Axial Perturbations of Extended Tolman III in the New $f(R,T)$ Theory

We investigate compact objects using the extended Tolman III equation of state in the framework of $f(R,T)=R+2\chi T$ gravity. In the matter sector of the Lagrangian density, we employ a new formalism based on the evaluation of the second-order derivative of the matter Lagrangian with respect to the metric \cite{Jayawiguna:2026das}. We analyze the corresponding observational signatures through axial gravitational perturbations including stellar stability and gravitational echo. We find that the effective potential in $f(R,T)$ gravity is wider and deeper than its counterpart in general relativity (GR). The quasinormal-mode (QNM) frequencies are calculated using the Bohr–Sommerfeld (BS) fitting and both the lower- and higher-order Wentzel–Kramers–Brillouin (WKB) approximations. For the ultracompact configurations considered, trapped modes can tunnel through the potential barrier, giving rise to gravitational-wave echoes. The imaginary parts of the QNM frequencies are found to be negative, $\mathrm{\Im }({\omega }_{n}M)<0$, indicating that the perturbations decay exponentially with time. Our results show that the echo train in GR contains more echoes than that in $f(R,T)$ gravity. Finally, assuming that the stellar object is a neutron star and adopting a stellar radius consistent with measurements by the Neutron Star Interior Composition Explorer (NICER), we find that the $f(R,T)$ gravity model yields a characteristic frequency consistent with the estimated frequency associated with GW170817 for $\beta =4.663\times {10}^{-1}$ and $\chi =0.96\pi$, whereas the corresponding GR model does not exhibit comparable agreement.