Relativistic gas in a Schwarzschild metric

A relativistic gas in a Schwarzschild metric is studied within the framework of a relativistic Boltzmann equation in the presence of gravitational fields, where Marle's model for the collision operator of the Boltzmann equation is employed. The transport coefficients of bulk and shear viscosities and thermal conductivity are determined from the Chapman-Enskog method. It is shown that the transport coefficients depend on the gravitational potential. Expressions for the transport coefficients in the presence of weak gravitational fields in the non-relativistic (low temperatures) and ultra-relativistic (high temperatures) limiting cases are given. Apart from the temperature gradient the heat flux has two relativistic terms. The first one, proposed by Eckart, is due to the inertia of energy and represents an isothermal heat flux when matter is accelerated. The other, suggested by Tolman, is proportional to the gravitational potential gradient and indicates that -- in the absence of an acceleration field -- a state of equilibrium of a relativistic gas in a gravitational field can be attained only if the temperature gradient is counterbalanced by a gravitational potential gradient.
Comment: Sign error in the potential in eq. (25) has propagated to eqs. (27)-(29), (31) and (35)-(42). As a consequence the transport coefficients become smaller in the presence of gravitational field, instead of larger as mentioned in the previous version