Numerical evolution of general Robinson-Trautman spacetimes: Code tests, wave forms, and the efficiency of the gravitational wave extraction

We present an efficient numerical code based on spectral methods to integrate the field equations of general Robinson-Trautmann spacetimes. The most natural basis functions for the spectral expansion of the metric functions are spherical harmonics. Using the values of appropriate combinations of the metric functions at the collocation points, we have managed to reduce expression swell when the number of spherical harmonics increases. Our numerical code runs with relatively little computational resources and the code tests have shown excellent accuracy and convergence. The code has been applied to situations of physical interest in the context of Robsinson-Trautmann geometries such as: perturbation of the exterior gravitational field of a spheroid of matter; perturbation of an initially boosted black hole; and the nonfrontal collision of two Schwarzschild black holes. In dealing with these processes we have derived analytical lower and upper bounds on the velocity of the resulting black hole and the efficiency of the gravitational wave extraction, respectively. Numerical experiments were performed to determine the forms of the gravitational waves and the efficiency in each situation of interest.

Figure 1

Plots of the maximum relative errors $\delta I$ after evolving the initial data (13, 14, 15) (from top to bottom). In all cases the exponential decay of the maximum error was observed as a result of the fast convergence of the code. The saturation due to the round-off error is shown for the two first cases and is achieved for different values of $N$.

Figure 2

Plots of the maximum values of the $L_2$ norm [see Eq. (19)] with respect to the truncation orders $N$ for the initial data (13, 14, 15) (from top to bottom). Again, spectral convergence is obtained.

Figure 3

Sequence of three-dimensional polar plots of $D(u,\theta ,\varphi )$ corresponding to the first set of initial data evaluated at $u=0$, 0.1, 0.4, 0.7, and to the third set of initial data evaluated at $u=0.002$, 0.004, 0.006, 0.008. The symmetric pattern in the first sequence suggests that no momentum is transferred to the remnant, while in the second sequence the lack of symmetry of the angular pattern indicates that the resulting black hole might be moving with respect to an asymptotic observer.

Figure 4

Illustration of the three situations representing (from top to bottom) orthogonal, oblique, and head-on collisions of two Schwarzschild black holes.

Figure 5

Illustration of the sequence of polar plots of $D$ projected on the planes $x=0$ ($u=0.02$, 0.10, 0.26) and $x=0.3$ ($u=0.02$, 0.10, 0.18) for the oblique collision. The angular pattern is dominated by lobes that indicate the directions of maximum magnitude of the gravitational wave emission. Note that the two dominant lobes open as the black hole is decelerated, forming a bremsstrahlung-like pattern.