.