Post-Carrollian Mechanics, Ideal Gas, and Gravity

TL;DR

This work defines post-Carrollian mechanics by applying the ultra-relativistic limit to tachyon theory, yielding $E_{PC}=\frac{m c^{3}}{v}$ and $\mathbf{P}_{PC}=\frac{1}{2}\frac{m c^{3}}{v^{2}}\hat{\mathbf{v}}$, and derives the associated energy-momentum relation $|\mathbf{P}_{PC}|=\frac{(E_{PC})^{2}}{2 m c^{3}}$. It then builds a thermodynamic description for a post-Carrollian ideal gas, obtaining a nonstandard equation of state with $\gamma=\frac{7}{6}$ and $PV/U=\frac{1}{6}$, and demonstrates a Carroll–Schrödinger framework emerging from the same post-Carrollian energy-momentum structure. In gravity, the weak-field, static, ultra-relativistic limit of Einstein equations with tachyon dust yields a Poisson-like equation for the post-Carrollian potential, giving a Newtonian-like potential $\phi=-\frac{M G_N}{r}$ but with an outward gravitational field $\mathbf{g}_{PC}=\nabla\phi$, implying a negative mass charge for post-Carrollian matter and a possible repulsive interaction with Newtonian matter. Overall, the paper reveals a consistent post-Carrollian regime with distinct dynamical laws, symmetry algebras, and gravitational behavior that may have implications for dark sector phenomenology and the study of ultra-relativistic limits in gravitational theories.

Abstract

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.

Figures (5)

• Figure 1: Theories of motion based on their velocities in comparison to the speed of light (see Table \ref{['tab:example1']}). At $v=0$, the relations \ref{['teee']}, \ref{['tmm']} indicate that the particle possesses energy but zero momentum, characterizing a particle at rest. Conversely, at $v=\infty$, the relations \ref{['tee']}, \ref{['tm']} reveal that the particle carries momentum while having zero energy, corresponding to a magnetic Carroll particle.
• Figure 2: This figure illustrates that the velocity of a tachyon is bounded within the range $c < v < \infty$, whereas the velocity of a post-Carroll particle is unbounded, ranging from $0 < v \leqslant \infty$. At high velocities ($v \gg c$), the post-Carroll particle curve coincides with the tachyon curve.
• Figure 3: This figure illustrates that the Newtonian gravitational field $\mathbf{g}_{_{N}}$ generated by Newtonian matter $M_{_{N}}$ is radially inward, while the post-Carrollian gravitational field $\mathbf{g}_{_\mathrm{PC}}$ produced by post-Carrollian matter $M_{_\mathrm{PC}}$ is radially outward. A Newtonian particle of mass $m_{_{N}}$ (with $m_{_{N}}\ll M_{_{N}}$) in the gravitational field $\mathbf{g}_{_{N}}$ experiences an attractive force $\mathbf{F}\!_{_{N}}$, aligned with the gravitational field $\mathbf{g}_{_{N}}$. Similarly, a post-Carrollian particle of mass $m_{_\mathrm{PC}}$ (with $m_{_\mathrm{PC}}\ll M_{_\mathrm{PC}}$) in the gravitational field $\mathbf{g}_{_\mathrm{PC}}$ experiences an attractive force $\mathbf{F}\!_{_\mathrm{PC}}$, but anti-aligned with the gravitational field $\mathbf{g}_{_\mathrm{PC}}$.
• Figure 4: The gravitational force behavior of different types of particles.
• Figure 5: This figure shows that the velocity of a relativistic particle is bounded within the range $0 \leqslant v < c$, while the velocity of a Newtonian particle is unbounded, ranging from $0 \leqslant v < \infty$. At low velocities ($v\ll c$), the Newtonian particle curve coincides with the relativistic particle curve.