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 can not be used to study properties of plane gravitational waves at null infinities , including memory effects and soft graviton theorems .