Penrose junction conditions extended: Impulsive waves with gyratons

We generalize the classical junction conditions for constructing impulsive gravitational waves by the Penrose “cut and paste” method. Specifically, we study nonexpanding impulses which propagate in spaces of constant curvature with any value of the cosmological constant (that is, Minkowski, de Sitter, or anti–de Sitter universes) when additional off-diagonal metric components are present. Such components encode a possible angular momentum of the ultrarelativistic source of the impulsive wave—the so-called gyraton. We explicitly derive and analyze a specific transformation that relates the distributional form of the metric to a new form which is (Lipschitz) continuous. Such a transformation automatically implies an extended version of the Penrose junction conditions. It turns out that the conditions for identifying points of the background spacetime across the impulse are the same as in the original Penrose cut and paste construction, but their derivatives now directly represent the influence of the gyraton on the axial motion of test particles. Our results apply both for vacuum and nonvacuum solutions of Einstein’s field equations and can also be extended to other theories of gravity.

Figure 1

Minkowski space is cut into two parts ${\mathcal{M}}^{-}$ and ${\mathcal{M}}^{+}$ along a plane null hypersurface $\mathcal{N}$. These parts are then reattached with an arbitrary warp in which points are shunted along the null generators of the cut and then identified. This generates an impulsive gravitational wave in flat space.

Figure 2

Left: An example of the impulsive wave for which the gyratonic profile $J$ is proportional to the Heaviside step function $\mathrm{\Theta }$. Right: Plot of the Heaviside step function $\mathrm{\Theta }$ and of the kink function $U_{+}$ (bold).