  The fundamental metallicity relation (FMR) is a postulated correlation
between galaxy stellar mass, star formation rate (SFR), and gas-phase
metallicity. At its core, this relation posits that offsets from the
mass-metallicity relation (MZR) at a fixed stellar mass are correlated with
galactic SFR. In this Letter, we quantify the timescale with which galactic
SFRs and metallicities evolve using hydrodynamical simulations. We find that
Illustris and IllustrisTNG predict that galaxy offsets from the star formation
main sequence and MZR evolve over similar timescales, are often anti-correlated
in their evolution, evolve with the halo dynamical time, and produce a
pronounced FMR. In fact, for a FMR to exist, the metallicity and SFR must
evolve in an anti-correlated sense which requires that they evolve with similar
time variability. In contrast to Illustris and IllustrisTNG, we speculate that
the SFR and metallicity evolution tracks may become decoupled in galaxy
formation models dominated by globally-bursty SFR histories, which could weaken
the FMR residual correlation strength. This opens the possibility of
discriminating between bursty and non-bursty feedback models based on the
strength and persistence of the FMR -- especially at high redshift.
