Deep inside a star compressed to its limit, particles behave like the strings of an unusual musical instrument. Their oscillations obey a complex equation that for nearly a century no one could solve exactly. The authors of the work have for the first time tackled this challenge: they found an exact solution for massive particles inside a Schwarzschild star (a sphere of perfect fluid with constant density).
The secret lay in a hidden symmetry of the problem. After a clever change of coordinates, dictated by the star's own geometry, the equation reduced to a known type — the Fuchsian (with singular points, as in many classical problems). The solution is written in terms of the general Heun function — a rare but powerful mathematical tool.
Near the star's center, where density tends to infinity (the so-called Buchdahl limit), the behavior of oscillations changes dramatically. For static modes (standing waves), the authors derived a regularity condition — otherwise the solution blows up. For dynamic modes, the amplitude grows while the oscillation step shrinks — the sound seems to rise in pitch and volume simultaneously as it approaches the singularity. These details were previously inaccessible in approximate calculations.
The Buchdahl limit is the maximum possible compactness of a star, at which the pressure at the center becomes infinite. For a typical neutron star, the radius is only 2.25 times its gravitational radius.
In Stanislaw Lem's novel "Solaris," the ocean of a living planet generated complex wave structures — perhaps their mathematics would resemble the solutions from this work.