  The origin of high-redshift quasars and their supermassive black hole engines
is unclear. One promising solution is the collapse of a primordial supermassive
star. Observational confirmation of this scenario may be challenging, but a
general relativistic instability supernova provides one avenue for such.
Previous studies have found that a general relativistic instability supernova
has a potentially decades-long plateau phase visible to JWST at high redshift.
In this work, we examine stars with mass just below the general relativistic
instability supernova mass range. These stars pulsate, ejecting a portion of
their envelopes. They then contract quasi-statically back to an equilibrium
temperature, at which point they again become unstable and pulsate once more.
Because each pulse consumes a small amount of the available nuclear fuel, there
exists the possibility of multiple pulsations. We present simulations of the
contracting phase, the pulsation, and the light-curve phase. We find that the
lower mass pulsating models are even brighter than the higher mass supernovae
because the pulsations occur in the late helium burning phase when the stars
have extremely large radii. The fact that the pulsations are more luminous and
occur in a wider mass range than the supernovae bodes well for observation.
