Post-Carrollian Mechanics, Ideal Gas, and Gravity

Energy and momentum in Newtonian mechanics have the familiar relations, ($\mathrm{E}=mv^2/2$) and ($\mathbf{P}=m\mathbf{v}$), derived from the non-relativistic limit of special relativity. In this study, we find the corresponding relations to formulate the so-called ``post-Carrollian mechanics'' by applying the ultra-relativistic limit to tachyon theory, resulting in ($\mathrm{E}={m\,c^{\,3}}/{v}$) and ($\mathbf{P}=\mathbf{\hat{v}}\,{m\,c^{\,3}}/{2\,v^{\,2}}$). Using these, we determine the energy-momentum relation and investigate the thermodynamics of an ideal gas composed of post-Carroll particles. Moreover, by applying the ultra-relativistic limit to Einstein's equations coupled to tachyon dust, we find the post-Carrollian gravitational potential. Finally, utilizing the geodesic equation, we determine the post-Carrollian gravitational field, which unlike the Newtonian case is found to be radially outward.