[Paper Review] "Nonrelativistic" kinematics: Particles or waves?

J. M. Houlrik, Germain Rousseaux|arXiv (Cornell University)|May 11, 2010

Relativity and Gravitational Theory18 references8 citations

TL;DR

This paper reveals that special relativity's kinematics extends beyond low-velocity Galilean physics to include a dual Carrollian superluminal limit, where waves exhibit particle-like behavior. By exploiting spacetime exchange symmetry in the SI definition of length via $c$, it shows that wave-particle duality—exemplified by the Landé paradox—is naturally resolved through dual Galilean and Carrollian kinematics, with Maxwell’s equations covariant under both symmetries.

ABSTRACT

The kinematics of particles refer to events and tangent vectors, while that of waves refer to dual gradient planes. Special relativity [1-3] applies to both objects alike. Here we show that spacetime exchange symmetry [7] implicit in the SIdefinition of length based on the universal constant c has profound consequences at low velocities. Galilean physics, exact in the limit c o \infty, is mirrored by a dual so-called Carrollian superluminal kinematics [4-6] exact in the limit c o 0. Several new results follow. The Galilean limit explains mass conservation in Newtonian mechanics, while the dual limit is a kinematical prerequisite for wavelike tachyonic motion [8, 9]. As an example, the Landé paradox [19, 20] of waveparticle duality has a natural resolution within special relativity in terms of superluminal, particlelike waves. It is emphasized that internal particle energy mc^2 can not be ignored, while kinetic energy leads to an extended Galilei group. We also demonstrate that Maxwell's equations have magnetic and electric limits covariant under Galilean and Carrollian symmetry.

Motivation & Objective

• To explore the full kinematic range of special relativity beyond the low-velocity limit, revealing a dual symmetry between Galilean and Carrollian physics.
• To resolve the Landé paradox in wave-particle duality by showing that superluminal, particlelike waves emerge naturally in the Carrollian limit.
• To demonstrate that Maxwell’s equations are covariant under both Galilean and Carrollian symmetries, with electric and magnetic limits corresponding to these dual limits.
• To establish that internal particle energy $mc^2$ must be retained in kinematic descriptions, even in nonrelativistic regimes.
• To clarify the role of spacetime exchange symmetry in the SI definition of length, showing its profound consequences at low and high velocities.

Proposed method

• Utilizes spacetime exchange symmetry $r_{lat} o ct, t o r_{lat}/c$ to derive dual transformations from the Galilean boost $\mathbf{r}' = \mathbf{r} - \mathbf{v}_0 t, t' = t$ to the Carroll transformation $\mathbf{r}' = \mathbf{r}, t' = t - \mathbf{v}_0 \cdot \mathbf{r}/c^2$.
• Applies first-order matrix expansions of the Lorentz transformation as $I - \beta L$, with $L = G + C$, where $G$ and $C = G^T$ are generators for Galilean and Carrollian boosts.
• Analyzes four-vectors for particles ($\gamma(\mathbf{v}, c)$) and waves ($({\bf k}, \omega/c)$), showing their transformation under $G$ and $C$ subgroups.
• Derives velocity transformations for $v \ll c$ (Galilean) and $v \gg c$ (Carrollian), showing $\gamma \to 0$ as $\beta \to \infty$ in the dual limit.
• Applies duality to Maxwell’s equations by transforming $\mathbf{D} \leftrightarrow \mathbf{B}, \mathbf{H} \leftrightarrow -\mathbf{E}$, revealing Galilean and Carrollian limits for electric and magnetic approximations.
• Uses the Lorenz gauge and four-potential $q(\mathbf{A}, \phi/c)$ to derive the Galilean Doppler shift $\omega = \mathbf{k} \cdot \mathbf{v}$, linking wave and particle kinematics.

Experimental results

Research questions

• RQ1 How does spacetime exchange symmetry in the SI definition of length extend the kinematic framework of special relativity beyond the low-velocity limit?
• RQ2 Can the Landé paradox of wave-particle duality be resolved within a unified kinematic framework of special relativity?
• RQ3 What are the kinematic consequences of the dual limits $c \to \infty$ (Galilean) and $c \to 0$ (Carrollian), and how do they relate to particle and wave motion?
• RQ4 How are Maxwell’s equations covariant under both Galilean and Carrollian symmetries, and what are the physical interpretations of their electric and magnetic limits?
• RQ5 What is the role of internal energy $mc^2$ in nonrelativistic kinematics, and how does it affect the extended Galilei group?

Key findings

• The Galilean limit ($c \to \infty$) explains mass conservation in Newtonian mechanics as a kinematic consequence of the $c \to \infty$ limit.
• The dual Carrollian limit ($c \to 0$) is a kinematic prerequisite for superluminal, particlelike wave motion, with $\gamma \sim 1/\beta$ vanishing as $\beta \to \infty$.
• The Landé paradox is resolved by showing that wave-particle duality arises naturally from dual kinematics: waves behave as particles in the Carrollian limit.
• Maxwell’s equations are covariant under both Galilean and Carrollian symmetries, with the electric limit ($cD \to 0$) corresponding to Galilean invariance and the magnetic limit ($cB \to 0$) to Carrollian invariance.
• The first-order Lorentz transformation is self-dual under spacetime exchange, and the Galilean and Carrollian transformations are exact duals of each other via $v_0 V_0 = c^2$.
• Sources in the equations transform as $\rho_s' = \rho_s$ (Galilean, timelike) or $\rho_s' = \rho_s - \mathbf{v}_0 \cdot \mathbf{j}_s / c^2$ (Carrollian, spacelike), showing dual continuity equations.