The Ultimate Stiffness Limit of Neutron Stars express
Just as in a crowded room each new person sharply increases the pressure on the walls, in a neutron star increasing density leads to a rise in internal resistance. But even this resistance has an absolute ceiling. Physicists describe it with the ratio of pressure to energy density (the quantity φ) — a sort of measure of matter's "stubbornness." From special relativity, φ cannot exceed 1 anyway (otherwise signals inside the star would outpace the speed of light). And general relativity, adding spacetime curvature, tightens the limit even further. The key ingredient is the stability condition: the star's mass must increase from center to edge without gaps. With it, φ does not exceed 0.385. This number works for any conceivable star filling, from ordinary neutrons to quark soup. Surprisingly, the same law applies to water in a glass — it's just realized at negligible fractions of a percent, so we don't feel it.
Original abstract
as written by the authors arXiv:2601.02980v1
The equation-of-state (EOS) parameter $$\phi \equiv P/\varepsilon$$, defined as the ratio of pressure to energy density, encapsulates the fundamental response of matter under extreme compression. Its value at the center of the most massive neutron star (NS), $$\x \equiv P_{\rm c}/\varepsilon_{\rm c}$$, provides an upper bound on the maximum attainable central EOS parameter of cold visible matter. Remarkably, owing to the intrinsically nonlinear structure of the EOS in General Relativity (GR), this bound lies far below the naive Special Relativity (SR) limit of unity. In this work, we refine the theoretical upper bound on $$\x$$ in a self-consistent manner by incorporating, in addition to the causality constraint from SR, the mass-sphere stability condition associated with the mass evolution pattern in the vicinity of the NS center. This condition is formulated within the intrinsic and perturbative analysis of the dimensionless Tolman--Oppenheimer--Volkoff equations (IPAD-TOV) framework. The combined constraints yield an improved bound, $$\x \lesssim 0.385$$, which is slightly above but fully consistent with the previously derived causal-only limit, $$\x \lesssim 0.374$$. We further derive an improved scaling relation for NS compactness and demonstrate its robustness across a broad set of 284 realistic EOSs, including models with first-order phase transitions, exotic degrees of freedom, continuous crossover behavior, and deconfined quark cores. Within the IPAD-TOV framework, the resulting bound on $$\x$$ provides a new EOS-insensitive probe of the microphysics of cold superdense matter compressed by strong-field gravity in GR.
A teaspoon of neutron star material weighs as much as Mount Everest.