  Supermassive black holes might form by direct collapse, with a supermassive
star (SMS) as progenitor. In this scenario, the SMS accretes at > 0.1 Msun/yr
until it collapses into a massive black hole seed due to the
general-relativistic (GR) instability. However, the exact mass at which the
collapse occurs is not known, as existing numerical simulations give divergent
results. Here, this problem is addressed analytically, which allows for ab
initio, reliable determination of the onset point of the GR instability, for
given hydrostatic structures. We apply the relativistic equation of radial
pulsations in its general form to the hydrostatic GENEC models already
published. We show that the mass of spherical SMSs forming in atomically cooled
haloes cannot exceed 500 000 Msun, in contrast to previous claims. On the other
hand, masses in excess of this limit, up to 10^6 Msun, could be reached in
alternative versions of direct collapse. Our method can be used to test the
consistency of GR hydrodynamical stellar evolution codes.
