Asymptotic Gravitational Radiation of Photon Rockets with $\Lambda$: Point, String, and Sheet Sources

Abstract

We study the existence of gravitational radiation at infinity for pure-radiation Robinson-Trautman metrics with one spacelike Killing vector and a cosmological constant $\Lambda$ of arbitrary sign. These metrics describe "photon rockets" with point, string, and sheet sources moving through spacetime while emitting null radiation. For $\Lambda\neq 0$, we use criteria for the existence of gravitational radiation based on the asymptotic super-Poynting vector. For point sources, we extend to $\Lambda\neq 0$ the known $\Lambda=0$ result that the Kinnersley rocket is the only rocket without gravitational radiation. However, for string and sheet sources, new possibilities without gravitational radiation arise. For $\Lambda>0$, we show that the vanishing of the canonical asymptotic super-Poynting vector---computed with respect to the unit normal to scri---exactly characterizes the absence of radiation. On the other hand, for $\Lambda<0$, we show that the appropriate criterion is the vanishing of the components normal to scri of the asymptotic super-Poynting vectors associated with any unit timelike vector tangent to scri or, equivalently, the proportionality between the Cotton-York tensor and the holographic stress tensor at scri. In particular, we provide explicit examples of metrics for which the Cotton-York and holographic stress tensors commute, yet gravitational radiation is present because the two tensors are not proportional. We also determine the principal null directions (PNDs) of the rescaled Weyl tensor at scri for both signs of $\Lambda$ and relate their geometry relative to the normal to scri to the tensorial criteria for gravitational radiation.