Mass inflation or scalaron rigidity: Cauchy horizons in metric f(R)

Abstract

We develop a double-null framework for Cauchy-horizon dynamics in metric $f(R)$ gravity that separates ordinary mass inflation from scalaron cancellation and genuine escape from the blueshift instability. Starting from the exact spherical Hawking-mass transport equation, we show that a regular nondegenerate Cauchy horizon undergoes mass inflation whenever the effective longitudinal source is eventually nonnegative, uniformly dominates the mixed channel, and has a divergent blueshift-weighted integral. Hence bounded mass requires sufficiently strong blueshift-integrable suppression of this source or an independent competing channel. We then analyze the weaker branch in which the scalaron cancels only the leading Price-tail contribution. Under non-cancelling transverse and trace asymptotics, this forces $|R|\to\infty$ and an asymptotically linear high-curvature theory, $f(R)/R\to F_->0$, where $F_-\equiv\lim_{v\to\infty}f_R$. The curvature coefficient is fixed explicitly by the transverse asymptotics. For regular model classes with a finite high-curvature $f_{RR}$ limit one obtains $f_{RR}\to0$. More sharply, every eventually viable branch with $f_{RR}>0$ is forced to $R\to-\infty$ and $Λ_\infty>-1$; the standard scalaron mass parameter then diverges, while an additional differentiable rate condition yields $m_{\rm sc}^2\sim F/(3f_{RR})\to+\infty$. In the Einstein frame the scalaron null kinetic contribution is nonnegative, so the Jordan-frame cancellation cannot be interpreted as negative scalaron null energy. The result is a local rigidity classification rather than a global theorem of strong cosmic censorship.