Dynamical systems approach to stellar modelling in $f(G, B)$ gravity

The novel proposal to invoke the split of the Ricci scalar into bulk and boundary terms in the gravitational action, opens up a new avenue of investigation into stellar dynamics. The Lagrangian contains functional forms of the bulk while the boundary terms do not contribute to the dynamics. The advantage of the proposition is that the stellar structure equations are up to order two thus the theory is not haunted by ghosts. We obtain explicitly the defining equations for the thermodynamical variables and the geometry for the pure quadratic case since the linear case amounts to general relativity. In trying to establish the vacuum geometry associated with the theory it turns out that two possible metrics emerge through the vanishing of the energy-momentum tensor. Next we analyse the isotropy equation and make the observation that it is autonomous. It is rare that this happens in astrophysical modelling. This behaviour prompted the use of dynamical systems to understand the stability properties of fixed points or fixed manifolds. It was necessary to choose a gauge in order to split the autonomous equation into a system from which we could plot a phase portrait and deduce the stability of solution trajectories. We find that the fixed curves were generally stable with nearby paths approaching the fixed curves.

💡 Research Summary

The paper investigates a modified gravity theory in which the Ricci scalar R is decomposed into a bulk term G and a boundary term B, leading to the action S=∫f(G,B)√−g d⁴x + S_m. Because the boundary contribution does not affect the field equations, the dynamics are governed solely by the bulk term, guaranteeing that the resulting equations of motion are at most second order in derivatives. This feature eliminates Ostrogradsky ghosts that typically plague higher‑order theories such as f(R).

After introducing the formalism, the authors specialize to a static, spherically symmetric spacetime written in isotropic Cartesian coordinates and then in standard spherical form. They compute the bulk scalar G for this metric and derive the general field equations for an arbitrary function f(G). When f(G)=G the equations reduce to the Einstein equations, confirming that the linear case reproduces General Relativity (GR).

The core of the work focuses on the pure quadratic model f(G)=α G², which represents the next level of complexity beyond GR. The authors impose pressure isotropy (radial pressure equals tangential pressure), which yields a master isotropy equation that, remarkably, is autonomous: it depends only on the metric potentials µ(r) and ν(r) and their derivatives, not explicitly on the radial coordinate. This autonomy is unusual in stellar modelling and motivates a dynamical‑systems treatment.

Two vacuum branches arise from the condition G=0. In the first branch the relation µ′=−2ν′ holds, leading to a metric function e^µ=C₂(r²+C₁)⁴. In the second branch µ′=0, giving a constant µ. Imposing the vacuum condition ρ=p_r=p_t=0 forces both µ′ and ν′ to vanish, so the only consistent vacuum solution is flat Minkowski space.

To analyse the autonomous isotropy equation, the authors introduce scale‑invariant variables (e.g., x=µ′+ν′, y=µ′−ν′) and rewrite the second‑order equation as a first‑order autonomous system. A gauge choice is required to split the equation cleanly; once fixed, the system becomes (x′,y′)=F(x,y). Fixed points and, more generally, fixed curves are identified by solving F(x,y)=0. Phase‑plane portraits reveal that the fixed curves act as attractors: trajectories starting from a wide range of initial conditions flow toward these curves, indicating that the corresponding metric configurations are dynamically stable.

The paper concludes that f(G,B) gravity offers a ghost‑free, second‑order framework for stellar structure, that the isotropy condition’s autonomy enables a powerful dynamical‑systems analysis, and that the quadratic model possesses stable fixed‑curve solutions. These results provide a new, global perspective on compact‑star modelling and suggest that further extensions (e.g., inclusion of matter equations of state or rotation) could be tackled within the same formalism.