Vacuum Spacetime With Multipole Moments: Gravitational Wave Observables, Black Hole Shadow and the Minimal Size Conjecture*

In this work, we explicitly construct the vacuum solution of Einstein's equations with prescribed multipole moments. By observing the behavior of the metric at small distances, we conjecture that for a sufficiently large multipole moment, there is a minimal size below which no object in nature can support such a moment. With the metric of such a spacetime, we discuss the shift of orbital frequencies with respect to those of the Kerr spacetime, for a test particle moving around an object with this set of multipole moments. These frequency shifts are required for the program of mapping out the spacetime multipole moments based on gravitational wave observations of extreme mass ratio inspirals. In addition, we analyze the shape of black hole shadow for various multipole moments and discuss the prospects of constraining the moments from shadow observations.