Three-Hair Relations for Rotating Stars: Nonrelativistic Limit

The gravitational field outside of astrophysical black holes is completely described by their mass and spin frequency, as expressed by the no-hair theorems. These theorems assume vacuum spacetimes, and thus they apply only to black holes and not to stars. Despite this, we analytically find that the gravitational potential of arbitrarily rapid rigidly rotating stars can still be described completely by only their mass, spin angular momentum, and quadrupole moment. Although these results are obtained in the nonrelativistic limit (to leading order in a weak-field expansion of general relativity, GR), they are also consistent with fully relativistic numerical calculations of rotating neutron stars. This description of the gravitational potential outside the source in terms of just three quantities is approximately universal (independent of equation of state). Such universality may be used to break degeneracies in pulsar and future gravitational wave observations to extract more physics and test GR in the strong-field regime.