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.
