Gravitational wave recoil in Robinson-Trautman spacetimes

We consider the gravitational recoil due to nonreflection-symmetric gravitational wave emission in the context of axisymmetric Robinson-Trautman spacetimes. We show that regular initial data evolve generically into a final configuration corresponding to a Schwarzschild black hole moving with constant speed. For the case of (reflection-)symmetric initial configurations, the mass of the remnant black hole and the total energy radiated away are completely determined by the initial data, allowing us to obtain analytical expressions for some recent numerical results that have appeared in the literature. Moreover, by using the Galerkin spectral method to analyze the nonlinear regime of the Robinson-Trautman equations, we show that the recoil velocity can be estimated with good accuracy from some asymmetry measures (namely the first odd moments) of the initial data. The extension for the nonaxisymmetric case and the implications of our results for realistic situations involving head-on collision of two black holes are also discussed.

Figure 1

The fraction $\Delta$ of the initial Bondi’s mass lost due to gravitational wave emission for the initial configuration (17), as a function of $y=1-ϵ$. The curve is in very good agreement with the one inferred from numerical results in 14, 15. Notice, however, that the nonextensive distribution function proposed in 14, 15 is merely an approximation for $y\approx 1$; see (20).

Figure 2

Final odd moments $q_n(\infty )$, $n=1$, 3, 5, as functions of the recoil velocity $v=\tanh \alpha$, as given by (22). Notice that, for a given $0<|v|<1$, one has $|q_1|>|q_3|>|q_5|>\cdots$.

Figure 3

Polar plot of some typical nonreflection-symmetric initial data $Q(0,x)$ of the family (27). The initial condition $a$, $b$, and $c$ correspond, respectively, to the parameters $\alpha =1/2$, $\beta =1$, $\gamma =0$; $\alpha =\beta =0$, $\gamma =-2/3$; and $\alpha =0$, $\beta =4$, $\gamma =3$. The dashed lines correspond to the associated gravitational radiation content (without scale); see Sec. 4.

Figure 4

Evolution of the modes $b_{\ell }(u)$ governed by (24) for the case $(a)$ of Fig. 3. $N=8$ was used, leading to an accuracy (controlled by the constant $q_0=2$) of ${10}^{-4}$. The final evolution state has $b_0=1.0197$ and $b_1=0.20017$ and, consequently, the recoil velocity is $v=0.19628$ and the radiated energy fraction $\Delta =0.05420$, calculated according to (26).

Figure 5

Plot of $v\times q_1(0)$ for some typical regular initial conditions. The dotted line is the curve predicted by (35). The detail depicts the plot of $q_1(0)\times q_1(\infty )$. The assumption of $q_1(\infty )=q_1(0)$ is, typically, a good approximation when $q_1(u)$ is the dominant moment.

Figure 6

Dependence of the recoil velocity $v$ with the infalling velocity $w$ of the two black holes with different masses, as predicted by (39), and some results from numerical calculations. Notice that $v\to -v$ if $\mu \to 1/\mu$.