Does the inertia of a body depend on its heat content?

Einstein's assertion that, based on the relativity principle, inertial mass is to be associated with all forms of energy, was contested by Planck, who showed that a transfer of heat is to be associated with a transfer of inertial mass. The equivalence between mass and energy is interpreted in terms of the equivalence of the rate of heating of a body and the rate of increase of its inertial mass. In an adiabatic process, where the proper mass remains constant, it is the heat content, and not the energy, which is conserved because the pressure, and not the volume, is Lorentz-invariant. There are two categories of relativistic quantities: inertial and thermodynamic, which are transformed into one another by the work necessary to keep the body in motion at a constant velocity. In a non-adiabatic process, the rate of heating is Lorentz-invariant, and it must always be greater than the power that it generates. Therefore, the amount of heat developed in the rest frame has a corresponding inertial mass.     

Thermodynamics of an ideal generalized gas: II

The property that power means are monotonically increasing functions of their order is shown to be the basis of the second laws not only for processes involving heat conduction, but also for processes involving deformations. This generalizes earlier work involving only pure heat conduction and underlines the incomparability of the internal energy and adiabatic potentials when expressed as powers of the adiabatic variable. In an L-potential equilibration, the final state will be one of maximum entropy, whereas in an entropy equilibration, the final state will be one of minimum L. Unlike classical equilibrium thermodynamic phase space, which lacks an intrinsic metric structure insofar as distances and other geometrical concepts do not have an intrinsic thermodynamic significance in such spaces, a metric space can be constructed for the power means: the distance between means of different order is related to the Carnot efficiency. In the ideal classical gas limit, the average change in the entropy is shown to be proportional to the difference between the Shannon and Rényi entropies for nonextensive systems that are multifractal in nature. The L potential, like the internal energy, is a Schur convex function of the empirical temperature, which satisfies Jensen's inequality, and serves as a measure of the tendency to uniformity in processes involving pure thermal conduction.     

Thermodynamics of an ideal generalized gas: I. Thermodynamic laws

The equations of state for an ideal relativistic, or generalized, gas, like an ideal quantum gas, are expressed in terms of power laws of the temperature. In contrast to an ideal classical gas, the internal energy is a function of volume at constant temperature, implying that the ideal generalized gas will show either attractive or repulsive interactions. This is a necessary condition in order that the third law be obeyed and for matter to have an electromagnetic origin. The transition from an ideal generalized to a classical gas occurs when the two independent solutions of the subsidiary equation to Lagrange's equation coalesce. The equation of state relating the pressure to the internal energy encompasses the full range of cosmological scenarios, from the radiation to the matter dominated universes and finally to the vacuum energy, enabling the coefficient of proportionality, analogous to the Grüeisen ratio, to be interpreted in terms of the degrees of freedom related to the temperature exponents of the internal energy and the absolute temperature expressed in terms of a power of the empirical temperature. The limit where these exponents merge is shown to be the ideal classical gas limit. A corollary to Carnot's theorem is proved, asserting that the ratio of the work done over a cycle to the heat absorbed to increase the temperature at constant volume is the same for all bodies at the same volume. As power means, the energy and entropy are incomparable, and a new adiabatic potential is introduced by showing that the volume raised to a characteristic exponent is also the integrating factor for the quantity of heat so that the second law can be based on the property that power means are monotonically increasing functions of their order. The vanishing of the chemical potential in extensive systems implies that energy cannot be transported without matter and is equivalent to the condition that Clapeyron's equation be satisfied.