  Observed supermassive black holes in the early universe have several proposed
formation channels, in part because most of these channels are difficult to
probe. One of the more promising channels, the direct collapse of a
supermassive star, has several possible probes including the explosion of a
helium-core supermassive star triggered by a general relativistic instability.
We develop a straightforward method for evaluating the general relativistic
radial instability without simplifying assumptions and apply it to population
III supermassive stars taken from a post Newtonian stellar evolution code. This
method is more accurate than previous determinations and it finds that the
instability occurs earlier in the evolutionary life of the star. Using the
results of the stability analysis, we perform 1D general relativistic
hydrodynamical simulations and we find two general relativistic instability
supernovae fueled by alpha capture reactions as well as several lower mass
pulsations, analogous to the puslational pair instability process. The mass
range for the events (2.6-3.0 $\times 10^4$ ${\rm M}_\odot$) is lower than had
been suggested by previous works (5.5 $\times 10^4$ ${\rm M}_\odot$) because
the instability occurs earlier in the star's evolution. The explosion may be
visible to, among others, JWST, while the discovery of the pulsations opens up
additional possibilities for observation.
