Scalaron excitation by topological vortices in quadratic $f(R)$ gravity on a BTZ black hole background

TL;DR

This paper studies how topological vortices excite a scalar degree of freedom in quadratic $f(R)$ gravity on a BTZ black hole background, revealing universal decay and stability properties of the scalar excitation.

Contribution

It demonstrates the excitation mechanism of the scalaron by localized vortices in three-dimensional quadratic $f(R)$ gravity and analyzes the resulting scalar profile and stability.

Findings

Scalaron is excited by vortices with universal decay $r^{-(1+ u)}$

The scalar excitation is linearly stable and has finite energy

Backreaction effects are parametrically suppressed

Abstract

In three spacetime dimensions, pure Einstein gravity admits no local propagating degrees of freedom, yet nontrivial gravitational backgrounds such as the BTZ black hole provide a natural arena to probe dynamical extensions of the theory. In quadratic $f(R)$ gravity the Ricci scalar becomes a propagating degree of freedom - the scalaron. We investigate how localized Maxwell-Higgs vortices excite this scalar mode in a static BTZ black-hole background. Working in the perturbative regime $\alpha \ll \ell^2$, the trace equation reduces to a massive Klein-Gordon equation for the curvature scalar sourced by the trace of the vortex energy-momentum tensor. Using the Sturm-Liouville structure of the radial operator, we construct the corresponding Green function and obtain the curvature profile generated by an arbitrary localized source. The induced excitation exhibits a universal asymptotic decay $R(r) \sim r^{-(1+\nu)}$, independent of the detailed vortex structure. The scalar excitation is linearly stable, carries finite energy, and produces parametrically suppressed backreaction, ensuring the smooth recovery of the Einstein limit. These results provide a concrete realization of how higher-curvature corrections activate the unique local gravitational degree of freedom in three dimensions and how localized sources excite this scalar mode in black-hole spacetimes.