Subluminal Moving Potentials

• Subluminal moving potentials are localized field configurations propagating below the speed of light, characterized by precise space–time coupling derived from analytic solutions like Bessel beams.
• They utilize both frequency and longitudinal wavenumber integration methods to construct non-diffracting, causality-preserving pulses applicable in optics, acoustics, and quantum systems.
• Experimental and theoretical analyses confirm that energy transport adheres to universal velocity bounds, enabling advanced manipulations in signal processing and material interactions without violating relativity.

Subluminal moving potentials refer to temporally and spatially localized field configurations, interfaces, or modulations that propagate at speeds strictly less than the invariant speed of light in vacuum, $c$. These phenomena arise naturally in wave-theoretic, quantum, classical, and relativistic contexts, and their paper encompasses exact analytic field solutions, spacetime measures, causal structure, physical realizability, and practical applications from optics to astrophysics. Their properties are intimately connected to Lorentz covariance, the spectral structure of the underlying fields, energy and information transport limits, and the design of fields or devices that manipulate electromagnetic, acoustic, and matter waves without violating causality.

1. Exact Localized Solutions and Construction Methodologies

Subluminal moving potentials are exemplified by exact analytic solutions to the scalar wave equation and Maxwell’s equations, notably constructed via superpositions of Bessel beams. For axially symmetric fields, the central form is given by

$$\Psi(\rho, z, t) = e^{-i b z/v} \int_{\omega_{-}}^{\omega_{+}} S(\omega) J_0(\rho k_{\rho}) e^{i (\omega/v) \zeta}\, d\omega,$$

where $\omega = v k_z + b$ ensures strict space–time coupling, $J_0$ is the Bessel function of order zero, and $k_{\rho}^2 = (\omega^2/c^2 - k_z^2)$. Integration can be carried out in the angular frequency or longitudinal wavenumber domains. For certain choices of the spectral function $S(\cdot)$, these integrals yield the MacKinnon solution—ball-like, non-diffracting, and free from non-causal components. This methodology allows for arbitrary spectral tailoring within the bounds $\omega_{\pm}$ determined by the velocity and the fixed $b$ parameter.

Two main strategies are used:

• Frequency integration (Method 1): The spectrum is Fourier-expanded within the finite interval, yielding a discrete mode expansion with explicit spatial-temporal profiles.
• Longitudinal wavenumber integration (Method 2): The approach is particularly robust for the $v \to 0$ limit, enabling construction of "Frozen Waves"—static, longitudinally patterned field distributions.

Higher-order Bessel beams generalize the solutions to non-axially symmetric cases, introducing angular momentum and azimuthal structure, with field representations involving $J_\nu(k_{\rho} \rho)e^{i\nu\phi}$.

2. Causal Structure, Spacetime Measures, and Transition Frameworks

Classical Lorentz transformations predict a divergence in the Lorentz factor $\gamma = (1 - v^2/c^2)^{-1/2}$ as $v \to c$. Recent generalizations resolve these divergences by introducing a small regularizing parameter $\epsilon^2$, so that, for subluminal velocities,

$\gamma_1 = (1 + \epsilon^2 - (v/c)^2)^{-1/2},$

which remains finite even at $v = c$ (Calvo-Mozo, 2013, Calvo-Mozo, 2014). Causal structure is recast by defining regimes indexed by $k$: for each, the top velocity $c_k$ defines $k$-null cones and $k$-timelike paths, preserving causality within each regime. In the subluminal domain ($k=1$), all dynamics are confined within the standard Minkowski light cone, with transitions between regimes (to superluminal, $k>1$) postulated to occur discretely in $v^2/c^2$ steps of order $\epsilon^2$.

The mathematical framework for generalized space–time measures is: $$x = \frac{x' + v t'}{\sqrt{1 + \epsilon^2 - (v/c)^2}}, \quad ct = \frac{ct' + (v/c) x'}{\sqrt{1 + \epsilon^2 - (v/c)^2}}$$ with modified transverse components, and each motion regime endowed with its own invariant cone structure to guarantee consistent causality.

3. Energy, Group Velocity, and Local Transport for Subluminal Pulses

For subluminal propagation-invariant electromagnetic fields, the local instantaneous energy transport velocity (from the Poynting vector $S$ and energy density $w$) along the major axis satisfies a universal and bounded relation (Saari et al., 2019): $$V = \frac{2v}{1 + (v/c)^2}, \qquad \left( \frac{V}{c} = \frac{2\beta}{1 + \beta^2} \right),$$ with $\beta = v/c$. This formula holds for both scalar and vectorial configurations, including realistic, finite-energy nearly propagation-invariant fields (e.g., pulsed Bessel beams, finite-energy Mackinnon solutions). It is significant that for $v < c$, $V > v$, but always $V \leq c$—energy can locally "leak" ahead of the group envelope, yet never violates causality. In the $v \to 0$ limit, $V \to 0$ (energy is stationary in stationary potentials); as $v \to c$, $V \to c$ (energy and envelope co-propagate at $c$).

Notably, experimental work with tightly focused or twisted light pulses in vacuum and matter has confirmed these predictions, quantifying subluminal group velocities varying by up to $0.1\%$ relative to $c$ in structured beam configurations (Bouchard et al., 2015). These results are accounted for by analyzing wavevector projections and field geometry—structured beams possess a significant transverse component, reducing the projected group velocity along the propagation axis (Tamburini et al., 2017, Bliokh, 2023).

4. Special Relativity, Lorentz Transformations, and the Structure of Solutions

Lorentz covariance is intrinsic: a subluminal localized potential can always be mapped to a stationary or monochromatic solution in a different inertial frame via a Lorentz boost with velocity $u = v$. For instance, the MacKinnon wavepacket is mapped under a double Lorentz transformation to its time-reversed counterpart, reflecting the spectral and temporal symmetry (Saari et al., 2020). For beams, a local null Doppler shift coincides with velocity composition dictated by

$\beta_e = \frac{2\beta}{1 + \beta^2}$

mirroring the speed transformation for successive boosts.

Special relativity thus demands that subluminal, luminal, and superluminal potentials are members of a broader family, each interrelated via relativistic transformations. The ball-like nature of subluminal pulses contrasts with the X-shaped (conical) patterns of superluminal solutions; these distinct morphologies are visual manifestations of their respective causal and velocity domains.

5. Physical Realizability, Stability, and Applications

Subluminal moving potentials have a rich range of physical realizations:

• Electromagnetic and optical pulses: Non-diffracting, distortion-free field propagation in homogeneous media enables applications in laser physics, communications, material processing, and optical trapping (0709.2372).
• Acoustic and matter waves: Acoustic bullets and stationary wavefields are constructed via analogous techniques, facilitating subsonic nondiffracting sound propagation and precise energy delivery in media.
• Quantum and relativistic systems: The modeling of subluminal particles described by space-like (tachyonic) equations shows, via properly defined spectral decompositions, that physical group velocities always remain subluminal—even for fields with imaginary mass parameters (Nanni, 2021).
• Moving refractive index interfaces: For subluminal (B < 1) moving mirrors or index modulations, incident waves experience frequency upshift and field amplitude enhancement (double Doppler effect). The reflection coefficient remains finite, and the interaction leads to compressed, upconverted field components, enabling novel sources of high-energy photons (Esirkepov et al., 2023).

The stability of these solutions is preserved under linear perturbations, and—when embedded in numerical codes (e.g., RMHD)—reconstruction in appropriately constrained variables (Lorentz-weighted velocities) ensures that all computed velocities remain subluminal everywhere in the computational domain (Balsara et al., 2016).

6. Broader Impact and Practical Ramifications

The theoretical flexibility of subluminal moving potentials—allowing arbitrary shaping via spectral engineering, coupled with rigorous maintenance of causality—enables their deployment in diverse fields, including:

• Optical communications (managing slow light and group velocity dispersion)
• Quantum information (OAM-based time buffering, photon arrival time control (Tamburini et al., 2017))
• Ultrafast electron microscopy (velocity-matched subluminal THz waves for attosecond pulse shaping (Volkov et al., 2022))
• Cosmology and modified gravity (constructing completely subluminal cosmological bouncing models within beyond-Horndeski frameworks, avoiding superluminal perturbative modes under broad circumstances (Mironov et al., 2019))
• Fundamental considerations in the causal structure of spacetime, potentially influencing approaches to quantum gravity and foundational physical theories (Calvo-Mozo, 2014)

Subluminal moving potentials thus serve as both a unifying analytic and conceptual framework, and a practical engineering tool for controlling wave propagation, energy transport, and signal information within strict causal and relativistic bounds. Their paper continues to inform progress in fields requiring precise control and understanding of moving, temporally structured potentials.