Non-Geodesic Field-Space Trajectories

• Non-geodesic field-space trajectories are curves in configuration space that deviate from free-fall geodesics due to external forces, non-minimal couplings, or intrinsic curvature.
• They are modeled by modifying the standard geodesic equation with an extra force term, allowing quantification of turning rates, acceleration effects, and variable mass phenomena.
• These trajectories have practical applications in multifield inflation, topological field theory, and robotic motion planning, where they reveal novel particle production and non-Gaussianity features.

Non-geodesic field-space trajectories are curves in configuration or field spaces that do not satisfy the geodesic (free-fall or extremal action) equation with respect to an intrinsic or induced metric. Unlike geodesics, which represent locally distance-minimizing or extremal paths, non-geodesic trajectories encode the effects of external forces, non-minimal couplings, or field-space curvature, introducing physically or mathematically significant deviations from free motion. Such trajectories appear throughout modern physics and geometry, including quantum field theory in curved spacetime, multifield inflation, non-minimally coupled gravity theories, topological field theory, and even metric geometry and robotics.

1. Mathematical Formulation and General Characterization

A general geodesic in a manifold (or field space) with metric $g_{ab}$ is a curve $x^a(s)$ obeying

$$\frac{d^2 x^a}{ds^2} + \Gamma^a_{bc} \frac{dx^b}{ds} \frac{dx^c}{ds} = 0 ,$$

where $s$ is an affine parameter and $\Gamma^a_{bc}$ are the Christoffel symbols.

Non-geodesic trajectories are solutions to deformed equations:

$$\frac{d^2 x^a}{ds^2} + \Gamma^a_{bc} \frac{dx^b}{ds} \frac{dx^c}{ds} = F^a(x, \dot{x}, s),$$

with $F^a$ a nonzero "force" term arising from acceleration (in coordinate space or field space), background couplings (e.g., non-minimal scalar coupling), or induced by a curved or otherwise structured configuration space.

Key features from the literature:

• In (O'Hara, 2010) generic non-geodesic curves break the exact differential relation underlying the Dirac or wave equation, introducing a trajectory-dependent factor $X(s) = \cos\theta(s)$ modifying the mass term: $\gamma_a \dot{x}^a \psi = m(s)\, X(s)\, \psi$.
• In field-theory applications, the deviation from geodesicity is often quantified by a turning rate, as in multifield inflation where the background trajectory's extrinsic curvature or rate of "turn" in field space measures the strength of non-geodesic effects (Fumagalli et al., 2019, Parra et al., 17 Oct 2024).
• In topological applications, trajectories arising as integral curves of a generic vector field need not be geodesics with respect to any Riemannian metric (Burghelea et al., 2011).

2. Physical and Geometric Origins of Non-Geodesic Trajectories

Non-geodesic field-space trajectories can arise due to:

1. External or Non-Gravitational Forces: In classical motion, deviation from geodesicity reflects the presence of non-gravitational forces acting on particles.
2. Non-Minimal Couplings in Gravitational Theories: In the chameleon Brans–Dicke model, explicit coupling of the scalar field to matter leads to a modified conservation law and, generically, particle trajectories depart from geodesics unless certain Lagrangian choices are made (Saaidi, 2012). For example,

$$\nabla_\mu T^{\mu\nu} = -\left(g^{\mu\nu} L_m + T^{\mu\nu}\right) \nabla_\mu \ln f(\phi)$$

produces force terms altering the geodesic equation.

1. Field-Space Geometry and Curvature: In multifield inflation with negatively curved field space (e.g., hyperbolic sigma-models), the background trajectory can have significant curvature, leading to a "bending parameter" $\eta_\perp$ that quantifies deviation from geodesic flow (Fumagalli et al., 2019, Parra et al., 17 Oct 2024).
2. Variable Particle Mass and Quasi-Geodesics: Mass variation along a worldline induces a parallel-force deviation, resulting in so-called "quasi-geodesics" where proper time is not an affine parameter (Faraoni et al., 2020).
3. Metric and Topological Structure of Configuration Spaces: In spaces where no genuine geodesics exist between some points (e.g., spaces with obstacles), one may define "near geodesics" as continuous deformations of true geodesics in the metric completion (Davis, 2020).

3. Non-Geodesic Trajectories in Quantum, Relativistic, and Statistical Dynamics

Quantum and Dirac Extensions

For free motion, the Dirac equation can be associated with geodesic trajectories:

$\gamma_a \dot{x}^a \psi = m \psi$

For arbitrary (non-geodesic) curves, the wave equation generalizes to (O'Hara, 2010):

$$\gamma_a \dot{x}^a \psi = m(s) \cos\theta(s)\, \psi$$

where $\theta(s)$ quantifies the lack of parallelism between $ds$ and its dual. The non-parallel "cross term" modifies the effective mass and embodies the influence of acceleration or external forces (cf. Lorentz force analogies).

A corresponding Hamilton–Jacobi formalism incorporates non-geodesic corrections via the addition of acceleration-dependent terms to the characteristic function $S$:

$dS = g^{\alpha\beta} p_\alpha\, dx_\beta$

Thermodynamics and Mass–Temperature Interplay

When considering scalar fields, non-geodesic motion translates to a mass parameter $M$ that depends on the parametrization, yielding a temperature

$T = k k_B M$

with $k_B$ Boltzmann's constant and $k$ a scale factor, establishing a direct relationship between particle acceleration (or deviation from geodesicity) and the effective temperature of the scalar field (O'Hara, 2010). Positive acceleration increases temperature, deceleration decreases it, and the geodesic limit gives constant temperature.

Non-Geodesic Propagation and Eikonalization

For massless fields in curved spacetime, the eikonal approximation reveals that, except in conformally flat backgrounds, rays corresponding to wave propagation do not generally follow null geodesics. Curvature terms (Ricci tensor/scalar) acquire leading-order corrections to the Hamilton–Jacobi equation:

$$\nabla_\mu T\, \nabla^\mu T = -\frac{R}{6} ~\text{(for scalars)}$$

implying that the transport of energy-momentum is along eikonal trajectories that can be timelike or spacelike, not strictly null (Mannheim, 2021).

4. Topological, Geometric, and Robotic Aspects

Trajectories of Generic Vector Fields

In Morse theory and its variants, one studies the topology induced by flow lines of generic (gradient-like) vector fields, which need not be geodesics. An explicit construction is provided for the canonical compactification of the space of trajectories, including "broken" ones, resulting in a moduli space with a manifold-with-corners structure (Burghelea et al., 2011).

Geodesic Complexity and Motion Planning

In the absence of genuine geodesics (e.g., in configuration spaces with obstacles), "near geodesics" are defined as families of paths that approximate geodesic behavior in the ambient or completed metric space. The concept of "geodesic complexity" is thereby extended to non-geodesic spaces, with concrete implications for robot motion planning and the topological complexity of motion planners (Davis, 2020).

Setting	Geodesic Trajectories	Non-Geodesic Trajectories
Riemannian geometry	Distance minimizers (free motion)	Forced motion, flows, constraints, external forces
Morse theory	Gradient flow lines (when metric is induced)	Generic vector field flows
Robotics	Shortest paths	Near geodesics, efficient non-minimal moves
Field theory	Free-particle motion	Interacting, coupled, or forcibly evolved fields

5. Non-Geodesic Trajectories in Modern Field Theory and Cosmology

Multifield Inflation and Hyper Non-Gaussianities

In multifield models, particularly in inflation with negatively curved field space, strong non-geodesic motion (large turning rate $\eta_\perp$) leads to significant dynamical effects. The entropic perturbations develop transient tachyonic instability (negative mass-squared), and effective field theory reductions yield an imaginary sound speed ($c_s^2 < 0$), resulting in exponential amplification of the curvature perturbation. This process produces "hyper non-Gaussianities"—enhanced bispectrum/trispectrum in flattened configurations (Fumagalli et al., 2019). The scaling is:

$$\langle \zeta^n \rangle / \langle \zeta^2 \rangle^{n-1} \propto [(1/|c_s|^2 + 1)x^3]^{n-2}$$

for $n$-point functions, where $x$ encodes the growth rate from the instability.

Particle Production from Non-Geodesic Field Space Motion

In models such as multifield inflation, non-geodesic (turning) trajectories induce non-adiabatic particle production. The Bogoliubov coefficients $|\beta_k|^2$ quantify the number density of excited modes:

$n_k(T) = |\beta_k(T)|^2$

In single-field slow-roll inflation, this introduces mild, logarithmic corrections set by the spectral index $n_s$, while non-geodesic coupling in multifield scenarios generates a steeper (power-law) scale dependence and higher particle excitation, with possible observable relics (Parra et al., 17 Oct 2024).

6. Classification, Examples, and Broader Implications

Non-geodesic field-space trajectories are realized in diverse contexts:

• Electrodynamics of light-like charges: In flat spacetime, charged massless particles with non-geodesic (accelerated, null) worldlines exhibit a field generalizing the Liènard–Wiechert solution, splitting into a Coulomb-like term and a radiation term; the geometric construction requires adapted (non-inertial) coordinates to capture these features (Boehmer et al., 2019).
• Defect geometry and elastic media: In the presence of defects (e.g., ball dislocations), the induced geometry leads to altered geodesic equations where circular orbits are forbidden, forcing test particle trajectories to be non-geodesic in a geometrically controlled way (Andrade et al., 2012).
• Conformal and magnetic trajectories: Conformal (non-Killing) vector fields generate non-geodesic curves with special curvature/torsion properties—distinct from geodesics except in degenerate limits—suggesting generalizations to rectifying curves and magnetic flows in geometric mechanics (Lopez et al., 24 May 2024).
• Loxodromes in pseudo-isotropic space: Non-geodesic loxodrome curves, cutting meridians at a constant angle, are described on rotational surfaces within pseudo-isotropic geometry and are not generally geodesic (distance minimizers), but have special navigation and geometric significance (Babaarslan et al., 24 Dec 2024).

7. Outlook and Open Problems

The paper of non-geodesic field-space trajectories yields fundamental insights into both the analytic and topological structure of dynamical systems. Open directions noted in the literature include:

• Topological invariants for non-geodesic flows (Burghelea et al., 2011, Davis, 2020): Extending Morse and de Rham theory to incorporate generic (non-geodesic) dynamics.
• Observational constraints on non-geodesic inflationary dynamics (Fumagalli et al., 2019, Parra et al., 17 Oct 2024): Further quantification and detection possibilities for hyper non-Gaussian and particle production signatures.
• Rigorous thermodynamic interpretation of field-space temperature-mass relations (O'Hara, 2010): Understanding the interface of quantum dynamics, geometry, and thermal properties.
• Non-geodesic effects in metric and pseudo-metric spaces for robotics and planning (Davis, 2020): Generalized motion planning leveraging near-geodesics where shortest paths fail to exist.
• Connections between quasi-geodesic motion and varying-mass/modified gravity scenarios in cosmology (Faraoni et al., 2020, Saaidi, 2012).

These threads reflect the centrality of non-geodesic trajectories as both a technical tool and a source of physical phenomena in modern mathematical physics, cosmology, and geometry.