High-energy collision of black holes in higher dimensions

We compute the gravitational wave energy $E_{\mathrm{rad}}$ radiated in head-on collisions of equal-mass, nonspinning black holes in up to ($D=8$)-dimensional asymptotically flat spacetimes for boost velocities $v$ up to about 90% of the speed of light. We identify two main regimes: weak radiation at velocities up to about 40% of the speed of light, and exponential growth of $E_{\mathrm{rad}}$ with $v$ at larger velocities. Extrapolation to the speed of light predicts a limit of 12.9% (10.1, 7.7, 5.5, 4.5)% of the total mass that is lost in gravitational waves in $D=4$ (5, 6, 7, 8) spacetime dimensions. In agreement with perturbative calculations, we observe that the radiation is minimal for small but finite velocities, rather than for collisions starting from rest. Our computations support the identification of regimes with super-Planckian curvature outside the black-hole horizons reported by Okawa, Nakao, and Shibata [Phys. Rev. D 83, 121501(R) (2011)].

Figure 1

The normalized difference between the analytic and numerical ADM mass, $|M-M_{\mathrm{num}}|/M$, as obtained from the initial data of the Appendix for $D=5$, 6, 7 and 8 and the different initial velocities is shown as $\text{black}\times \text{symbols}$. The $\text{red}+\text{symbols}$ likewise denote deviations in the expected energy balance between the total ADM mass $M$, the horizon mass $M_{\mathrm{AH}}$ of the merger remnant BH and the radiated GW energy, i.e., $|M-E_{\mathrm{rad}}-M_{\mathrm{AH}}|/M$.

Figure 2

The Hamiltonian constraint along the collision axis for a binary with $v=0.6$ in $D=8$ dimensions. Note that only the range $x\ge 0$ is shown, as the second BH and the range $x<0$ are incorporated through reflection symmetry across the origin. The times $t=9.6R_S$ and $t=80R_S$ correspond to the infall and postmerger stages of the collision. The high-resolution results (red dashed curves) have been amplified by a factor $Q^3$, $Q=96/64$, to approximately match the low-resolution results, indicating convergence at about third order. The loss of convergence at $\sim {10}^{-13}$ is due to roundoff error.

Figure 3

The energy released in gravitational radiation in the collision of a BH binary starting with $v=0.6$ in $D=8$ dimensions. (Upper panel) The results obtained for the different resolutions and the prediction obtained from third-order Richardson extrapolation. (Lower panel) The differences between the individual simulations. The high versus medium resolution differences have been amplified by factors $Q_3=1.947$ and $Q_4=2.692$ expected for third- and fourth-order convergence. The results indicate convergence in between and we estimate the uncertainty using the more conservative third-order extrapolation to the continuum limit.

Figure 4

The energy $E_{\mathrm{rad}}$ radiated in gravitational waves from the head-on collision of two equal-mass nonspinning BHs with initial velocity $v$ in $D$ spacetime dimensions. The fits have been computed from data with $v\ge 0.4$ assuming a functional relation $\log E_{\mathrm{rad}}=a_0+a_{1}v$. The results have been rewritten to facilitate easy reading of the limit $E_{\mathrm{rad}}(v\to 1)$.

Figure 5

The normalized Kretschmann scalar $\mathcal{K}/{\mathcal{K}}_p$ at times $t=12.8R_S$ (left panel) and $t=13.3R_S$ (right panel) in the collision of a binary with $v=0.85$ in $D=6$ dimensions. The light-blue lines show the apparent horizon. At $t=12.8R_S$ two regions where $\mathcal{K}>1$ form, one above and one below the collision axis, indicating that super-Planckian curvature may become visible outside the BH horizon. At $t=13.3$ a common horizon has formed and engulfed this region.