Can One Modified Gravity Rule Work Across a Star’s Different Matter Phases?

One star, one gravitational function

Models of hybrid stars consider a possible transition from hadronic matter, containing particles such as neutrons, to quark matter deeper inside. A theoretical question follows: if gravity couples directly to properties of matter, can the same gravitational function describe both phases consistently?

Samuel Isidoro dos Santos Júnior and Rafael C. R. de Lima address this in a preprint submitted to arXiv on November 3, 2026, and described as prepared for submission to JCAP. Their study establishes mathematical constraints within a specified model. It does not report observations of quark cores or evidence that gravity differs from general relativity.

The model is written as f(R,T) = R + h(T). Here R describes spacetime curvature, while T is the trace of the matter stress tensor. With the authors’ conventions, T = ρ − 3p, where ρ is total energy density and p is pressure. The additional function h therefore depends on a combination of density and pressure, not simply on position inside a star.

Why separate solutions are not enough

The authors require the material stress tensor to be conserved separately. In their static setup, this requirement determines how h changes for a chosen equation of state—the relationship between pressure and density.

But solving that problem independently for hadronic and quark matter does not guarantee a consistent combined model. Because h belongs to the gravitational action, it must remain one fixed function wherever either phase occurs.

The first test concerns shared values of T. Different matter states can have the same trace, even at different densities. At every shared trace value, both phases must demand compatible values and derivatives of the same function. This is a comparison in trace space, not merely a comparison of conditions immediately across the physical boundary.

For a hadronic polytrope, whose pressure follows a power law in energy density, paired with an affine quark equation of state, whose pressure is linear in density with an offset, the authors prove a strong restriction: if their occupied trace ranges overlap over an open interval, h must be constant on both ranges. A constant h gives general relativity with a constant geometric contribution; pure general relativity corresponds to h = 0.

A boundary must also balance effective pressure

The second test applies at the sharp interface between phases. The paper assumes a cold Maxwell transition, with equal material pressure and chemical potential, no matter flux across the boundary, and no surface layer or surface tension.

Those material equilibrium conditions are not sufficient. Spacetime responds to an effective stress that includes the modification to gravity. Writing q = h'(T), the extra pressure contribution depends on q(ρ + p) − h/2. That combination must be continuous across the interface for a shell-free match.

The authors illustrate the distinction by shifting an additive constant in the quark-phase reconstruction. This leaves bulk material conservation and the material transition conditions unchanged, but disrupts effective-pressure matching. A valid solution for either phase alone can therefore fail when the phases are joined.

A construction that works locally

When the occupied trace ranges are separated, the shared-trace obstacle disappears. The authors show that suitable bulk solutions can satisfy the interface condition and be connected by a function with continuous first and second derivatives.

Their explicit example uses a fifth-degree polynomial to bridge the unoccupied gap in trace values. This bridge is not a physical layer of intermediate matter: the chosen configuration never samples those values. Another configuration might, so the function cannot simply be retuned from star to star.

The examples use synthetic, dimensionless matter data, not an astrophysical fit. The construction demonstrates local compatibility at one interface, not a complete star with a regular center, envelope and exterior.

Two limits on the result

The paper also proves that if the same hadronic polytrope extends to zero density, requiring h's derivative to remain bounded near T = 0 removes its nonconstant part throughout that branch. With an asymptotically flat vacuum exterior, h must be zero there.

A separate restriction concerns time dependence. Wherever h'(T) is nonzero, separate material conservation forces each moving fluid element to retain its density. Ordinary compressional oscillations are therefore unavailable under these assumptions; static consistency does not establish dynamical viability.

All these conclusions depend on a stipulated prescription for how the perfect fluid couples to the metric, called the energy-density closure. Other prescriptions require a new analysis. Realistic equations of state, a complete hybrid-star solution and compatible perturbations remain open. The contribution is a consistency framework—not a demonstrated new kind of star.