Rigorous results are proved which are directly applicable to the problem of writing the thermodynamic properties of an ideal relativistic Bose gas in terms of modified Bessel functions of the second kind. The results are particularly important when the Bose gas is at, or below, its condensation temperature.
The condensation temperature and the specific-heat anomaly of an ideal relativistic Bose gas are discussed. For a high concentration of bosons with small rest mass there is a marked departure from the normal nonrelativistic theory. The heat capacity Cv remains continuous at the condensation temperature unless the rest mass vanishes.
The quantum statistical mechanics of an ideal relativistic Bose gas of massive particles is discussed. The thermodynamic functions of the system are obtained from the exact expression for the logarithm of the grand partition function. The correct procedure for carrying out the non-relativistic and ultra-relativistic limits is presented. The limitations and errors of previous calculations which relied on the binomial expansion of the density of states are pointed out. It is shown that the behaviour of the system in the limit of zero temperature is dominated by the non-relativistic effects.
For the relativistic Bose gas in the gravitational field,the Bose-Einstein condensation(BEC) and the thermodynamic properties are investigated using the local density approximation(LDA).The thermodynamic quantities are calculated in the non-relativistic(NR) and ultra-relativistic(UR) limits,including the transition temperature,the ground state fraction,the energy and the heat capacity of the system.We also discussed the continuity of the heat capacity at the critical temperature.It is found that the effect of the relativistic results in the increase of the gap of the heat capacity.