A first-order formulation of f(R) gravity in spherical symmetry
Philippe G. LeFloch, Filipe C. Mena
TL;DR
The paper develops a first-order, Bondi-type formulation of $f(R)$ gravity in spherical symmetry, coupled to a (possibly massive) scalar field, to study global evolution and causal structure. It introduces an augmented conformal formulation and derives a two-equation integro-differential system for $(\phi,\rho)$, proving equivalence to the full $f(R)$ system under center regularity and establishing Hawking-mass monotonicity. The framework yields concrete expressions for the metric scalars and provides a robust path toward rigorous geometric analysis and numerically robust simulations of spherical collapse in modified gravity, including the GR limit $f(R)\to R$. By proving center-regularity results, mass monotonicity, and an invariant-domain property, the work isolates the essential evolution and constraint content on two null hypersurfaces and sets the stage for further mathematical and computational advances in $f(R)$ cosmology and gravitational collapse.
Abstract
We develop a first-order formulation of the field equations in f(R) gravity governing the global evolution of a (possibly massive) scalar field under spherical symmetry. Our formulation allows us to pose the characteristic initial value problem and to establish several properties of solutions. More precisely, we work in generalized Bondi-Sachs coordinates and prescribe initial data on an asymptotically Euclidean, future light cone with vertex at the center of symmetry, and we identify the precise regularity conditions required at the center. Following and extending Christodoulou's approach to the Einstein-massless scalar-field system, we recast the f(R) field equations as an integro-differential system of two coupled, first-order, nonlocal, nonlinear hyperbolic equations, whose principal unknowns are the scalar field and the spacetime scalar curvature. In deriving this reduced two-equation system, we identify the regularity conditions at the center of symmetry and impose natural assumptions on the scalar-field potential and on the function f(R) governing the gravitational Lagrangian density. As an application, we prove the monotonicity of the Hawking mass in this setting and formally analyze the singular limit in which the integrand f(R) of the action approaches R, corresponding to the Einstein-Hilbert action. Hence, the formulation isolates the essential evolution and constraint content on the future domain of dependence of two null hypersurfaces and is designed to facilitate subsequent advances in geometric analysis and robust numerical simulations of spherical collapse in modified gravity.