Covariant phase space analysis of static spherical vacua in power-law $f(R)$ gravity

Abstract

We formulate the static, spherically symmetric vacuum equations of power-law metric $f(R)$ gravity as a covariant autonomous system using the $(1+1+2)$ formalism. The finite critical sets include the scale-invariant Clifton--Barrow solution, a Minkowski endpoint shared by the Schwarzschild family, and degenerate boundaries of the normalized chart. The Clifton--Barrow equilibrium is an outward-radial attractor over a positive interval of the power-law parameter, whereas the Schwarzschild endpoint is a saddle of the polynomially extended system. Poincaré compactification, supplemented by secondary and weighted charts, resolves several asymptotic regimes. In particular, normally attracting sheets at large normalized curvature occur in the positive-curvature sector and are separated from the negative-curvature Clifton--Barrow equilibrium. Metric reconstruction also identifies a selected scalar-flat family with a regular throat, one asymptotically flat end, and a curvature-singular end. These results organize the regular reduced chart and the boundary regimes resolved here; they do not constitute a complete classification of static spherical vacua. All stability statements concern the static radial flow.