Khan and Penrose, and separately Szekeres, presented solutions of colliding impulsive and sandwich plane gravitational waves, respectively. They only considered equal strength waves, which would impart no momentum to test particles in their path on the plane of collision at the time. We had earlier probed both spacetimes using the pseudo-Newtonian (\( \psi \)N) formalism. In neither case was it really clear what parameter represented the “strength” of the wave. In this paper we identify the required parameter and extend the colliding sandwich plane gravitational wave solution to unequal strengths. Since the impulsive plane waves of Khan and Penrose correspond to a limit of the sandwich waves with the pulse duration tending to zero, we have used the procedure to obtain the generalization of the Khan–Penrose solution to unequal strengths. The resulting spacetimes are exact solutions of vacuum field equations.

Khan 和 Penrose 以及 Szekeres 分别提出了碰撞脉冲引力波和夹层平面引力波的解决方案。他们只考虑了相同强度的波，这不会向当时碰撞平面上路径上的测试粒子传递动量。我们之前使用伪牛顿 ( \(\psi\) N) 形式主义探索了两个时空。在这两种情况下，都不清楚什么参数代表了波的“强度”。在本文中，我们确定了所需的参数，并将碰撞夹层平面引力波解扩展到不等强度。由于 Khan 和 Penrose 的脉冲平面波对应于脉冲持续时间趋向于零的夹层波的极限，因此我们使用该过程来获得 Khan-Penrose 解对不等强度的推广。由此产生的时空是真空场方程的精确解。