The Boltzmann equation in general relativistic rotating systems: Cooling of rotating neutron stars

We derive the general relativistic Boltzmann equation for massless particles in a stationary axisymmetric metric. The energy and angles of the particles are written in the comoving frame in order to use the local expressions for the collision terms. The zero and first-order moment equations in both spectral integrated and nonintegrated form are obtained. From these we derive the thermal equilibrium condition and the diffusion approximation in a relativistic rotating axisymmetric star. We check the equations by recovering the spherically symmetric case and the heat conduction approximation (equilibrium diffusion limit) for axisymmetry. The effects of rotation on the cooling of rotating neutron stars are studied using Hartle's formalism for the rotational deformation together with the cooling equation we derive.