In general relativity, freely-falling test objects follow geodesics of the background spacetime in which they live. In a sense, this feature is a mere rephrasing of Einstein’s equivalence principle. In 1968, Brandon Carter showed that the geodesic motion of objects orbiting a Kerr black hole was integrable, in the sense of Hamiltonian mechanics, by discovering a fourth constant of motion that now bears his name. This “universality” of geodesic free fall is, however, but an approximation: In general, two different bodies will follow two distinct paths, depending on how they spin and deform.

The structure of neutron stars is determined by the so-called TOV equations of general relativity. Knowledge of the pressure-energy density relation is sufficient to determine the neutron star mass-radius (M-R) relation. Recent observations from X-ray telescopes, radio timing of pulsars, and gravitational wave observations, have provided several constraints on the masses and radii of neutron stars.

An open question in the search for ultra-high-energy cosmic-ray (UHECR) sources is whether they are few and prominent or if a large population of sources collectively contributes to the diffuse flux. Motivated by this question, we investigated whether the latest UHECR data are consistent with originating from a population of sources that exhibit substantial diversity in terms of the cosmic-ray spectra they produce. We found that the fit to the cosmic ray data requires sources that are intrinsically very similar in terms of the maximum energy reached.