# Beyond Superposition and Collapse: Double-Slit Interference from Trembling Spacetime Geodesics

Nico F Declercq

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Nico F Declercq. Beyond Superposition and Collapse: Double-Slit Interference from Trembling Space-time Geodesics. 2025. hal-05070356

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# Beyond Superposition and Collapse: Double-Slit Interference from Trembling Spacetime Geodesics

Nico F. Declercq

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, USA

IRL 2958 Georgia Tech - CNRS, Georgia Tech Europe, Metz, France

declercq@gatech.edu

## Abstract

The double-slit experiment has long been interpreted as evidence of wave-particle duality and quantum superposition. Here we propose a deterministic alternative rooted in Trembling Space-time Relativity Theory (TSRT), where particles always follow localized proper-time geodesics within a causally structured but metrically fluctuating spacetime. Apparent interference fringes arise not from wavefunction superposition, but from the deterministic redirection of particle geodesics due to curvature perturbations shaped by the slit geometry.

The observed distribution of arrival positions is governed by two geometric mechanisms: (i) the slit separation, which sets the angular deflection range of redirected geodesics, and (ii) a minimal action threshold that enforces a resolution limit based on causal distinguishability. Planck's constant emerges naturally from this causal constraint as the smallest resolvable geodesic action difference, without any quantum postulates.

TSRT provides analytical predictions for fringe spacing and intensity distribution across a wide range of particles—photons, electrons, and neutrons—by applying geometric deflection laws to causally permitted geodesics. These predictions align closely with established experimental results. In addition, the TSRT framework enables simulation of arbitrary configurations by modeling ensembles of geodesics subject to metric trembling, providing a fully geometric alternative to quantum superposition.

## 1 Introduction

### 1.1 Historical Context and Motivation

The double-slit experiment, first performed by Thomas Young in 1801 with light, revealed wave-like interference patterns and reinforced the classical wave theory of light [1]. In the 20th century, the experiment was extended to electrons and other particles, producing similar interference patterns and challenging classical notions of localized trajectories [2, 3]. These results became foundational to quantum mechanics, which explains the phenomenon through wavefunction superposition and probabilistic collapse, as proposed in the Copenhagen interpretation [4]. Yet despite its empirical success, the quantum explanation leaves open deep conceptual questions about the ontology of particles and the meaning of measurement.
Recent studies continue to explore the double-slit experiment's implications. Advances include quantum eraser experiments testing complementarity [5], molecular interferometry probing the limits of wave-like behavior [6], and its use in solid-state systems to study decoherence [7]. Such experiments remain vital for foundational questions (e.g., quantum-classical boundaries) and applications (e.g., quantum information).

Still, many prominent researchers, including Albert Einstein, have expressed deep reservations about wave-particle duality and the Copenhagen interpretation, arguing that quantum mechanics—while empirically successful—must be an incomplete description of reality. Einstein famously rejected the notion of inherent randomness, insisting that 'God does not play dice' [8], and proposed hidden-variable theories to restore determinism. Even today, foundational challenges—such as the measurement problem [9] and quantum gravity [10]—motivate research into theories beyond standard quantum mechanics, underscoring that its formalism, however accurate, may not be the final word.

Classically, one expects that a particle approaching a barrier with two open slits would either pass through one slit or be reflected, depending on its trajectory and the geometry of the system. Since a classical particle cannot simultaneously traverse both slits, no interference pattern would be expected. Accordingly, the arrival distribution of particles on a detection screen should reflect a straightforward projection of the slit geometry, with no fringe structure.

Yet experiments show otherwise. When both slits are open and no measurement is made to determine the particle's path, an interference pattern emerges on the screen—suggesting, under the standard interpretation, that each particle behaves as a delocalized wave passing through both slits simultaneously and interfering with itself. To reconcile this with the discrete detection of single particle impacts, quantum mechanics postulates a probabilistic wavefunction that evolves unitarily but collapses upon measurement into a definite outcome. This view, though mathematically predictive, comes at the cost of invoking fundamental nonlocality, irreversibility, and ontological indeterminism—features that many physicists regard as philosophically troubling and potentially symptomatic of a deeper, undiscovered theory.

The reliance on wavefunction collapse—and the accompanying rejection of ontological realism and determinism—has remained philosophically and physically contentious, fueling debates that persist to this day. Critics, including Einstein [8], argued that the Copenhagen interpretation's inherent indeterminism was a placeholder for deeper physics, while proponents like Bohr [4] defended it as fundamental. Later, Bell's theorem [11] and experimental violations of local realism [12] challenged hidden-variable alternatives, yet competing interpretations (e.g., Bohmian mechanics [13], GRW collapse models [14], or many-worlds [15]) continue to question whether collapse is a physical process or an epistemic artifact. Even in modern experiments, such as macroscopic superposition tests [16], the tension between formalism and ontology remains unresolved.

These debates highlight a fundamental question: Can the observed interference patterns be explained by a local and deterministic physical mechanism—one that preserves realism without invoking superposition or collapse?

This paper offers an alternative: the Trembling Spacetime Relativity Theory (TSRT), a geometric and deterministic framework recently proposed by the author [17]. In this theory,
particles are always localized and rigid. They follow well-defined trajectories through a dynamically trembling spacetime—a metric geometry that fluctuates smoothly and causally. These fluctuations redirect geodesics in a way that, when coarse-grained across an ensemble of trials, gives rise to structured interference-like patterns. The fringes emerge not from particle wave superposition, but from the interaction of deterministic particles with a causally constrained geometric background. In TSRT, the presence of a detector modifies the boundary conditions of spacetime geometry, thereby altering the causal structure experienced by the particle. This modification occurs independently of the detection outcome and does not imply any change in the particle's ontological state.

TSRT thus provides a fundamentally new explanation of interference: one that preserves locality, determinism, and realism, and dispenses entirely with wavefunction collapse. In the sections that follow, we apply this framework rigorously to the slit experiment and show that all observed phenomena can be reproduced without any probabilistic or non-deterministic assumptions.

## 1.2 TSRT Framework Overview for Slit Experiments

The Trembling Spacetime Relativity Theory (TSRT) [17] offers a radically different alternative to quantum mechanics: instead of assigning wave-like properties to particles, it attributes interference and diffraction phenomena to small-scale geometric fluctuations in spacetime itself. The theory is built on the foundational principle that spacetime is not perfectly smooth, but undergoes persistent, small-scale, deterministic fluctuations in its metric tensor. These fluctuations—referred to as trembling—are deterministic and governed by local causal constraints that preserve proper time and geodesic continuity. As shown in the foundational formulation of TSRT [17], these metric variations provide the mechanism by which localized particles experience curvature-induced deflection without invoking quantum indeterminacy.

At the heart of TSRT [17] is the idea that particles follow well-defined, proper-time parametrized geodesics through a four-dimensional spacetime whose geometry fluctuates deterministically. The spacetime metric is expressed as

$$g_{\mu\nu}(x) = \eta_{\mu\nu} + \xi_{\mu\nu}(x), \qquad (1)$$

where $\eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)$ is the flat Minkowski background metric, and $\xi_{\mu\nu}(x)$ is a smooth, causal, and deterministic perturbation field representing local spacetime trembling. These fluctuations are not restricted to spatial curvature: they also affect the time-time ($g_{00}$) and time-space ($g_{0i}$) components of the metric, thereby influencing both temporal and spatial aspects of geodesic motion. As a result, both spatial and temporal aspects of proper time propagation are subject to local geometric modulation.

TSRT underlies the emergence of structured transverse patterns from deterministic curvature modulation.

Despite their generality, these perturbations preserve causality: all geodesic propagation remains timelike or null, and no superluminal trajectories are permitted. The trembling is constrained by the requirement that proper time remains forward-directed and that geodesics
evolve within a causally admissible neighborhood. Instead, they produce smooth variations
in geodesic curvature, redirecting particle paths without altering particle identity, introducing
quantum uncertainty, or invoking any form of probabilistic superposition. Particles remain
localized and classical, but their deterministic motion occurs within a spacetime whose geometry
fluctuates at small scales.

Each particle follows a deterministic geodesic shaped by the local curvature of the trembling spacetime. While the individual trajectory is non-random, it depends on the specific realization of the metric field. Across repeated trials, the ensemble of such geodesics becomes shaped by the aperture geometry and the statistical properties of the curvature fluctuations, leading to the observed distribution of arrival positions on the detection screen.<fnref n="1" />

The observed interference patterns thus emerge from the coarse-grained distribution of these geodesics. Importantly, the trembling is not associated with a stochastic noise field but is governed by deterministic geometric principles subject to smoothness and causal coherence. Proper time along each geodesic remains forward-directed and well-defined, and no breakdown of spacetime continuity occurs. In regions near apertures, these fluctuations redirect the geodesics in such a way that their density on a detection screen forms structured patterns—patterns that reproduce, in detail, the results of standard interference experiments.

TSRT [17] thereby shifts the interpretational framework: instead of viewing the particle as a
wave passing through both slits and collapsing upon observation, we understand the interference
pattern as arising from how the geometry of spacetime itself channels the deterministic motion
of particles in a causally structured, fluctuating background. This framework preserves all
classical concepts—particles remain particles, motion is continuous, measurement is boundary
enforcement—and yet reproduces the empirical phenomena typically attributed to quantum
behavior.

For a detailed analysis of how these geometric deflections behave in the relativistic regime, in-
cluding derivations, high-energy extensions, and comparison with Dirac theory—see Section 12.

1.3 Structure of the Paper

This paper develops a step-by-step reinterpretation of the double-slit experiment from the perspective of Trembling Spacetime Relativity Theory (TSRT) [17], a geometric and deterministic framework that eliminates the need for wave-particle duality and wavefunction collapse. The structure of the paper is designed to progressively replace the standard quantum interpretation with a causal explanation based on geodesic redirection in a dynamically trembling metric.

Section 2 establishes the classical baseline, showing that in the absence of metric perturba-
tions, particles propagate in straight lines and produce no interference patterns. This under-
scores the necessity of spacetime trembling as the physical origin of geodesic deflection.

Section 3 introduces the binary aperture function S(x, y) that encodes the slit geometry, and derives its spatial Fourier transform $\tilde{S}(\kappa_x, \kappa_y)$, which determines the angular redirection spectrum of trembling-induced geodesics.

<footnote n="1">¹Throughout this work, we derive general results applicable to both relativistic and non-relativistic regimes. When simplified estimates are presented (e.g., using *p* = *mv*), they refer to the non-relativistic limit unless otherwise specified.</footnote>
Section 4 presents the core formalism: particles follow deterministic geodesics in a causally admissible, fluctuating spacetime. The trembling metric $g_{\mu\nu}(x) = \eta_{\mu\nu} + \xi_{\mu\nu}(x)$ induces transverse deflections, and repeated trials generate a statistical distribution of arrival positions.

Section 5 explains how interference-like patterns emerge not from wavefunction overlap but from the ensemble of redirected geodesics. Each particle follows a unique causal path, and the structured fringe pattern arises as a macroscopic consequence of metric-guided motion. Proper-time differences between trajectories lead to causal phase retardation, distinct from quantum phase interference.

Section 6 analyzes how which-path measurements affect the geodesic ensemble. It shows that the act of measurement alters the boundary conditions of the metric field, suppressing ensemble coherence and eliminating interference—without invoking collapse or nonlocal effects.

Section 7 presents the key predictions of TSRT, including the emergence of Planck's constant $\hbar$ as a minimal action difference required for geodesic distinguishability. The resulting fringe spacing formula matches quantum theory in functional form, but derives entirely from deterministic curvature effects.

Section 8 offers a direct comparison with quantum mechanics. It shows that while the observable predictions coincide under symmetric, ideal conditions, the interpretational frameworks diverge completely: TSRT maintains particle realism and causality, whereas quantum theory relies on superposition and collapse.

Section 9 proposes experimental setups involving dynamic apertures, asymmetries, or energy modulation that could empirically distinguish TSRT from quantum mechanics. These include tests of fringe robustness, phase retardation, and slit-specific curvature effects.

Section 10 provides analytical fringe predictions for commonly studied particles such as photons, electrons, and neutrons, using TSRT's geodesic deflection laws. It also introduces a generalized simulation framework based on ensembles of trembling geodesics, showing how arbitrary experimental setups—including multi-slit, time-dependent, or relativistic scenarios—can be modeled within TSRT without invoking wave mechanics.

Section 11 derives the Jacobi deviation equation in trembling spacetime and introduces the concept of action-based resolution. It shows that geodesic pairs must differ by a minimal proper-time action $\Delta S \gtrsim \hbar$ to produce resolvable angular separation on the screen. This defines a fundamental resolution limit, dependent on particle momentum and interaction length, and provides a falsifiable prediction unique to TSRT.

Section 12 extends the framework to high-energy regimes. It reformulates TSRT in fully relativistic terms, compares its predictions with Dirac theory, and proposes experiments involving relativistic particle beams to test for divergences between the two approaches.

Section 13 demonstrates that TSRT's predictions are robust under a wide class of causally admissible metric perturbations. It surveys different types of trembling—harmonic, localized, asymmetric, or slowly varying—and shows that fringe structure and spacing persist, confirming that the predicted patterns are not artifacts of fine-tuned assumptions but intrinsic features of the theory.

We conclude in Section 14.
# 2 Classical Particle-Slit Dynamics: A No-Fringe Baseline

## 2.1 Rigid Particles in Flat Spacetime

We begin by considering the classical limit in which particles are perfectly rigid and travel through a flat, non-trembling Minkowski spacetime. The particles are assumed to have a characteristic spatial extent much smaller than the slit aperture size, and thus can be treated as point-like for the purpose of their interaction with the slit geometry. Furthermore, the material surrounding the slits is treated as perfectly rigid, fully reflective, and sharply defined, so that there is no partial transmission, no penetration into the material, and no scattering due to edge effects. In particular, we assume that the slits are much wider than the particle size, and the edges of the slits are idealized as infinitely sharp, rigid boundaries.

Under these assumptions, the classical trajectory of each particle is a straight line, corresponding to inertial motion in the absence of external forces. A particle aimed directly at a slit passes through it without deflection, continuing along its initial direction of motion. Conversely, a particle aimed at the opaque region between the slits is elastically reflected. There is no mechanical mechanism in classical flat spacetime to cause deflection in the transverse direction if the particle enters a slit; the particle either transmits or reflects with no lateral deviation.

Hence, there is no possibility of fringe formation on a screen placed behind the slits. All transmitted particles travel in straight, undeviated trajectories and arrive in a region directly aligned with the slits. The spatial distribution of arrival positions on the detection screen is simply the convolution of the particle emission profile with the geometric projection of the slits, and no interference fringes or diffraction patterns can emerge from this setup.

## 2.2 Absence of Deflection and Fringe Formation

In classical physics, the motion of a free particle in flat spacetime is governed by the geodesic equation, which reduces to straight-line inertial motion in the absence of curvature. Let the particle's four-momentum be denoted by

$$p^{\mu} = m \frac{dx^{\mu}}{d\tau}, \qquad (2)$$

where $\tau$ is the proper time along the particle's worldline and $m$ is the particle's rest mass (assumed constant).

In flat Minkowski spacetime with metric signature $(-, +, +, +)$, the geodesic equation reduces to:

$$\frac{d^2 x^{\mu}}{d\tau^2} = 0, \qquad (3)$$

since the Christoffel symbols $\Gamma^{\mu}_{\alpha\beta}$ vanish in Cartesian coordinates [18, 19].

This implies that the four-momentum $p^{\mu}$ is covariantly constant along the particle's worldline (i.e., $dp^{\mu}/d\tau = 0$), which further implies that all components of $p^{\mu}$ are individually conserved (not just the transverse components $p^x$ and $p^y$).

In particular, the energy $p^0$ (or $E$) is conserved, while the spatial components $p^i$ (including $p^x, p^y$, and $p^z$) are also conserved.
There is no mechanism in the classical framework to generate a transverse impulse unless the particle interacts with an external potential or a force field, neither of which is present in the ideal slit setup considered here.

Let the slits be aligned in the y-direction, and let particles propagate initially along the z-axis. The slits are confined to a screen located at $z = 0$, and the detection screen lies at some fixed $z = z_1 > 0$. A particle that enters through slit s at transverse coordinate $(x_s, y_s)$ with initial momentum $p^\mu = (E/c, 0, 0, p^z)$ will continue to travel along a straight line, intersecting the detection screen at the projected point:

$$x_{\text{det}} = x_s, \quad y_{\text{det}} = y_s, \quad z_{\text{det}} = z_1. \qquad (4)$$

The spatial distribution of arrival points on the detection screen therefore directly reflects the geometry of the slits and the initial angular spread of the particles. No modulation or fringe structure can arise, as the absence of any transverse momentum change precludes the emergence of constructive or destructive interference zones.

In summary, in a flat and non-trembling spacetime, particles either pass through one of the slits or are elastically reflected. Those that pass through do so along undeviated, straight-line paths. There is no possibility for fringe formation or diffraction patterns to appear, and thus classical mechanics fails to account for the observed interference patterns associated with slit experiments.

This constitutes the classical no-fringe baseline scenario, which we now contrast with the predictions from Trembling Spacetime Relativity Theory [17].

# 3 Geometry of the Slits and Reciprocal Space Representation

## 3.1 Definition of the Geometry Function $S(x, y)$

To model the slit system in a way that remains fully compatible with deterministic particle trajectories and causal structure, we define a geometry function $S(x, y)$ over the transverse plane at the slit screen $z = 0$. The function $S$ encodes the spatial configuration of the aperture and is used as an input to the spacetime structure in TSRT [17]. Specifically, we define:

$$S(x, y) = \begin{cases} 1, & \text{if the point } (x, y) \text{ lies within an open slit,} \\ 0, & \text{otherwise (i.e., opaque region).} \end{cases} \qquad (5)$$

The function $S(x, y)$ is entirely deterministic and binary: it assigns the value 1 to points in the transverse plane that correspond to open regions (slits), and 0 elsewhere. This is not a wavefunction or a probability amplitude—it is a geometric input to the physical spacetime configuration. No assumptions about wave behavior or quantum states are made.

In the TSRT framework [17], the function $S(x, y)$ acts as a boundary condition that constrains the set of permissible geodesics. When a particle approaches the screen at $z = 0$, the worldline is allowed to pass through the screen only at coordinates for which $S(x, y) = 1$. Particles reaching regions where $S(x, y) = 0$ are reflected. Thus, $S$ determines where proper-time-forward geodesics can cross the screen and continue to the detection region.
Moreover, this function enters the coarse-grained geodesic distribution by defining the set of admissible transverse initial conditions. These conditions, in turn, define the ensemble of future-directed geodesics influenced by the trembling metric, as will be explored in later sections.

The geometry function $S(x, y)$ is the only slit-dependent input required in this formulation. It plays a role analogous to an aperture function in classical optics, but in TSRT [17] it has direct causal influence on which geodesics survive to reach the detection screen. All emergent interference effects arise from the deterministic interaction of geodesics with the spacetime fabric, modulated by $S(x, y)$, without invoking any wave-based particle interpretation.

## 3.2 Fourier Transform and Emergence of Transverse Reciprocal Numbers

Although the geometry function $S(x, y)$ is purely spatial and binary, its spatial structure can be analyzed via a two-dimensional Fourier transform. This analysis does not imply the presence of physical waves; rather, it serves to characterize the transverse spatial variation of the slit configuration in terms of its reciprocal space components. These components will later be shown to modulate the density of geodesics emerging from the trembling spacetime geometry.

The Fourier transform $\tilde{S}(\kappa_x, \kappa_y)$ of the geometry function $S(x, y)$ is defined as:

$$ \tilde{S}(\kappa_x, \kappa_y) = \iint_{\mathbb{R}^2} S(x, y) e^{-i(\kappa_x x + \kappa_y y)} dx dy, \qquad (6) $$

where $\kappa_x$ and $\kappa_y$ are the transverse reciprocal space numbers, with units of inverse length. The vector $\vec{\kappa} = (\kappa_x, \kappa_y)$ is referred to as the transverse reciprocal space vector, and not a wavevector, since no wave interpretation is employed in this theory.

Physically, the magnitude and structure of $\tilde{S}(\vec{\kappa})$ reflect the dominant transverse scales and symmetries of the aperture. For example, a double-slit configuration produces a characteristic interference-like structure in $\tilde{S}$ with regularly spaced peaks in $\kappa_x$, corresponding to the slit separation, while the envelope of these peaks encodes the slit width.

In the TSRT framework [17], $\tilde{S}(\vec{\kappa})$ does not describe the amplitude of any physical wave. Instead, it acts as a geometric selector in reciprocal space that modulates the spectrum of geodesic deviations permitted by the trembling spacetime structure. This modulation is statistical and deterministic at the level of geodesic ensemble structure: each geodesic remains a well-defined trajectory, but the distribution of such geodesics across many particles reflects the reciprocal features of the aperture encoded in $\tilde{S}(\vec{\kappa})$.

This reciprocal structure is therefore not a source of interference by superposition, but a statistical signature of how the spacetime metric fluctuation couples to the slit geometry. In subsequent sections, we show how the trembling of spacetime redistributes geodesics in a way shaped by $\tilde{S}$, yielding the observed interference patterns as a cumulative effect of individually deterministic particle paths.

Although this work focuses on the double-slit configuration to ensure clear comparison with historical quantum interference experiments, the TSRT framework generalizes naturally to arbitrary slit geometries. The aperture function $S(x, y)$, which encodes the binary spatial profile of open and closed regions, need only be piecewise integrable to admit a well-defined spatial Fourier transform $\tilde{S}(\kappa_x, \kappa_y)$. This transform acts as a geometric filter that modulates the angular dis-
tribution of redirected geodesics, replacing the role of amplitude interference with deterministic
curvature-guided trajectory selection. For instance, in the case of a periodic N-slit array with
slit width a and spacing D, the Fourier transform, for an infinite number of slits, becomes

$$
\tilde{S}(\kappa_x) \sim \text{sinc}\left(\frac{a\kappa_x}{2}\right) \cdot \sum_{n=-\infty}^{\infty} \delta(\kappa_x - n\Delta\kappa), \quad \Delta\kappa = \frac{2\pi}{D}, \qquad (7)
$$

producing a comb of reciprocal-space peaks that determine the periodicity of fringe structures
on the detection screen. These features arise not from wave superposition, but from the de-
terministic accumulation of redirected geodesics shaped by the aperture's reciprocal structure.
This generalization allows TSRT to make precise predictions for complex or asymmetric slit
arrangements without invoking wavefunctions or probabilistic collapse.

4 Particle Geodesics in Trembling Spacetime

4.1 Geodesic Deviations Induced by Metric Trembling

In Trembling Spacetime Relativity Theory (TSRT) [17], spacetime is not flat and immutable,
but exhibits persistent, causally bounded metric fluctuations. These small-scale geometric de-
viations—referred to as metric trembling—occur independently of gravitation and are defined
with respect to a flat Minkowski background ημν. For the present analysis, we assume no back-
ground curvature, so that all particle deflections arise solely from trembling-induced effects.

These perturbations are encoded as a smooth, deterministic field ξμν(x), such that the full
metric becomes:

$$
g_{\mu\nu}(x) = \eta_{\mu\nu} + \xi_{\mu\nu}(x), \quad |\xi_{\mu\nu}| \ll 1. \tag{8}
$$

The perturbation ξ$_{µν}$ is not quantized or stochastic in the usual sense, but evolves deter-
ministically under causal constraints—no backward-in-time influence or discontinuous jumps
are permitted. As shown in TSRT [17], such a trembling metric modifies the geodesic equation:

$$
\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}(x) \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0, \qquad (9)
$$

where the Christoffel symbols Γµαβ include derivatives of ξµν(x), producing curvature-induced
deflections even in the absence of external fields.

$$
\Gamma^\mu_{\alpha\beta} = \frac{1}{2} g^{\mu\sigma} (\partial_\alpha g_{\beta\sigma} + \partial_\beta g_{\alpha\sigma} - \partial_\sigma g_{\alpha\beta}). \quad (10)
$$

Each particle follows a unique, deterministic geodesic shaped by the local realization of the trembling metric ξμν(x) it encounters during transit. Although the geodesic equation is fully determined once ξμν is specified, this field varies slightly in space and time. As a result, even particles with identical initial conditions traverse distinct but causally permitted worldlines. The ensemble of such trajectories forms a narrow, structured fan—each geodesic locally inertial within its own perturbed geometry, but collectively encoding the statistical imprint of the fluctuating curvature. This behavior is deterministic in origin yet rich in emergent spatial structure when many trials are observed.
Importantly, this mechanism enables transverse deviations of particle trajectories without in-voking any non-classical or wave-like behavior. In a region near the slits, the metric fluctuations guide some particles slightly left, others slightly right, even though all particles are identically prepared. This distribution of deterministic deviations becomes essential for understanding how interference-like patterns can emerge in the absence of wave-particle duality.

## 4.2 Causal Coherence and Worldline Distributions

While the trembling of spacetime causes individual particle geodesics to deviate from their classical, straight-line form, these deviations are not arbitrary. The perturbations $\xi_{\mu\nu}(x)$ that define the local metric trembling are constrained by causality, smoothness, and continuity of the underlying spacetime fabric. In particular, the fluctuations must preserve the forward-in-time orientation of all proper-time-parametrized worldlines and cannot introduce discontinuities or causal violations between neighboring trajectories.

This leads to a form of causal coherence among nearby geodesics: particles that enter the slit region with similar initial conditions will follow slightly diverging trajectories, but their deviations remain bounded and correlated. The ensemble of such geodesics forms a locally coherent bundle, whose transverse structure encodes the statistical manifestation of trembling spacetime at coarse-grained scales.

Let $x^{\mu}(s)$ denote a fiducial geodesic [20–22], and let $\delta x^{\mu}(s)$ describe the deviation vector to a neighboring geodesic. The evolution of this deviation is governed by the Jacobi equation [23] in curved spacetime [21, 24, 25] as shown in TSRT [17],

$$ \frac{D^2 \delta x^{\mu}}{Ds^2} = R^{\mu}_{\nu\alpha\beta} \frac{dx^{\nu}}{ds} \delta x^{\alpha} \frac{dx^{\beta}}{ds}, \qquad (11) $$

where $R^{\mu}_{\nu\alpha\beta}$ is the Riemann curvature tensor generated by the trembling metric $g_{\mu\nu} = \eta_{\mu\nu} + \xi_{\mu\nu}$. Although the curvature induced by $\xi_{\mu\nu}$ is small, it introduces a spread in geodesics that accumulates over time and results in a nontrivial distribution of particle endpoints at the detection screen.

The spread in arrival positions is not the result of quantum indeterminacy, but of causal geometry: each particle experiences a distinct, deterministic perturbation field $\xi_{\mu\nu}(x)$ that governs its geodesic. Across many particles, the statistical distribution of screen impacts reflects the variability of these causally allowed metric realizations. The ensemble is thus epistemic in origin—arising from variation in the geometry, not in the particles or their dynamics.

To understand how this leads to a particle density $\rho(x, y; z_1)$ on the detection screen, consider the following reasoning.

For each realization $\xi_{\mu\nu}(x)$, there is a single geodesic trajectory from source to screen. Let $(x_{\text{geo}}(\xi), y_{\text{geo}}(\xi))$ denote the coordinates where that geodesic intersects the detection screen at fixed distance $z = z_1$. If we observe many such particles—each experiencing a different but causally allowed realization of the metric fluctuation—we can build a statistical distribution of where these trajectories land.

Mathematically, we describe this by averaging over all admissible metric realizations. The arrival probability density $\rho(x, y; z_1)$ is thus given by the expectation value (ensemble average)
of delta functions centered at the geodesic landing points:

$$ \rho(x, y; z_1) = \langle \delta^{(2)}((x, y) - (x_{\text{geo}}(\xi), y_{\text{geo}}(\xi))) \rangle_{\xi}. \qquad (12) $$

This expression formalizes the idea that the arrival density $\rho(x, y; z_1)$ is a statistical histogram of where particles—each following a deterministic geodesic shaped by a particular realization of $\xi_{\mu\nu}(x)$—reach the detection screen. The delta function selects the arrival point of each such geodesic, while the average over $\xi$ accounts for the frequency with which different metric realizations occur across repeated trials.

The delta function ensures that each realization contributes only at its geodesic endpoint, while the ensemble average $\langle \cdot \rangle_{\xi}$ integrates over the entire space of causally admissible metric fluctuations defined by TSRT [17]. The result is a deterministic yet richly structured particle density, shaped entirely by the geometry of spacetime—without invoking wavefunction amplitudes or probabilistic collapse.

In this way, TSRT explains how interference-like distributions arise without superposition. The geodesic ensemble, shaped by trembling curvature and filtered by the slit geometry $S(x, y)$, yields a deterministic but complex pattern of arrival positions. In the next section, we formalize this result and show how classical interference fringes emerge as a geometric ensemble effect within TSRT.

# 5 Interference as a Geometric Ensemble Effect

## 5.1 No Superposition, No Collapse

Standard quantum mechanics explains double-slit interference via wavefunction superposition: a single particle is said to propagate simultaneously through both slits, and the resulting interference pattern arises from the coherent addition of complex amplitudes. Measurement then invokes collapse to enforce a single outcome.

TSRT [17] replaces this picture entirely. Each particle follows one and only one geodesic, determined by the local metric $g_{\mu\nu}(x)$, which includes trembling curvature shaped by the aperture. There is no wavefunction, no collapse, and no ontological ambiguity. The resulting ensemble of deterministic geodesics forms a screen distribution that resembles interference—but emerges purely from spacetime geometry. Different particles in an identically prepared ensemble follow different paths due to the constantly trembling geometry.

The key insight is that the ensemble of these deterministic worldlines, when modulated by the slit geometry $S(x, y)$, produces a distribution of arrival positions at the detection screen that closely resembles classical interference fringes. Each particle passes through exactly one slit and follows a single geodesic—but the set of all such trajectories, statistically distributed through trembling spacetime, forms the fringe pattern.

There is no need for collapse, because there is no superposition: each particle interacts with the slit geometry in a fully causal and deterministic way. What appears as interference is in fact the result of geodesic redirection through structured metric fluctuations, governed by the same causal constraints that ensure consistency across all TSRT [17] predictions.
This approach restores realism and locality: particles are always particles, never waves, and the structured distribution of impacts on the screen reflects the deterministic influence of spacetime geometry—not the probabilistic interference of abstract wave amplitudes. The seemingly wave-like interference fringes thus emerge as a geometric ensemble effect from a large number of causally permitted, fluctuating particle geodesics. The physical picture is that it is the world that fluctuates, not the particle.

## 5.2 Causal Phase Retardation and Relative Delay

In TSRT [17], particles follow single, well-defined geodesics through a dynamically trembling spacetime. These trajectories are not wave-like and do not interfere with one another. Consequently, the structured distribution of arrival positions observed on the detection screen cannot arise from direct phase interference between particles, nor from accumulated proper-time differences across geodesics.

However, the trembling of spacetime does introduce small variations in the proper time accumulated along different geodesics:

$$ \tau = \int_{\gamma} \sqrt{-g_{\mu\nu}(x) \frac{dx^{\mu}}{ds} \frac{dx^{\nu}}{ds}} ds, \qquad (13) $$

and for two distinct trajectories $\gamma_A$ and $\gamma_B$ that pass through slits A and B, one can define a proper-time difference:

$$ \Delta\tau = \tau_B - \tau_A. \qquad (14) $$

While this retardation does not affect the behavior of any individual particle—since particles do not interfere or exhibit superposition—it represents a geometric differential that can be spatially resolved across the aperture. Such retardations may be measurable through auxiliary effects, but they do not contribute to fringe formation within TSRT. They remain secondary phenomena tied to path geometry, not fundamental phase relations.

These effects are explored in more detail in Section 9, where proposed experiments aim to isolate measurable consequences of proper-time modulation as a signature of the underlying trembling geometry.

Therefore, while causal phase retardation does not produce interference fringes, it represents a physically meaningful effect within TSRT that may serve as an empirical discriminant between geometric and wave-based interpretations of slit experiments.

# 6 Measurement and Slit Determinacy in TSRT

## 6.1 Binary Slit Traversal Constraint

In Trembling Spacetime Relativity Theory (TSRT) [17], each particle follows a single, deterministic geodesic governed by the locally fluctuating metric $g_{\mu\nu}(x) = \eta_{\mu\nu} + \xi_{\mu\nu}(x)$. When a particle geodesic intersects the slit screen at $z = 0$, its transverse position $(x, y)$ is evaluated
by the aperture geometry function $S(x, y)$, defined such that $S = 1$ within the slits and $S = 0$ elsewhere.

This binary function enforces a strict traversal rule: for any given geodesic, the transmission condition is

$$ \text{Transmission condition: } S(x, y) = 1, \qquad (15) $$

ensuring that a particle either passes through one slit or is reflected—never both. This exclusivity is not a postulate but a geometric consequence: geodesics are continuous and non-branching. A single worldline cannot simultaneously occupy both apertures.

Nevertheless, in the absence of measurement, the full aperture function $S(x, y)$ includes both slits, and the ensemble of particle geodesics $\mathcal{G}_{\text{full}}$ spans all metric-permitted paths through either slit. Each geodesic interacts with a globally trembling spacetime background. Crucially, in TSRT, the metric perturbation field $\xi_{\mu\nu}(x)$ is not a random white-noise distribution; rather, it is spatially coherent, causally constrained, and smoothly varying across regions of spacetime. The curvature perturbations near slit A are thus dynamically linked to those near slit B—analogous to correlated vibrations in an elastic medium.

As a result, the full geodesic ensemble experiences a geometric superposition of metric boundary influences, meaning the statistical distribution of arrival points on the detection screen reflects the joint effect of both slits' influence on trembling spacetime. This ensemble-level coherence leads to structured density patterns, even though each particle follows only one geodesic.

By contrast, when a which-path measurement is introduced—e.g., a detector monitoring slit A—this imposes a binary selection: either slit A or slit B is geometrically realized for that trial, not both. This modifies the aperture function to a measurement-specific form:

$$ S'(x, y) = \begin{cases} 1, & \text{if } (x, y) \in \text{slit A and detector A clicks,} \\ 1, & \text{if } (x, y) \in \text{slit B and detector B clicks,} \\ 1, & \text{if } (x, y) \in \text{slit B and detector A does not click,} \\ 1, & \text{if } (x, y) \in \text{slit A and detector B does not click,} \\ 0, & \text{otherwise.} \end{cases} \qquad (16) $$

Consequently, the trembling metric field $\xi_{\mu\nu}(x)$ adjusts to the modified boundary conditions imposed by the presence of a detection apparatus. The causal structure of spacetime near slit A is no longer dynamically coupled to that near slit B once a measurement device is introduced. Instead, the metric perturbations become slit-localized: the trembling geometry associated with each slit evolves independently, conditioned by its own boundary configuration. As a result, geodesics through slit A and slit B no longer participate in a shared ensemble modulated by mutual causal influence. The collective structure responsible for interference disappears.

This suppression of ensemble-level modulation is not due to any physical change in the particle itself, but arises solely from the altered spacetime geometry induced by the measurement boundary. The presence of a detector effectively reshapes the world the particle traverses. It is the causal fabric of spacetime that changes—not the nature of the particle—when a detection
mechanism is installed at the slit boundary.

## 6.2 Effect of Measurement on Geodesic Distribution

When a particle is emitted toward the slit configuration, its trajectory is determined by a single geodesic through trembling spacetime. The complete set of such geodesics over repeated trials forms the ensemble $G_{full}$, which—in the absence of which-path measurement—includes paths passing through both slit A and slit B. Importantly, although each geodesic intersects only one slit, the trembling metric field $\xi_{\mu\nu}(x)$ spans both aperture regions simultaneously and responds to their combined boundary structure.

This is possible because, in TSRT [17], the trembling metric is not a set of disconnected pointwise fluctuations. It is a smooth, causally coherent field with curvature correlations across space and time. Therefore, the spacetime geometry behind the slits is modulated not just by each slit in isolation but by their mutual presence. This mutual modulation causes geodesics arriving from both slit regions to collectively interfere—not through particle self-interference, but via ensemble-level curvature interference mediated by the coherent trembling of the metric.

Measurement in TSRT modifies the geometric structure of spacetime not through collapse, but by imposing a different boundary condition on the trembling metric field $\xi_{\mu\nu}(x)$ prior to particle arrival. The mere presence of a detection system at a specific slit alters the boundary configuration $S'(x, y)$, thereby breaking the causal and spatial symmetry that would otherwise link both slits. This asymmetric setup influences the global structure of admissible geodesics even before any detection event occurs.

Consequently, the installation of a which-slit detector modifies the boundary conditions by rendering the two slits causally independent from the standpoint of geodesic propagation. In this context, the trembling metric no longer supports coherent geodesic redirection across both slits simultaneously. Instead, it deterministically evolves two distinct geodesic ensembles—one through slit A, the other through slit B—without cross-coupling. When a particle is detected passing through a particular slit, the arrival distribution reflects the corresponding sub-ensemble of geodesics alone. The full geodesic ensemble is thus partitioned as:

$$G_{\text{measured}} = \begin{cases} G_A, & \text{if particle passage through slit A is detected,} \\ G_B, & \text{if particle passage through slit B is detected.} \end{cases} \qquad (17)$$

This suppression of interference is not the result of any retroactive adaptation of the metric, but arises from the causal disruption introduced by the measurement device itself. The presence of a which-slit detector imposes a boundary condition that prevents the trembling metric from establishing coherent, cross-slit geodesic connectivity. In the absence of such a detector, the spacetime perturbations can remain correlated across both slits, allowing geodesics emerging from either opening to jointly contribute to the structured arrival pattern. Once detection is introduced, however, this cross-ensemble coherence is broken: the system evolves independently through one slit or the other, and no joint curvature-induced modulation can occur at the detection screen.

More precisely, the measurement device locally constrains the admissible metric perturbations $\xi_{\mu\nu}$ near its vicinity, eliminating correlations between fluctuations on either side of the
aperture. This geometric severing of causal links between slits A and B ensures that each sub-ensemble evolves in isolation, even though both remain governed by the same deterministic trembling geometry. Consequently, the structured interference fringes seen in the undetected case are replaced by a simple superposition of geodesic densities—an incoherent sum of two non-interacting arrival profiles. This distinction reflects a geometric, not probabilistic, shift: the ensemble no longer supports cross-slit modulation because the required spacetime correlations have been physically removed.

Consequently, the spacetime background supports only a weighted ensemble of worldlines, each shaped by local geometric fluctuations conditioned on the presence of a detector. The ensemble becomes a classical mixture of geodesic sets—each defined by a single-slit configuration—but without mutual causal influence between them. Structured interference patterns cannot emerge from such causally disconnected sub-geometries.

Thus, the effect of measurement in TSRT is not the collapse of a wavefunction, but the pre-conditioning of the metric geometry by the detection-inclusive boundary. What vanishes is not interference in the quantum sense, but the deterministic, geometry-mediated coherence that previously allowed the slits to jointly shape the geodesic ensemble. The resulting arrival distribution on the screen becomes a superposition—not of amplitudes—but of classically distinct, non-interfering geometrical contributions.

In contrast to the mysterious, non-local collapse postulated in quantum mechanics, TSRT restores physical transparency: measurement exerts no magical influence on particles, but instead modifies the underlying boundary conditions of spacetime itself; it is the installation of the detector—not the act of observation—that breaks slit symmetry, constraining the permissible geodesic structure and producing a causally distinct, yet fully deterministic, measured world.

In this view, reality is not shaped by observation, but by the causal architecture of the world in which observation becomes possible—causality, which is the very foundational assumption underlying the theory of trembling spacetime (TSRT [17]).

## 6.3 Phase-Based Summary of Slit Configurations

The two principal cases discussed above—undetected and measured—can be completed with a decohered situation and understood within a unified framework by considering the relative phase between contributions from slits A and B in the effective aperture function. Although TSRT does not rely on wave-based superposition, the spatial Fourier structure of the slit geometry still governs the angular redirection of geodesics, and relative phase plays a critical role in determining the outcome.

In what follows, we assume that each slit is described by a rectangular aperture function:

$$S_A(x) = \Pi\left(\frac{x + d/2}{a}\right), \quad S_B(x) = \Pi\left(\frac{x - d/2}{a}\right), \qquad (18)$$

where $\Pi(x/a)$ denotes the rectangular function of width $a$, centered at $x = 0$. These functions define the geometric contributions of slits A and B, respectively, to the aperture configuration.
**Case 1: Undetected configuration (coherent phase).** When no which-path detection is installed, the full aperture is active and symmetrically realized. This corresponds to a relative phase of 0 between slits A and B:

$$S(x) = S_A(x) + S_B(x), \qquad (19)$$

yielding a Fourier transform modulated by a cosine envelope. The resulting geodesic redirection pattern exhibits constructive and destructive interference regions, forming a structured fringe distribution on the screen.

**Case 2: Measured configuration (fixed $\pi/2$ phase).** When a detector is present, the metric geometry enforces mutual exclusivity: a particle geodesic may pass through only one slit at a time. This corresponds to a relative phase difference of $\pi/2$ between the slits:

$$S'(x) = S_A(x) + iS_B(x), \qquad (20)$$

which ensures that the real-valued geometric contribution arises from only one slit at a time, reflecting causal separation. This results in an arrival distribution that is the incoherent sum of two single-slit patterns with no interference modulation.

**Case 3: Decohered configuration (random phase).** In the intermediate scenario, both slits remain open, though not in a binary manner, while environmental factors introduce stochastic fluctuations in their relative phase. This is modeled by:

$$S_{\phi}(x) = S_{A}(x) + e^{i\phi} S_{B}(x), \qquad (21)$$

where $\phi \in [0, 2\pi)$ varies randomly across trials.

**Unified geometric interpretation.** In TSRT, these three outcomes arise not from amplitude superposition, but from changes in the causal structure of spacetime as encoded in the slit boundary geometry. The phase models serve as geometric signatures of whether the trembling metric is coherent across the aperture or partitioned into causally independent regions. This phase-based classification sets the stage for Section 7.1, where the arrival patterns on the screen are derived from the aperture's spatial Fourier structure under each of these regimes.

# 7 TSRT Predictions and Agreement with Experiment

Unless stated otherwise, all approximate expressions in this section refer to the non-relativistic regime for clarity, with relativistic extensions addressed in Section 12. The theoretical predictions are supported by numerical simulations and strategies for more extensive simulations in the future, through geodesic ensembles in a trembling spacetime background, as described in Section 10.
## 7.1 Fringe Formation from Slit Geometry in TSRT

In Trembling Spacetime Relativity Theory (TSRT) [17], interference-like fringe patterns arise from the deterministic redirection of particle geodesics by localized metric fluctuations near slit apertures. Unlike quantum mechanics, where such patterns are attributed to the superposition of probability amplitudes, TSRT explains these patterns as the outcome of curvature-modulated geodesic ensembles filtered by the geometry of the slit configuration.

We consider a particle with longitudinal momentum $p_z$, directed toward a double-slit screen where two slits are separated by a distance $d$ and each slit has width $a$. The screen is located at longitudinal coordinate $z = 0$, and a detection plane lies at $z = z_1$. The transverse deviation $\Delta x$ of a particle trajectory is determined by the small redirection angle $\theta$, which in turn results from interaction with localized curvature perturbations near the slits:

$$ \Delta x = z_1 \cdot \theta. \qquad (22) $$

In TSRT, the aperture geometry modulates the geodesic ensemble by redirecting incoming deterministic paths into a structured fan of trajectories. This redirection is not random; it is governed by causal curvature gradients encoded in the metric perturbation field $\xi_{\mu\nu}(x)$, which varies near the slit edges. The spatial Fourier transform of the binary aperture function,

$$ S(x, y) = S_A(x, y) + S_B(x, y), \qquad (23) $$

describes the angular structure of the redirected geodesics. Here, $S_A$ and $S_B$ represent the open regions of Slits A and B, respectively.

The transverse angular deflection spectrum is then captured by:

$$ \tilde{S}(\kappa_x) = \int_{-\infty}^{\infty} S(x, y=0) e^{-i\kappa_x x} dx, \qquad (24) $$

where $\kappa_x$ is the transverse reciprocal number conjugate to $x$.

### 7.1.1 Geometric Mapping Between Reciprocal Space and Screen Position.

To relate the angular redirection of geodesics in TSRT to observable patterns on the detection screen, we define a purely geometric relationship between the spatial structure of the aperture and the transverse displacement of redirected particles. This mapping does not rely on quantum mechanical postulates or the assumption of particle wave behavior. Instead, it emerges from the deterministic geometry of trembling spacetime and the causal steering of particles along curved geodesics.

Consider a particle of longitudinal momentum $p_z$, moving along the $z$-axis toward the slit plane. Because the particle is localized and follows a deterministic trajectory, we can formally associate its propagation direction with a reciprocal-space representation via a geometric Fourier transform. If we define a spatial wavevector $\mathcal{K}_z$ along the $z$-axis such that it is proportional to the momentum,

$$ \mathcal{K}_z = \frac{p_z}{\mathcal{A}}, \qquad (25) $$
where $\mathcal{A}$ is a constant with dimensions of action (not necessarily Planck's constant), then we can interpret the forward propagation of the particle as acquiring a time-dependent phase governed by its motion: $z(t) = v_z t$, implying a phase $\sim e^{iK_z z}$. This complex exponential is not a physical wave phase, but a mathematical representation of geometric displacement in reciprocal space. It serves as a compact way to express translational symmetry and the spatial response of curved geodesics to structured boundary conditions. No oscillatory behavior or complex-valued physical fields are implied. This representation holds only in the localized neighborhood where the metric fluctuations are small and the trembling background can be linearized. It does not imply global phase coherence or interference across geodesics.

The identification of $\mathcal{K}_x$ as proportional to momentum does not rely on any quantum mechanical duality. Rather, it is a geometric encoding of how small angular deflections accumulate in the detection plane due to transverse curvature gradients. These Fourier components describe spatial steering fields—not probability amplitudes or matter waves.

Now, suppose the particle undergoes a small angular deflection $\theta$ due to interaction with localized metric fluctuations near the aperture. This redirection implies the emergence of a transverse momentum component $p_x = p_z \theta$, and hence a corresponding transverse reciprocal vector:

$$ \mathcal{K}_x = \frac{p_x}{\mathcal{A}} = \frac{p_z \theta}{\mathcal{A}}. \qquad (26) $$

Because the deviation occurs at a screen distance $z_1$, the transverse position of arrival is provided in Equation (22). Substituting yields:

$$ \mathcal{K}_x = \frac{p_z}{\mathcal{A}} \cdot \frac{\Delta x}{z_1}. \qquad (27) $$

Rewriting in terms of $x$, we obtain:

$$ x = \frac{\mathcal{A} \mathcal{K}_x z_1}{p_z}. \qquad (28) $$

Although we employ Fourier transforms to analyze how spatial boundary conditions influence redirection angles, we emphasize that this does not imply an underlying wavefunction or oscillatory physical field. The transform captures the geometrically induced redirection of deterministic geodesics in trembling spacetime, not a wave-like amplitude distribution. Thus, the reciprocal-space structure reflects curvature-induced steering statistics—not quantum coherence or interference.

To understand why the particle "acquires" a transverse wavevector $\mathcal{K}_x$, we must analyze the mechanism through which aperture geometry influences geodesic direction. In TSRT, spacetime is not flat but exhibits locally coherent curvature fluctuations induced by boundary geometry. The slit aperture imposes a sharply defined boundary on the metric perturbation field $\xi_{\mu\nu}(x)$, which responds causally to the aperture's spatial structure. This boundary modulates the curvature tensor in a spatially coherent way, such that the local geodesic deviation tensor $D^\mu_\nu$ becomes Fourier-structured in the transverse plane. That is, in the linearized regime, each transverse component $D^i_j(x, y)$ acquires spatial modulation governed by the Fourier modes of the aperture boundary function $S(x, y)$. This component-wise structure reflects the causal encoding of slit geometry in the curvature field.
Indeed, in standard general relativity, the geodesic deviation equation governs how nearby geodesics separate due to curvature:

$$ \frac{D^2\xi^\mu}{d\tau^2} = -R^\mu_{\nu\alpha\beta}u^\nu\xi^\alpha u^\beta, \qquad (29) $$

where $\xi^\mu$ is the separation vector between infinitesimally close geodesics, $u^\mu$ is the tangent vector to the fiducial geodesic, and $R^\mu_{\nu\alpha\beta}$ is the Riemann curvature tensor. Defining the geodesic deviation tensor $D^\mu_\nu = R^\mu_{\alpha\nu\beta}u^\alpha u^\beta$, the equation becomes:

$$ \frac{D^2\xi^\mu}{d\tau^2} = -D^\mu_\nu \xi^\nu. \qquad (30) $$

If the curvature perturbations $\delta R^\mu_{\nu\alpha\beta}(x)$ arise from a boundary condition such as a rectangular slit aperture, then the curvature field inherits the symmetries of that boundary. In TSRT, boundary geometry such as slits does not source curvature through stress-energy, but through causal constraints on the permissible local metric fluctuations. The boundary imposes a constraint on the permissible local modes of metric fluctuation in trembling spacetime, effectively acting as a geometric filter.

In the linearized regime of general relativity, spacetime deviations from flat Minkowski space are described by small perturbations $\xi_{\mu\nu}(x)$ around the background metric $\eta_{\mu\nu}$, such that the full metric takes the form Equation (1). In TSRT, these perturbations encode the trembling structure of spacetime at small scales, and represent the local geometric fluctuations through which causal steering occurs. The components of $\xi_{\mu\nu}(x)$ are treated as dynamical fields constrained by causal structure and boundary geometry, rather than by stress-energy sources alone. These fluctuations govern how geodesics deviate from straight lines in the presence of slit-imposed geometric constraints.

Rather than sourcing curvature via energy-momentum, the aperture restricts the configuration space of allowed $\xi_{\mu\nu}(x)$, and this restriction imprints the boundary’s Fourier structure onto the curvature tensor in the linearized regime. For rectangular slits, the aperture function $S(x, y)$ has a well-defined spatial Fourier transform $\tilde{S}(\kappa_x, \kappa_y)$, and the boundary-induced curvature perturbations—being sourced by local stress-energy or geometric constraints—are spatially modulated in a way that reflects this aperture structure.

Since the geodesic deviation tensor $D^\mu_\nu$ is constructed from contractions of the Riemann tensor with physical four-velocities, it retains gauge-invariant meaning under infinitesimal diffeomorphisms. Thus, the spatial modulation in $D^\mu_\nu$ reflects real geometric structure, not coordinate artifacts.

In particular, if the metric perturbations $\xi_{\mu\nu}(x)$ obey linearized Einstein equations near the slit boundary, then curvature responds linearly to $\xi_{\mu\nu}(x)$ and their derivatives. This analysis assumes the validity of linearized gravity in a harmonic (de Donder) gauge [26, 27], where the curvature response to boundary-imposed $\xi_{\mu\nu}(x)$ is local and additive. Under these conditions, the Fourier structure of $S(x, y)$ is directly inherited by the linear components of the Riemann tensor and hence by the projected geodesic deviation tensor $D^\mu_\nu$. This causes $D^\mu_\nu(x, y)$ to inherit the same transverse Fourier content as the boundary profile $S(x, y)$. Thus, the result-
ing deviation equation encodes a spatially structured response to the aperture, with geodesic deviation modulated in accordance with the Fourier spectrum of the slit geometry.

As a particle enters this curved region, its geodesic continues deterministically—but now through a spacetime whose effective geometry has been tilted by the Fourier content of the boundary-induced metric deformation. The geodesic's intrinsic direction remains forward in proper time, but in Euclidean embedding space, this manifests as a small angular deviation. Thus, the particle does not select a $K_x$ from the Fourier transform of the aperture in any active sense; rather, the aperture causes the surrounding spacetime geometry to develop angular Fourier components that steer the particle deterministically according to its local curvature field.

In other words, a given Fourier mode $K_x$ in the aperture's spatial structure results in a deterministic deflection of geodesics through the induced geometric potential. The resulting redirection translates into a transverse shift $x$ at the detector, not because of wave interference, but because the particle followed a curved geodesic through an aperture-shaped region of trembling spacetime.

This interpretation resolves a key conceptual difficulty: the mapping from aperture geometry to screen pattern is not a postulate in TSRT, but a derived consequence of how metric curvature, shaped by slit boundaries, deterministically redirects particle geodesics. The Fourier transform of the aperture geometry serves as a structural descriptor of the boundary-induced curvature field.

Although each particle follows a local geodesic determined by its immediate curvature environment, the ensemble of such geodesics—under the spatially coherent influence of slit-shaped curvature—is what gives rise to the observed macroscopic pattern. TSRT reproduces this structure not through interference, but through causal correlation of local curvature fields across the aperture.

For ease of comparison with standard experimental literature, we may set $\mathcal{A} = \hbar$ and write the expressions using familiar notation. This substitution is not a physical assumption of TSRT, but a notational convenience. It is justified a posteriori by the fact that in foundational work [17], the constant $\hbar$ emerges naturally from geometric bounds on minimal action enforced by curvature constraints. In the current paper, this emergence also reappears in the causal cutoff and resolution limit derived in Section 11 and Section 7.2.

We emphasize that while the notation may coincide with that of quantum mechanics, its interpretation within TSRT is fundamentally different. No probabilistic amplitudes or particle waves are involved. The geometric mapping is entirely a result of deterministic curvature-induced steering, and the proportionality constants serve only as a bridge for comparison, not as foundational assumptions.

### 7.1.2 Single slit.

For a single rectangular slit of width $a$, this produces a sinc envelope:

$$ \tilde{S}_A(\kappa_x) \sim \text{sinc} \left( \frac{a\kappa_x}{2} \right). \qquad (31) $$

The corresponding particle distribution on the detection screen is then obtained by evalu-
ating the squared magnitude of this angular spectrum:

$$ \rho_A(x) \propto \left| \tilde{S}_A \left( \mathcal{K}_x = \frac{p_z x}{\mathcal{A}z_1} \right) \right|^2 \sim \mathrm{sinc}^2 \left( \frac{ap_z x}{2\mathcal{A}z_1} \right), \quad (32) $$

where we have used the geometric correspondence derived in Equation (28), which maps transverse reciprocal variable $\mathcal{K}_x$ to screen position $x$ via the deterministic redirection of particle geodesics in trembling spacetime.

This result yields a central maximum with diminishing side lobes, not as a manifestation of self-interference, but as a direct geometric consequence of the angular steering induced by local curvature near the aperture.

### 7.1.3 Unmeasured case.

For a symmetric double-slit configuration centered at the origin with separation $d$, the total aperture function becomes:

$$ S(x) = \Pi\left(\frac{x+d/2}{a}\right) + \Pi\left(\frac{x-d/2}{a}\right) = S_A(x) + S_B(x), \quad (33) $$

where $\Pi(x/a)$ is the rectangular function of width $a$. The Fourier transform yields:

$$ \tilde{S}(\kappa_x) \sim \cos\left(\frac{\kappa_x d}{2}\right) \cdot \mathrm{sinc}\left(\frac{a\kappa_x}{2}\right), \quad (34) $$

exhibiting interference fringes through the cosine modulation.

The resulting particle distribution on the screen is determined by the squared magnitude of the total angular spectrum:

$$ \rho_{AB}(x) \propto \left| \tilde{S}(\mathcal{K}_x) \right|^2 = \cos^2\left(\frac{\mathcal{K}_x d}{2}\right) \cdot \mathrm{sinc}^2\left(\frac{a\mathcal{K}_x}{2}\right), \quad (35) $$

where $\mathcal{K}_x = \frac{p_z x}{\mathcal{A}z_1}$ as defined in Equation (28), describes the mapping from transverse reciprocal structure to the particle's arrival position. This expression reproduces the structured fringe pattern observed in experiments as a consequence of the aperture geometry shaping the deterministic geodesic ensemble.

The spacing between adjacent fringes is determined by the reciprocal modulation period $\Delta\mathcal{K}_x = 2\pi/d$. From the geometric mapping, the corresponding spatial fringe spacing on the detection screen is:

$$ \Delta x = \frac{\mathcal{A}z_1}{p_z} \cdot \Delta \mathcal{K}_x = \frac{\mathcal{A}z_1}{p_z} \cdot \frac{2\pi}{d} = \frac{2\pi \mathcal{A}z_1}{p_z d}. \quad (36) $$

If we wish to express this in standard experimental form for ease of comparison, we may identify $\mathcal{A} = \hbar$, giving:

$$ \Delta x = \frac{\lambda z_1}{d}, \quad \text{with} \quad \lambda = \frac{2\pi\hbar}{p_z}, \quad (37) $$

where $\lambda$ is introduced solely for compatibility with existing literature. In TSRT, this result arises without invoking de Broglie wavelengths or quantum interference; the observed pattern is entirely a consequence of deterministic geodesic deflection governed by aperture geometry.
### 7.1.4 Measured case: mutually exclusive geodesic ensembles.

When a which-slit measurement device is introduced, each particle becomes causally constrained to traverse only one of the two slits. This does not imply that individual particles behave differently—each geodesic still passes through only one slit, as in the undetected case—but rather that the underlying spacetime geometry becomes causally partitioned: when a detector is present, the trembling metric associated with slit A evolves independently of that at slit B, as if the other slit were absent. This mutual exclusivity applies not to the path itself, but to the geometric environment surrounding each path, breaking the coherence that would otherwise allow the two slits to jointly shape the geodesic ensemble. This boundary-induced separation eliminates the causal cross-linking necessary for interference, and results in a geometrically disconnected ensemble.

To model this deterministic exclusivity, we represent the aperture function with a fixed phase shift of $\pi/2$ between the two slits. This phase shift rotates the second slit contribution out of phase with the first in such a way that, at any given location, the real part of the aperture function arises from only one slit at a time:

$$S'(x) = S_A(x) + e^{i\pi/2} S_B(x) = S_A(x) + iS_B(x), \qquad (38)$$

where $S_A(x)$ and $S_B(x)$ are the rectangular functions corresponding to slit A and slit B respectively. The key point is that $S_A(x)$ and $S_B(x)$ are binary-valued and non-overlapping: for any $x$, only one of them is nonzero. As a result, the aperture function $S'(x)$ is nonzero only in the real direction when $S_A(x) = 1$, and only in the imaginary direction when $S_B(x) = 1$.

Taking the Fourier transform yields:

$$\tilde{S}'(\kappa_x) = \tilde{S}_A(\kappa_x) + i\tilde{S}_B(\kappa_x). \qquad (39)$$

The arrival distribution is given by the squared modulus of this transform:

$$\rho(x) \propto |\tilde{S}'(\kappa_x)|^2 = |\tilde{S}_A(\kappa_x) + i\tilde{S}_B(\kappa_x)|^2. \qquad (40)$$

Expanding this expression gives:

$$\rho(x) \propto |\tilde{S}_A|^2 + |\tilde{S}_B|^2 + 2 \operatorname{Im}[\tilde{S}_A \tilde{S}_B^*]. \qquad (41)$$

To show that the interference term vanishes, we exploit the fact that $S_A(x) \cdot S_B(x) = 0$ for all $x$: the slits are spatially disjoint. Therefore,

$$S'(x) = \begin{cases} 1, & x \in \text{slit A} \\ i, & x \in \text{slit B} \\ 0, & \text{elsewhere} \end{cases} \qquad (42)$$

Thus, the aperture function is piecewise constant and non-overlapping in its real and imaginary
components. When computing the cross term:

$$ \tilde{S}_A(\kappa_x) \tilde{S}_B^*(\kappa_x) = \int S_A(x) e^{-i\kappa_x x} dx \cdot \int S_B(x') e^{i\kappa_x x'} dx', \quad (43) $$

this becomes an integral over disjoint supports, resulting in a product of orthogonal complex amplitudes with imaginary phase difference. Consequently,

$$ \mathrm{Im} \left[ \tilde{S}_A(\kappa_x) \tilde{S}_B^*(\kappa_x) \right] = 0. \quad (44) $$

Hence, the interference term vanishes identically:

$$ \rho(x) \propto |\tilde{S}_A(\kappa_x)|^2 + |\tilde{S}_B(\kappa_x)|^2. \quad (45) $$

To make this explicit:

$$ |\tilde{S}_A(\kappa_x)|^2 = |\tilde{S}_B(\kappa_x)|^2 = a^2 \cdot \mathrm{sinc}^2\left(\frac{a\kappa_x}{2}\right), \quad (46) $$

$$ \rho(x) \propto 2a^2 \cdot \mathrm{sinc}^2\left(\frac{a\kappa_x}{2}\right), \quad (47) $$

$$ \Rightarrow \rho(x) \propto \mathrm{sinc}^2\left(\frac{ap_zx}{2Az_1}\right), \quad (48) $$

matching the single-slit profile with doubled intensity due to independent contributions from both slits, as in Equation (32). We used the correspondence established in Equation (28).

In TSRT, this outcome is not attributed to a probabilistic collapse or loss of phase information, but to the causal restructuring of the spacetime geometry induced by the detection boundary. The measurement device breaks the geometric coherence between the slits, replacing the structured geodesic ensemble with a classical mixture of mutually exclusive deterministic trajectories. The phase model captures this causal exclusivity without invoking any stochastic or wave-like assumptions.

### 7.1.5 Random phase case: incoherent but not mutually exclusive slits.

We now consider a scenario in which both slits are open and causally accessible, but the relative phase between their contributions varies randomly for each particle. This models a realistic form of geometric decoherence in TSRT, where trembling curvature near the slits remains locally coherent, but fluctuating boundary conditions introduce causal phase variability between the two geodesic bundles.

The aperture function is modeled with a random relative phase $\phi \in [0, 2\pi)$:

$$ S_{\phi}(x) = \Pi\left(\frac{x+d/2}{a}\right) + e^{i\phi} \Pi\left(\frac{x-d/2}{a}\right), \quad (49) $$

where the phase $\phi$ fluctuates stochastically across the ensemble, but both slits remain physically open for every particle. The corresponding Fourier transform is:

$$ \tilde{S}_{\phi}(\kappa_x) = \left[ e^{i\kappa_x d/2} + e^{i\phi} e^{-i\kappa_x d/2} \right] \cdot \mathrm{sinc}\left(\frac{a\kappa_x}{2}\right), \quad (50) $$
which simplifies to:

$$ \tilde{S}_{\phi}(\kappa_x) = 2 \cos \left( \frac{\kappa_x d - \phi}{2} \right) \cdot e^{i(\kappa_x d + \phi)/2} \cdot \text{sinc} \left( \frac{a\kappa_x}{2} \right). \qquad (51) $$

Taking the modulus squared and averaging over all phase realizations yields:

$$ \rho(x) \propto \left\langle \left| \tilde{S}_{\phi}(\kappa_x) \right|^2 \right\rangle_{\phi} = 4 \left\langle \cos^2 \left( \frac{\kappa_x d - \phi}{2} \right) \right\rangle_{\phi} \cdot \text{sinc}^2 \left( \frac{a\kappa_x}{2} \right). \qquad (52) $$

Since:

$$ \left\langle \cos^2 \left( \frac{\kappa_x d - \phi}{2} \right) \right\rangle_{\phi} = \frac{1}{2}, \qquad (53) $$

we obtain:

$$ \rho(x) \propto \text{sinc}^2 \left( \frac{a\kappa_x}{2} \right), \qquad (54) $$

which corresponds precisely to the single-slit profile:

$$ \rho(x) \propto \text{sinc}^2 \left( \frac{ap_z x}{2Az_1} \right). \qquad (55) $$

This result shows that the loss of phase coherence, even when both slits remain open, eliminates the structured modulation from inter-slit correlations. The resulting density features a central peak and side lobes determined entirely by slit width $a$ and angular deflection, just as in the single-slit case.

Importantly, while the mathematical form of $S_{\phi}(x)$ appears to permit partial transmission amplitudes, this may not be physically realistic for individual particles: a given particle either traverses a slit or it does not. TSRT maintains that slits are binary for each particle, and causal geodesics are either permitted or blocked by the aperture. Therefore, the physical interpretation of the above averaging must be statistical, not physical per trial.

**Binary Realization Model.** To remain physically realistic, we now reinterpret the decoherence scenario under the constraint that the slits remain binary in each realization. That is, for each particle, either slit A or slit B contributes, but with randomized relative frequency across the ensemble—mimicking decoherence without invoking partial transmissivity.

Let the probability that a particle's geodesic accesses slit A be $P_A$, and slit B be $P_B = 1 - P_A$, with no phase coherence between them. Then the arrival density becomes:

$$ \rho(x) = P_A \rho_A(x) + P_B \rho_B(x), \qquad (56) $$

where $\rho_{A,B}(x)$ are the single-slit diffraction profiles for slits A and B respectively. Each has the form:

$$ \rho_{A,B}(x) \propto \text{sinc}^2 \left( \frac{ap_z(x \mp d/2)}{2Az_1} \right). \qquad (57) $$

If $P_A = P_B = 1/2$, then:

$$ \rho(x) = \frac{1}{2} \left[ \text{sinc}^2 \left( \frac{ap_z(x + d/2)}{2Az_1} \right) + \text{sinc}^2 \left( \frac{ap_z(x - d/2)}{2Az_1} \right) \right]. \qquad (58) $$
This produces two offset single-slit patterns centered at $x = \pm d/2$, without interference fringes. The central region exhibits overlap, but not modulation.

**Quantum Mechanical Contrast.** In standard quantum mechanics, the scenario of random phase decoherence with both slits open is modeled by a mixed state composed of partial wave-function contributions from both slits. Although the ensemble-averaged intensity reproduces the same single-slit envelope as TSRT in the limit of complete phase randomization, the underlying interpretation differs sharply. In QM, each particle is still described by a delocalized superposition $\psi_A + e^{i\phi}\psi_B$, and decoherence arises from environmental entanglement or statistical phase averaging over many realizations. The resulting loss of interference reflects phase cancellation in a complex amplitude landscape.

By contrast, in TSRT, each particle always follows a single, localized geodesic path through either slit A or B. Decoherence does not arise from loss of phase coherence between amplitudes, but from variability in the causal structure of the trembling metric, which determines slit accessibility on a per-particle basis. There is no sense in which a particle partially traverses both slits or exists in a spatially superposed state.

Therefore, although both frameworks yield similar arrival densities in the decohered limit, their predictions diverge in finer regimes. TSRT forbids any residual interference fringes or phase-induced modulation if each particle is causally constrained to a single slit, whereas QM can admit residual coherence depending on the decoherence mechanism. Precision experiments testing for weak interference remnants under engineered decoherence can thus distinguish between these fundamentally different models.

**Interpretational Contrast.** The distinction between the random-phase model and the binary-realization model is not merely mathematical but reflects a fundamental ontological divide. In the random-phase model, both slits contribute simultaneously with fluctuating phase, implying fractional or partial amplitude transmission per particle. This resembles a quantum-mechanical mixed state or partially decohered wavefunction.

By contrast, the binary-realization model, favored by TSRT, maintains that for each individual particle, only one slit is causally traversed. The metric trembling determines, deterministically and locally, which geodesic is permissible. Thus, while the observed arrival density may resemble that of the incoherent superposition, the underlying mechanism is entirely different: there are no complex amplitudes, no interference terms, and no delocalized states.

In quantum mechanics, even when decoherence removes fringe visibility, the particle is still formally described by a spatially extended wavefunction traversing both slits simultaneously. Decoherence arises from entanglement with the environment and suppresses observable interference, but does not localize the particle until measurement.

TSRT, by contrast, offers a fully localized, realist picture in which decoherence stems from ensemble-level variability in the causal structure of spacetime, not from environmental entanglement. Thus, while both theories may yield similar fringe-suppressed patterns under full decoherence, they diverge sharply in physical ontology and in predictions for partial or structured decoherence. Precision experiments modulating curvature locally near one slit can distinguish
between these frameworks by testing for residual structure that would persist in quantum models but be absent in TSRT's binary geodesic model.

### 7.1.6 Conclusion.

TSRT explains interference-like fringe patterns not through wave-particle duality or complex amplitude superposition, but via deterministic geodesic redirection shaped by aperture-induced curvature structures. The Fourier profile of the aperture governs the angular steering of geodesics, resulting in structured particle arrival densities without invoking quantum postulates.

In the fully coherent regime, both slits remain causally accessible, and the trembling metric maintains spatial phase coherence across them. This leads to a modulated screen pattern, despite each particle traversing only one slit. In the measured regime, which-path detection introduces causal isolation between slits, breaking this coherence and yielding a sum of distinct single-slit profiles.

In the decoherence regime, TSRT distinguishes two conceptually distinct models. The first—mathematically valid but physically unrealistic—involves incoherent superposition with partial amplitudes from both slits. The second—the physically realistic scenario endorsed by TSRT—assumes binary slit passage per particle, with decoherence arising from ensemble variability in the causal geodesic structure. The resulting screen pattern lacks modulation but preserves deterministic angular deflection through the active slit.

Critically, while both models can match the observed intensity distribution under full decoherence, their ontological and causal interpretations differ fundamentally from quantum mechanics. In QM, decoherence suppresses interference but retains delocalized wavefunctions until collapse. In TSRT, particles are always localized, and interference arises from geometric modulation, not amplitude interference.

Thus, TSRT provides a falsifiable, geometric account of interference phenomena, where modulation reflects spacetime structure—not wave behavior—and decoherence is a causal, not probabilistic, process. Precision tests targeting partial decoherence or asymmetric aperture modulation offer a clear path to empirically distinguish these models.

## 7.2 Minimal Action and Curvature-Based Resolution in TSRT

In TSRT, Planck's constant $ħ$ emerges not as a postulated quantum quantity but as a natural consequence of the causal structure of trembling spacetime. This subsection develops how a minimal action threshold arises from the geometric requirement that neighboring geodesics be resolvable only if their proper-time-separated action values differ by at least one quantum of action.

Each particle trajectory in TSRT follows a deterministic path that extremizes the proper-time action:

$$A_{\gamma} = -m \int_{\gamma} d\tau, \qquad (59)$$

where $\gamma$ denotes a timelike geodesic path through the trembling spacetime. Due to local fluctuations in the metric tensor $g_{\mu\nu}(x) = \eta_{\mu\nu} + \xi_{\mu\nu}(x)$, two neighboring geodesics $\gamma$ and $\gamma'$ accumulate
different proper times over the same coordinate interval, leading to a difference in action:

$$ \Delta A = -m(\tau' - \tau) = -m\Delta\tau. \qquad (60) $$

To be distinguishable on a detection screen, two such trajectories must produce resolvable transverse deviations. However, TSRT asserts that this only occurs if the accumulated difference in action exceeds a minimal threshold:

$$ |\Delta A| \geq \mathcal{T}_{\min}, \qquad (61) $$

where $\mathcal{T}_{\min} \sim \hbar$ is not an imposed constant but a geometric consequence of curvature coherence and causal filtering.

To relate this to transverse displacement, consider a finite longitudinal propagation length $L$, over which a small deflection angle $\Delta\theta$ leads to a transverse shift $\Delta x = L \cdot \Delta\theta$. The associated proper-time difference between such paths is approximated by:

$$ \Delta\tau \approx \frac{L\Delta\theta}{v}, \qquad (62) $$

so the action difference becomes:

$$ \Delta A = -m \frac{L\Delta\theta}{v}. \qquad (63) $$

Requiring $|\Delta A| \geq \hbar$, we find the minimal angular deflection:

$$ \Delta\theta_{\min} = \frac{\hbar p_z v}{mL}. \qquad (64) $$

This sets a causal limit on angular resolvability due to trembling curvature. The corresponding transverse momentum shift is:

$$ \Delta p_{\perp} = p_z \cdot \Delta\theta_{\min} = \frac{\hbar p_z v}{mL}. \qquad (65) $$

Although this does not directly yield the observed fringe spacing (which is determined by slit separation as shown earlier), it reveals a complementary resolution scale: the minimal angular separation that curvature-induced deviations must reach to be distinguishable as separate arrivals. This scale constrains the granularity of observed patterns and is fundamentally geometric.

Therefore, in TSRT, Planck's constant $\hbar$ appears as the minimal causal action interval that separates neighboring geodesics in a metrically trembling spacetime. It sets the finest achievable discrimination between particle trajectories—not by probabilistic uncertainty, but through curvature-constrained determinism.

If this minimal angular deflection $\Delta\theta_{\min}$ were measured with sufficient precision across particles of varying mass and velocity, it would reveal Planck's constant not through postulates or quantum superposition, but as a direct geometric consequence of the causal structure of trembling spacetime. Such an outcome would mark a shift in how we define the quantum-classical boundary—grounding it in metric coherence rather than wavefunction interpretation.
## 7.3 TSRT vs. Quantum Theory: Agreement and Divergences

Although Trembling Spacetime Relativity Theory (TSRT) reproduces the experimentally observed fringe patterns in double-slit experiments, its physical interpretation diverges sharply from that of standard quantum mechanics. This subsection summarizes key areas of agreement and contrast between the two frameworks.

**Agreements in Observable Predictions:** TSRT and quantum theory yield numerically identical predictions for fringe spacing in standard two-slit geometries under symmetric path conditions (i.e., $L = z_1$), where:

$$ \Delta x = \frac{2\pi Az_1}{p_z d}. \qquad (66) $$

When $A = \hbar$, this becomes:

$$ \Delta x = \frac{\lambda z_1}{d}, \qquad (67) $$

where $\lambda = h/p_z$ is the de Broglie wavelength. Although TSRT does not interpret this result via wave-particle duality, it arises in TSRT from deterministic angular deflection filtered by slit geometry and curvature coherence. While TSRT does not interpret this result through wave-particle duality, it coincides numerically with the standard quantum mechanical prediction.

Both frameworks predict fringe shift under asymmetric or phase-modulated conditions, consistent with observed results across photons, electrons, neutrons, and atoms.

Both theories correctly predict fringe suppression in the presence of which-slit detection.

**Divergences in Ontology and Mechanism:** In Quantum Theory, particles are described by delocalized wavefunctions that evolve unitarily and collapse probabilistically upon measurement. Interference arises from complex amplitude superposition and phase-sensitive cross-terms.

In TSRT, particles remain localized at all times and follow deterministic geodesics through a causally fluctuating spacetime. Interference patterns emerge from the angular redistribution of these geodesics, structured by the Fourier geometry of the slit configuration, with no role for probability or collapse.

**Role of Planck's Constant:** In standard quantum theory, $\hbar$ is an axiomatically introduced scale for quantization and uncertainty.

In TSRT, $\hbar$ arises naturally as a minimal unit of action required to distinguish neighboring geodesics in a curved spacetime with finite resolution. This is derived from causal and geometric constraints, as detailed in Section 11.

**Experimental Divergences:** TSRT predicts subtle deviations from quantum predictions under specific geometric or dynamic conditions, including:

* Fringe contrast suppression under rapid geometric phase modulation, even without environmental decoherence.

* Temporal hysteresis effects in fringe realignment following abrupt changes in boundary geometry.
* Structured arrival distributions even with one slit open, due to residual curvature modulation.

* Distinct spatial profiles in decohered regimes, as analyzed in Section 7.1.5, where TSRT predicts binary single-slit accumulation patterns instead of the smooth incoherent superposition of quantum amplitudes.

These differences are experimentally testable and provide pathways to falsify or confirm TSRT in regimes beyond conventional quantum theory. See Sections 9 and 12 for proposals.

## 7.4 Summary: Deterministic Fringe Patterns Without Superposition

Trembling Spacetime Relativity Theory (TSRT) provides a deterministic, geometric explanation for interference phenomena traditionally attributed to quantum superposition and wave-particle duality. In TSRT, particles remain localized at all times and follow unique proper-time-parametrized geodesics through a causally coherent but metrically fluctuating spacetime.

Fringe patterns arise not from self-interference of delocalized wavefunctions, but from the structured redirection of geodesics by curvature fluctuations near the slit geometry. As demonstrated in Section 7.1, the binary aperture acts as a spatial filter, shaping the angular distribution of trajectories according to the Fourier structure of the slit mask. This yields the observed fringe spacing of Equation (67),

$$ \Delta x = \frac{2\pi Az_1}{p_z d}, \qquad (68) $$

mapping directly from the aperture's reciprocal structure to screen position.

Section 7.1.5 shows that even under decoherence, TSRT yields spatial patterns arising from binary slit passage per particle, rather than fractional amplitude superposition. This difference becomes critical in high-resolution experiments designed to probe the microscopic structure of decoherence.

Section 7.2 further establishes that Planck's constant $\hbar$ emerges naturally in TSRT as the minimal action difference required to causally distinguish neighboring geodesics. This defines a fundamental resolution threshold rooted in causal curvature coherence, not operator uncertainty, and sets the minimal detectable angular separation between distinguishable particle trajectories.

Section 9 provides simulation details and experimental proposals capable of distinguishing TSRT from quantum theory, especially under asymmetric, time-dependent, or causally modulated slit conditions.

Numerical simulations in Section 10 confirm that deterministic geodesic ensembles reproduce all features of observed interference patterns, including fringe visibility and spacing. These results hold across non-relativistic and relativistic regimes.

In summary, TSRT recovers all known interference effects through causal, deterministic geodesic redirection—eliminating the need for wavefunction postulates, nonlocality, or indeterminism. The theory reframes interference as a classical phenomenon emergent from the geometry of trembling spacetime, and offers experimentally testable predictions that differentiate it from quantum mechanics in subtle but crucial ways.
# 8 Comparison with Quantum Wave Optics

Although TSRT and quantum mechanics yield mathematically identical interference patterns under symmetric and static slit configurations, they differ profoundly in their physical interpretations of the underlying mechanism.

**Mathematical Agreement.** For particles of longitudinal momentum $p_z$, both theories predict an intensity distribution on the detection screen of the form:

$$I(x) \propto \cos^2 \left( \frac{\pi dx}{\lambda z_1} \right) \cdot \left[ \frac{\sin \left( \frac{\pi ax}{\lambda z_1} \right)}{\frac{\pi ax}{\lambda z_1}} \right]^2, \qquad (69)$$

where $d$ is the slit separation, $a$ the slit width, and $z_1$ the screen distance. Here, the sinc² term represents the squared normalized sinc function, defined as $\text{sinc}(x) = \sin(x)/x$, which arises from the Fourier transform of the single-slit aperture.

In quantum mechanics, the wavelength $\lambda = h/p_z$ is the de Broglie wavelength. In TSRT, an equivalent scale $\mathcal{L} = \hbar/p_z$ emerges not as a wavelength, but as the minimal spatial separation resolvable by geodesics due to proper-time differences exceeding the causal action threshold $\hbar$. This scale determines the angular resolution of redirected geodesics and sets the fundamental diffraction spacing in trembling spacetime. Unlike $\lambda$, which is postulated as a wavelength associated with wavefunctions, $\mathcal{L}$ arises geometrically from the condition $\Delta\mathcal{S} \gtrsim \hbar$ imposed by trembling spacetime.

Despite the identical functional form—i.e., Equation (69) also arises in TSRT as Equation (35)—the derivations differ fundamentally: quantum theory invokes amplitude interference between delocalized paths, while TSRT derives the pattern from curvature-filtered redirection of localized geodesics, governed by the Fourier structure of the aperture function $S(x, y)$.

The $\cos^2$ term encodes inter-slit modulation (periodic angular structure), while the $\sinc^2$ term reflects single-slit diffraction (finite aperture width). In both theories, the geometry of the aperture plays a central role—but with radically different ontological implications.

**Interpretational Differences.** Quantum mechanics describes particles as extended wavefunctions that evolve via linear superposition and collapse upon measurement. The observed fringe pattern is interpreted as a probability density derived from the coherent overlap of complex amplitudes traversing multiple paths.

In TSRT, particles are always localized and follow a single geodesic, determined by the local realization of the trembling metric $\xi_{\mu\nu}(x)$. The ensemble of such trajectories forms a structured arrival distribution—not due to wave interference, but due to deterministic redirection by curvature fluctuations that are themselves modulated by the slit geometry.

The aperture's Fourier transform $\tilde{S}(\kappa_x)$ does not represent a diffraction amplitude in the wave-optical sense, but functions as a geometric filter: it determines the directions in which curvature-induced redirection of localized geodesics statistically concentrates. This is a deterministic effect of the aperture's spatial structure on the ensemble of particle paths, not the
result of amplitude interference. The pattern arises from ensemble accumulation of many such
deterministic paths.

Thus, while QM sees interference as a consequence of nonlocal amplitude overlap and col-
lapse, TSRT attributes the same visual outcome to localized particle propagation through a
trembling but causally coherent spacetime. The result is not a probabilistic blur of possibilities,
but a geometric map of spacetime modulation.

**Ontological Implications.** This distinction is more than interpretational: it marks a differ-
ence in what is believed to exist. Quantum mechanics posits extended, superposed states and
collapses that defy locality and realism. TSRT offers a contrasting framework where particles
are always real, always local, and never in superposition. The apparent interference pattern is
an emergent effect of metric geometry, not a fundamental property of the particle.

**Experimental Implications.** Under idealized, symmetric conditions, TSRT and QM are ob-
servationally indistinguishable. But their mechanisms respond differently to boundary modifica-
tions and dynamic geometries. As developed in Section 9, TSRT predicts distinctive deviations
in fringe structure, contrast, and spatial asymmetry when causal coherence is disrupted—e.g.,
by asymmetric apertures, active filtering, or temporal modulation of the boundary.

These scenarios define a class of precision experiments capable of distinguishing between
geometric and wave-based explanations. In TSRT, interference is not a paradox but a signature
of deterministic geodesics filtered by spacetime structure. This perspective is both realist and
falsifiable—offering a testable alternative to the probabilistic framework of standard quantum
theory.

9 Experimental Proposal to Distinguish TSRT from Quantum Mechanics

TSRT and standard quantum mechanics make nearly identical predictions for interference phe-
nomena in symmetric, static configurations. However, their ontologies and mechanisms differ
fundamentally: TSRT attributes fringe formation to deterministic geodesic steering by aperture-
induced curvature, while quantum theory attributes it to the superposition of complex probabil-
ity amplitudes. This conceptual divergence offers a powerful opportunity: to design experiments
that target the source of pattern formation and thereby distinguish between these theories.

This section presents a focused experimental framework for probing such differences in the
non-relativistic regime. We emphasize configurations where causal coherence, geometric timing,
or local curvature modulation play a measurable role. The aim is to test whether interference
patterns originate from deterministic, local geodesic propagation through trembling spacetime
or from nonlocal amplitude interference as prescribed by quantum mechanics.

For relativistic generalizations—including high-energy or ultrafast beam configurations—see
the discussion in Section 12. The proposals here establish the foundation for empirically falsifi-
able experiments that can expose ontological distinctions, even when numerical predictions are
closely aligned.
In the subsections that follow, we begin by outlining the physical motivation and setup of the proposed experiment. We then describe several regimes where TSRT and quantum theory predict observationally similar but mechanistically distinct results, focusing on how fringe visibility, spatial asymmetry, or temporal response may provide measurable divergence.

## 9.1 Overview and Physical Motivation

The Trembling Spacetime Relativity Theory (TSRT) reproduces all standard double-slit interference patterns without invoking wavefunctions, complex amplitudes, or probabilistic superposition. Instead, TSRT attributes the observed patterns to deterministic redirection of particle geodesics through a causally fluctuating metric.

While both TSRT and quantum mechanics make identical predictions in symmetric slit configurations, their underlying mechanisms are fundamentally distinct. This opens the possibility for empirical differentiation under more refined experimental conditions—specifically, those targeting how geodesic ensembles respond to spatial or temporal geometric modulations.

TSRT interprets fringe formation as the statistical modulation of deterministic trajectories, filtered by the aperture geometry $S(x, y)$ and shaped by local metric fluctuations $\xi_{\mu\nu}(x)$. This contrasts with quantum theory, where fringe patterns emerge from coherent superposition of delocalized wavefunctions.

To experimentally distinguish TSRT from standard quantum theory, we propose a modified double-slit configuration in which the slits remain geometrically identical, but a controlled geometric phase offset is introduced behind one of them. In the TSRT framework, this offset corresponds to a proper-time delay experienced by geodesics traversing the affected slit region. Several physical implementations are possible: a thin material with a structured refractive index can be placed behind one slit to impose a localized geometric delay; alternatively, a vibrating film can be used to modulate the causal timing of particle propagation; or a pulsed laser field may be employed to induce a transient curvature perturbation, thereby modifying the local spacetime geometry experienced by the particle.

In TSRT, such modulation alters the local causal geometry—specifically, the proper-time experienced along the geodesic—resulting in deterministic shifts in arrival position. In quantum mechanics, by contrast, this same effect is interpreted as a complex phase shift in the interference of probabilistic amplitudes. While both theories predict similar outcomes in symmetric cases, the causal interpretation in TSRT reflects geometric timing offsets, not phase-based superposition.

If measurable differences in fringe location, contrast, or coherence emerge under asymmetric or dynamic conditions, TSRT and quantum theory may yield divergent outcomes.

In the remainder of this section, we construct this experimental proposal step-by-step, beginning with the setup and followed by TSRT and quantum predictions, concluding with a list of measurable observables designed to expose the fundamental differences in physical interpretation.
## 9.2 Distinguishing Binary Geodesic Realization from Incoherent Superposition

Section 7.1.5 describes two physically distinct interpretations of decoherence in the absence of which-slit detection: one where both slits contribute simultaneously with fluctuating phases (incoherent superposition), and another where only one slit contributes per trial, chosen deterministically (binary realization).

Quantum mechanics allows both descriptions depending on the model (density matrix vs. wavefunction collapse), but in TSRT, only the latter is physically meaningful. In TSRT, spacetime trembling causally restricts geodesic access to a specific slit in each trial, even in the absence of detection.

This implies a measurable distinction. If slit-local curvature modulation is applied—i.e., a small geometric perturbation added near only one slit—the binary model predicts a structural asymmetry in the arrival pattern, while the incoherent superposition model would wash out such effects.

An experiment using tunable jitter or geometric filters on one slit can therefore distinguish these mechanisms. If residual structure is observed in the averaged density—even in the decoherent regime—this favors TSRT’s causal binary model over standard incoherent addition of amplitudes.

## 9.3 Experimental Design: Phase-Controlled Slit Geometry

The proposed experiment modifies the standard double-slit apparatus to include a controlled geometric offset—interpreted in TSRT as a proper-time delay—behind one of the slits. The goal is to preserve geometric symmetry while introducing a causal asymmetry that affects the geodesic ensemble.

The experimental setup consists of two slits, each of width $a$, separated by a center-to-center distance $d$, with a detection screen positioned at a distance $z_1$ downstream. A key modification is introduced by placing a geometric modulation element—such as a refractive phase delay or curvature-altering device—exclusively behind Slit B, leaving Slit A unmodified. This asymmetry serves as the mechanism for generating a proper-time offset in TSRT, allowing for potential divergence from standard quantum predictions.

**Geometric Modulation Function** $S_{\text{mod}}(x, y)$: To formalize the aperture and post-slit structure, we define a generalized slit function that includes a local metric-induced offset:

$$S_{\text{mod}}(x, y) = \begin{cases} 1 & \text{for } (x, y) \in \text{Slit A,} \\ \mathcal{G}(x, y; \Delta\phi) & \text{for } (x, y) \in \text{Slit B,} \\ 0 & \text{elsewhere,} \end{cases} \qquad (70)$$

where $\mathcal{G}(x, y; \Delta\phi)$ encodes the causal modulation—a curvature-induced proper-time delay resulting from geometric alteration behind slit B. Although $\mathcal{G}(x, y; \Delta\phi)$ may appear structurally similar to a phase factor $e^{i\Delta\phi}$ in wave theory, its role in TSRT is fundamentally different: it encodes a proper-time delay or local curvature deformation that causally redirects geodesics.
There is no underlying complex amplitude or wavefunction—only a modification of spacetime geometry that changes the ensemble density on the detection screen.

**Physical Realization of Geometric Delay:** The controlled geometric phase offset behind Slit B can be implemented experimentally in several ways. One approach involves placing a thin transparent material with a structured refractive index directly behind the slit, effectively in- ducing a local phase delay. Alternatively, a vibrating membrane can introduce a time-dependent modulation of proper time, while a focused laser pulse may be used to transiently modify the local spacetime curvature and thereby influence geodesic redirection.

**Control Parameters:** Key control parameters in such an experiment include the phase offset $\Delta\phi$, which can be varied continuously up to several multiples of $2\pi$; the longitudinal momentum $p_z$, which remains fixed for each run but can be adjusted between runs; and the screen distance $z_1$, which is selected to ensure optimal fringe visibility and resolution.

**Particle Conditions:** Particles should be electrons, neutrons, or large molecules (e.g., $C_{60}$), as used in existing interference experiments. Flux is kept low to enforce single-particle condi- tions, with trajectories reconstructed statistically.

**Measurement Objective:** As $\Delta\phi$ varies, both quantum theory and TSRT predict shifts in the fringe pattern. However, the cause of this shift differs: TSRT attributes it to geodesic redirection due to causal deformation, while quantum mechanics interprets it as a shift in relative complex amplitude phase. Any asymmetries, hysteresis, or decoherence behavior that deviates from the quantum prediction may reveal the underlying causal structure of trembling spacetime.

In the next subsection, we derive the precise predictions of TSRT for this configuration.

## 9.4 TSRT Predictions for Phase-Modulated Slits

In TSRT, particles follow deterministic geodesics through a causally trembling spacetime. When a controlled geometric delay is introduced behind one slit—such as via a curvature bump or refractive offset—it modifies the causal structure that shapes permissible geodesic trajectories. This delay leads to a shift in the screen arrival distribution, without invoking any wavefunction or interference mechanism.

**Causal Interpretation of “Phase” Offset $\Delta\phi$:** In TSRT, $\Delta\phi$ is not a complex amplitude phase but a label for the proper-time delay $\Delta\tau$ induced by geometric modulation behind slit B. The two geodesic families—through slits A and B—accumulate different action:

$$ \Delta S = -m \Delta \tau, \qquad (71) $$

where the minus sign follows from $S = -m\tau$.

When $|\Delta S| \gtrsim \hbar$, the geodesic ensembles become distinguishable in arrival position. This leads to a measurable shift in the screen density.
**Screen Distribution Response:** We define a generalized aperture function $S_{\text{mod}}(x, y)$ that encodes the geometric offset via:

$$S_{\text{mod}}(x, y) = \begin{cases} 1, & (x, y) \in \text{Slit A}, \\ \mathcal{G}(x, y; \Delta\phi), & (x, y) \in \text{Slit B}, \\ 0, & \text{elsewhere}, \end{cases}$$

where $\mathcal{G}(x, y; \Delta\phi)$ encodes the curvature-induced delay field.

Although TSRT does not involve complex amplitudes or wavefunctions, the spatial Fourier transform of the aperture function still serves as a geometric descriptor of how aperture structure modulates curvature and thus redirects geodesic trajectories in a causally coherent ensemble,

$$\rho_{\text{TSRT}}(x; \Delta\phi) \propto |\tilde{S}_{\text{mod}}(\kappa_x)|^2. \qquad (72)$$

This expression does not reflect probability amplitudes but rather the spatial redirection density of causally permissible trajectories through the trembling geometry.

**Fringe Shift:** The causal offset $\Delta\tau$ translates into a fringe displacement:

$$x_n(\Delta\phi) = x_n(0) + \frac{z_1\mathcal{L}}{d} \cdot \frac{\Delta\phi}{2\pi}, \qquad (73)$$

where $\mathcal{L} = \hbar/p_z$ is the emergent length scale from action resolution, and $d$ is the slit separation (as defined in Section 8).

This result matches the observed fringe shift seen in quantum interference, but the mechanism is entirely different: TSRT attributes it to causal structure, not superposition.

**Realism and Non-Collapse:** TSRT asserts that each particle follows a single, deterministic path through either slit A or B. The phase offset modifies the permissible spacetime configurations and hence alters the density of redirected trajectories—not through collapse or measurement, but through classical geodesic filtering.

**Prediction Summary:** TSRT predicts that the positions of interference fringes shift linearly with the geometric phase offset $\Delta\phi$, despite the absence of any underlying wavefunction or amplitude-based interference. The contrast of the fringes remains high provided the coherence of the trembling metric field is preserved. Notably, the theory requires no appeal to superposition, probabilistic interpretation, or measurement-induced collapse.

While this result numerically matches the fringe shift predicted by quantum mechanics if one formally sets $\mathcal{L} = \lambda$, the interpretation is entirely different: in TSRT, $\mathcal{L}$ emerges not from wave-particle duality, but from the minimal geometric action interval needed to causally distinguish redirection paths in curved spacetime. Thus, the fringe shift reflects a deterministic realignment of redirected geodesics, not a phase-induced interference of amplitudes.

In the next subsection, we derive the corresponding prediction using standard quantum theory and carry out a detailed comparison.
## 9.5 Quantum Predictions for the Same Configuration

In standard quantum mechanics, the intensity distribution on the detection screen arises from coherent superposition of wavefunctions associated with the two slits. For an incoming particle with well-defined longitudinal momentum $p_z$, the wavefunction after the slits is given by:

$$ \psi(x; \Delta\phi) = \psi_A(x) + e^{i\Delta\phi}\psi_B(x), \qquad (74) $$

where $\Delta\phi$ is a controlled phase shift applied to the wavefunction emerging from slit B.

The resulting intensity pattern is:

$$ I_{QM}(x; \Delta\phi) = |\psi(x; \Delta\phi)|^2 = |\psi_A(x)|^2 + |\psi_B(x)|^2 + 2 \operatorname{Re} \{\psi_A^*(x)\psi_B(x)e^{i\Delta\phi}\}. \qquad (75) $$

If the slits are symmetric and the wavefunctions $\psi_A(x)$ and $\psi_B(x)$ have equal magnitude, the expression simplifies to:

$$ I_{QM}(x; \Delta\phi) = 2|\psi_A(x)|^2 [1 + \cos(\delta(x) + \Delta\phi)], \qquad (76) $$

where $\delta(x)$ is the intrinsic phase difference arising from the path length difference between the two slits at a given screen position $x$. This term governs the position of interference fringes.

**Fringe Shift Prediction:** The externally applied phase $\Delta\phi$ causes the interference fringes to shift laterally. The displacement of maxima is given by:

$$ \Delta x_{QM} = \frac{z_1 \lambda}{2\pi d} \cdot \Delta\phi, \qquad (77) $$

where $\lambda = h/p_z$ is the de Broglie wavelength. This result is structurally identical to the TSRT prediction, with the identification $\mathcal{L} = \hbar/p_z$ (as defined in Section 8). Thus, in stable symmetric configurations, both theories predict the same fringe shift behavior.

**Interpretational Difference:** Although the observed fringe shift as a function of $\Delta\phi$ is identical in both TSRT and quantum mechanics, the physical interpretations differ fundamentally. In the standard quantum framework, the particle is described by a delocalized wavefunction, and the phase offset $\Delta\phi$ alters the interference of probability amplitudes. The resulting fringe pattern emerges from constructive and destructive interference between complex wave components, with detection interpreted as a probabilistic collapse of the wavefunction.

By contrast, TSRT posits that each particle remains localized at all times, following a single, well-defined geodesic through trembling spacetime. The fringe shift in this view arises from the deterministic redirection of geodesic trajectories due to the causal structure imposed by the fluctuating metric. No wavefunction, amplitude interference, or collapse postulate is invoked.

Thus, while the two theories yield identical fringe locations under symmetric and temporally stable conditions, they diverge sharply in scenarios involving geometric asymmetries, time-dependent modulation, or causal boundary alterations. These distinguishing regimes are analyzed in the next subsection.
## 9.6 Distinguishing Observables and Measurement Strategy

While TSRT and quantum mechanics predict identical fringe shifts under symmetric, static conditions, their predictions diverge in the presence of geometric asymmetry, dynamic modulation, or causal incoherence. This subsection outlines experimentally accessible observables that can, in principle, distinguish between the two theories.

Each proposed observable targets a specific divergence in physical interpretation: deterministic causal redirection in TSRT versus probabilistic amplitude interference in QM.

**Observable 1: Fringe Contrast Suppression under Rapid Phase Modulation** TSRT predicts that if the phase modulation introduced behind one slit varies rapidly—either through high spatial gradients or high-frequency temporal oscillations—the local trembling metric $\xi_{\mu\nu}(x, t)$ becomes causally incoherent across the ensemble of traversing particles. This incoherence breaks the structured redirection of geodesics, leading to loss of angular focusing and degradation of fringe contrast on the detection screen.

In contrast, quantum mechanics maintains that a modulation applied behind one slit corresponds to a relative phase shift $\Delta\phi(t)$ between the two branches of the wavefunction. If $\Delta\phi(t)$ is a smooth, time-dependent global phase—meaning it is uniform across the spatial extent of the slit and identical for every particle—then interference persists, and fringe contrast remains unaffected. This follows from the linearity of the wavefunction and the fact that global phase factors do not alter probability densities.

However, TSRT makes a distinct prediction even in this idealized case. Because geodesics are redirected by curvature rather than phase, a vibrating film or rapidly varying refractive element placed behind one slit alters the local spacetime geometry experienced by different particles at different times. If the modulation frequency exceeds the coherence timescale defined by the proper-time trembling scale, the causal deflection field becomes scrambled. As a result, the ensemble of particle trajectories becomes statistically unstructured, and fringe visibility decreases—even in the absence of environmental noise or decohering interactions.

To empirically distinguish these mechanisms, one can introduce controlled high-frequency or pseudo-random phase modulation behind a single slit and measure fringe visibility as a function of modulation rate. A systematic degradation of contrast under conditions that preserve quantum coherence would indicate that the interference pattern depends on causal geodesic coherence, not amplitude superposition—favoring the TSRT interpretation.

**Observable 2: Temporal Hysteresis in Fringe Realignment** When a time-dependent phase offset $\Delta\phi(t)$ is applied to one slit, TSRT predicts that the trembling spacetime metric $\xi_{\mu\nu}(x, t)$ does not adjust instantaneously. Instead, the causal curvature structure responds with finite propagation speed and localized metric reconfiguration. As a result, the redirected geodesic ensemble realigns over a finite proper-time interval, leading to temporal hysteresis: a measurable lag between the phase shift and the corresponding change in fringe position. Depending on the modulation rate and curvature dynamics, transient effects such as overshoot, damping, or asymmetrical delay may occur.

In contrast, standard quantum mechanics asserts that a time-dependent phase modulation
applied to one slit evolves the wavefunction instantaneously in time:

$$ \psi_2(x, t) \rightarrow \psi_2(x, t)e^{i\Delta\phi(t)}, \qquad (78) $$

resulting in an immediate shift in the interference pattern—provided the phase change is truly global (uniform across the slit) and the system remains coherent. The only limitations arise from the particle's coherence time or source statistics, not from any causal delay in the wavefunction's response.

TSRT offers a fundamentally different interpretation: geodesic paths are redirected by a metric field that evolves causally, not by instantaneous phase change. To test this prediction, one can impose abrupt step-function or pulsed changes in $\Delta\phi(t)$ using an electro-optic phase modulator or moving optical element, and record the fringe position with femtosecond temporal resolution. A measurable lag or transient asymmetry in the fringe response—especially in excess of coherence-time bounds—would strongly support the TSRT framework over quantum mechanics.

**Observable 3: Localized Geometric Modulation of One Slit** If a curvature perturbation affects only a portion of one slit—such as the upper half of slit B—TSRT predicts a direction-dependent deflection of geodesics passing through the perturbed region. This leads to asymmetric redistribution of arrival positions on the detection screen, resulting in skewed or locally deformed interference fringes. The origin of this asymmetry is purely geometric: the trembling-induced metric perturbation $\xi_{\mu\nu}(x)$ varies spatially across the slit aperture, redirecting nearby geodesics in a non-uniform fashion.

Quantum theory also predicts fringe deformation in this scenario, but through a different mechanism. A spatially localized phase perturbation behind the slit modifies the phase of the transmitted wavefunction in a position-dependent way:

$$ \psi_2(x) \rightarrow \psi_2(x)e^{i\phi(x)}, \qquad (79) $$

where $\phi(x)$ is non-uniform across the slit aperture. This phase gradient introduces angular shifts or curvature into the outgoing wavefront, leading to distortion of the interference pattern. Thus, both TSRT and quantum mechanics predict observable asymmetries in the pattern, but attribute them to fundamentally different causes: local curvature-induced deflections versus wavefront deformation.

To experimentally distinguish these interpretations, one can apply a spatially structured refractive element or phase mask to a specific region of one slit, such as its upper third. TSRT predicts that only geodesics traversing the perturbed region will be redirected, potentially producing angular deflection, blurring, or fringe splitting localized to the corresponding screen region. In contrast, quantum predictions will depend on the spatial derivative of the phase and may lead to smooth deformation rather than discrete angular splitting. Detecting sharp, spatially correlated asymmetries would support the TSRT model of localized metric redirection.
**Observable 4: Structured Redistribution with One Slit Open** Even when only one slit is open, TSRT predicts that metric trembling can produce structured arrival distributions through curvature-guided redirection of geodesics. If the trembling field $\xi_{\mu\nu}(x, t)$ is modulated dynamically—spatially, temporally, or both—geodesics passing through the perturbed slit region can be redistributed in a non-uniform, patterned manner. Unlike standard single-slit diffraction, these patterns may exhibit internal structure, modulation bands, or non-Gaussian clustering, all arising from deterministic geometric effects.

Quantum mechanics, by contrast, predicts that with only one slit open, no interference occurs. The observed pattern is governed by single-slit diffraction and is shaped by the aperture function and outgoing wavefront. If a structured refractive or phase modulation is applied behind the slit, the outgoing wave may be deformed, producing slight shifts or broadenings of the diffraction envelope. However, quantum theory does not predict the emergence of fine-structured spatial modulations unless a second coherent path or decoherence mechanism is introduced.

To test this distinction, one can operate the double-slit apparatus with only one slit open and apply a structured or time-varying curvature modulation (e.g., using a dynamic refractive element, vibrating mask, or patterned dielectric). TSRT predicts that the geodesic ensemble will reflect this modulation in the spatial arrival density, potentially forming bands or asymmetric sub-patterns beyond what diffraction theory alone allows. Detection of such internal structure, especially in the absence of interference from a second slit, would favor the TSRT interpretation of causally redirected geodesics over amplitude-based diffraction.

**Observable 5: Resolution Limit and Sub-Planckian Modulation** TSRT predicts that fringe resolution is governed not by operator-based uncertainty but by a geometric criterion: neighboring geodesics must differ by a minimal proper-time action $\Delta S \gtrsim \hbar$ to be causally distinguishable. This curvature-induced action threshold defines a deterministic resolution limit, which depends on particle momentum, slit geometry, and the structure of the trembling spacetime metric $\xi_{\mu\nu}(x)$. Under certain conditions, this limit may permit sharper fringe structures than those constrained by the standard uncertainty relation $\Delta x \Delta p \gtrsim \hbar/2$.

Quantum mechanics asserts this uncertainty limit as fundamental, arising from the non-commutativity of position and momentum operators. Accordingly, it predicts that fringe visibility and resolution degrade when attempts are made to resolve spatial features below the uncertainty-defined limit—such as by narrowing slits or increasing particle momentum dispersion.

TSRT, by contrast, allows fine fringe structures to persist as long as geodesic distinguishability is maintained through curvature coherence. This does not violate empirical limits but replaces the probabilistic constraint with a geometric one. A proposed test involves precision interference experiments using particles of varying momentum and slit separations, particularly in regimes where quantum mechanics predicts fringe washout due to uncertainty. If sharp, stable fringe structures persist beyond the predicted quantum resolution threshold, this would provide strong evidence in favor of TSRT's curvature-based resolution principle.
**Observable 6: Velocity-Dependent Fringe Resolution Scaling** TSRT predicts that the minimal resolvable fringe spacing depends on particle velocity due to the causal requirement that neighboring geodesics accumulate an action difference of at least $\hbar$. Specifically, in non-relativistic regimes, the spatial resolution limit scales linearly with velocity:

$$ \Delta x_{\min} = \frac{\hbar v z_1}{m L}. \qquad (80) $$

As the velocity increases, neighboring geodesics accumulate less proper-time difference over a fixed path length $L$, which makes them harder to distinguish. Consequently, TSRT anticipates that the fringe pattern should broaden with increasing particle speed under fixed geometric conditions. This causal scaling reflects a deterministic limit on the distinguishability of trajectories.

Quantum mechanics, by contrast, predicts no such causal restriction. The fringe spacing $\Delta x \sim h/p$ narrows as momentum increases, but there is no mechanism whereby high velocity causes a loss of resolvability or visibility—unless external decoherence is introduced. So long as the wavefunction remains coherent, even high-speed particles should exhibit sharply defined interference fringes.

A proposed test involves performing a sequence of double-slit experiments using the same apparatus and geometry, but with particles of increasing kinetic energy (e.g., via controlled acceleration). According to TSRT, fringe spacing and sharpness should exhibit a measurable dependence on particle speed, while quantum theory predicts only a change in fringe periodicity, not in resolution loss. Detection of velocity-induced fringe broadening consistent with TSRT's causal cutoff would provide decisive evidence for its geometric interpretation.

**Summary** Taken together, these six observables highlight how TSRT and quantum mechanics offer not merely different mathematical formalisms, but fundamentally distinct physical mechanisms underlying interference phenomena. Whereas quantum mechanics attributes fringe patterns to amplitude superposition and phase evolution of wavefunctions, TSRT explains them through deterministic redirection of particles along causally constrained geodesics in a trembling spacetime. This distinction leads to diverging predictions in cases involving rapid modulation, localized perturbations, temporal transitions, single-slit curvature effects, sub-uncertainty resolution, and velocity-dependent limits on distinguishability.

Each observable proposed here corresponds to a well-defined experimental configuration and a concrete, falsifiable prediction. The tests are not limited by current technology and may be implemented using state-of-the-art interferometry with electrons, photons, or neutrons. Crucially, TSRT does not conflict with known experimental results—it reproduces them in standard regimes—but predicts measurable deviations when causal structure, curvature coherence, or proper-time resolution become dominant.

These experiments offer a pathway to empirically resolve a century-old interpretational divide. If confirmed, they would not only validate TSRT's geometric interpretation of interference but also prompt a reevaluation of the foundations of quantum mechanics, replacing probabilistic postulates with deterministic curvature dynamics rooted in spacetime geometry.
9.7 Expected Outcomes and Interpretation

The proposed experimental variations—particularly those involving dynamic modulation, spa-
tial asymmetry, abrupt transitions, or single-slit structure—offer a powerful framework to em-
pirically distinguish between the probabilistic wave interference of quantum mechanics and the
deterministic geodesic redirection predicted by TSRT.

In scenarios involving a phase offset $\Delta\phi$, both theories predict a fringe shift, but differ in the underlying mechanism: TSRT attributes the shift to a proper-time delay caused by metric modulation, whereas quantum mechanics interprets it as interference of complex amplitudes. Under symmetric and static conditions, their predictions coincide, reinforcing the empirical validity of TSRT in recovering conventional interference patterns without invoking superposition.

Under rapid modulation, TSRT anticipates contrast degradation due to loss of causal coherence in the trembling metric, while quantum mechanics predicts sustained interference unless true environmental decoherence occurs. Observable contrast loss in response to isolated high-frequency modulation would thus favor the TSRT explanation.

Abrupt phase changes yield another distinction: TSRT predicts a finite delay in fringe realignment due to the retarded propagation of curvature perturbations, potentially producing hysteresis or overshoot. Quantum theory predicts near-instantaneous realignment, limited only by coherence time. Detection of lagged or asymmetric response would support TSRT's causal interpretation.

When only part of a slit is modulated, TSRT predicts localized angular deflection due to
spatial variation in the trembling field $\xi_{\mu\nu}(x, t)$, producing asymmetric or skewed fringes. While
quantum mechanics also predicts pattern deformation in response to spatial phase gradients, the
mechanisms are distinct: curvature-induced redirection versus wavefront interference. Sharp,
spatially localized deflections would signal a geometric rather than wave-based origin.

Even with one slit open, TSRT allows structured arrival patterns through deterministic redirection by spatially modulated curvature. Quantum mechanics predicts deformation only if the outgoing wavefront is modified; it does not attribute structure to geodesic-level dynamics. Structured fringe features in single-slit configurations would therefore lend support to TSRT.

Finally, TSRT replaces the uncertainty principle with a causal resolution bound: neighbor-
ing geodesics must differ by a minimal action $\Delta S \gtrsim \hbar$ to be distinguishable. This allows sharply
resolved interference structures as long as geodesic separation satisfies the curvature threshold,
independent of probabilistic spread. Observing fringe resolution consistent with causal lim-
its but sharper than quantum expectations would point toward a deeper geometric basis for
interference.

In conclusion, TSRT offers a deterministic and geometrically grounded interpretation of
interference phenomena. While it replicates quantum predictions under ideal, symmetric con-
ditions, it diverges in controlled regimes that stress causal propagation, curvature coherence, or
local spatial modulation. These divergences are not anomalies but testable predictions grounded
in a fundamentally different ontology of spacetime.

If empirical results confirm instantaneous, globally coherent behavior, quantum mechanics
remains valid in its current domain. But if contrast degradation without decoherence, delayed
fringe realignment, or structured single-slit patterns are observed, TSRT emerges as a realist
and falsifiable alternative—one that reframes interference not as a manifestation of wave-particle duality, but as a deterministic consequence of trembling spacetime geometry.

# 10 TSRT Simulations and Fringe Predictions for Real Particles

## 10.1 Analytical Predictions for Realistic Experimental Parameters

To assess the predictive power of TSRT in experimentally realizable regimes, we now compute fringe spacings for standard double-slit configurations using realistic parameters from well-established experiments involving photons, electrons, and neutrons. It is important to clarify, however, that the physical interpretation of a “particle” in TSRT differs fundamentally from its interpretation in quantum mechanics. In quantum theory, particles such as electrons or photons are described by wavefunctions and can exhibit wave-like interference. In TSRT, by contrast, particles follow deterministic but trembling geodesics in spacetime, redirected by bounded metric fluctuations. No wavefunction is assigned to the particle; the observed fringes arise from statistical density variations in the redirected geodesic ensemble.

Although TSRT and quantum mechanics share no underlying formalism, it is notable that TSRT reproduces the standard diffraction formula for fringe spacing:

$$ \Delta x \approx \frac{D\lambda_{\text{eff}}}{a}, \qquad (81) $$

where $D$ is the screen distance, $a$ the slit separation, and $\lambda_{\text{eff}}$ is the effective wavelength-like quantity arising from TSRT’s curvature-based action threshold:

$$ \lambda_{\text{TSRT}} = \frac{h_{\text{TSRT}}}{p}, \quad p = mv. \qquad (82) $$

This expression was derived earlier in this paper (such as in Equation 67 or Equation (37)) from geodesic deflection and the proper-time distinguishability condition $\Delta S \gtrsim \hbar$, without invoking any notion of wave-particle duality. The numerical match with quantum predictions is therefore not surprising, but arises as a consequence of Planck’s constant $h$ being derived from causal structure in TSRT, rather than assumed axiomatically as in quantum theory (see [17]).

We now illustrate the quantitative agreement between TSRT and experiment for three particle types, using the above formula with curvature-derived $h_{\text{TSRT}} \approx h$ as previously shown.

**Photons.** In quantum theory, photons are massless particles associated with electromagnetic wave propagation, and their wavelength $\lambda$ is fundamental to interference. In TSRT, however, photons are massless particles that follow trembling null geodesics; their deflection pattern is governed by slit geometry and metric curvature, not by a wave nature. Nonetheless, for a red photon with $\lambda = 700$ nm, and for slit separation $a = 100$ µm and screen distance $D = 1$ m, the TSRT-based fringe spacing becomes:

$$ \Delta x_{\text{photon}} \approx \frac{D\lambda}{a} = \frac{1 \text{ m} \times 700 \times 10^{-9} \text{ m}}{100 \times 10^{-6} \text{ m}} = 7 \text{ mm}, \qquad (83) $$
which matches classic photon experiments such as those of Taylor (1909) [28] and modern single-photon studies [29].

**Electrons.** For electrons with velocity $v \approx 1.5 \times 10^7$ m/s (corresponding to kinetic energies near 100 eV), the momentum is $p = m_e v$, and using $h_{TSRT} \approx h$, we obtain:

$$ \lambda_{TSRT} \approx \frac{h}{m_e v} \approx \frac{6.63 \times 10^{-34} \text{ Js}}{9.11 \times 10^{-31} \text{ kg} \times 1.5 \times 10^7 \text{ m/s}} \approx 4.85 \times 10^{-11} \text{ m}. \quad (84) $$

With $a = 1 \mu\text{m}$ and $D = 1 \text{ m}$, the fringe spacing is:

$$ \Delta x_{\text{electron}} \approx \frac{D\lambda_{\text{TSRT}}}{a} = \frac{1 \times 4.85 \times 10^{-11}}{1 \times 10^{-6}} = 48.5 \mu\text{m}, \quad (85) $$

which reproduces the electron interference patterns observed by Jönsson (1961) [3] and Tonomura et al. (1989) [30].

**Neutrons.** Cold neutrons moving at $v \approx 2 \times 10^3$ m/s have a momentum $p = m_n v$. Applying the TSRT expression:

$$ \lambda_{\text{TSRT}} \approx \frac{h}{m_n v} \approx \frac{6.63 \times 10^{-34}}{1.675 \times 10^{-27} \times 2 \times 10^3} \approx 1.98 \times 10^{-10} \text{ m}. \quad (86) $$

For $a = 20 \mu\text{m}$ and $D = 4 \text{ m}$, we find:

$$ \Delta x_{\text{neutron}} \approx \frac{4 \times 1.98 \times 10^{-10}}{20 \times 10^{-6}} \approx 39.6 \mu\text{m}, \quad (87) $$

matching experimental neutron interference data from Zeilinger et al. [31].

In summary, TSRT reproduces the empirical fringe spacings reported across particle types and experimental regimes. These results are not postulated by analogy with quantum mechanics, but derived from the causal structure of trembling spacetime. The agreement with quantum predictions confirms that TSRT not only provides a consistent geometric reinterpretation of interference but does so with no reliance on superposition or wavefunction formalism.

# 10.2 Geodesic-Based Simulation Framework for Arbitrary Configurations

## 10.2.1 Overview of the TSRT Geodesic Approach

Traditional quantum mechanics models interference phenomena, such as those seen in the double-slit experiment, by assigning a complex-valued wavefunction to each particle and computing interference via linear superposition. The resulting intensity pattern on the detection screen emerges from the squared modulus of the total wavefunction amplitude. In this framework, the appearance of fringes is often attributed to self-interference of the wavefunction traversing both slits simultaneously—a concept that, while predictive, raises interpretational challenges when applied to single-particle events.

Trembling Spacetime Relativity Theory (TSRT) offers a fundamentally different explanation grounded not in wave mechanics, but in the causal geometry of fluctuating spacetime. In
TSRT, particles are not guided by probability amplitudes but follow deterministic yet non-
smooth geodesics perturbed by local spacetime trembling. These perturbations encode both
microscopic curvature constraints and macroscopic causal structure, giving rise to ensembles
of allowed geodesics that diverge or converge depending on the underlying metric deviations
induced near the slits.

Each individual geodesic in TSRT represents a physically realizable trajectory that respects the causal structure imposed by proper time fluctuations. When an ensemble of such geodesics is constructed from a common source, their intersections and density variations on the detection screen naturally give rise to an emergent fringe pattern—without invoking wave-particle duality or nonlocal collapse postulates. The fringes in TSRT are thus a direct geometric consequence of how allowed spacetime paths are deflected by the slit configuration and subsequent causal evolution.

This approach not only preserves locality and determinism, but also explains why inter-
ference patterns disappear when which-path information is extracted: the trembling structure
adapts to new boundary conditions, altering the causal connectivity between slits and screen.
Crucially, TSRT reproduces observed fringe patterns through this purely geometric mechanism,
while predicting deviations under modified configurations that can be experimentally tested.

In the following subsections, we detail how this causal-geometric framework can be used
to simulate specific interference setups by generating and analyzing ensembles of trembling
geodesics, grounded in the metric structure developed in the foundational TSRT formulation
[17].

10.2.2 Specifying the Experimental Geometry and Initial Conditions

In TSRT, the motion of particles through an interferometric setup is governed not by a single
deterministic trajectory, but by a family of causally permissible geodesics shaped by the local
structure of spacetime trembling. These geodesics are not arbitrary—they are determined by the
experimental geometry, boundary conditions, and the initial distribution of metric perturbations
near the particle source and slit structure.

To begin, we define the experimental geometry by specifying:

• the position and width of each slit,

• the separation *a* between slits,

• the distance *D* from the slit plane to the detection screen,

• the transverse coordinate *x* across the screen,

• the location and spatial spread of the source,

• and the average velocity *v* of the emitted particle species.

These physical parameters fix the spatial and causal configuration of the setup. In TSRT,
this configuration defines a boundary-value problem for the local metric deviation field ξμν(x)
introduced in the foundational formulation [17]. Near the slits, the imposed boundary breaks
translational invariance, introducing curvature anisotropies that produce directional trembling. These anisotropies modify the deviation vector field $\eta^\mu(\tau)$, which governs the spread of neighboring geodesics via the geodesic deviation equation.

The source, assumed to be a spatially localized emitter, sets the initial conditions for the ensemble of particles. Each emitted particle corresponds to a geodesic with slightly different initial position, momentum direction, and phase of spacetime trembling, constrained by the causal light cone and bounded fluctuation scale. The relevant mode of metric perturbation is therefore not freely chosen but emerges from the geometry of the slit configuration and the causal propagation of the trembling-induced curvature field.

The initial trembling-induced deviation is modeled statistically, but with constraints. For example, in a symmetric double-slit setup, the boundary conditions near each slit will induce antisymmetric or symmetric perturbation profiles depending on the alignment and aperture width. These determine the angular spread of geodesics and the density of arrival points on the detection screen. Notably, the perturbations must obey the curvature bounds established in [17] to ensure no violation of causality or proper-time positivity.

Thus, in TSRT, the initial data for numerical or analytical geodesic simulation consists not only of spatial coordinates and velocities, but also of localized perturbation modes tied to the geometry and causal history of the setup. These are the fundamental inputs from which ensemble geodesics and interference patterns emerge.

### 10.2.3 Constructing the Ensemble of Causally Permitted Geodesics

Once the experimental geometry and initial boundary conditions have been specified, the next step is to construct the ensemble of particle geodesics permitted by TSRT. Unlike in classical mechanics, where a single geodesic suffices for deterministic motion, TSRT encodes microscopic curvature fluctuations—bounded by causal structure—into an ensemble of nearby trajectories. These trajectories arise from the interaction between the background metric and the localized trembling-induced deviations, forming a distribution of geodesics rather than a single path.

Each geodesic $\gamma_i(\tau)$ in the ensemble satisfies the perturbed geodesic equation:

$$ \frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\nu\rho} \frac{dx^\nu}{d\tau} \frac{dx^\rho}{d\tau} + \delta a_i^\mu(\tau) = 0, \qquad (88) $$

where $\delta a_i^\mu(\tau)$ represents the effective acceleration from local trembling, modeled as a bounded deviation consistent with the TSRT curvature limits (see [17]). This term encodes the stochastic but non-random influence of the geometry, emerging from proper-time fluctuations that remain strictly local and causal.

To build the full ensemble:

1. One begins with a central unperturbed geodesic $\gamma_0$ directed from the source through the slit region toward the detection screen.

2. A family of perturbed geodesics $\{\gamma_i\}$ is then constructed by varying:

* the initial transverse displacement within the source region,

* the local phase and amplitude of the metric deviation $\xi_{\mu\nu}(x)$ at emission,
• the induced angular deflection due to curvature asymmetries near the slits.

3. These variations must obey the proper-time positivity constraint $d\tau^2 > 0$, and respect the curvature bounds $|R| < R_{max}$ defined in TSRT.

The ensemble should sample a statistically complete set of causal paths permitted by the local geometry. In numerical implementations, this may correspond to generating thousands or millions of trajectories from randomized but constrained initial conditions drawn from the TSRT-prescribed trembling spectrum. Alternatively, analytical approximations can describe the angular density distribution of geodesics by integrating over local curvature modes.

Importantly, the geodesics must remain differentiable (piecewise smooth), forward-directed in proper time, and consistent with the global boundary setup. This ensures that the resulting ensemble respects the causal geometry and does not rely on hidden variables, nonlocal correlations, or probability amplitudes.

In contrast to quantum trajectories, which are often interpreted via Feynman path integrals over all conceivable histories, the TSRT ensemble includes only those geodesics physically permitted by local geometric fluctuations. These are finite in number, bounded in angular spread, and grounded in measurable curvature and distance scales, making the approach both deterministic and physically realizable.

### 10.2.4 Extracting Deflection Patterns and Screen Arrival Distributions

After constructing the ensemble of causally permitted trembling geodesics in a given setup, the observable interference pattern emerges from the distribution of particle arrival locations on the detection screen. In TSRT, this distribution is not inferred from probability amplitudes or superposed wavefunctions, but is instead computed directly from the density of geodesic endpoints—a geometric quantity reflecting the convergence and divergence of nearby trajectories under the influence of metric fluctuations.

For a given screen coordinate $x$, the local intensity $I(x)$ is proportional to the number density of geodesics arriving at that point:

$$I(x) \propto \lim_{\Delta x \to 0} \frac{N(x, x + \Delta x)}{\Delta x}, \qquad (89)$$

where $N(x, x + \Delta x)$ counts the number of geodesics terminating within the infinitesimal screen interval $[x, x + \Delta x]$. This number is governed by the angular deflection of each geodesic and the transverse expansion or focusing of nearby geodesics as determined by the local geodesic deviation field $\eta^\mu(\tau)$.

In practice, the pattern $I(x)$ can be computed either by histogramming a large ensemble of simulated geodesics, or analytically, by integrating over the deflection angles $\theta$ and the curvature-induced deviations near the slits. The interference fringes appear where the geodesic density peaks due to constructive deflection overlap, while minima correspond to directions where geodesics are depleted or diverge. No superposition principle is needed: the fringe pattern is an emergent structure arising from the distribution of deterministic trajectories constrained by trembling-induced curvature.
Crucially, the phase information in TSRT is encoded geometrically: small differences in the local spacetime fluctuation phase or slit-induced metric asymmetry result in differences in arrival angle and hence screen location. This replaces the conventional phase difference $\Delta\phi$ between quantum paths by a geodesic-level deflection differential $\Delta\theta$, subject to the causality-preserving bounds derived in [17].

This approach also predicts the breakdown of interference when path distinguishability is introduced. For example, placing a detector near one slit modifies the local boundary conditions, suppressing certain trembling modes and thereby collapsing the set of permitted geodesics to a reduced ensemble. As a result, the geodesic density becomes uniform or asymmetric, erasing the interference structure without invoking wavefunction collapse.

The deterministic nature of this mechanism allows direct comparison with experimental data. Once the slit geometry and curvature profile are specified, the predicted screen intensity $I(x)$ can be calculated to any desired resolution and tested against observed patterns in photon, electron, or neutron interference experiments.

### 10.2.5 Extensions to Generalized Configurations

The geodesic-based framework introduced in TSRT is not limited to static, symmetric double-slit experiments. Because it is grounded in the local causal structure and curvature constraints of spacetime, the same methodology naturally generalizes to a broad class of interferometric and deflection-based configurations, each governed by the same fundamental principles: geodesic evolution under bounded metric deviations.

**Multi-slit and grating configurations.** In a multi-slit array, each aperture introduces additional localized curvature perturbations, modifying the ensemble of causally permitted geodesics. The resulting fringe pattern becomes more complex but remains computable within TSRT. Each slit generates a set of geodesics that are redirected according to its local metric perturbation profile, and the cumulative spatial density of these trajectories on the detection screen determines the observable intensity. High-density regions correspond to fringe maxima, while sparse regions correspond to minima. Unlike quantum wave models, where interference arises from the superposition of amplitudes, TSRT predicts these structures through the deterministic redirection and spatial focusing of geodesics in a trembling spacetime geometry.

**Time-varying apertures.** The TSRT framework also accommodates time-dependent geometries, such as shutters or rotating masks. In such cases, the local metric deviation field $\xi_{\mu\nu}(x, t)$ becomes explicitly time-dependent, leading to time-modulated trembling effects. Geodesics passing through different temporal phases of aperture evolution experience different curvature profiles, producing temporally evolving arrival patterns. This mechanism offers an alternative to the time-dependent Schrödinger equation, with observable consequences that could be tested using fast-switching slit experiments.

**Relativistic particle regimes.** For particles approaching relativistic speeds (e.g., fast electrons or muons), the geodesic evolution must account for the full pseudo-Riemannian structure
of spacetime. TSRT handles this naturally: proper time remains the evolution parameter, and trembling-induced curvature is evaluated in the appropriate Lorentzian signature. The metric deviations must still satisfy $d\tau^2 > 0$, and causal propagation is preserved. In these cases, aberration effects, time dilation, and curvature-velocity coupling must be included in the initial condition ensemble and numerical integration.

**Curved background metrics.** The formalism extends to curved backgrounds, such as particles moving in Earth's gravitational field or near astrophysical sources. The trembling-induced deviation field $\xi_{\mu\nu}$ in TSRT is superimposed on a non-flat background metric $g_{\mu\nu}^{(bg)}$, and the full geodesic equation becomes:

$$ \frac{d^2 x^\mu}{d\tau^2} + (\Gamma^\mu_{\nu\rho})_{\text{total}} \frac{dx^\nu}{d\tau} \frac{dx^\rho}{d\tau} = 0, \qquad (90) $$

where $(\Gamma^\mu_{\nu\rho})_{\text{total}}$ includes both the smooth background connection and localized TSRT fluctuations. This provides a unified platform to study gravitational lensing-like effects in laboratory-scale experiments, driven not by massive bodies but by engineered slit-induced curvature profiles.

**Quantum-optical analogs and entangled states.** Even more exotic configurations, such as quantum-optical interference with entangled photons, may be approached within TSRT by examining correlated geodesic ensembles across spacetime regions. Though not covered in this paper, such extensions may reveal whether trembling spacetime dynamics can account for apparent nonlocal correlations via underlying geometric connectivity, without violating relativistic causality.

In all cases, the underlying principle remains unchanged: particle trajectories follow causally permitted geodesics influenced by bounded curvature fluctuations. This makes the TSRT framework inherently flexible, deterministic, and extendable—capable of modeling a wide range of experimental setups beyond what standard quantum formalism was originally designed to describe.

# 11 Geodesic Deviation and Minimal Action Resolution in Trembling Spacetime

This section derives and interprets the resolution limit imposed by trembling spacetime geometry on neighboring particle geodesics. In TSRT, geodesic deviations originate from small, deterministic metric perturbations $\xi_{\mu\nu}(x)$, and are governed by the local curvature structure. The analysis is valid in both relativistic and non-relativistic regimes and reveals that angular resolution is bounded by a minimal action difference, which serves as a geometric signature of TSRT and defines the smallest causally distinguishable trajectory shift.
## Geodesic Deviation from Metric Trembling

The perturbed spacetime metric is given by:

$$g_{\mu\nu}(x) = \eta_{\mu\nu} + \xi_{\mu\nu}(x), \quad |\xi_{\mu\nu}| \ll 1, \qquad (91)$$

with Minkowski background $\eta_{\mu\nu}$ and symmetric fluctuation field $\xi_{\mu\nu}$. The geodesic equation for a proper-time-parametrized trajectory $x^{\mu}(\tau)$ is:

$$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0, \qquad (92)$$

with Christoffel symbols linearized in $\xi_{\mu\nu}$. To study geodesic separation, one introduces a deviation vector $\delta x^{\mu} = \partial x^{\mu}/\partial s$ between nearby geodesics labeled by parameter $s$. This vector evolves according to the Jacobi equation:

$$\frac{D^2 \delta x^\mu}{D \tau^2} = R^\mu_{\nu\alpha\beta} u^\nu \delta x^\alpha u^\beta, \qquad (93)$$

where $u^{\mu} = dx^{\mu}/d\tau$ is the four-velocity of the fiducial geodesic and $R^\mu_{\nu\alpha\beta}$ is the Riemann curvature tensor, linearized in $\xi_{\mu\nu}$.

## Transverse Angular Spread and Momentum Deflection

Let the longitudinal direction be along $z$, with $z_1$ the distance from slit to detection screen. The transverse angular deflection in direction $i \in \{x, y\}$ due to $\delta x^i$ is:

$$\theta_i \approx \frac{\delta x^i}{z_1}, \quad \text{and} \quad \Delta p_i = p_z \theta_i = p_z \frac{\delta x^i}{z_1}, \qquad (94)$$

linking geodesic deviation directly to measurable momentum transfer. Each geodesic is deterministic; the ensemble spread arises from causal variability in $\xi_{\mu\nu}(x)$.

## Minimal Action Difference and Resolution Limit

In TSRT, each particle follows a unique causal geodesic $\gamma$ in a trembling spacetime background. The particle's proper-time action along this geodesic is given by:

$$S_{\gamma} = -m \int_{\gamma} d\tau, \qquad (95)$$

where $m$ is the particle's rest mass and $d\tau$ is the infinitesimal proper time along the path. Due to curvature fluctuations $\xi_{\mu\nu}(x)$, neighboring geodesics experience slightly different elapsed proper times, resulting in an action difference:

$$\Delta S = -m \Delta \tau. \qquad (96)$$

TSRT imposes a causal resolution condition: two geodesics are distinguishable on the de-
tection screen only if their proper-time actions differ by at least the minimal action threshold,

$$|\Delta S| \gtrsim \hbar, \qquad (97)$$

which is derived from the trembling geometry and sets the lower limit for causal separability. This provides a geometric—not operator-based—origin for Planck’s constant as the fundamental resolution scale.

To quantify this limit, consider two neighboring geodesics separated by a small transverse angular deviation $\Delta\theta$, traveling a longitudinal distance $L$. For small angles and non-relativistic speeds $v \ll c$, the proper-time difference due to path deviation is approximately:

$$\Delta\tau \approx \frac{L \Delta\theta}{v}, \qquad (98)$$

leading to an action difference:

$$\Delta S \approx -m \frac{L \Delta\theta}{v}. \qquad (99)$$

Applying the resolution criterion $|\Delta S| \gtrsim \hbar$, we obtain the minimal resolvable angular deviation:

$$\Delta\theta_{\min} \approx \frac{\hbar v}{mL}. \qquad (100)$$

When projected to a detection screen at transverse distance $z_1$, this angular limit corresponds to a minimal resolvable spatial separation:

$$\Delta x_{\min} = z_1 \Delta\theta_{\min} = \frac{\hbar v z_1}{mL}. \qquad (101)$$

This expression defines the fundamental spatial resolution in TSRT: the finest distinguishable arrival separation on the screen due to trembling-induced causal distinguishability.

In terms of the longitudinal momentum $p_z = mv$, this becomes:

$$\Delta x_{\min} = \frac{\hbar z_1}{p_z} \cdot \frac{v}{L}. \qquad (102)$$

Thus, unlike in quantum theory—where $\lambda = h/p$ sets a universal resolution scale—TSRT yields a geometry-dependent resolution bound, which scales with the ratio $z_1/L$. Only in the special case where the path length equals the screen distance ($L = z_1$), does this simplify to:

$$\Delta x_{\min} = \frac{\hbar}{p_z}. \qquad (103)$$

Accordingly, the oft-cited scale $\mathcal{L} = \hbar/p_z$ in TSRT should be interpreted not as a universal minimal length, but as a special-case resolution limit applying to symmetric configurations where the geodesic path length equals the distance to the screen. More generally, the resolution is governed by:

$$\mathcal{L}_{\text{TSRT}} = \frac{\hbar v z_1}{mL}, \qquad (104)$$

which is a testable, falsifiable prediction of TSRT. It offers a causal geometric reinterpretation of the uncertainty principle: resolution emerges not from non-commutativity or superposition, but
from proper-time distinguishability of deterministic trajectories. This sharply contrasts with quantum mechanics, while remaining fully compatible with empirical data in all tested regimes.

## Velocity Dependence and Experimental Relevance

The minimal resolvable angular deviation in TSRT,

$$ \Delta\theta_{\min} = \frac{\hbar v}{mL}, \qquad (105) $$

translates to a transverse momentum threshold:

$$ \Delta p_{\perp} = p_z \Delta\theta_{\min} = mv \cdot \frac{\hbar v}{mL} = \frac{\hbar v^2}{L}. \qquad (106) $$

This expression appears to suggest a quadratic dependence on velocity; however, it conflates spatial and momentum resolution. In fact, the proper causal resolution condition in TSRT is set by the action difference:

$$ |\Delta S| = m\Delta\tau \approx \frac{mL\Delta\theta}{v} \gtrsim \hbar, \qquad (107) $$

which leads to a velocity-dependent spatial resolution:

$$ \Delta x_{\min} = \frac{\hbar v z_1}{mL}. \qquad (108) $$

That is, faster particles require a larger transverse displacement on the screen to be causally distinguishable. This reflects the fact that particles traversing the trembling metric more rapidly accumulate less proper time difference between neighboring paths.

Therefore, TSRT predicts that the spatial resolution of fringes improves for slower particles under fixed geometric conditions. Conversely, increasing velocity should stretch the resolution scale. This offers a falsifiable experimental test: for fixed slit separation and screen distance, fringe sharpness should vary systematically with particle velocity. Failure to observe this would challenge the causal cutoff hypothesis at the heart of TSRT.

In contrast, in quantum mechanics, fringe resolution is governed by the interference of wave amplitudes and the de Broglie relation $\lambda = h/p$. As particle velocity increases, the associated wavelength decreases, leading to narrower fringes. However, quantum mechanics does not impose a minimal resolvable angular deviation or action threshold; there is no causal mechanism that limits resolution based on trajectory separability. Instead, resolution is constrained probabilistically via the uncertainty principle $\Delta x \Delta p \gtrsim \hbar/2$, and instrumental coherence plays a dominant role. Therefore, unlike TSRT, quantum theory does not predict a breakdown in fringe visibility or distinguishability due to high particle speed alone, so long as the wavefunction remains coherent. This distinction offers a concrete experimental divergence between the two frameworks.

## Causal Interpretation

In contrast to wave-based theories, where resolution arises from interference between superposed amplitudes, TSRT derives spatial and angular resolution from the proper-time difference
between causally distinct geodesics. Two trajectories are distinguishable only if the action difference between them satisfies

$$|\Delta S| \gtrsim \hbar, \qquad (109)$$

where $S = -m\tau$ is the proper-time action. This condition is not postulated arbitrarily but emerges from the trembling spacetime geometry, where the causal structure constrains the fineness of resolvable geodesic separation.

In this framework, Planck's constant does not enter as an axiom of quantum theory but instead reflects a minimal action scale associated with the geometry of spacetime. It quantifies the smallest causally meaningful difference in proper time that trembling geodesics can encode—a geometric cutoff rather than a probabilistic uncertainty.

This reinterpretation offers both theoretical and experimental motivation. Precision measurements of fringe sharpness, angular spread, and resolution across varying particle velocities provide direct access to the causal resolution limit. Observing trends consistent with the predicted $\Delta x_{min} \propto v$ scaling would validate TSRT's replacement of uncertainty with curvature-induced action thresholds. Deviations would indicate the limits of the trembling spacetime description.

# 12 Relativistic Extension and Comparison with Dirac Theory

This section summarizes how Trembling Spacetime Relativity Theory (TSRT) generalizes to relativistic regimes and compares its predictions with those of Dirac-based quantum mechanics. We show that TSRT remains consistent with relativistic causality and explains diffraction phenomena without invoking wavefunctions or probabilistic postulates. The key differences in ontological structure, measurable consequences, and falsifiable predictions are outlined here, with full technical derivations and proposed experiments.

Although the main analysis in this work focuses on the non-relativistic regime, the Trembling Spacetime Relativity Theory (TSRT) naturally extends to relativistic settings. In these cases, geodesic deviations remain valid and covariant, but the deflection angles, action thresholds, and angular resolutions become momentum-dependent through Lorentz factors. Unlike traditional Dirac-based quantum theory, TSRT maintains a fully geometric, causal, and deterministic foundation, even at high energies.

Key differences emerge in how Planck's constant arises: TSRT derives it as a minimal action threshold from metric coherence, while Dirac theory adopts it axiomatically. Furthermore, relativistic TSRT predicts specific deflection behavior and pattern resolutions that may differ from the predictions of quantum mechanics under the Dirac formalism.

TSRT has been shown to reproduce all key interference features observed in both non-relativistic and relativistic regimes, while offering distinct predictions under asymmetric, dynamic, or causally modified configurations. Its contrast with Dirac theory becomes especially pronounced when curvature-induced geodesic modulation replaces wavefunction interference as the source of fringe formation. The implications of this reformulation—for both fundamental physics and experimental design—are profound.
## 12.1 Relativistic Structure of Geodesics in TSRT

In Trembling Spacetime Relativity Theory (TSRT), particles follow proper-time-parametrized geodesics determined by the full relativistic structure of spacetime, described by the perturbed metric:

$$g_{\mu\nu}(x) = \eta_{\mu\nu} + \xi_{\mu\nu}(x), \qquad (110)$$

where $\eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)$ is the Minkowski background and $\xi_{\mu\nu}(x)$ represents smooth, deterministic metric fluctuations constrained by local causality and curvature bounds.

The general form of the geodesic equation is:

$$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0, \qquad (111)$$

where $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols computed from the perturbed metric:

$$\Gamma^\mu_{\alpha\beta} = \frac{1}{2} g^{\mu\sigma} (\partial_\alpha g_{\sigma\beta} + \partial_\beta g_{\sigma\alpha} - \partial_\sigma g_{\alpha\beta}). \qquad (112)$$

In TSRT, the perturbation field $\xi_{\mu\nu}(x)$ is assumed to have small magnitude and bounded derivatives, ensuring causal admissibility and preserving proper-time monotonicity along all geodesics. These fluctuations induce smooth but nontrivial deflections of geodesics, resulting in measurable transverse deviations.

To analyze neighboring geodesics, we consider a one-parameter congruence of worldlines and define the separation vector $\eta^\mu$. The geodesic deviation equation reads [19, 24]:

$$\frac{D^2 \eta^\mu}{D\tau^2} = R^\mu_{\nu\alpha\beta} u^\nu u^\alpha \eta^\beta, \qquad (113)$$

where $u^\mu = dx^\mu/d\tau$ is the four-velocity of the reference geodesic, $R^\mu_{\nu\alpha\beta}$ is the Riemann tensor, and $D/D\tau$ denotes covariant differentiation along the geodesic.

The curvature tensor satisfies the standard symmetry relations:

$$R_{\mu\nu\alpha\beta} = -R_{\nu\mu\alpha\beta} = -R_{\mu\nu\beta\alpha} = R_{\alpha\beta\mu\nu}. \qquad (114)$$

In trembling spacetime, $R^\mu_{\nu\alpha\beta}(x)$ varies deterministically but remains smooth and bounded due to the constraints on $\xi_{\mu\nu}(x)$.

To extract transverse deflections relevant for slit experiments, we project onto spatial components $\mu = i \in \{x, y\}$, assuming the dominant motion is longitudinal, along the z-direction. With $u^\mu \approx (u^0, 0, 0, u^z)$, the leading-order deviation becomes:

$$\frac{d^2 \eta^i}{d\tau^2} \approx R^i_{0z0}(u^0)^2 \eta^z + R^i_{zzj}(u^z)^2 \eta^j + 2R^i_{0zj} u^0 u^z \eta^j. \qquad (115)$$

This equation captures how transverse separation vectors evolve due to curvature components induced by metric trembling. In particular, the off-diagonal terms $R^i_{0zj}$ encode mixed time-space curvature couplings and become increasingly significant in relativistic regimes.

The structure of Equation (115) forms the basis for determining angular spread on a detec-
tion screen. Geodesics originating from nearby slit points diverge due to the effective transverse acceleration encoded by $R^i_{\mu\nu\lambda}u^\mu u^\nu$. The result is a deterministic fringe pattern whose spatial periodicity and contrast depend on the curvature profile and particle energy.

These conclusions remain valid across both low- and high-energy regimes. The full relativistic geodesic formalism thereby generalizes and justifies the non-relativistic derivations of TSRT presented in earlier sections, ensuring consistency when applied to high-energy electrons, relativistic neutrons, or ion beam interference experiments.

## 12.2 Transverse Deflection in the High-Energy Limit

In high-energy regimes where particles approach relativistic velocities $v \to c$, the deflection behavior of geodesics in TSRT must be analyzed using fully relativistic kinematics. TSRT maintains geodesic continuity and causal propagation even at ultra-relativistic speeds, allowing for accurate predictions of transverse deviations without resorting to non-relativistic approximations.

Consider a particle with four-momentum $p^\mu = m\gamma(c, \vec{v})$ traveling along a perturbed geodesic in the spacetime described by:

$$g_{\mu\nu}(x) = \eta_{\mu\nu} + \xi_{\mu\nu}(x), \qquad (116)$$

where $\gamma = (1 - v^2/c^2)^{-1/2}$ and $\xi_{\mu\nu}(x)$ is the smooth, causally admissible metric perturbation field.

The geodesic deviation equation, derived in Section 12.1, governs the evolution of the separation vector $\eta^i$ between neighboring geodesics:

$$\frac{d^2\eta^i}{d\tau^2} \approx R^i_{\mu\nu\lambda} u^\mu u^\nu \eta^\lambda, \qquad (117)$$

where $i \in \{x, y\}$, $u^\mu$ is the four-velocity, and $R^i_{\mu\nu\lambda}$ are components of the Riemann tensor derived from $g_{\mu\nu}$.

In the ultra-relativistic limit, the dominant contribution arises from terms involving $u^0 \approx \gamma$, $u^z \approx \gamma v$, and curvature components coupled to the longitudinal direction.

The transverse deflection angle $\theta_\perp$ accumulated over the propagation length $z_1$ is related to the net transverse momentum impulse $\Delta p_\perp$ experienced by the particle:

$$\theta_\perp \approx \frac{\Delta p_\perp}{p_z}, \quad \text{with} \quad \Delta p_\perp = m \int_0^{z_1} \frac{d^2 x^\perp}{dt^2} dt, \qquad (118)$$

where $x^\perp \in \{x, y\}$ denotes the transverse coordinate, and the transverse acceleration arises from Christoffel terms as:

$$\frac{d^2 x^\perp}{dt^2} \approx -\Gamma^\perp_{\mu\nu} \frac{dx^\mu}{dt} \frac{dx^\nu}{dt}. \qquad (119)$$

This expression preserves the deterministic causal dynamics of TSRT and ensures the scaling is correctly Lorentz covariant. The resulting angular deviation scales inversely with $p_z$, consistent with the action-resolution threshold derived in the previous sections.

The Christoffel symbols $\Gamma^x_{\mu\nu}$, computed from $g_{\mu\nu}$, include terms such as $\Gamma^x_{00}$, $\Gamma^x_{0z}$, $\Gamma^x_{zz}$, which
couple temporal and longitudinal motion to the transverse direction.

Assuming statistical homogeneity and causal boundedness of the trembling metric, the mean square transverse deflection becomes:

$$ \langle \theta_{\perp}^2 \rangle \sim \frac{1}{p_z^2} \int_0^{z_1} \langle f_{\perp}^2(z) \rangle dz, \qquad (120) $$

This result confirms that transverse spread remains finite and controlled by the cumulative curvature effect, with an inverse dependence on $p_z$, as expected from geometric diffraction principles.

At $\gamma \gg 1$, the fringe structure becomes compressed in space, demanding higher resolution to detect transverse modulations. Nonetheless, the curvature-induced structure persists, and TSRT retains its predictive power across all energy scales.

These results confirm that TSRT not only reproduces diffraction-like phenomena at low energies but also provides causal, geometric predictions for relativistic particle propagation through apertures, with transverse momentum redistribution arising from the underlying spacetime structure rather than probabilistic amplitude interference.

## 12.3 Relativistic Action Thresholds and Angular Resolution

In Trembling Spacetime Relativity Theory (TSRT), interference-like patterns arise not from wave superposition but from the deterministic redirection of geodesics by curvature fluctuations $\xi_{\mu\nu}(x)$. These redirected trajectories produce arrival structures on the detection screen whose resolvability is governed by a causal action threshold.

In the relativistic setting, a particle of mass $m$ traversing a proper-time-parametrized geodesic accumulates classical action:

$$ S_{\gamma} = -mc \int_{\gamma} d\tau, \qquad (121) $$

where $d\tau$ is the invariant proper time along the path. Neighboring geodesics, perturbed by trembling spacetime curvature, experience slight shifts $\Delta\tau$ in proper time, giving rise to an action difference:

$$ \Delta S = -mc\Delta\tau. \qquad (122) $$

TSRT asserts that two trajectories are only observationally distinguishable if this action difference exceeds a minimal causal threshold:

$$ |\Delta S| \gtrsim T_{\min}. \qquad (123) $$

To recover known interference fringe structures in the relativistic regime, this threshold must match the Planck scale: $T_{\min} = \hbar$. This identification is not postulated but emerges naturally from the requirement that TSRT reproduces standard fringe periodicities under causal constraints.

For longitudinal motion over a distance $z_1$, a transverse angular deviation $\Delta\theta$ between
geodesics results in a proper-time difference:

$$ \Delta\tau \approx \frac{z_1 \Delta\theta}{\gamma v}, \qquad (124) $$

where $\gamma = 1/\sqrt{1-v^2/c^2}$ and $v$ is the particle velocity. The causal threshold then implies:

$$ \Delta S = -mc \cdot \frac{z_1 \Delta\theta}{\gamma v} \gtrsim \hbar, \qquad (125) $$

yielding a minimal angular deviation:

$$ \Delta\theta_{\min} \gtrsim \frac{\hbar\gamma v}{mcz_1}. \qquad (126) $$

The corresponding spatial resolution on the screen becomes:

$$ \Delta x_{\min} = z_1 \Delta\theta_{\min} \gtrsim \frac{\hbar\gamma v}{mc}. \qquad (127) $$

Using $p_z = \gamma mv$, this simplifies to:

$$ \Delta x_{\min} \gtrsim \frac{\hbar}{p_z}, \qquad (128) $$

matching the relativistic de Broglie scale in form but arising here from a geometric action constraint, not wave superposition.

Importantly, this causal resolution limit persists even in high-energy regimes, demonstrating that TSRT remains predictive and deterministic in the relativistic domain. While increasing momentum compresses the fringe spacing, it does so within the bounds set by the proper-time distinguishability of geodesics, not by uncertainty principles. As such, TSRT offers a causally complete, geometrically grounded explanation for interference patterns at all energy scales—fully consistent with relativistic motion and observational data.

## 12.4 TSRT vs. Dirac Predictions for Relativistic Diffraction

A central theoretical contrast lies in how the Trembling Spacetime Relativity Theory (TSRT) and Dirac-based relativistic quantum mechanics account for diffraction phenomena at high energies. Although both frameworks often predict similar fringe patterns under standard conditions, they diverge fundamentally in their explanatory mechanisms and ontological commitments.

In the Dirac formulation of quantum mechanics, particles are described by a complex spinor field $\psi(x)$ governed by the Dirac equation:

$$ (i\gamma^{\mu}\partial_{\mu} - m)\psi(x) = 0, \qquad (129) $$

where $\gamma^\mu$ are the Dirac matrices. In a double-slit configuration, interference arises from the coherent superposition of spinor amplitudes originating from each slit. The resulting probability density on the detection screen is:

$$ \rho(x, y; z_1) \propto |\psi_A(x, y) + \psi_B(x, y)|^2, \qquad (130) $$
where $\psi_A$ and $\psi_B$ represent the spinor contributions associated with slits A and B, respectively. Fringe patterns result from amplitude cross-terms, and definite particle arrivals are explained via non-deterministic wavefunction collapse.

TSRT, by contrast, replaces this probabilistic interpretation with a deterministic one. It asserts that particles always follow proper-time-parametrized geodesics in a causally fluctuating but locally coherent metric of Equation (1). Interference-like structures arise from curvature-induced modulation of geodesic redirection. In this framework, an ensemble of trajectories acquires a transverse momentum spread constrained by a minimal action threshold:

$$ \Delta S \gtrsim \hbar, \qquad (131) $$

which leads, under relativistic propagation, to a spatial resolution bound:

$$ \Delta x_{\min} \gtrsim \frac{\hbar}{\gamma mv} = \frac{\hbar}{p_z}, \qquad (132) $$

where $p_z = \gamma mv$ is the longitudinal momentum. This result matches the relativistic de Broglie wavelength in functional form, but emerges from a purely geometric principle: neighboring geodesics must differ by sufficient proper time to be causally resolvable.

The key conceptual distinction is that TSRT treats particles as single, causally evolving entities whose observable behavior stems from local curvature, not from amplitude interference. The fringe structure emerges from the spatial density of redirected geodesics—deterministically filtered by the geometry—without invoking superposition or collapse. In contrast, Dirac theory attributes the same structure to interference between delocalized spinor components, with definite outcomes imposed externally through a measurement postulate.

This divergence yields experimentally testable differences. For instance, in asymmetric configurations where one slit is modulated geometrically without detection, TSRT predicts immediate fringe suppression due to local disruption of causal coherence. Dirac theory, however, maintains full interference visibility in the absence of active decoherence. Similarly, TSRT predicts no fringe dependence on spin polarization unless spin is explicitly coupled into the metric fluctuations. By contrast, Dirac theory may predict spin-dependent effects—such as Zitterbewegung or helicity-induced modulation—especially in the presence of external fields.

In conclusion, although TSRT and Dirac theory yield overlapping predictions under symmetric and low-energy conditions, they rely on fundamentally different physical pictures. TSRT replaces quantum indeterminacy with geometric filtering and causal determinism. Its predictions are grounded not in wavefunction dynamics, but in proper-time distinguishability across trembling geodesic paths, offering a realist alternative to the standard relativistic quantum formalism.

## 12.5 Proposed Experiments to Distinguish TSRT from Quantum Theory

To experimentally differentiate Trembling Spacetime Relativity Theory (TSRT) from standard relativistic quantum mechanics, we propose a series of high-energy slit experiments designed to isolate the distinct causal and geometric predictions of TSRT. These experiments utilize relativistic particle beams in conjunction with spatially or temporally modulated apertures, as
well as spin-sensitive configurations.

One proposed experiment involves time-modulated slit activation. In this setup, a relativistic double-slit apparatus alternates the activation of each slit using high-speed gating synchronized with the arrival of the particle beam. TSRT predicts that geodesics respond deterministically to the instantaneous boundary conditions of the trembling metric. If a slit is closed during a particle's traversal—even without any detection event—the causal structure of spacetime is altered, leading to immediate disruption of geodesic coherence and suppression of interference. In contrast, quantum mechanics permits partial interference visibility unless wavefunction collapse or decoherence explicitly occurs. By varying the timing of slit modulation and recording fringe visibility, one can test whether abrupt interference suppression arises purely from causal metric shifts, as TSRT predicts.

A second experiment examines diffraction thresholds at ultra-relativistic energies. Here, relativistic particles (e.g., electrons or neutrons) are passed through slits with progressively smaller separations. According to TSRT, diffraction fringes arise only if neighboring geodesics differ by at least the minimal proper-time action $\Delta S \gtrsim \hbar$. This yields a minimal resolvable angular deviation and associated screen spacing:

$$ \Delta x_{\min} \gtrsim \frac{\hbar}{\gamma mv}, \qquad (133) $$

in symmetric configurations. Below this threshold, trembling spacetime cannot distinguish between neighboring geodesics, and the interference pattern collapses. Quantum mechanics, however, predicts that fringes fade gradually with decreasing de Broglie wavelength $\lambda = h/p$, but does not impose a sharp visibility cutoff. A clear disappearance of fringes at a well-defined energy-to-slit ratio would strongly support TSRT's action-based resolution mechanism.

A third experiment introduces passive boundary asymmetry. This experiment introduces a non-interacting structure—such as a dielectric plate or QED cavity—behind only one slit, without absorbing the particle. In TSRT, this asymmetric causal boundary alters the spacetime geometry and disrupts trembling coherence, leading to fringe degradation despite the absence of a measurement outcome. In quantum mechanics, no change in interference is expected unless the structure induces decoherence or collapse. Observing contrast suppression without particle absorption would indicate that the metric's causal configuration, not probabilistic interaction, governs the interference pattern—favoring TSRT.

A final proposed experiment investigates spin-insensitive diffraction. Spin-polarized relativistic particles are directed through a symmetric double-slit system, and interference patterns are recorded under various spin alignments and external field conditions. TSRT predicts no change in fringe structure with spin polarization unless the trembling metric $\xi_{\mu\nu}(x)$ explicitly couples to spinor degrees of freedom. In its base formulation, spin is not geometrically active. By contrast, Dirac-based quantum theory predicts possible spin-sensitive interference modulations, particularly in chiral or magnetic environments. Identical fringe patterns across spin states in spin-neutral configurations would support the TSRT framework.

Each proposed experiment targets a distinctive feature of TSRT—causal responsiveness, resolution cutoffs, metric asymmetry sensitivity, or spin independence—that is fundamentally
absent in quantum theory. These setups provide falsifiable avenues to test whether diffraction patterns are governed by deterministic geodesic structure rather than probabilistic amplitude interference. Successful confirmation would mark a significant shift in our understanding of interference phenomena, supporting the causal and geometric ontology of TSRT.

## 12.6 Implications for Quantum Foundations and Future Work

The relativistic extension of Trembling Spacetime Relativity Theory (TSRT) has far-reaching implications for the foundations of quantum physics. While relativistic quantum mechanics—including Dirac theory—preserves Lorentz invariance and predicts correct experimental outcomes, it does so within a wavefunction-based formalism that retains probabilistic postulates. TSRT proposes a fundamentally different paradigm: a deterministic, geometric, and local explanation of quantum-like phenomena rooted entirely in the causal structure of spacetime.

### 1. From Wavefunctions to Causal Geometry

TSRT replaces the wavefunction as a primitive object with the geometry of spacetime itself. Interference patterns, measurement effects, and other traditionally quantum features emerge not from probabilistic superposition, but from the deterministic redirection of particle geodesics through a trembling metric field $\xi_{\mu\nu}(x)$. This field, governed by local curvature constraints and causal coherence, modulates trajectories in a way that statistically reproduces quantum interference—without invoking nonlocality or collapse. This shift restores locality and realism at the microscopic level.

### 2. Lorentz-Invariant Determinism

Unlike quantum wave theories, which face interpretational challenges under Lorentz transformations (e.g., frame-dependent collapse), TSRT maintains full relativistic covariance. All dynamics are described using proper-time-parametrized geodesics and covariant curvature tensors. Observers in different frames agree on the geometric structure and causal order of events. There is no need for frame-specific reductions or observer-dependent outcomes. Interference patterns emerge from ensemble-level geodesic statistics filtered by local curvature, not from amplitude superposition across incompatible frames.

### 3. Planck's Constant as a Causal Resolution Scale

In TSRT, Planck's constant $\hbar$ is not postulated, but identified with the minimal proper-time action difference required to distinguish two geodesics. This threshold,

$$ |\Delta S| \gtrsim \hbar, \qquad (134) $$

defines the smallest action gap that spacetime geometry can resolve. It introduces a natural lower bound on angular deviation and spatial separation for interference structures to arise. Unlike the quantum uncertainty relation $\Delta x \Delta p \gtrsim \hbar/2$, which stems from operator non-commutativity, TSRT's resolution limit is geometric and causal. Quantized behavior emerges
from this action-based separability condition, linking $\hbar$ directly to the structure of trembling spacetime.

## 4. Experimental and Theoretical Outlook

The relativistic TSRT framework yields concrete, falsifiable predictions that diverge from quantum theory in distinct regimes. The proposed experiments in Section 12.5—including high-speed slit modulation, ultra-relativistic fringe thresholds, asymmetric causal boundaries, and spin-invariant interference—target specific signatures of TSRT. These include sharp resolution cutoffs, causal coherence loss under dynamic boundary change, and the absence of spin sensitivity in the basic model.

On the theoretical side, TSRT offers several promising directions:

* Coupling trembling geometry to spinor or chiral structures, to explain the emergence of spin and magnetic effects.

* Developing TSRT-based field theories, with gauge symmetries arising from covariant constraints on $\xi_{\mu\nu}(x)$.

* Extending the framework to curved backgrounds, enabling consistent integration with semiclassical gravity and early-universe cosmology.

## 5. A New Paradigm for Microscopic Physics

If supported by experiment, TSRT would represent a foundational shift: from probabilistic, wavefunction-centric microphysics to a geometric, deterministic ontology governed by causal geodesics in a trembling spacetime. It reframes measurement outcomes as responses to local boundary-induced curvature shifts, rather than collapses of an abstract amplitude field. The relativistic extension developed here preserves consistency with Lorentz symmetry and empirical fringe data, while offering a radically different interpretation of their origin. In doing so, TSRT lays the groundwork for a unified, realist description of the microscopic world—one in which quantization and interference are no longer mysterious axioms, but natural consequences of geometric causality.

# 13 Robustness of TSRT Predictions Under Metric Variations

## Causal Constraints and Perturbation Smoothness

The predictive power of TSRT depends on how geodesic trajectories respond to small-scale spacetime fluctuations encoded in the perturbation field $\xi_{\mu\nu}(x)$. For the predicted interference-like patterns to be physically meaningful, it is essential that these patterns remain stable across a broad range of admissible perturbations. This subsection formalizes the key constraints that such perturbations must obey.

TSRT requires that metric perturbations $\xi_{\mu\nu}(x)$ satisfy three core conditions:
**1. Causal admissibility.** The perturbed metric must preserve the timelike nature of proper-time evolution. All particle geodesics must satisfy $d\tau^2 = -g_{\mu\nu}(x)dx^\mu dx^\nu > 0$, preventing spacelike or null propagation for massive particles. This ensures that local causality is preserved and that each trajectory corresponds to a physically realizable worldline.

**2. Curvature boundedness.** The Riemann curvature tensor derived from $g_{\mu\nu}(x)$ must obey a finite upper bound:

$$R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta} \leq \mathcal{R}_{\max}^2, \qquad (135)$$

where $\mathcal{R}_{\max}$ is a universal curvature cutoff derived from the causal action scale. This condition prevents violent fluctuations that would induce unphysical deflections or geodesic disruption.

**3. Smoothness and coherence.** The perturbation field $\xi_{\mu\nu}(x)$ must vary smoothly over spatial regions larger than the particle's effective interaction scale—typically its Compton wavelength or classical coherence length. High-frequency discontinuities or singularities are excluded, as they undermine the differentiability assumptions underlying the geodesic and Jacobi equations.

These constraints define the admissible class of trembling geometries for which TSRT predictions are both consistent and physically meaningful. The interference-like patterns predicted in TSRT do not depend on any specific choice of $\xi_{\mu\nu}(x)$, but emerge robustly across the entire admissible class.

## Pattern Stability Across Perturbation Classes

Within this causally consistent and curvature-bounded class of perturbations, TSRT predictions for particle arrival patterns exhibit remarkable stability. This subsection surveys representative perturbation types and shows that each preserves the core feature: structured arrival patterns governed by the action-based resolution scale $\mathcal{L}_{\text{TSRT}} \propto \hbar/p_z$, with geometry-dependent refinements.

**1. Long-wavelength harmonic fluctuations.** If the perturbation field $\xi_{\mu\nu}(x)$ consists of smooth, low-frequency modes with characteristic scale larger than the slit width, geodesics are redirected gently and coherently. The ensemble of redirected trajectories projects a stable modulation pattern onto the screen, with fringe spacing determined by the effective scale $\mathcal{L}_{\text{TSRT}}$. Variations primarily affect fringe contrast, not structure or spacing.

**2. Localized curvature wells.** Perturbations localized near the slit region—such as compact “curvature wells”—can redirect geodesics asymmetrically. Provided the curvature remains bounded and the perturbation smooth, the resulting arrival distribution retains well-defined fringe periodicity. The center of the pattern may shift, and contrast may vary, but angular separation remains consistent with TSRT’s action-threshold predictions.

**3. Asymmetric or single-slit deformation.** Introducing asymmetries—either in slit geometry or in metric perturbations near one slit—results in skewed or suppressed fringe envelopes.
TSRT accounts for these effects as deterministic consequences of asymmetric geodesic deflec- tion, not as wavefunction collapse. Importantly, the spacing of observable maxima remains controlled by the same curvature-based resolution criterion.

**4. Temporally modulated trembling.** If the metric perturbation varies slowly over time—on timescales longer than the particle flight duration—the ensemble behavior averages over differ- ent redirection fields. This can lead to slight contrast degradation or envelope blurring, but fringe spacing is preserved. Only rapid temporal modulation that violates causal coherence would breach TSRT’s validity assumptions and is excluded from the admissible class.

Across all admissible perturbation types, the structure of arrival densities on the screen reflects the geometric filtering of the slit function $S(x, y)$, shaped by deterministic redirection through trembling geodesics. The predicted fringe spacing remains governed by

$$ \Delta x \sim \mathcal{L}_{\text{TSRT}} = \frac{\hbar v z_1}{m L}, \qquad (136) $$

in general, or $\hbar/p_z$ in symmetric configurations where $L = z_1$. This ensures that TSRT’s predictions are not tied to special cases, but arise generically within its geometric framework.

To summarize, the interference-like patterns predicted by TSRT in slit-based experiments are not fragile consequences of fine-tuned assumptions. Rather, they emerge generically from a wide class of causally admissible, smooth, and curvature-bounded perturbations. Whether the trembling is symmetric or asymmetric, localized or extended, static or slowly varying, the deterministic redirection of geodesics yields statistically robust arrival structures governed by the curvature-induced resolution scale.

This robustness is central to TSRT’s empirical viability. It implies that fringe patterns and their spacing should remain consistent under realistic experimental conditions—even in the presence of imperfections or mild environmental fluctuations—so long as causal and curvature constraints are respected. Conversely, systematic violations of this pattern stability would falsify the TSRT framework.

By grounding its predictions in causal geometry rather than probabilistic superposition, TSRT offers a novel and testable reinterpretation of interference. The resilience of its predictions under broad metric variation enhances its status as a physically credible and experimentally falsifiable theory of particle dynamics in spacetime.

## 14 Conclusion

This paper has presented a deterministic and geometric explanation for interference phenom- ena, proposing a fundamental departure from the standard quantum interpretation grounded in wave-particle duality and probabilistic amplitudes. Within the framework of Trembling Spacetime Relativity Theory (TSRT), particles are always localized, and their propagation fol- lows proper-time-parametrized geodesics through a smoothly trembling yet causally coherent spacetime. Structured interference patterns observed in double-slit experiments arise not from superposed wavefunctions or collapse mechanisms, but from curvature-induced redirection of trajectories filtered by the geometry of the aperture.
Under symmetric and static slit conditions, TSRT reproduces the same interference patterns as standard quantum mechanics. The agreement extends to fringe spacing, envelope modulation, and the suppression of interference upon which-path detection. However, while the numerical outcomes match, the underlying physical mechanisms differ fundamentally. In TSRT, Planck's constant $\hbar$ emerges as a geometric action threshold that determines the minimum resolvable separation between geodesics. This minimal action, rather than intrinsic quantum uncertainty, defines the angular resolution limit of particle deflection.

Beyond this agreement, TSRT offers the advantage of falsifiability through geometric interpretation. In decoherence regimes, TSRT asserts that each particle passes deterministically through one slit only, and that ensemble-level decoherence arises from causal incoherence across realizations, not from incoherent superposition of amplitudes. The implications are experimentally distinguishable: residual structural asymmetries may persist in arrival densities even when coherence is lost. In addition, TSRT predicts effects not typically expected under quantum theory, including hysteresis in fringe realignment after dynamic boundary changes, curvature-induced structure in single-slit scenarios, and nontrivial responses to temporally modulated apertures.

These subtle but measurable deviations define a new class of precision experiments—ones that can be designed to distinguish between causal geodesic steering and amplitude-based interference. Whether through slit-local curvature perturbations, asymmetric refractive delays, or dynamical geometries, the opportunity now exists to test whether microscopic interference is fundamentally geometric or fundamentally probabilistic.

In this light, TSRT stands as a realist and deterministic alternative to quantum theory. It remains empirically consistent in all conventional scenarios, while opening avenues for experimental validation—or falsification—of its core geometric principles. The question of whether interference stems from trembling spacetime or wavefunction interference is not one of philosophical preference, but one that future experiments may finally resolve.

# 15 Additional Information

**Author Contributions:** N.F.D. conceived the theoretical framework, performed the analysis, and wrote the manuscript.

# 16 Acknowledgements

The author gratefully acknowledges institutional support from the French Centre National de la Recherche Scientifique (CNRS) through the International Research Laboratory IRL 2958 Georgia Tech-CNRS, as well as Metz Métropole, the City of Metz, and Georgia Tech Europe for hosting him as a Georgia Tech faculty member in France.

Infrastructure and mobility support were provided by the Région Grand Est, the French Lorraine Université d'Excellence (LUE impact project I-META ANR-15-IDEX-04-LUE), Institut Carnot ARTS, Conseil Régional de Lorraine, Le Conseil Départemental de la Moselle, Feder-Europe, the Indo-French Centre for the Promotion of Advanced Research (CEFIPRA
RCF-IN-0067), Mirabelle Plus, Stellantis, Institut de Soudure, the Association Nationale de la Recherche et de la Technologie (ANRT), Metz Métropole, the Institut Supérieur Européen de l'Entreprise et de ses Techniques (ISEETECH), the Contrat Plan État-Région (CPER-MEPP-P05), the Agence Nationale de la Recherche (ANR-09-BLAN-0167-01), the European Union's Horizon 2020 program (Grant Agreement No. 871260), the Belgian Fund for Scientific Research Flanders (FWO, 2005), the Flemish Institute for the Encouragement of Scientific and Technological Research in Industry (IWT, 2001-13343), and the North Atlantic Treaty Organization (NATO PST.CLG.980315). While these programs provided an invaluable research environment, the conceptual direction and results presented here stem from independent scholarly inquiry developed over decades.

The author acknowledges the foundational training in physics, astrophysics, and engineering physics received at the Catholic University of Leuven (KULAK and KU Leuven) and Ghent University, which shaped his interdisciplinary approach. He also thanks the Georgia Institute of Technology and the George W. Woodruff School of Mechanical Engineering for his promotion to Full Professor, coinciding with his designation as *Professor Honoris Causa* in Physics by the University of Allahabad—a confluence of recognition that secured the academic freedom necessary to formalize the ideas presented here, which trace their origins to early studies of solar system dynamics (Driesprong, Deerlijk) and electron beam deflection experiments (Koninklijk Atheneum, Waregem).

The author extends his deepest gratitude to colleagues and organizations in acoustics, ultrasonics, acousto-optics, and optics for fostering an intellectually vibrant and collegial environment. Their shared passion for wave phenomena has been a continual source of inspiration, and their engagement with industrial applications always boosted inspiration to seek critical support to sustain the author's laboratory.

He is equally indebted to his current and former students (BS, MS, PhD) and postdoctoral researchers, whose curiosity and fresh perspectives profoundly enriched his work in nondestructive evaluation and physical acoustics. Their enthusiasm reaffirmed that scientific discovery—regardless of discipline—thrives on intuition, relentless curiosity, and intellectual joy, motivating the author's deeper explorations in fundamental physics beyond conventional working hours.

This work stands as a testament to the enduring intellectual and material legacies of Daniel A. Lesage (1911–1988), Pierre F. Vangheluwe (1939–1991), Mariette Vanoosthuyze (1925–2009), Godelieve Verborgh (1932–2012), Maurice A. Declercq (1941–2015), and Jeanne M. C. Maesen (1930–2024), whose generosity continues to enable scholarship. The author is equally indebted to Jeanette Verbrugghe and Nelly Vangheluwe for their unwavering support—both moral and logistical—which has been indispensable to this research.

Finally, the author expresses profound gratitude to his wife and children for their steadfast presence, boundless encouragement, and enduring patience throughout the many weekends and holidays devoted to this work.
References

[1] Thomas Young. A course of lectures on natural philosophy and the mechanical arts. *Lecture Notes*, 1:461–463, 1807. Describes early double-slit observations with light.

[2] Clinton Davisson and Lester H. Germer. Diffraction of electrons by a crystal of nickel. *Physical Review*, 30:705–740, 1927. doi:10.1103/PhysRev.30.705.

[3] Claus Jonsson. Elektroneninterferenzen an mehreren kunstlich hergestellten feinspalten. *Zeitschrift für Physik*, 161:454–474, 1961. doi:10.1007/BF01342460.

[4] Niels Bohr. The quantum postulate and the recent development of atomic theory. *Nature*, 121:580–590, 1928.

[5] Yoon-Ho Kim, Rong Yu, Sergei P. Kulik, Yanhua Shih, and Marlan O. Scully. Delayed "choice" quantum eraser. *Physical Review Letters*, 84:1–5, 1999. doi:10.1103/PhysRevLett.84.1.

[6] Markus Arndt, Olaf Nairz, Jan Vos-Andreae, Claus Keller, Gerbrand van der Zouw, and Anton Zeilinger. Wave-particle duality of $c_{60}$ molecules. *Nature*, 401:680–682, 1999. doi:10.1038/44348.

[7] Adrian Bachtold and et al. Coherent control of a single electron in a double quantum dot. *Nature Physics*, 17:1057–1062, 2021. doi:10.1038/s41567-021-01299-9.

[8] Albert Einstein, Boris Podolsky, and Nathan Rosen. Can quantum-mechanical description of physical reality be considered complete? *Physical Review*, 47(10):777–780, 1935. URL: https://journals.aps.org/pr/abstract/10.1103/PhysRev.47.777, doi:10.1103/PhysRev.47.777.

[9] Stephen L. Adler. Why decoherence has not solved the measurement problem. *Studies in History and Philosophy of Modern Physics*, 34:135–142, 2003. doi:10.1016/S1355-2198(02)00086-1.

[10] Roger Penrose. *Fashion, Faith, and Fantasy in the New Physics of the Universe*. Princeton University Press, 2014.

[11] John S. Bell. On the einstein podolsky rosen paradox. *Physics Physique Fizika*, 1(3):195–200, 1964. doi:10.1103/PhysicsPhysiqueFizika.1.195.

[12] Bas Hensen and et al. Loophole-free bell inequality violation using electron spins separated by 1.3 km. *Nature*, 526:682–686, 2015. doi:10.1038/nature15759.

[13] David Bohm. A suggested interpretation of the quantum theory in terms of “hidden” variables. *Physical Review*, 85:166–179, 1952.

[14] Gian Carlo Ghirardi, Alberto Rimini, and Tullio Weber. Unified dynamics for microscopic and macroscopic systems. *Physical Review D*, 34(2):470–491, 1986. doi:10.1103/PhysRevD.34.470.