Numerical relativity for $D$ dimensional space-times: Head-on collisions of black holes and gravitational wave extraction

Higher dimensional black holes play an exciting role in fundamental physics, such as high energy physics. In this paper, we use the formalism and numerical code reported in [1] to study the head-on collision of two black holes. For this purpose we provide a detailed treatment of gravitational wave extraction in generic $D$ dimensional space-times, which uses the Kodama-Ishibashi formalism. For the first time, we present the results of numerical simulations of the head-on collision in five space-time dimensions, together with the relevant physical quantities. We show that the total radiated energy, when two black holes collide from rest at infinity, is approximately $(0.089\pm 0.006)\%$ of the center of mass energy, slightly larger than the 0.055% obtained in the four-dimensional case, and that the ringdown signal at late time is in very good agreement with perturbative calculations.

Figure 1

Left panel: ${\mathcal{G}}_{tt}$ calculated from Eq. (2.16) for $D=4$, at different extraction radii. This quantity has been shifted in time to account for the different extraction radii and rescaled by the corresponding $R_{\mathrm{ex}}$. The late-time behavior is shown in the inset. Right panel: time evolution of areal radius [cf. (2.21)] extracted at the radii $R_{\mathrm{ex}}=30r_S$ (black solid line), $R_{\mathrm{ex}}=40r_S$ (red dashed line) and $R_{\mathrm{ex}}=50r_S$ (green dashed-dotted line).

Figure 2

Left panel: Time derivatives of the $l=2$ modes of the KI function $\Phi$ (black solid line), and of the Zerilli function $\overline{\Phi }$ (red dashed line) extracted for model $\mathrm{HD}{4}_f$ at $R_{\mathrm{ex}}=30r_S$. The KI function has been rescaled by a constant factor [cf. (3.1)] which accounts for the different normalizations of both formulations. Right panel: comparison of the second time derivative ${\Phi }_{,tt}$ with the outgoing Newman-Penrose scalar ${\Psi }_4$ for the same model. The KI wave function has been rescaled according to Eq. (3.2).

Figure 3

The $l=2$ component of the KI wave function ${\Phi }_{,t}$ extracted at the radii $R_{\mathrm{ex}}=30r_S$ (black solid line), $R_{\mathrm{ex}}=40r_S$ (red dashed line) and $R_{\mathrm{ex}}=50r_S$ (green dashed-dotted line). They have been shifted in time by the corresponding $R_{\mathrm{ex}}$.

Figure 4

Convergence analysis of the $l=2$ component of ${\Phi }_{,t}$ extracted at $R_{\mathrm{ex}}=30r_S$. We plot the differences between the low and medium resolution (black solid line) and medium and high resolution (red dashed line) run. The latter is rescaled by the factor $Q=1.58$ expected for 4th order convergence [14].

Figure 5

Energy flux (left panel) and radiated energy (right panel) for the $l=2$ mode extracted at $R_{\mathrm{ex}}=50r_S$ from the KI wave function ${\Phi }_{,t}$ (black solid curve) and the Newman-Penrose scalar ${\Psi }_4$ (red dashed curve).

Figure 6

Left panel: Time evolution of the areal radius $r$ in units of the extraction radius averaged over coordinate spheres at $R_{\mathrm{ex}}=20r_S$ (black solid), $40r_S$ (red dashed) and $60r_S$ (green dash-dotted curve). Right panel: Deviation of the metric component $R_{\mathrm{ex}}^2/r_S^2{\mathcal{G}}_{tt}$ calculated from Eq. (2.16) at the same extraction radii and shifted in time to account for differences in the propagation time of the wave signal.

Figure 7

Estimates for the time it takes for two equal-mass BHs to collide in $D=5$. The first estimate is given by the time $t_{\mathrm{CAH}}$ elapsed until a single common apparent horizon engulfs both BHs (diamonds), the second estimate is obtained by using the trajectory of the BHs, i.e., the time $t_{\mathrm{traj}}$ at which their separation has decreased below the Schwarzschild radius (circles). Finally, these numerical results are compared against a simple Newtonian estimate, given by Eq. (4.4) [blue solid line].

Figure 8

Left panel: The $l=2$ component of the KI wave form for model $\mathrm{HD}5e_f$ extracted at radii $R_{\mathrm{ex}}/r_S=20$, 40 and 60 and shifted in time by $R_{\mathrm{ex}}/r_S$. Right panel: The $l=2$ and $l=4$ mode of the KI function for the same simulation, extracted at $R_{\mathrm{ex}}/r_S=60$. For clarity, the $l=4$ component has been rescaled by a factor of 100.

Figure 9

Convergence analysis of the $l=2$ component of the KI function generated by model HD5e extracted at $R_{\mathrm{ex}}=60r_S$. The difference between the medium and high resolution wave forms has been amplified by the factors 1.97 (red dashed line) and 2.33 (green dashed-dotted line) indicating third and fourth order convergence.

Figure 10

The $l=2$ components of the KI function as generated by a head-on collision of BHs with initial (coordinate) distance $d=6.37r_S$ (black solid line) and $d=10.37r_S$ (red dashed line). The wave functions have been shifted in time such that the formation of a common apparent horizon corresponds to $\Delta t=0$ and taking into account the time it takes for the waves to propagate up to the extraction radius $R_{\mathrm{ex}}=60r_S$.

Figure 11

Energy flux in the $l=2$ component of the KI wave function ${\Phi }_{,t}$, extracted at $R_{\mathrm{ex}}=60r_S$, for models $\mathrm{HD}5e_f$ (black solid line) and HD5f (red dashed line) in Table . The fluxes have been shifted in time by the extraction radius $R_{\mathrm{ex}}=60r_S$ and the time $t_{\mathrm{CAH}}$ at which the common apparent horizon forms.

Figure 12

Left panel: Fraction of the center of mass energy, $E_{\mathrm{rad}}/M$, radiated in the $l=2$ mode of the KI function shifted in time such that the origin of the time axis corresponds to the formation of a common apparent horizon. Right panel: Fraction of the center of mass energy $1-M_{AH}/M$ radiated during the collision, estimated using apparent horizon information. The oscillations in this diagnostic quantity have a frequency comparable to the $l=2$ quasinormal mode frequency.