The instabilities of the self-gravitating, relativistic, ideal gas to all temperature and density regimes are studied in the case of static, spherically symmetric spacetime. For an ideal gas, thermal energy is the only one that can halt gravitational collapse. However, since thermal energy gravitates as well, it can also cause gravitational collapse at high energies. Hence, we anticipate to find two instabilities for a bounded sphere that contains relativistic gas. One at low energies, where thermal energy becomes too weak to halt gravity, and another at high energies, where gravity becomes too strong to be halted by thermal pressure even if it is high, as well. The two energy limits correspond also to radius limits. So that, stable static configurations exist only in between two marginal radii for any fixed energy with negative gravothermal (thermal plus gravitational) energy. For positive gravothermal energy, there is only one maximum energy and one minimum radius. All these turning points of stability are found to depend on the total number of particles, i.e. on the rest mass. Relativistic, ultimate limits of rest mass, total mass and radius are found. Regarding the rest mass, stable equilibria exist only for: Mrest < 0.35 Ms, where Ms is the Schwarzschild mass. For total mass M and radius R, the relativistic, ultimate limit for stability is 2GM/Rc2 < 0.44.
Kinematic Hilbert space in QECT is constructed via von Neumann infinite dimensional tensor product of point Hilbert spaces. The Hilbert space of this kind is too big therefore we need to pick up its certain subspace which mimics underlying manifold structure of spatial section in better way. This enables us to define measures with values in operator algebra (bounded and unbounded cases are considered) which are absolutely continuous with respect to Lebesgue - coordinate measure over spatial section. This general construction helps us to define Volume and Area operators in QECT.