[Paper Review] Superluminal velocities and nonlocality in relativistic mechanics with scalar potential

Hrvoje Nikolić|arXiv (Cornell University)|Jun 10, 2010

Mechanical and Optical Resonators7 references3 citations

TL;DR

This paper develops a relativistic classical mechanics framework incorporating a scalar potential that dynamically modifies particle mass, allowing superluminal velocities when mass squared becomes negative and enabling nonlocal interactions in a Lorentz-covariant way. The formalism uses a reparameterization-invariant action with a scalar parameter analogous to Newtonian time, and it demonstrates that relativistic Bohmian mechanics for spin-0 particles is a special case of this generalized theory.

ABSTRACT

Even though the usual form of relativistic mechanics does not allow superluminal particle velocities and nonlocal interactions, these features are not forbidden by relativity itself. To understand this on a deeper level, we study a generalized form of relativistic mechanics in which the particle is influenced not only by the usual tensor (gravitational) and vector (electromagnetic) potentials, but also by the scalar potential. The scalar potential promotes the mass squared M^2 to a dynamical quantity. Negative values of M^2, which lead to superluminal velocities, are allowed. The generalization to the many-particle case allows a nonlocal scalar potential, which makes nonlocal interactions compatible with relativity. Particle trajectories are parameterized by a scalar parameter analogous to the Newton absolute time. An example in which all these general features are explicitly realized is provided by relativistic Bohmian mechanics.

Motivation & Objective

• To demonstrate that relativity does not inherently forbid superluminal particle velocities or nonlocal interactions.
• To develop a generalized relativistic mechanics formalism that includes scalar potentials, extending beyond standard tensor and vector potentials.
• To show that the scalar potential dynamically alters mass squared, allowing negative values that correspond to tachyonic (superluminal) motion.
• To establish a manifestly Lorentz-covariant description of nonlocal interactions using a scalar parameter analogous to Newtonian time.
• To show that relativistic Bohmian mechanics for spin-0 particles emerges as a special case of this generalized formalism.

Proposed method

• Formulates relativistic dynamics using an action invariant under general reparameterizations of the particle trajectory, introducing a scalar parameter s as the evolution parameter.
• Introduces a scalar potential that promotes the mass squared M² to a dynamical quantity, allowing M² < 0 and thus superluminal motion.
• Derives equations of motion from a Hamilton-Jacobi-like formalism, where particle trajectories are determined by the gradient of a relativistic action S.
• Defines a generalized proper time ds using a conformal metric derived from the scalar potential, ensuring ds² > 0 even for spacelike or lightlike trajectories.
• Extends the formalism to many-particle systems, allowing nonlocal scalar potentials that preserve Lorentz covariance.
• Demonstrates that the relativistic Bohmian mechanics equations for spin-0 particles satisfy the derived equations of motion, confirming consistency.

Experimental results

Research questions

• RQ1 Can superluminal particle velocities be consistently described within a relativistic framework without violating causality?
• RQ2 Is it possible to construct a Lorentz-covariant theory of nonlocal interactions using a scalar potential?
• RQ3 How does the inclusion of a scalar potential modify the mass and dynamics of relativistic particles?
• RQ4 What is the role of the scalar parameter s in parameterizing particle trajectories, and how does it relate to proper time?
• RQ5 Can relativistic Bohmian mechanics be derived as a special case of a more general relativistic classical mechanics with scalar potentials?

Key findings

• The scalar potential allows the mass squared M² to become negative, leading to superluminal particle velocities while preserving Lorentz covariance.
• The generalized proper time s is defined via a conformal metric, ensuring ds² > 0 even for spacelike or lightlike trajectories, generalizing the concept of proper time.
• In the many-particle case, the scalar potential can be nonlocal yet still compatible with relativistic covariance, enabling nonlocal interactions without violating relativity.
• The equations of motion for relativistic Bohmian mechanics of spin-0 particles are shown to be a special case of the proposed formalism, providing a deeper classical foundation.
• The scalar parameter s acts as a relativistic analog of Newtonian absolute time, parameterizing trajectories in a way that preserves manifest Lorentz invariance.
• For free particles with constant positive masses, the generalized proper time ds² reduces to a weighted average of individual proper time differentials, ds² = ∑ w_a dτ_a² with w_a = m_a / ∑ m_b.