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We identify combinations of observables for rotating neutron stars that canone day bear on the question of whether there can be first order phasetransitions in the neutron matter therein. We employ the Hartle-Thorne theoryfor stationary, rotating neutron stars at conventional angular velocities (inthe pulsar and millisecond pulsar ranges) and extract three-dimensionalsections of the ellipticity or the dynamical angular momentum as function ofthe star's mass and angular velocity. An eventual first order phase transitionin the equation of state (EoS) leaves a clear ridge (nonanalyticity) in theseobservables, akin to the sudden kink in popular mass-radius diagrams for staticstars. Finally, we observe that static neutron stars in General Relativity (GR)will fail to be compact enough for the light ring's position at r=3M to beoutside the star, except for the most extreme equations of state. The outerlight ring of a rotating star might however be formed unless the EoS softenstoo much, and its eventual detection can then be used to constrain the EoS (orthe gravity theory).

Ridges in rotating neutron-star properties due to first order phase transitions