Singularities of plane gravitational waves and their memory effects

Similar to the Schwarzschild coordinates for spherical black holes, the Baldwin, Jeffery and Rosen (BJR) coordinates for plane gravitational waves are often singular, and extensions beyond such singularities are necessary, before studying asymptotic properties of such spacetimes at the null infinity of the plane, on which the gravitational waves propagate. The latter is closely related to the studies of memory effects and soft graviton theorems. In this paper, we point out that in the BJR coordinates all the spacetimes are singular physically at the focused point $u = u_s$, except for the two cases: (1) $\alpha =1/2, \; \forall \; \chi_n$; and (2) $\alpha =1, \; \chi_i = 0\; (i = 1, 2, 3)$, where $\chi_n$ are the coefficients in the expansion $\chi \equiv \left[{\mbox{det}}\left(g_{ab}\right) \right]^{1/4} = \left(u - u_s\right)^{\alpha}\sum_{n = 0}^{\infty}\chi_n \left(u - u_s\right)^n$ with $\chi_0 \not= 0$, the constant $\alpha \in (0, 1]$ characterizes the strength of the singularities, and $g_{ab}$ denotes the reduced metric on the two-dimensional plane orthogonal to the propagation direction of the wave. Therefore, the hypersurfaces $u= u_s$ already represent the boundaries of such spacetimes, and the null infinity does not belong to them. As a result, they cannot be used to study properties of plane gravitational waves at null infinities, including memory effects and soft graviton theorems.