Virial theorem for rigidly rotating matter

The scaling property of the thermodynamic free energy ($\Phi$) of a system at global equilibrium has been examined using a real-time method known as the virial theorem. We demonstrate these scaling properties through a derived relation based on the general structure of equal-time commutators among Poincare charges and their densities. This relation is applicable to any renormalizable fields with spin $\leq 1$, excluding gauge fields. In this particular study, we investigate a rigidly rotating solution ($\Omega = \text{const}$) at global equilibrium for massless fermionic matter. It has been shown that the applicability of a hydrodynamic description requires a hierarchy $\Omega R \gg \Omega \beta_{0}$, where $R$ is the radius of the cylindrical-shaped rotating matter and $\beta_{0} = 1/T_{0}$ is the inverse temperature on the rotation axis. Consequently, the thermodynamic free energy $\Phi$ depends on the angular velocity through the product $\Omega R$, a dependency that extends to other thermodynamic variables as well. These findings are consistent with recent lattice QCD simulation results. Furthermore, we compute the moment of inertia for massless fermions and estimate the light quark contribution to the total moment of inertia of the Quark-Gluon Plasma (QGP) produced in heavy-ion collisions.