Neutron stars are the final chord in the evolution of massive suns. Born in the cataclysm of a supernova, they pack a mass comparable to the Sun's into a sphere the size of an earthly metropolis. Predicted by Fritz Zwicky and discovered by Jocelyn Bell Burnell as mysteriously pulsing radio sources, these objects have teased astrophysicists for decades: we know their masses with good precision, but radii slipped away like shadows on an X-ray image. The whole difficulty lies in the enigmatic equation of state governing matter at densities trillions of times that of water. And now, an unexpected measurer steps onto the stage — spacetime causality itself.

Imagine that sound is a flexible tailor's tape, which the master-causality draws through the entire star. The tape can't move faster than light: otherwise the world would mix up causes and effects, like a suit stitched from patches in the wrong order.

This principle works like a strict cosmic tailor. If the tension of the fabric (pressure) is spread over a large volume, maintaining shape requires that signals about local changes travel from center to edge quickly enough. A star too puffy for its mass would demand superluminal "tailor's stitches," which is forbidden. Calculations based on general relativity equations (Tolman–Oppenheimer–Volkoff equations) with a single hard constraint, causality, yield an elegant linear formula. The radius of a 2.0 solar mass star cannot exceed the radius of a canonical 1.4 solar mass star, multiplied by 1.16 and reduced by 1.1 km: \[ R_{2.0} \le 1.16\,R_{1.4} - 1.1\,\text{km}. \] This is not just a theoretical refinement — the limit is saturated by real matter configurations in which a phase jump occurs just below the crust, and sound accelerates right up to the light barrier.

Interestingly, if a star violated this limit, sound in its interior would race faster than light: we would hear the thunder before seeing the lightning flash. Nature, like a strict pattern cutter, does not permit such liberties.

The tailor's measurement immediately proved useful for real data. The NICER observatory measured the radii of three millisecond pulsars: PSR J0437−4715, PSR J0614−3329, and the more massive PSR J0740+6620. When the causality constraint was imposed on their posterior distributions, the joint probability remained tiny — only 7.5% of the original. The lion's share of the pruning fell on the "heavyweight" J0740+6620: without causality, its median radius freely roamed around 12.5 km, but with the constraint it shrank to ~11.4 km. Our tailor's tape simply wouldn't accept sizes that contradict the foundation of relativity.

Such an approach takes us beyond Bayesian games with model priors. It proves that the speed of sound in cores must cross the conformal threshold \(c_s^2 = 1/3\), and, judging by the data, it comes right up to the absolute limit \(c_s^2 = 1\). This is a strong hint that at the center of a neutron star, matter ceases to be merely neutron: there, perhaps, quark soup or even more exotic states are born. Think about it: a reduction of the radius by just half a kilometer changes the core pressure by a factor of ten, and thus the very nature of matter. It's not excluded that deviations from the predicted line will point to the presence of dark matter in the interior. In the future, the same causal yardstick can be applied without changes to the analysis of tidal deformabilities in gravitational wave signals detected by LIGO/Virgo. With each new merger, the tape will be pulled ever tighter, sketching out the true shape of the equation of state.

We are witnessing a rare case when a philosophical principle, known since the creation of special relativity, turns into a working astrophysical tool. No adjustable parameters, only the universal prohibition on violating the order of events. Perhaps the next generation of X-ray telescopes, STROBE-X or eXTP, will be able to measure the diameter of a distant star with an accuracy of hundreds of meters, turning the tailor's pattern into a geodetic compass. And, along the way, answer the main question: what is the densest observable matter actually made of?