Ultrarelativistic boost of the black ring

We investigate the ultrarelativistic boost of the five-dimensional Emparan-Reall nonrotating black ring. Following the classical method of Aichelburg and Sexl, we determine the gravitational field generated by a black ring moving “with the speed of light” in an arbitrary direction. In particular, we study in detail two different boosts along axes orthogonal and parallel to the plane of the ring circle, respectively. In both cases, after the limit one obtains a five-dimensional impulsive $pp$-wave propagating in Minkowski spacetime. The curvature singularity of the original static spacetime becomes a singular source within the wave front, in the shape of a ring or a rod according to the direction of the boost. In the case of an orthogonal boost, the wave front contains also a remnant of the original disk-shaped membrane as a component of the Ricci tensor (which is everywhere else vanishing). We also analyze the asymptotic properties of the boosted black ring at large spatial distances from the singularity, and its behavior near the sources. In the limit when the singularity shrinks to a point, one recovers the well-known five-dimensional analogue of the Aichelburg-Sexl monopole solution.

Figure 1

The profile function $H_{\perp }(x_2,\xi )$ given by Eq. (31). The upper picture represents the values taken by $H_{\perp }$ over the plane $(x_2,\xi )$, whereas the lower one displays curves along which $H_{\perp }$ is constant. This is the case of a static black ring boosted to the speed of light in a direction orthogonal to the plane $(\xi ,\psi )$, which contains the ring circle. The coordinates $(x_2,\xi ,\psi )$ span spatial sections of the impulsive wave front $u=0$, cf. Eq. (27). Here the coordinate $\psi$ is suppressed, since it just describes the orbits of a Killing vector field. The profile function $H_{\perp }$ diverges (only) at the ring singularity $x_2=0$, $\xi =L$, as indicated by the thick point(s) in the pictures. In addition, there is a disk membrane within the ring, i.e. at $x_2=0$, $\xi <L$ [cf. Eq. (34)], which manifests itself as a jump in $\partial H_{\perp }/\partial x_2$. This is drawn above as a thick line.

Figure 2

Plot of the profile function $H_{||}(y_2,\eta )$ given by Eq. (53). This is the case of a static black ring boosted to the speed of light along a direction contained within the plane of the ring circle, i.e. $(\xi ,\psi )$. The coordinates $(y_2,\eta ,\varphi )$ span spatial sections of the impulsive wave front $u=0$, cf. Eq. (51). The Killing coordinate $\varphi$ is suppressed in the figures. The profile function $H_{||}$ diverges (only) at the rod singularity $\eta =0$, $|y_2|\le L$, as indicated by the thick line in the pictures.

Figure 3

Plot of the profile function $H(x_2,y_2,z_2)$ given by Eqs. (60, 61). The integration in Eq. (60) has been performed numerically. The coordinates $(x_2,y_2,z_2)$ span spatial sections of the impulsive wave front $u=0$, cf. Eq. (59). Since this plot corresponds to a boost in a general direction $z_1$, the function $H$ is not axially symmetry. We have thus depicted representative plots of the values taken by $H$ over different sections of the three space $(x_2,y_2,z_2)$. In the left figure, in particular, it is evident the ellipse-shaped singularity, cf. Eq. (72).