Semianalytical estimates of scattering thresholds and gravitational radiation in ultrarelativistic black hole encounters

Ultrarelativistic collisions of black holes are ideal gedanken experiments to study the nonlinearities of general relativity. In this paper we use semianalytical tools to better understand the nature of these collisions and the emitted gravitational radiation. We explain many features of the energy spectra extracted from numerical relativity simulations using two complementary semianalytical calculations. In the first calculation we estimate the radiation by a “zero-frequency limit” analysis of the collision of two point particles with finite impact parameter. In the second calculation we replace one of the black holes by a point particle plunging with arbitrary energy and impact parameter into a Schwarzschild black hole, and we explore the multipolar structure of the radiation paying particular attention to the near-critical regime. We also use a geodesic analogy to provide qualitative estimates of the dependence of the scattering threshold on the black hole spin and on the dimensionality of the spacetime.

Figure 1

Energy spectrum for the dominant (quadrupolar, i.e. $l=2$) component of the gravitational radiation computed from NR simulations of the head-on collision of two equal-mass BHs (from 4). The collision speed in the center-of-mass frame, $\beta =v/c$, is indicated in the legend. The energy spectrum is roughly flat (independent of frequency) up to the quasinormal mode (QNM) frequencies (marked by vertical lines), after which it decays exponentially. All quantities are normalized to the Arnowitt-Deser-Misner (ADM) mass of the system $M_{\mathrm{ADM}}$.

Figure 2

Critical impact parameter versus reduced energy $E$ for different values of $j$ in a Kerr background.

Figure 3

Critical impact parameter for $E=10$ as a function of $j$ for Myers-Perry BHs in different dimensions $D$. The inset zooms along the $y$ axis to show the local minima more clearly.

Figure 4

Contour plots of the critical impact parameter in the $(v,j)$ plane for corotating orbits ($j>0$) and counterrotating orbits ($j<0$) in dimensions $D=4$, $D=5$, $D=6$, and $D=7$.

Figure 5

The system before and after the collision. ${\xi }_1$ is defined in Eq. (22).

Figure 6

Energy per unit solid angle and per unit frequency emitted in the directions $\widehat{\mathbf{k}}={\mathbf{e}}_{\mathbf{x}}$, ${\mathbf{e}}_{\mathbf{y}}$, ${\mathbf{e}}_{\mathbf{z}}$, $({\mathbf{e}}_{\mathbf{x}}+{\mathbf{e}}_{\mathbf{y}}+\sqrt{2}{\mathbf{e}}_{\mathbf{z}})/2$ by equal-mass binaries with $E=3$, as a function of $\omega /\Omega$.

Figure 7

Normalized energy spectrum per unit solid angle emitted in the directions $\widehat{\mathbf{k}}={\mathbf{e}}_{\mathbf{y}}$ (thick lines) and $\widehat{\mathbf{k}}={\mathbf{e}}_{\mathbf{z}}$ (thin lines) as a function of $M\omega$ for several values of $b/M$ (as indicated in the legend) and $E=3$. As indicated by Eq. (40) the ZFL for these two different directions is the same and approximately equal to $0.18013M^2$.

Figure 8

Normalized energy spectrum per solid angle emitted in the directions $\widehat{\mathbf{k}}={\mathbf{e}}_{\mathbf{x}}$, ${\mathbf{e}}_{\mathbf{y}}$, ${\mathbf{e}}_{\mathbf{z}}$, $({\mathbf{e}}_{\mathbf{x}}+{\mathbf{e}}_{\mathbf{y}}+\sqrt{2}{\mathbf{e}}_{\mathbf{z}})/2$ as a function of $\omega /\Omega$ in the extreme-mass ratio case.

Figure 9

Trajectories for different values of $L_z/L_{\mathrm{crit}}$ (as indicated in the legend) and $E=3$. The black circle of radius 2 marks the location of the horizon.

Figure 10

Spectra for $l=m=2,\dots ,6$ for nonrelativistic infalls ($E=1$, left column) and kinetic-energy dominated infalls ($E=3$, right column). The top row refers to a radial infall, and the middle (bottom) row to an infall with $L/L_{\mathrm{crit}}=0.9$ ($L/L_{\mathrm{crit}}=0.9999$).

Figure 11

Spectra for $l=2$ and $|m|\le l$ for nonrelativistic infalls ($E=1$, left column) and kinetic-energy dominated infalls ($E=3$, right column). The top row refers to a radial infall, and the bottom row to an infall with $L/L_{\mathrm{crit}}=0.9$.

Figure 12

Infall from rest: spectra for $l=m=2$ as $L_z/L_{\mathrm{crit}}\to 1$. In the inset: logarithmic divergence of the total radiated energy in the same limit.

Figure 13

Multipolar energy distribution for $E_l/E^2$ for $E=1$ (left) and $E=100$ (right). For large $E$ the radiated energy scales like $E^2$, not like ${\mu }^2$. Higher multipoles contribute (relatively) more when $E$ is large. The scaling with $l$ is well approximated by an exponential for collisions from rest and by a power law for relativistic collisions.

Figure 14

Total energy radiated $M/(\mu E{)}^2{E}^{\mathrm{rad}}$ rescaled by the particle energy squared, as a function of $L_z/L_{\mathrm{crit}}$. In extrapolating for $l>6$, the fitting coefficients in Eqs. (65, 69) were obtained either including (empty symbols) or excluding (filled symbols) the $l=2$ data. The difference is visible only in the limit $L_z/L_{\mathrm{crit}}\to 1$.

Figure 15

Ratio $M{E}^{\mathrm{rad}}/J\mathrm{rad}$ for different values of $E$. The meaning of filled and empty symbols is the same as in Fig. 14. Oohara and Nakamura [51] find that this ratio is $\simeq 0.15\pm 0.01$ for $1\lesssim L_z\lesssim 3.9$ when $E=1$ and $L_{\mathrm{crit}}=4$, but we expect that $E^{\mathrm{rad}}/J^{\mathrm{rad}}\simeq (M{\Omega }_{\mathrm{scat}})$ as $L_z/L_{\mathrm{crit}}\to 1$ (see text).

Figure 16

Total radiated angular momentum (obtained by extrapolation) as a function of $L_z/L_{\mathrm{crit}}$. The meaning of filled and empty symbols is the same as in Fig. 14.

Figure 17

Multipolar components of the radiated angular momentum $J_l/E^2$ for $E=1$ (left) and $E=100$ (right). For large $E$ the radiated angular momentum scales like $(\mu E{)}^2$. Higher multipoles contribute (relatively) more when $E$ is large. The scaling with $l$ is logarithmic for collisions from rest, but it acquires corrections for relativistic collisions. By symmetry, no angular momentum is radiated when $L_z=0$.

Figure 18

Spectra of the linear momentum radiated for infalls from rest ($E=1$) and ultrarelativistic infalls ($E=100$). Different lines refer to different truncation choices in the sum of Eq. (59). In the head-on case (top row) the only nonzero component of the linear momentum is $P_x$; for $L/L_{\mathrm{crit}}=0.9$ (middle and bottom rows) both the $P_x$ and $P_y$ components are nonzero. To test convergence we sum Eq. (59) up to different values of $l_{\max }$, as indicated in the legend.

Figure 19

Left: $l=m=2$ component of the energy spectrum of the gravitational radiation emitted in the collision of two equal-mass BHs having speed $\beta =v/c\simeq 0.75$ in the center-of-mass frame [4, 6]. All quantities are normalized to the ADM mass of the system $M_{\mathrm{ADM}}$. Right: $l=m=2$ component of the energy spectrum of the gravitational radiation emitted by point particles falling into Schwarzschild BHs of mass $M$ with energy $E=1.5$.