⚛️ general relativity

Extremal curves of Perlick's proper time in Weyl geometry

This paper investigates the extremization of Perlick's proper time in non-integrable Weyl geometry, deriving generalized Euler-Lagrange equations that reveal a nonlocal acceleration dependent on remaining proper time and demonstrating how local Lorentz-force dynamics emerge in a specific double-scaling limit where Weyl non-integrability vanishes.

Original authors: E. Rodrigues, F. Dahia, I. P. Lobo, C. Romero

Published 2026-09-17

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CC BY 4.0

Original authors: E. Rodrigues, F. Dahia, I. P. Lobo, C. Romero

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the grand architecture of the universe, space and time are not merely a static stage where events unfold; they are a dynamic fabric that can stretch, curve, and twist. For over a century, physicists have relied on a specific mathematical framework called pseudo-Riemannian geometry to describe this fabric, a system where the distance between two points is fixed and absolute, regardless of the path taken to get there. However, there exists a more flexible, older idea known as Weyl geometry. In this version, the very length of a ruler can change as it moves through space and time, depending on where it has been and the route it took. This path-dependence introduces a new kind of curvature, distinct from the bending of space itself, which measures how much the scale of things shifts during transport. While this concept was once proposed as a way to unify gravity and electromagnetism, it has largely remained a mathematical curiosity. The question that has long lingered is how a particle would actually move in such a shifting environment. If the rules of length are fluid, how does a clock tick, and what path does a free-falling object follow?

A team of researchers from Brazil has recently tackled this question by examining the motion of particles in a non-integrable Weyl geometry, a setting where the changing of lengths cannot be simplified into a single, smooth function. They focused on a specific definition of time proposed by physicist Volker Perlick, which offers a way to measure time that remains consistent even as the geometry of space stretches and shrinks. Unlike the standard way of measuring time, which simply adds up the distance traveled, Perlick's method accumulates the history of how the geometry has changed along the particle's entire path. The researchers set out to find the specific curves that a particle would take to maximize this accumulated time, essentially asking: "What is the most efficient path for a clock in a universe where the rules of size are constantly shifting?"

The results of their investigation reveal a surprising departure from established expectations. In standard physics, the path that maximizes time is a geodesic, a straight line in curved space that follows the natural curvature of the geometry. The researchers found that in this flexible Weyl universe, the path that maximizes Perlick's time does not generally match the standard paths predicted by the geometry's connection. Instead, the particle's motion is governed by a new equation that includes an extra force-like term. This term depends on the "length curvature," a measure of how much the geometry fails to be consistent, and it pushes the particle away from the standard path. The difference between the two paths is not a minor correction; it is a fundamental consequence of the geometry's non-integrability, meaning the two types of motion only coincide when the geometry is perfectly consistent and the length changes can be undone.

Perhaps the most striking feature of this new equation is its nonlocal nature. In ordinary physics, the acceleration of an object at any given moment depends only on the forces acting on it right then and there. In this new framework, the acceleration at a specific point depends on the entire remaining journey the particle has yet to take. The equation requires knowledge of the final destination of the particle's path to determine how it moves right now. This creates a "terminal" character to the motion, where the future endpoint of the trajectory influences the present behavior. The researchers showed that this dependence is not an artifact of how they chose to measure time, but a genuine property of the variational principle itself. The particle's path is determined by a boundary-value problem, where the start and end points are fixed, and the curve in between is shaped by the need to optimize the total time, including the influence of the destination.

To understand how this abstract behavior plays out in a real scenario, the team modeled a simple universe with a constant, weak field of this shifting geometry. They compared three different types of motion: the straight path of a particle in empty space, the path predicted by the standard Weyl geometry, and the path that maximizes Perlick's time. They found that while the standard Weyl path bends in one direction, the Perlick path initially bends in the opposite direction, driven by the influence of the distant endpoint. The further away the destination is, the stronger this initial deviation becomes. This counterintuitive result highlights that simply moving the destination point further away does not make the effect disappear; instead, it amplifies the nonlocal influence, causing the particle to deviate even more from the standard path.

However, the researchers discovered a way to recover familiar physics from this complex behavior through a specific mathematical limit. If they simultaneously weakened the shifting geometry field and moved the destination point infinitely far away, while keeping the product of these two factors constant, the strange nonlocal effects began to fade. In this double-scaling limit, the complex equation simplified dramatically. The nonlocal dependence on the future endpoint vanished, and the remaining force term transformed into the familiar Lorentz force, which describes how charged particles move in electromagnetic fields. This suggests that the electromagnetic force, which governs the motion of electrons and protons, could be viewed as a limiting case of this more fundamental, nonlocal geometry. The finite strength of the electromagnetic interaction emerges from the balance between a vanishingly weak geometric field and an infinitely distant boundary.

The study concludes that the quest to understand motion in a Weyl universe leads to a new kind of geometry where the path of a particle is not just a reaction to local conditions but a solution to a global optimization problem. While the equations governing this motion are nonlocal and depend on the future, they can be mathematically rewritten to look like local equations if one introduces an auxiliary variable that carries the memory of the endpoint. This work does not claim that our universe is definitely a Weyl geometry, but it demonstrates that the familiar laws of electromagnetism can emerge naturally from the extremization of a specific type of time in a non-integrable geometric setting. It bridges the gap between abstract geometric principles and the concrete forces observed in nature, showing that the local dynamics we experience might be a shadow of a deeper, nonlocal structure where the past and future are inextricably linked in the definition of a particle's path.

Technical Summary: Extremal Curves of Perlick's Proper Time in Weyl Geometry

Problem Statement
In Weyl geometry, the comparison of lengths at different spacetime points is path-dependent, characterized by a metric tensor $g_{\mu\nu}$ and a Weyl 1-form $\sigma_\mu$ . While the autoparallels of the Weyl connection represent the standard notion of "straight" lines in this geometry, the operational definition of proper time for an observer requires a Weyl-invariant prescription. Perlick proposed such a definition, where the proper time functional includes an accumulated contribution of the Weyl 1-form along the worldline. The central problem addressed in this paper is to determine the curves that extremize Perlick's proper time functional and to investigate whether these extremals coincide with the autoparallels of the Weyl connection, as is the case in pseudo-Riemannian geometry.

Methodology
The authors employ a variational approach to derive the equations of motion for Perlick's proper time.

Functional Formulation: They define Perlick's proper time as a functional $S[x^\alpha(s)]$ involving a local Lagrangian part $L$ (related to the metric norm) and a nonlocal part $l$ (the integral of the Weyl 1-form).
Generalized Variational Principle: By varying the functional with respect to the curve $x^\alpha(s)$ , they derive a generalized Euler–Lagrange equation. Unlike standard local Lagrangians, this derivation accounts for the nonlocal exponential factor dependent on the history of the curve.
Specialization to Weyl Geometry: The authors specialize the general result to Weyl geometry, identifying the Weyl length curvature $F_{\mu\nu} = \partial_\mu \sigma_\nu - \partial_\nu \sigma_\mu$ .
Parametrization and Analysis: They analyze the resulting integro-differential equation using the Weyl proper time $\tau$ as the parameter. This simplifies the tangential terms and allows for an explicit evaluation of the nonlocal functional $J(t)$ .
Limits and Approximations: The study includes:
- A weak-field analysis in a 1+1 dimensional Minkowski background with a constant Weyl field.
- A comparison between Minkowski geodesics, Weyl autoparallels, and Perlick extremals.
- A "double-scaling limit" where the Weyl coupling $\lambda \to 0$ and the terminal proper time $\tau_b \to \infty$ simultaneously, keeping their product finite.

Key Contributions and Results

Non-Coincidence with Autoparallels: The primary result is that the extremals of Perlick's proper time do not generally coincide with the autoparallels of the Weyl connection. The difference is governed by the Weyl length curvature $F_{\mu\nu}$ . In the integrable case (where $\sigma_\mu$ is an exact form and $F_{\mu\nu}=0$ ), the two notions of motion coincide, recovering the standard correspondence.
Nonlocal Dynamics: The equation of motion for Perlick extremals is inherently nonlocal. In the Weyl proper-time parametrization, the acceleration at a point $\tau$ depends explicitly on the remaining proper time to the endpoint of the variational interval, $\tau_b - \tau$ . The equation takes the form:
$\bar{D}u^\lambda/d\tau = \frac{\rho^2}{2(\tau_b - \tau)} F^\lambda_{\ \beta} u^\beta$
where $\bar{D}$ is the covariant derivative associated with the Weyl connection. This "terminal character" implies that the local evolution depends on the future boundary of the variational problem.
Weak-Field Behavior: In a constant Weyl field, the authors show analytically that Perlick extremals deviate from both Minkowski geodesics and Weyl autoparallels. Crucially, increasing the terminal time $\tau_b$ at a fixed field strength does not suppress the nonlocal effect; rather, it enhances the deviation, as the correction term scales with $\tau_b$ .
Emergence of Lorentz Force: The paper identifies a specific double-scaling limit where the Weyl field strength $\lambda \to 0$ and the terminal time $\tau_b \to \infty$ such that $\lambda \tau_b / 2 = q/m$ remains constant. In this limit, the direct Weyl connection corrections vanish, but the curvature-dependent term remains finite. The resulting equation of motion converges exactly to the relativistic Lorentz-force equation:
$Du^\lambda/d\tau = \frac{q}{m} F^{\text{em}\lambda}_{\ \ \ \beta} u^\beta$
This demonstrates that local electromagnetic-like dynamics can emerge as a limiting regime of the nonlocal Perlick dynamics.

Significance and Claims
The paper claims to uncover a nontrivial relationship between proper-time extremization, Weyl non-integrability, and local force dynamics.

Geometric Insight: The results suggest that Perlick's definition of proper time induces a new, non-local geometry where "geodesics" are distinct from affine geodesics. The nonlocality is an intrinsic feature of the boundary-value formulation of the variational problem and cannot be removed by reparametrization.
Physical Interpretation: The emergence of the Lorentz force in the double-scaling limit provides a potential geometric link between Weyl's original unification attempt and electromagnetism. It suggests that a finite local interaction strength ( $q/m$ ) can arise from the compensation between a vanishingly weak local Weyl field and the integrated effect of a distant boundary.
Caveats: The authors are modest regarding the physical interpretation of the double-scaling limit. They note that while the mathematical limit yields the Lorentz force, the physical origin of the fixed scale $q/m$ (derived from the arbitrary variational endpoint $\tau_b$ ) requires a physical prescription not provided by the variational problem itself. The terminal character of the equation is presented as a property of the boundary-value problem rather than a claim of retrocausal physics, though it distinguishes these trajectories from those generated by standard initial-value equations.

In summary, the work establishes that extremizing Perlick's proper time leads to nonlocal equations of motion that differ from Weyl autoparallels, with the difference vanishing only in the integrable case. Furthermore, it demonstrates that standard local electromagnetic dynamics can be recovered as a specific scaling limit of this nonlocal theory.

Technical Summary: Extremal Curves of Perlick's Proper Time in Weyl Geometry

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