  This paper presents general relativistic numerical simulations of uniformly
rotating polytropes. Equations are developed using MSQI coordinates, but taking
a logarithm of the radial coordinate. The result is relatively simple
elliptical differential equations. Due to the logarithmic scale, we can resolve
solutions with near-singular mass distributions near their center, while the
solution domain extends many orders of magnitude larger than the radius of the
distribution (to connect with flat space-time). Rotating solutions are found
with very high central energy densities for a range of adiabatic exponents.
Analytically, assuming the pressure is proportional to the energy density
(which is true for polytropes in the limit of large energy density), we
determine the small radius behavior of the metric potentials and energy
density. This small radius behavior agrees well with the small radius behavior
of large central density numerical results, lending confidence to our numerical
approach. We compare results with rotating solutions available in the
literature, which show good agreement. We study the stability of spherical
solutions: instability sets in at the first maximum in mass versus central
energy density; this is also consistent with results in the literature, and
further lends confidence to the numerical approach.
