Braking index of the frequently glitching PSR J0537$-$6910

PSR J0537$-$6910 is the most frequently glitching pulsar, exhibiting large glitches at a rate of roughly thrice per year. It displays a negative long-term effective braking index $n' = -1.234(9)$ and roughly constant positive frequency second derivatives extending all the way from one glitch to the next. We use the published RXTE and NICER timing data (i) to explore whether `persistent shifts' (non-relaxing parts of the glitch $Δ\dotν$ in the spin-down rate, like those observed in the Crab pulsar) can explain the negative effective braking index; (ii) to ascribe part of the inter-glitch relaxation at constant frequency second derivative $\ddotν$ to internal torques due to the coupling between the observed surface and the interior of the neutron star like deduced for the Vela pulsar; (iii) to demonstrate, as a proof of concept, that after taking account of the persistent shifts and internal torques, the true braking index $n$ associated with the pulsar braking torque is $n\sim 3 $, and (iv) to explore various empirical or model dependent correlations between glitch parameters and the time to the next glitch. The average persistent shift derived for PSR J0537$-$6910 is in agreement with the persistent shift values for the Crab pulsar. By applying the crustquake model, we infer a broken plate size of $D=80$ m, similar to the value obtained for the minimum glitch size observed in the Crab pulsar. The inter-glitch $\ddotν$ values inferred for the internal torques are commensurate with those in the Vela pulsar. Delineating the effects of persistent shifts and inter-glitch relaxation due to internal torques as observed in the Crab, Vela and other pulsars, we obtained a true braking index value $n=2.75(47)$ for PSR J0537$-$6910, similar to braking indices observed from other young pulsars. PSR J0537$-$6910 is not a likely source of gravitational waves.