A new framework for the relativistic transformations of thermodynamic variables using momentum as the thermodynamic potential

Abstract

A generalised relativistic transformation for thermodynamic variables is derived in this study using the basic energy–momentum relationship of special relativity. We posit that momentum undergoes changes akin to a time coordinate and treat it as a thermodynamic potential analogous to energy potential. Additionally, we presume that momentum transforms similarly to a time coordinate. We analyse two mutually exclusive conditions to simplify generalised transformations. In one condition, the transformations are as follows: volume \( V = \gamma V' \), internal energy \( U = \gamma U' \), temperature \( T = \gamma T' \) and pressure \( P = P' \), where \( \gamma \) represents the Lorentz factor. The primed variables correspond to the moving frame, while the unprimed variables correspond to the stationary frame. The other condition yields \( V = V'/\gamma \), \( U = U'/\gamma \), \( T = T'/\gamma \), \( P = P' \). Since the first law of thermodynamics is an energy conservation statement and Maxwell and other thermodynamic relationships are mathematical constructs based on the first law, it is expected that such relationships should remain invariant in all frames for relativistic thermodynamic transformations. We demonstrate that the ideal gas equation, Maxwell relationships and other thermodynamic relationships (for example, \( (\partial U/\partial V)_T = -P + T(\partial P/\partial T)_V \)) remain invariant under these two sets of transformations. Furthermore, we show that, although the ideal gas equation and Maxwell relationships remain invariant for many transformations reported earlier, \( (\partial U/\partial V)_T = -P + T(\partial P/\partial T)_V \) remains invariant only for the Sutcliffe transformation (\( V = V'/\gamma \), \( U = \gamma U' \), \( T = \gamma T' \), \( P = \gamma ^2 P' \)). We establish that when \( U \), heat \( Q \) and work \( W \) transform similarly, all thermodynamic relationships remain invariant, and such a formalism is mathematically consistent.