Emmy's letter to Santa Claus (and a reply): Vaidya geometries and scalar fields with null gradients

英文摘要：

Emmy, a child prodigy, writes a letter to Santa Claus asking for a Vaidya solution sourced by a massless scalar field with null gradient for Christmas. Santa writes back explaining why such a toy cannot be built and offers an exact plane wave sourced by a scalar field with lightlike gradient as a replacement.