  We launch a fully relativistic study of the formation of supermassive black
holes via the collapse of supermassive stars. Here we initiate our
investigation by analyzing the secular evolution of supermassive stars up to
the onset of dynamical instability and collapse. We focus on the effects of
rotation, assumed uniform, and general relativity. We identify the critical
configuration at which radial instability sets in and determine its structure
in detail. We show that the key nondimensional ratios R/M, T/|W| and J/M^2 (T
is the rotational kinetic energy and W is the gravitational potential energy)
for this critical configuration are universal numbers, independent of the mass,
spin, radius or history of the star. We compare results from an approximate,
analytic treatment with a fully relativistic, numerical calculation and find
good agreement. We solve analytically for the time evolution of these
parameters up to the onset of instability. Cooling by photon radiation drives
the evolution, which is accompanied by mass, angular momentum and entropy loss.
The critical configuration serves as initial data for a future relativistic,
hydrodynamical, 3D simulation of the collapse of an unstable supermassive star.
Since this implosion starts from a universal critical configuration, the
collapse is also uniquely determined and should produce a universal
gravitational waveform. In this paper we briefly speculate on the possible
outcome of this collapse and asses to what extent it offers a promising route
to forming a supermassive black hole.
