# Causal Entanglement from Deterministic Geodesic Correlation in Trembling Spacetime

Nico F Declercq

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# Causal Entanglement from Deterministic Geodesic Correlation in Trembling Spacetime

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

## 1 Abstract

Quantum entanglement is widely regarded as the most puzzling feature of quantum theory, apparently violating local realism through nonlocal correlations between spacelike-separated particles. Although violations of Bell inequalities are experimentally confirmed, their standard theoretical explanation relies on wavefunction collapse or nonlocal hidden variables.

In this work, we present an alternative explanation based on Trembling Spacetime Relativity Theory (TSRT). In this framework, intrinsic metric fluctuations generate proper-time-synchronized geodesic ensembles whose deterministic trajectories give rise to statistical correlations across spacetime. Measurement outcomes are reinterpreted as geometric responses to these fluctuating backgrounds, without invoking quantum collapse or nonlocal influence.

We show that the Clauser-Horne-Shimony-Holt (CHSH) inequality can be violated deterministically within a locally causal geometric structure. These violations emerge as macroscopic imprints of shared spacetime curvature, not quantum superposition. The resulting correlation function reproduces quantum predictions at short ranges but decays with separation—yielding testable deviations in long-baseline and gravitationally modulated experiments. TSRT thus offers a realist, Lorentz-invariant, and falsifiable account of entanglement grounded in the causal structure of spacetime.

## 2 Introduction

Quantum entanglement, a cornerstone of quantum mechanics, has evolved from a foundational puzzle into a pivotal resource for cutting-edge technologies. Recent advances have positioned entanglement as the backbone of revolutionary applications, including secure quantum communication [1], distributed quantum computing [2], and ultra-precise quantum sensing [3]. The successful demonstration of satellite-based entanglement distribution over 1200 km [4] and loophole-free Bell tests [5] underscores its technological viability. Moreover, entanglement is now harnessed in quantum networks [6] and quantum-enhanced metrology [7], promising transformative impacts across cryptography, computation, and imaging. As research accelerates toward scalable implementations, entanglement-based protocols are poised to redefine information processing in the post-classical era.
Quantum entanglement, however, remains one of the deepest puzzles in modern physics. First raised by Einstein, Podolsky, and Rosen ('EPR') in 1935 as part of their critique of quantum mechanics [8], the phenomenon implies that the measurement outcomes of spatially separated particles can exhibit correlations that exceed what classical realism and locality would allow. The EPR paradox suggested that quantum mechanics must be incomplete, and posited the existence of "elements of reality" that determine measurement outcomes independently of observation.

In 1964, John Bell transformed this conceptual argument into a quantitative test [9]. He derived an inequality that any local hidden-variable theory must satisfy. The most widely used formulation, due to Clauser, Horne, Shimony, and Holt (CHSH) [10], provides a practical criterion for distinguishing between local realist models and quantum predictions. The CHSH inequality serves as the standard framework to test local realism, predicting a maximum bound of $S = 2$ under classical theories, and up to $S = 2\sqrt{2}$ under quantum mechanics [11].

Violations of Bell inequalities have been experimentally confirmed in a wide range of settings, from optical photon pairs [12], to long-distance fiber links [13], and most recently via satellite-based platforms [4]. These results are widely interpreted as proof that nature is either nonlocal or fundamentally indeterministic. Beginning with the landmark experiments of Aspect et al. in the 1980s [12], empirical violations of Bell inequalities have consistently upheld the predictions of quantum theory.

However, these conclusions rest on assumptions about the structure of spacetime and the independence of experimental settings. Standard interpretations assume a static, unstructured spacetime background in which measurement choices are treated as free variables, uncorrelated with any hidden factors that might influence outcomes. But if spacetime itself exhibits deterministic, causally propagating fluctuations, these assumptions may fail—and with them, the logical foundation of Bell's theorem.

While quantum mechanics successfully predicts the violation of Bell inequalities through the concept of entanglement, it does so by invoking inherently nonlocal dynamics—specifically, wavefunction collapse across spacelike intervals. This has long raised conceptual concerns, including Einstein's critique of "spooky action at a distance." The purpose of this work is to resolve these issues within a deterministic and causal framework rooted in geometric structure, offering a reformulation of entanglement not as a fundamental nonlocality, but as an emergent phenomenon from trembling spacetime.

This work, therefore, introduces an alternative explanation of entanglement based on the Trembling Spacetime Relativity Theory (TSRT) [14,15]. This model builds on the foundational TSRT framework<fnref n="1" /> developed by the author [14,15], where trembling metric fluctuations were shown to produce proper-time-synchronized geodesic motion that mimics quantum statistical behavior. In this framework, spacetime is a dynamic geometric field undergoing microscopic fluctuations that are deterministic and synchronized in proper time. These metric perturbations influence both the trajectories of particles and the configurations of measuring devices, embedding subtle correlations between distant systems through their shared causal structure.

<footnote n="1">¹Unlike hidden variable models, which attribute outcomes to unobservable internal states, TSRT attributes correlation to deterministic curvature fluctuations in spacetime geometry, maintaining full consistency with relativistic causality.</footnote>
This is in contrast to quantum mechanics, where traditional interpretations explain these correlations through wavefunction collapse or invoke nonlocal hidden variable theories, as formalized in the EPR argument [8] and debated in Bell's theorem [9].

Within TSRT, we derive the entanglement correlation function

$$C_{\text{TSRT}}(\Delta\theta, L) = \left(\frac{L_c}{L_c + L}\right)^n \exp\left(-\frac{L^2}{\lambda_c^2}\right) \cos(\Delta\theta), \quad (1)$$

from first principles, without invoking superposition, wavefunction collapse, or nonlocal signaling. In this expression, the cosine factor $\cos(\Delta\theta)$ reproduces the standard angular dependence predicted by quantum mechanics for spin-entangled systems [10, 11], while the prefactor—including both the rational decay and the Gaussian suppression—emerges uniquely from the trembling spacetime geometry of TSRT. These modulation terms quantify the decay of geodesic alignment with increasing spatial separation, and are derived in detail in later sections.

Our findings suggest that quantum correlations can be fully accounted for by local, deterministic interactions with a trembling spacetime manifold. This undermines the assumptions of Bell's theorem, particularly measurement independence and background staticity, and reframes entanglement as a geometric, testable property of causal spacetime structure rather than a uniquely quantum mystery.

## 3 Entanglement in Standard Quantum Mechanics

In the formalism of standard quantum mechanics, entanglement arises when two or more subsystems share a common wavefunction that cannot be factorized into separate states for each subsystem. A simple example is the spin singlet state of two spin-$\frac{1}{2}$ particles:

$$|\Psi\rangle = \frac{1}{\sqrt{2}} (|\uparrow\rangle_A \otimes |\downarrow\rangle_B - |\downarrow\rangle_A \otimes |\uparrow\rangle_B), \quad (2)$$

where the outcomes of spin measurements on each subsystem are perfectly anti-correlated, despite spatial separation. The core feature of such states is that the measurement result on one subsystem instantaneously determines the state of the other, independent of their separation distance.

In the quantum mechanical case, the expectation value of outcome correlations between two spin measurements is given by $E(a, b) = \cos(\theta_{ab})$, reflecting the projection of one measurement axis onto the other. This result emerges from the inner product of two spin-1/2 state vectors and is fundamental to quantum predictions of entanglement.

According to the standard interpretation, these correlations are not due to classical signaling or common causes, but are intrinsic to the structure of the quantum state itself. The formalism predicts that for certain measurement settings, the correlation function between binary outcomes $A, B \in \{-1, +1\}$ is given by [9, 16]:

$$E_{\text{QM}}(\theta) = -\cos(\theta), \quad (3)$$
where $\theta = \theta_A - \theta_B$ is the angle between measurement settings.

Bell's theorem shows that no local hidden-variable theory can reproduce these predictions if it respects both realism and measurement independence. The experimental confirmation of such quantum correlations—most notably by Aspect et al. [12] and more recent loophole-free tests [5, 17, 18]—has led many to conclude that nature is either nonlocal or fundamentally indeterministic.

However, these conclusions depend critically on the interpretation of the quantum state as a real, physical object and on the absence of any shared causal influence that might account for the correlations. In particular, the assumption of measurement independence—that detector settings are freely chosen and uncorrelated with any hidden variables—is essential for deriving Bell inequalities. If this assumption fails, then the observed violations no longer rule out locality or determinism.

Standard quantum mechanics does not address the ontological origin of the entangled state or the mechanism by which measurement settings and outcomes become correlated. Instead, the wavefunction serves as a computational tool, with collapse postulated as a non-dynamical process triggered by measurement. This interpretational ambiguity leaves open the possibility that entanglement correlations may arise from deeper, causally structured dynamics not captured by the standard formalism.

In the following sections, we develop such a framework based on trembling spacetime geometry. Rather than rejecting locality or realism, we reconsider the physical basis for Bell's assumptions and derive entanglement correlations from deterministic geodesic behavior embedded in the causal structure of spacetime.

# 4 Bell's Theorem and Its Domain of Applicability

Bell's theorem establishes that no physical theory satisfying both locality and measurement independence can reproduce all the predictions of quantum mechanics. It is formulated in terms of correlations between binary outcomes $A, B \in \{-1, +1\}$ measured at distant detectors, with settings $a$ and $b$. Under the assumptions of local realism, the joint outcome statistics are governed by hidden variables $\lambda$, with a factorized joint distribution:

$$P(A, B|a, b, \lambda) = P(A|a, \lambda) \cdot P(B|b, \lambda). \qquad (4)$$

The core additional assumption is that the distribution $\rho(\lambda)$ of hidden variables is statistically independent of the measurement settings:

$$\rho(\lambda|a, b) = \rho(\lambda). \qquad (5)$$

This last condition—known as measurement independence or statistical independence—reflects the belief that the detector settings can be freely chosen, uncorrelated with any properties of
the particles or hidden variables.<fnref n="2" /> Given this assumption, Bell derived the inequality:

$$|S| = |E(a, b) - E(a, b') + E(a', b) + E(a', b')| \le 2, \qquad (6)$$

where each $E(a, b)$ is the expectation value of the product AB at given settings.

Quantum mechanics predicts and experiments confirm violations of this inequality, reaching values up to $2\sqrt{2}$, the so-called Tsirelson bound [20]. These results are widely interpreted as evidence that nature must abandon either locality, realism, or the freedom of measurement choice.

However, Bell's theorem is not universally applicable; it holds only under specific assumptions about the nature of spacetime and the statistical independence of measurement settings from any hidden influences. Central to the theorem is the condition $\rho(\lambda|a, b) = \rho(\lambda)$, which assumes that the hidden variables $\lambda$ are uncorrelated with the detector settings $a$ and $b$. This assumption fails if both the measurement settings and particle properties are influenced by a shared causal background—such as deterministic fluctuations in the underlying spacetime geometry.

Moreover, Bell's formalism presumes a static and unstructured spacetime in which all statistical variables are defined independently of the geometry. It does not account for dynamically evolving or causally correlated fluctuations in the spacetime metric. As a result, the Bell inequality no longer holds in any framework—such as TSRT—where the geometry of spacetime itself coherently governs both particle trajectories and detector configurations.

Thus, Bell's inequality is not a universal constraint on all deterministic, local theories. It is a constraint only on those theories that assume a background-independent statistical independence between settings and hidden variables. The Trembling Spacetime Relativity Theory (TSRT) introduces precisely the kind of causal geometric correlations that invalidate this assumption, without invoking nonlocality or abandoning realism.

In the next sections, we explore how Bell-type correlations can arise from shared spacetime fluctuations in TSRT, providing a local, deterministic, and causal account of entanglement that lies outside the applicability domain of Bell's theorem.

# 5 Measurement Dependence from Metric Fluctuations

The assumption of measurement independence is central to the derivation of Bell's inequality. It asserts that the statistical distribution of hidden variables $\lambda$ influencing measurement outcomes is uncorrelated with the choice of detector settings $a$ and $b$. Mathematically, this is expressed as:

$$\rho(\lambda|a, b) = \rho(\lambda), \qquad (7)$$

implying that the process determining the measurement axes is causally disconnected from the hidden variables carried by the particles.

<footnote n="2">²This assumption is closely related to what is known in the literature as the “freedom-of-choice loophole” [19], which explores whether measurement settings might be statistically correlated with hidden variables. TSRT offers a deterministic mechanism that violates this assumption causally rather than conspiratorially.</footnote>
In Trembling Spacetime Relativity Theory (TSRT), however, this assumption does not hold. The underlying geometry of spacetime is not static or independent of the experiment—it trembles deterministically in a way that affects all physical systems embedded within it. These fluctuations in the metric field $\xi_{\mu\nu}(x)$ influence both the particles and the detectors via proper-time synchronized geodesic perturbations. As a result, the effective detector settings and the particle properties are both shaped by the same causal structure:

$$ \theta_A = \theta_A^0 + \delta\theta_A(\xi(x_A)), \quad \theta_B = \theta_B^0 + \delta\theta_B(\xi(x_B)), \quad \lambda = \lambda[\xi(x)], \qquad (8) $$

where $\delta\theta_A$, $\delta\theta_B$, and $\lambda$ are all functionals of the local realization of the trembling metric field.

This shared geometric origin leads to statistical correlations between settings and hidden variables:

$$ \rho(\lambda|\theta_A, \theta_B) \neq \rho(\lambda), \qquad (9) $$

even though all influences remain local and relativistically causal. No information or signal travels between measurement stations faster than light, and no retrocausal mechanism is invoked. Instead, the apparent dependence arises because both detectors and particles follow geodesics perturbed by a common spacetime field.

The resulting alignment of detector settings and particle properties is not conspiratorial or superdeterministic. Rather, it reflects the fact that both systems respond to coherent geometric fluctuations in the surrounding spacetime. In TSRT, this causal connection is entirely physical and grounded in the structure of relativistic geometry.

Consequently, Bell's derivation—which critically depends on the statistical independence of measurement settings from hidden variables—is no longer valid within TSRT, since this assumption is geometrically violated by the causal influence of metric fluctuations. The observed violations of the CHSH inequality no longer imply nonlocality. They reflect the failure of an assumption that holds only in theories with a fixed, uncorrelated spacetime background. In TSRT, the background itself is dynamic and correlated, and these correlations naturally give rise to entanglement-like statistics.

This opens the door to a fully deterministic, realist interpretation of quantum correlations—one that remains strictly local but no longer constrained by the assumptions of Bell's theorem.

# 6 Entanglement as Causal Metric Correlation

Entanglement is traditionally regarded as an intrinsically nonlocal feature of quantum systems. In standard interpretations, measurement outcomes at spacelike-separated locations exhibit statistical correlations that defy explanation by local hidden-variable models. Bell's theorem formalizes this apparent conflict between locality and realism, leading many to conclude that quantum mechanics must either violate relativistic causality or abandon determinism.

Trembling Spacetime Relativity Theory (TSRT) offers an alternative view. In this framework, quantum entanglement is not an expression of nonlocality, but the observable signature of coherent causal structure in the fluctuating geometry of spacetime itself. Specifically, entangled
systems exhibit strong correlations because both the particles and the detectors are embedded within the same deterministic, trembling spacetime. Their behavior is modulated by shared metric fluctuations that propagate causally, respecting proper-time synchronization and local relativistic constraints.

Detector settings, typically represented by orientation angles $\theta_A$ and $\theta_B$, are not externally imposed parameters, but emergent quantities shaped by geodesic deformation. The trembling metric $\xi_{\mu\nu}(x)$ perturbs both particle trajectories and detector alignments in a coordinated manner:

$$ \theta_A = \theta_A^0 + \delta\theta_A(\xi(x_A)) = \theta_A^0 + \delta\theta_A(x_A), \qquad (10) $$

$$ \theta_B = \theta_B^0 + \delta\theta_B(\xi(x_B)) = \theta_B^0 + \delta\theta_B(x_B), \qquad (11) $$

where $\delta\theta_A$ and $\delta\theta_B$ represent small angular deviations induced by local metric components. Because $\xi_{\mu\nu}$ is a causally propagating field, the perturbations at distant locations are statistically correlated, even though no superluminal influence occurs. This implies that:

$$ \langle \delta\theta_A \delta\theta_B \rangle = f(|x_A - x_B|), \qquad (12) $$

where the function $f(L)$ decays with spatial separation but remains nonzero over macroscopic distances due to long-range coherence in the trembling geometry.

These correlations affect both the measured particle outcomes and the detector configurations, producing the same angular-dependent statistics predicted by quantum theory—but without invoking wavefunction collapse, superposition, or nonlocal interaction. In TSRT, the wavefunction plays no fundamental ontological role. It is a statistical proxy summarizing the alignment of ensembles of trembling geodesics.

Thus, what is commonly interpreted as “entanglement” is recast in TSRT as the deterministic consequence of metric-induced geodesic alignment. The observed joint measurement statistics reflect shared causal history encoded in the structure of spacetime. Proper-time synchronization ensures that these correlations do not violate locality or enable signaling, even though they persist across spacelike separations.

This reformulation displaces the mystery of entanglement from the abstract domain of Hilbert space into the physical domain of geometric causality. It aligns observed phenomena with a realist, local, and deterministic description of nature—one rooted in the trembling fabric of spacetime itself.

# 7 Revisiting the Aspect Experiment in Causal Geometry

The landmark experiments by Alain Aspect and collaborators in the early 1980s [12] are widely considered decisive empirical confirmations of quantum mechanics. In particular, Aspect et al. demonstrated CHSH violations using entangled photon pairs and rapidly switching polarizers, ensuring spacelike separation between measurement events and effectively closing the locality loophole. Their findings remain a cornerstone of experimental quantum foundations [12]. By demonstrating statistically significant violations of Bell's inequality under conditions that
enforce spacelike separation between measurement events, these experiments are commonly
interpreted as ruling out all local realist theories.

However, this interpretation depends critically on a key assumption: that measurement settings are freely chosen and uncorrelated with any hidden variables carried by the particles. This is the assumption of measurement independence, and it is foundational to Bell's theorem. In the Trembling Spacetime Relativity Theory (TSRT), this assumption fails—not due to conspiratorial determinism, but because both the detector settings and the particle trajectories are influenced by the same causally structured fluctuations in the spacetime metric.

In TSRT, spacetime is not a fixed background. The metric $g_{\mu\nu}(x) = \eta_{\mu\nu} + \xi_{\mu\nu}(x)$ fluctuates deterministically across all regions of space and time. These fluctuations are proper-time synchronized and propagate within the light cone, maintaining strict locality. Yet because both detectors and particles are geodesically embedded in this trembling geometry, they are not independent subsystems. Their behavior is coordinated by a shared causal structure.

In Aspect's experiment, the measurement settings at each detector were modulated using fast, random switching mechanisms to enforce spacelike separation. Under standard assumptions of special relativity and quantum theory, this precludes any form of subluminal or luminal communication between the detectors that could explain observed correlations. Consequently, quantum mechanics resorts to the notion of entanglement and nonlocal wavefunction collapse to account for the statistical violations of Bell inequalities. This gives rise to the well-known paradox of “spooky action at a distance”.

In TSRT, no such paradox arises. The observed correlations are not the result of instantaneous influences or faster-than-light information exchange, but instead emerge from the common causal origin of the two measurement events within a trembling spacetime background. More precisely, the local configurations of the measuring devices—including the final values of their random switching settings—are not fundamentally random, but are causally influenced by deterministic fluctuations of the underlying spacetime metric. These fluctuations originate in a shared region of the past light cone of both detectors and propagate deterministically along geodesics shaped by trembling-induced geometry. Thus, statistical correlations observed in the measurement outcomes are not caused by direct communication between the detectors, but by a common causal trembling structure that influences both settings and outcomes.

To express this more precisely, we model the local trembling-induced variation in the detector orientation as a sum over deterministic spacetime modes. These modes arise from the proper-time synchronized fluctuations of the metric along each geodesic and are constrained to respect the causal past of both measurement events. The angular setting at each detector is thus influenced by the same underlying spacetime geometry, leading to naturally correlated outcomes without any exchange of information at or above the speed of light.

To model this, TSRT replaces stochastic randomness with deterministic but complex metric fluctuations, whose spacetime structure is encoded in mode functions arising from trembling geodesics. In the geometry of trembling spacetime, small deviations in measurement settings due to metric fluctuations at locations $x_A$ and $x_B$ can be expressed as:
$$ \theta_A = \theta_A^0 + \sum_n \alpha_n J_n(k_n x_A) \cos(\omega_n t + \phi_n), \quad (13) $$

$$ \theta_B = \theta_B^0 + \sum_n \beta_n J_n(k_n x_B) \cos(\omega_n t + \psi_n), \quad (14) $$

Here, $\theta_A^0$ and $\theta_B^0$ denote the intended baseline settings of the analyzers or polarizers, serving as reference orientations in the absence of metric fluctuations. The functions $J_n$ represent Bessel functions of the first kind, which naturally arise in the TSRT formalism as solutions to trembling-mode wave equations in cylindrical symmetry. The parameters $k_n$ and $\omega_n$ are the spatial and temporal frequencies associated with each trembling mode, while $\phi_n$ and $\psi_n$ are phase shifts determined by the local interaction between the fluctuating spacetime metric and the measurement apparatus. Finally, $\alpha_n$ and $\beta_n$ are the amplitudes of each mode at the two detector locations, reflecting the local coupling strength of the metric fluctuation to each detector.

These expressions show that both measurement settings are influenced by structured fluctuations rooted in causal geometry. Since these fluctuations originate in the overlapping causal past of the detectors and propagate deterministically, they preserve causal order while still giving rise to correlations that match those predicted by quantum entanglement. There is no need to invoke any wavefunction collapse or superluminal influence: the statistical match with quantum theory emerges from the deterministic geometry of trembling spacetime.

This reformulation makes the outcome of the Aspect experiment a direct consequence of the causal structure of TSRT, not of nonlocal quantum events. Although these expressions appear complex, they represent fully causal and deterministic geodesic deviations, not probabilistic noise or quantum uncertainty. The measured outcomes at each detector then depend on the local orientation angle $\theta_i$ and the internal particle alignment set by the same trembling geometry. This shared influence violates the statistical independence assumed in Bell's theorem, but not relativistic causality.

From the TSRT viewpoint, the observed violations of the CHSH inequality in Aspect's data are not surprising. Rather than requiring nonlocal action or wavefunction collapse, they arise from local, deterministic correlations embedded in the fabric of spacetime. The same trembling field shapes both the detectors' instantaneous orientations and the angular response of the entangled particles—yielding statistical alignment consistent with quantum predictions.

Importantly, TSRT provides a testable prediction: as the spatial separation between detectors increases, the causal coherence of the trembling metric declines. This leads to a smooth suppression of entanglement strength, which departs from the distance-invariant predictions of quantum mechanics. The precise form of this suppression is captured by the trembling CHSH bound (derived in Section 8), and is a central target for future experimental verification.

In summary, the Aspect experiment does not rule out all local realist theories—it rules out those that lack a mechanism for correlated geometry. TSRT preserves both realism and locality by introducing such a mechanism: a trembling metric field that enforces causal structure and generates entanglement correlations through deterministic alignment of geodesics.
# 8 Emergent Bell Inequality Violations from Causal Spacetime Trembling

## 8.1 From Metric Fluctuations to Angular Deviations

In the standard quantum mechanical picture, Bell inequality violations are attributed to entangled wavefunctions that remain nonlocally correlated across spacelike separations. Measurement is assumed to trigger an instantaneous collapse of the wavefunction, yielding statistical correlations that exceed classical bounds. Within the TSRT framework, by contrast, no such nonlocal mechanism is required. Instead, correlations emerge causally and deterministically from fluctuations in the spacetime metric that synchronize the motion of particles and detectors through proper-time-aligned geodesics.

Let $x_A$ and $x_B$ denote the spacelike-separated spacetime locations of two measurement devices, A and B. In a trembling spacetime, the effective measurement angle at each location is influenced not only by the nominal detector orientation, but also by the local deformation of the geodesic frame due to metric fluctuations. Specifically, as introduced in Equations (11) and (14), we write:

$$ \theta_A = \theta_A^0 + \delta\theta_A(x_A), \qquad (15) $$

$$ \theta_B = \theta_B^0 + \delta\theta_B(x_B), \qquad (16) $$

where $\theta_A^0$ and $\theta_B^0$ are the intended, macroscopic detector settings, and $\delta\theta_A$, $\delta\theta_B$ are local angular deviations induced by spacetime perturbations $\xi_{\mu\nu}(x)$.

These perturbations originate from a spectrum of trembling modes—small, causally propagating metric fluctuations that are synchronized in proper time. Each angular deviation $\delta\theta_i(x)$ arises as a functional of the local fluctuation field, and is deterministic in nature, though practically unknown in detail. Importantly, because the spacetime geometry itself evolves smoothly and coherently, the deviations at $x_A$ and $x_B$ are not statistically independent. Instead, they are partially correlated via their shared causal past, which imprints synchronized structure onto both geodesic evolutions.

We define the relative angular difference as:

$$ \Delta\theta = \theta_A - \theta_B = \theta_A^0 - \theta_B^0 + \delta\theta_A - \delta\theta_B. \qquad (17) $$

It is this fluctuating angular difference $\Delta\theta$ that enters the statistical analysis of Bell correlations. In the next subsection, we derive its probability distribution and connect it to the structure of trembling geodesics.

## 8.2 Probability Distribution of Detector Misalignment

To compute measurable correlations in the TSRT framework, we must evaluate the statistical properties of the angular difference $\Delta\theta = \theta_A - \theta_B$ between the two detector orientations, as perturbed by trembling spacetime. Although the underlying fluctuations $\delta\theta_A$ and $\delta\theta_B$ are deterministic, they arise from a large ensemble of trembling modes with varying spatial profiles,
frequencies, and proper-time phases. As a result, the effective angular difference $\Delta\theta$ behaves as a pseudo-random variable across repeated experiments.

Given the high dimensionality and proper-time synchronization of the trembling spectrum, we model the fluctuations in $\Delta\theta$ as a zero-mean Gaussian random variable:

$$P(\Delta\theta) = \frac{1}{\sqrt{2\pi\sigma^2(L)}} \exp\left(-\frac{\Delta\theta^2}{2\sigma^2(L)}\right), \qquad (18)$$

where the variance $\sigma^2(L)$ encodes the statistical spread of the angular difference as a function of detector separation $L = |x_A - x_B|$.

The Gaussian form is not assumed as a fundamental property of TSRT, but rather justified as a first-order approximation on the basis of three key arguments. First, from a central limit perspective, the angular difference $\Delta\theta$ arises as a sum over many mode contributions with different frequencies and phases. Even though each mode is deterministic, their aggregate effect approximates Gaussian behavior in the ensemble limit. Second, the proper-time coherence of TSRT fluctuations implies that the metric field evolves smoothly and without abrupt discontinuities, favoring compact and symmetric distributions such as the Gaussian. Third, the Gaussian form enables closed-form expressions for the correlation function, allowing direct comparison between TSRT predictions and experimental results.

It is important to emphasize, however, that this distribution is a tractable ansatz rather than a strict derivation. In principle, $P(\Delta\theta)$ could exhibit non-Gaussian features—such as anisotropy, skewness, or multimodality—arising from curvature-induced structure in the trembling spectrum. Detecting such deviations in future experiments would offer valuable insights into the finer details of trembling spacetime geometry.

In the present work, we proceed with the Gaussian approximation to facilitate the derivation of the TSRT correlation function, while recognizing that this assumption may ultimately be refined through more detailed geometric modeling or high-precision data.

## 8.3 Derivation of the Correlation Function

In Bell-type experiments, each measurement yields binary outcomes $A, B \in \{\pm 1\}$, which depend on the local detector orientations. In the TSRT framework, the effective orientations at each detector are influenced by the local trembling of spacetime, and the relative angle between them is given by the stochastic variable $\Delta\theta$. To determine the expected product of outcomes, we must average over the distribution $P(\Delta\theta)$ of this relative angle.

Assuming that the outcome correlation at fixed $\Delta\theta$ follows the standard deterministic relation<fnref n="3" /> $\langle AB \rangle_{\Delta\theta} = \cos(\Delta\theta)$, as it does for polarization-entangled photons or spin-$\frac{1}{2}$ particles in both quantum mechanics and TSRT, the full expectation is given by:

$$C(L) = \int_{-\infty}^{\infty} \cos(\Delta\theta) P(\Delta\theta) d(\Delta\theta). \qquad (19)$$

<footnote n="3">³For spin-1/2 particles or polarization-entangled photons, this follows from the quantum mechanical prediction for the singlet state [10]. See also [21] for a pedagogical derivation.</footnote>
Substituting the Gaussian form of the probability distribution of Equation (18),

$$P(\Delta\theta) = \frac{1}{\sqrt{2\pi}\sigma^2(L)} \exp\left(-\frac{\Delta\theta^2}{2\sigma^2(L)}\right), \qquad (20)$$

into Equation (19), we obtain:

$$C(L) = \int_{-\infty}^{\infty} \cos(\Delta\theta) \cdot \frac{1}{\sqrt{2\pi}\sigma^2(L)} \exp\left(-\frac{\Delta\theta^2}{2\sigma^2(L)}\right) d(\Delta\theta). \qquad (21)$$

This integral is a standard result from harmonic analysis of Gaussian kernels [22–24] and yields:

$$C(L) = \exp\left(-\frac{1}{2}\sigma^2(L)\right). \qquad (22)$$

To derive the final correlation function predicted by TSRT, we must now account for two contributions to the angular difference that governs outcome correlations between the two detectors: the nominal misalignment of the intended measurement settings, and the fluctuation-induced deviation due to trembling spacetime.

Let the intended orientations of the two detectors be $\theta_A^0$ and $\theta_B^0$, respectively. These are the angular settings selected by the experimenters in the absence of spacetime fluctuations. However, due to proper-time synchronized metric perturbations, the effective detector angles are given by Equation (16), where $\delta\theta_A$ and $\delta\theta_B$ are deterministic deviations induced by local projections of the trembling metric field at the spacetime positions of detectors A and B.

The effective angular difference between the two measurement directions is thus:

$$\Delta\theta_{\text{eff}} = \theta_A - \theta_B = (\theta_A^0 - \theta_B^0) + (\delta\theta_A - \delta\theta_B). \qquad (23)$$

Now, the statistical expectation value of the correlation between the outcomes $A, B \in \{\pm 1\}$ depends on the ensemble average over many realizations of the trembling field. This corresponds to averaging over the fluctuation-induced deviations $\delta\theta_A - \delta\theta_B$, which we denote $\delta\Delta\theta$. Under the assumption of a zero-mean Gaussian distribution for these relative fluctuations with variance $\sigma^2(L)$, we can compute:

$$C(L, \Delta\theta) = \langle \cos(\Delta\theta + \delta\Delta\theta) \rangle. \qquad (24)$$

To compute the correlation between the binary measurement outcomes $A, B \in \{\pm 1\}$, we consider how the trembling spacetime geometry affects the effective detector orientations.

As established earlier, the actual angles $\theta_A$ and $\theta_B$ experienced by the two detectors are not just the experimenter's chosen settings $\theta_A^0$ and $\theta_B^0$, but include additional deviations caused by local metric fluctuations. That is:

$$\theta_A = \theta_A^0 + \delta\theta_A, \qquad (25)$$

$$\theta_B = \theta_B^0 + \delta\theta_B. \qquad (26)$$
Therefore, the effective angular difference between the two detectors becomes:

$$ \Delta\theta_{\text{eff}} = \theta_A - \theta_B = \underbrace{(\theta_A^0 - \theta_B^0)}_{\text{nominal}} + \underbrace{(\delta\theta_A - \delta\theta_B)}_{\text{fluctuating}}. \qquad (27) $$

We now define:

$$ \Delta\theta = \theta_A^0 - \theta_B^0, \quad \delta\Delta\theta = \delta\theta_A - \delta\theta_B, \qquad (28) $$

so that:

$$ \Delta\theta_{\text{eff}} = \Delta\theta + \delta\Delta\theta. \qquad (29) $$

The measured correlation between outcomes is then governed by the average value of the cosine of this total angular difference:

$$ C(L, \Delta\theta) = \langle \cos(\Delta\theta_{\text{eff}}) \rangle = \langle \cos(\Delta\theta + \delta\Delta\theta) \rangle. \qquad (30) $$

This is the point at which we must evaluate the average (or expectation value) over the fluctuating quantity $\delta\Delta\theta$, which changes from event to event due to the trembling nature of spacetime.

To proceed, we assume that the metric fluctuations $\delta\theta_A$ and $\delta\theta_B$ are deterministic but vary across spacetime in a way that, when coarse-grained over many realizations, effectively follow a zero-mean Gaussian distribution. Hence, the difference $\delta\Delta\theta$ is modeled as a Gaussian random variable with zero mean and variance $\sigma^2(L)$, where L is the separation between the detectors:

$$ \delta\Delta\theta \sim \mathcal{N}(0, \sigma^2(L)). \qquad (31) $$

The expectation of the perturbed cosine follows from the characteristic function of the Gaussian distribution [22, 23]:

$$ \langle \cos(\Delta\theta + \delta\Delta\theta) \rangle = \cos(\Delta\theta) \cdot \exp\left(-\frac{1}{2}\sigma^2(L)\right). \qquad (32) $$

This is valid because the cosine function is symmetric and oscillatory, but if we average it over a Gaussian distribution centered at zero, the positive and negative deviations around the mean tend to cancel out the higher-frequency parts of the cosine wave. The result is that the amplitude of the cosine is suppressed by a factor that depends on the spread (variance) of the Gaussian. The wider the Gaussian (i.e., the more uncorrelated the metric fluctuations), the more the cosine gets damped. The exponential term captures precisely that damping effect.

Thus, the full correlation function becomes:

$$ C(L, \Delta\theta) = \cos(\theta_A^0 - \theta_B^0) \cdot \exp\left(-\frac{1}{2}\sigma^2(L)\right), \qquad (33) $$

where the cosine term $\cos(\theta_A^0 - \theta_B^0)$ reflects the purely geometric contribution to the correlation—the alignment of detector settings in the absence of fluctuations; and where the exponential factor $\exp(-\frac{1}{2}\sigma^2(L))$ accounts for the damping of this correlation due to the trembling of the spacetime metric, which increases with spatial separation.
This expression encapsulates the core physical insight of TSRT: the correlations observed in Bell-type experiments are not manifestations of nonlocal quantum effects or stochastic wave-function collapse, but arise from deterministic angular alignment shaped by the causal geometry of trembling spacetime. The familiar cosine dependence reflects the geometric configuration of measurement settings, while the exponential damping term captures the loss of coherence due to spatially dependent metric fluctuations. Together, they reveal entanglement not as a quantum mystery, but as an emergent signature of proper-time-synchronized curvature dynamics within a fully local and causal relativistic framework.

Equation (33) thus establishes the foundation for all subsequent CHSH predictions within TSRT. It shows that quantum-like correlations can be preserved at short range, but naturally diminish at large separations due to causal limitations in metric coherence. In the next section, we specify the functional form of $\sigma^2(L)$ based on geometric principles.

## 8.4 Geometric Modulation and Causal Cutoff

The correlation function in Equation (33) depends critically on the variance $\sigma^2(L)$ of the angular difference between detector settings, which in TSRT reflects the decorrelation of geodesic trembling as a function of spatial separation. To model this behavior, we must specify how $\sigma^2(L)$ evolves with distance.

A natural and physically grounded choice is to assume a power-law growth of variance with detector separation:

$$ \sigma^2(L) = \left(\frac{L}{L_c}\right)^n, \qquad (34) $$

where $L_c$ is a characteristic coherence length beyond which trembling modes begin to decorrelate substantially, and $n > 0$ is a geometric exponent determined by the curvature structure and spectrum of the trembling spacetime fluctuations. Larger values of $n$ correspond to more rapid loss of coherence, for instance in regions of strong curvature or complex mode structure.<fnref n="4" />

Substituting this into the exponential factor from Equation (33), we obtain the power-law modulated correlation function:

$$ C(L, \Delta\theta) = \cos(\Delta\theta) \cdot \exp \left[ -\frac{1}{2} \left( \frac{L}{L_c} \right)^n \right]. \qquad (35) $$

For analytic convenience and to enable a clearer connection to rational decay behavior, we may approximate this exponential form with a normalized rational function that captures the same qualitative behavior:

$$ \exp \left[ -\frac{1}{2} \left( \frac{L}{L_c} \right)^n \right] \approx \left( \frac{L_c}{L_c + L} \right)^n. \qquad (36) $$

<footnote n="4">⁴The light cone structure of spacetime imposes a strict upper bound on the coherence length $L_c$. Suppose a pair of detectors is fabricated in the same lab at time $t = 0$, transported to distant locations over time $t_1$, and used to measure entangled particles at time $t_2$. Then the longest possible causal correlation length allowed by TSRT is bounded by $L_c^{\max} = c(t_2 - 0)$, the distance light could travel from the shared origin to both detectors before the measurement. In practice, all real-world influences—transportation, thermalization, calibration—propagate slower than light, so the actual coherence length must satisfy $L_c < L_c^{\max}$, guaranteeing that trembling-induced correlations remain fully confined within the light cone. This enforces relativistic causality and excludes any possibility of superluminal influence.</footnote>
This approximation is not exact but is well-behaved across experimentally accessible ranges of $L$, and it avoids artificial divergences as $L \to \infty$. Using this, we define an effective geometric modulation factor:

$$ \mathcal{A}_{\text{geom}}(L) = \left( \frac{L_c}{L_c + L} \right)^n . \qquad (37) $$

However, this rational form still permits residual correlations to persist at arbitrarily large separations, which would violate relativistic causality if trembling coherence were not bounded by a light-cone structure. Because TSRT models all trembling-induced coherence as propagating through causal geodesics at or below the speed of light (as established in the foundational formulation [14]), any correlations arising from such metric fluctuations must vanish outside the mutual light cone of the involved events. While the power-law decay with distance effectively models the gradual loss of coherence due to geodesic decorrelation, it remains a smooth, asymptotic function and does not alone ensure that correlations vanish outside relativistically permitted domains. To consistently enforce the causal limitation derived from the light-cone-bounded propagation of metric fluctuations in TSRT [14], we supplement the power-law decay with a Gaussian suppression factor that sharply attenuates correlations beyond the causal reach, i.e.,

$$ \mathcal{A}(L) = \left( \frac{L_c}{L_c + L} \right)^n \exp \left( -\frac{L^2}{\lambda_c^2} \right), \qquad (38) $$

where $\lambda_c$ defines a causal cutoff scale associated with the maximal reach of proper-time synchronized trembling modes. This exponential damping does not arise from a new physical postulate, but serves as a conservative regularization consistent with TSRT's core principle: that all metric-induced correlations must remain confined within the shared causal past of the detectors and particles. The Gaussian factor ensures rapid suppression of correlations near the edge of causal connectivity and provides a physically transparent way to enforce light-cone boundedness in modeling the decay of trembling coherence.

This formulation guarantees that trembling-induced correlations remain strictly confined within relativistically admissible regions, as required by the finite propagation speed of metric fluctuations in TSRT [14]. The Gaussian cutoff introduces a physically meaningful parameter $\lambda_c$, representing the effective causal reach of proper-time-synchronized metric coherence—a scale that may, in principle, be constrained or measured through long-baseline entanglement experiments.

Inserting the modulation amplitude from Equation (38) into the base correlation expression of Equation (33) yields the final TSRT prediction:

$$ C_{\text{TSRT}}(L, \Delta\theta) = \mathcal{A}(L) \cos(\Delta\theta), \qquad (39) $$

which reproduces the short-range behavior of quantum theory, in the case of small detector separation where $L \ll L_c$ and $L \ll \lambda_c$, while providing a causal mechanism for the decay of entanglement-like correlations at large separation, in the case of $L \gg L_c$ or $L \gg \lambda_c$, where proper-time-synchronized trembling becomes incoherent across geodesics.
## 8.5 CHSH Quantity from TSRT Correlations

The Clauser-Horne-Shimony-Holt (CHSH) parameter [10] provides a standard criterion for testing violations of local realism in bipartite measurement scenarios. It is defined as:

$$S = |E(a, b) - E(a, b') + E(a', b) + E(a', b')|, \qquad (40)$$

where $E(a, b)$ denotes the expectation value of the product of outcomes measured along orientations $a$ and $b$, and $a', b'$ are alternate detector settings. In quantum mechanics, this parameter can reach a maximum of $S_{QM} = 2\sqrt{2}$ for certain angular configurations of the detectors.

In the TSRT framework, each expectation value $E(a, b)$ is given by the correlation function:

$$E_{\text{TSRT}}(a, b; L) = \mathcal{A}(L) \cos(\theta_a - \theta_b), \qquad (41)$$

where $\theta_a$ and $\theta_b$ are the nominal angles at detectors A and B, and $\mathcal{A}(L)$ is the trembling-induced modulation factor derived in Equation (38).

Substituting Equation (41) into the CHSH definition (40), we find:

$$S_{\text{TSRT}}(L) = \mathcal{A}(L) |\cos(\theta_a - \theta_b) - \cos(\theta_a - \theta_{b'}) + \cos(\theta_{a'} - \theta_b) + \cos(\theta_{a'} - \theta_{b'})| \qquad (42)$$

$$= \mathcal{A}(L) \cdot S_{\text{QM}}(\{\theta\}), \qquad (43)$$

where $S_{QM}(\{\theta\})$ denotes the value the CHSH parameter would take under standard quantum mechanics for the same set of angular settings $\{\theta\}$, assuming perfect entanglement and no spatial or causal modulation. This decomposition shows that TSRT preserves the angular structure of quantum mechanics but introduces a modulation in overall correlation strength as a function of separation $L$.

In particular, when the angular settings are chosen to maximize the CHSH quantity in quantum mechanics as derived in Section 8.6—namely, when $S_{QM} = 2\sqrt{2}$ under standard assumptions of perfect entanglement and nonlocal wavefunction collapse—the TSRT prediction becomes:

$$S_{\text{TSRT}}(L) = 2\sqrt{2} \cdot \mathcal{A}(L), \qquad (44)$$

where the prefactor $2\sqrt{2}$ emerges purely as a consequence of the angular dependence inherited from the deterministic geometry of TSRT, and the modulation factor $\mathcal{A}(L)$ captures the decay of trembling-induced coherence with spatial separation.

Crucially, this expression is derived entirely within the TSRT framework—without invoking quantum wavefunctions, probabilistic collapse, or Hilbert space postulates. Instead, it arises from proper-time-synchronized geodesic alignment in a causally fluctuating spacetime metric.

The result reveals a profound correspondence: at short separations $L \ll L_c$, the TSRT prediction converges to the quantum mechanical maximum, reproducing the same numerical value as standard entanglement experiments. However, as the separation increases, the trembling coherence encoded in $\mathcal{A}(L)$ decays, leading to a gradual suppression of the CHSH parameter. This behavior stands in stark contrast to quantum theory, which predicts distance-independent
violations in ideal conditions.

Thus, Equation (44) constitutes the trembling CHSH bound—a key falsifiable prediction of TSRT. It offers not only a deterministic and causal account of entanglement-like correlations, but also a specific, testable deviation from standard quantum mechanics at large scales.

## 8.6 Optimal Angular Configuration and Maximal Violation

In the standard quantum mechanical treatment of Bell-type experiments, the CHSH parameter $S$ attains its maximum value of $2\sqrt{2}$ for a specific set of detector orientations known as the standard optimal configuration [10]. This configuration exploits the trigonometric structure of the correlation function $E(a, b) = \cos(\theta_a - \theta_b)$ to maximize constructive interference between measurement outcomes across the four combinations of detector settings.

The optimal angles are:

$$ \theta_a = 0, \quad \theta_{a'} = \frac{\pi}{2}, \quad \theta_b = \frac{\pi}{4}, \quad \theta_{b'} = -\frac{\pi}{4}. \qquad (45) $$

Substituting these into the quantum CHSH expression in Equation (46),

$$ S_{QM}(\{\theta\}) = |\cos(\theta_a - \theta_b) - \cos(\theta_a - \theta_{b'}) + \cos(\theta_{a'} - \theta_b) + \cos(\theta_{a'} - \theta_{b'})|, \quad (46) $$

yields:

$$ S_{QM} = \left| \cos\left(\frac{\pi}{4}\right) - \cos\left(-\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{2} - \frac{\pi}{4}\right) + \cos\left(\frac{\pi}{2} + \frac{\pi}{4}\right) \right| \quad (47) $$

$$ = \left| \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \right| = 2\sqrt{2}. \qquad (48) $$

In the TSRT framework, the same optimal configuration is used to evaluate the modulated CHSH expression:

$$ S_{TSRT}(L) = 2\sqrt{2} \cdot \mathcal{A}(L), \qquad (49) $$

where $\mathcal{A}(L)$ is the geometric attenuation factor derived in Equation (38).

Thus, TSRT reproduces the quantum mechanical maximum at small separations, where $\mathcal{A}(L) \approx 1$, but predicts a smooth decay of the CHSH parameter as detector separation increases. This spatial dependence is absent from standard quantum theory, making it a key distinguishing feature of the trembling spacetime interpretation.

The standard optimal configuration therefore plays a central role in testing TSRT. It provides a fixed angular baseline for experiments while allowing the modulation $\mathcal{A}(L)$ to be isolated as a function of separation, curvature, or gravitational potential.

## 8.7 Critical Separation and Breakdown of Quantum Behavior

In both classical and quantum theories, the CHSH parameter serves as a binary diagnostic: values $S \le 2$ are consistent with local realist models, while values $S > 2$ imply a violation of Bell's inequality. In TSRT, the CHSH parameter is not fixed but varies with the spatial separation $L$ between detectors due to the scale-dependent coherence of trembling-induced correlations.
As shown in Equation (44), the CHSH parameter predicted by TSRT under the optimal angular configuration is:

$$S_{\text{TSRT}}(L) = 2\sqrt{2} \cdot \mathcal{A}(L), \qquad (50)$$

where the geometric modulation factor $\mathcal{A}(L)$ is given by:

$$\mathcal{A}(L) = \left( \frac{L_c}{L_c + L} \right)^n \exp \left( -\frac{L^2}{\lambda_c^2} \right). \qquad (51)$$

To determine the distance beyond which the TSRT-predicted CHSH value no longer violates Bell's inequality, we set:

$$S_{\text{TSRT}}(L^*) = 2. \qquad (52)$$

Solving for the critical separation $L^*$ yields:

$$\left( \frac{L_c}{L_c + L^*} \right)^n \exp \left( -\frac{(L^*)^2}{\lambda_c^2} \right) = \frac{1}{\sqrt{2}}. \qquad (53)$$

This transcendental equation defines the threshold distance $L^*$ at which TSRT transitions from entanglement-type correlations ($S > 2$) to classical correlations ($S \le 2$). The exact value of $L^*$ depends on three key parameters:

* the coherence length $L_c$, which determines the scale over which trembling geodesics remain statistically aligned;

* the causal cutoff scale $\lambda_c$, which enforces a maximum range of influence consistent with relativistic propagation;

* and the geometric exponent $n$, which characterizes the curvature-dependent rate of decoherence.

For small $L$, the modulation factor satisfies $\mathcal{A}(L) \approx 1$, recovering the quantum mechanical maximum $S_{\text{TSRT}}(L) \approx 2\sqrt{2}$. At large $L$, the exponential decay dominates, driving $S_{\text{TSRT}}(L)$ below the classical bound. The prediction of a finite $L^*$ at which this transition occurs provides a sharp, falsifiable deviation from quantum theory, which predicts no such decay in the absence of noise or decoherence.

Detecting the violation threshold $L^*$ experimentally would offer direct insight into the causal coherence structure of spacetime, and would constitute a pivotal test of the TSRT model. In the next section, we explore how these predictions compare to current data and how future experiments might detect the scale-dependent breakdown of quantum behavior.

## 8.8 Causal Interpretation and Experimental Significance

In the standard quantum picture, violations of Bell's inequality are interpreted as evidence that nature is either fundamentally nonlocal or indeterministic. However, this interpretation assumes that measurement settings and particle properties are statistically independent and that spacetime provides a static, structureless background. Trembling Spacetime Relativity Theory (TSRT) challenges this view by embedding both detectors and particles within a common, causally evolving geometric substrate.
In TSRT, correlations between measurement outcomes arise from deterministic geodesic alignment within a fluctuating metric field. These fluctuations are proper-time synchronized and spatially coherent over finite intervals, but they are strictly local and respect causal structure. As a result, CHSH violations do not require superluminal communication or wavefunction collapse. Instead, they reflect coordinated trembling of the spacetime fabric itself, which affects both detectors and particles via shared geodesic histories.

This framework leads to a crucial and testable prediction: Bell inequality violations should diminish with increasing spatial separation, altitude, or curvature. Specifically:

* Increasing separation $L$ reduces coherence between detector geodesics, weakening their alignment and suppressing the correlation amplitude.

* Increasing altitude introduces gravitational potential differences that desynchronize proper time across detectors, degrading trembling-induced alignment.

* Increasing curvature alters the spectral structure of trembling modes, enhancing geodesic divergence and accelerating the loss of coherence.

These effects are absent from standard quantum mechanics, which predicts scale-invariant entanglement in ideal conditions. Therefore, experimental observation of spatial or gravitational dependence in Bell violations would directly falsify the quantum interpretation and support TSRT.

Current satellite-based experiments, such as the Micius mission [4,25,26], have already tested Bell inequalities over distances exceeding 1200 kilometers and observed quantum-like violations. From the TSRT perspective, such results constrain the parameters $L_c$ and $\lambda_c$, indicating that the trembling coherence persists at least up to the observed range. These data do not disprove TSRT, but rather set empirical lower bounds on its geometric correlation scales.

To decisively test the model, future experiments must extend beyond current baselines. Suggested configurations include:

* long-baseline ground-to-satellite or satellite-to-satellite entanglement distribution over distances greater than $L^*$,

* variable-altitude Bell tests to probe gravitational effects on trembling synchronization,

* inertial or curved trajectory configurations to induce differential geodesic evolution,

* and time-resolved geolocation of detector orientations to monitor metric-induced drift.

Such experiments could reveal systematic reductions in Bell violation strength correlated with spacetime structure, offering a direct probe of causal geometry. In doing so, they would move the study of entanglement beyond interpretive debates and into the domain of falsifiable geometric physics.

TSRT thus provides a novel interpretative lens through which existing results remain valid but acquire new causal meaning. More importantly, it opens a pathway toward a new generation of high-precision experiments, capable of testing not only the validity of quantum mechanics, but the very fabric of spacetime itself.
# 9 Reconstructing Entanglement: Causal Geometry Beyond Bell's Theorem

## 9.1 Limits of Bell's Theorem and the Role of Causal Structure

Bell's theorem stands as one of the most profound results in the foundations of physics. It rigorously shows that no physical theory respecting both locality and realism can reproduce all the statistical predictions of quantum mechanics, provided a third condition—measurement independence—also holds. In its standard form, the theorem assumes:

1. **Locality:** Measurement outcomes at one location cannot be influenced by settings or outcomes at spacelike-separated regions.

2. **Realism:** Measurement outcomes reflect pre-existing properties of the system, determined by hidden variables $\lambda$.

3. **Measurement Independence:** The hidden variables $\lambda$ are statistically independent of the measurement settings $a, b$, i.e.,

$$ \rho(\lambda | a, b) = \rho(\lambda). \qquad (54) $$

Violations of Bell inequalities in numerous experiments—beginning with those of Aspect et al. [12] and continuing with modern satellite-based tests [4]—are commonly taken as proof that at least one of these assumptions must fail. The dominant interpretation in quantum mechanics sacrifices locality, accepting that entanglement correlations require nonlocal effects such as wavefunction collapse across spacelike intervals.

However, Bell's theorem does not claim that quantum mechanics proves nonlocality per se. Rather, it shows that if all three assumptions above are jointly valid, then the CHSH inequality must hold. The observed violations therefore falsify the conjunction of these assumptions, but not any individual one uniquely. This distinction is crucial: the theorem is not a physical impossibility proof against realism or locality, but a conditional logical result.

Trembling Spacetime Relativity Theory (TSRT) embraces this nuance. It preserves both locality and realism by questioning a different premise: the statistical independence of measurement settings from the underlying structure of spacetime. Within TSRT, measurement outcomes and detector settings are not randomly assigned but are both shaped by the same deterministic, causally coherent field: the trembling spacetime metric. As a result, the independence condition $\rho(\lambda | a, b) = \rho(\lambda)$ no longer holds—not through hidden conspiracies, but as a natural consequence of geometric causality.

Thus, the traditional formulation of Bell's theorem cannot be applied wholesale to TSRT. Its derivation becomes inapplicable because one of its foundational premises is systematically violated by the very structure of the physical model. This opens the door to a new interpretation of entanglement—one in which Bell inequality violations are not paradoxes requiring nonlocality, but natural outcomes of causal geometry.
In the next subsection, we examine precisely how TSRT causally violates measurement independence without invoking superdeterminism or retrocausality, thereby preserving the empirical successes of quantum theory while replacing its conceptual mysteries with geometric clarity.

## 9.2 TSRT and the Failure of Measurement Independence

In the conventional view, the measurement settings a and b in Bell experiments are assumed to be freely chosen, independent of any hidden variables $\lambda$ that determine the measurement outcomes. This assumption—often called 'measurement independence' or 'statistical independence'—is formalized by the condition:

$$ \rho(\lambda | a, b) = \rho(\lambda), \qquad (55) $$

meaning that the joint distribution of hidden variables and measurement settings factorizes. It reflects the idea that experimentalists are free to choose their settings in ways uncorrelated with the internal state of the system under test.

In Trembling Spacetime Relativity Theory (TSRT), this assumption no longer holds—but for reasons fundamentally different from those invoked in superdeterministic models or retrocausal interpretations. TSRT does not posit a conspiratorial coordination between experimenters' choices and particle properties. Instead, it reveals that the assumption of independence breaks down naturally in the presence of a dynamically fluctuating spacetime geometry.

The core mechanism is proper-time synchronized trembling of the metric tensor $\xi_{\mu\nu}(x)$, which propagates causally and deterministically. These trembling fluctuations arise from geometric constraints on local curvature and evolve according to relativistically consistent equations. Crucially, they influence both the motion of particles and the physical configuration of measurement devices—such as the effective alignment angles of polarizers or spin analyzers.

As a result, the variables determining the detector orientations and the variables governing particle outcomes are both shaped by the same spacetime field. While the metric fluctuations do not enforce exact equality between these quantities, they induce statistical alignment through shared causal ancestry. This implies that the apparent freedom of detector settings is constrained—not by direct control or communication, but by the mutual influence of the same trembling geodesic structure.

In this framework, the joint distribution becomes:

$$ \rho(\lambda | a, b) \neq \rho(\lambda), \qquad (56) $$

not due to a breakdown of experimental free will, but because the settings a and b are emergent quantities influenced by local curvature and proper-time synchronized fields. The TSRT violation of measurement independence is thus a consequence of physical geometry, not philosophical reinterpretation.

Importantly, this causal connection does not entail any superluminal signaling or conflict with special relativity. The correlations arise from the past light cones of the detectors and particles, where the trembling metric imprints deterministic modulations. All causal effects are confined to the local light-cone structure, preserving both Lorentz invariance and relativistic
causality.

In summary, TSRT replaces the assumption of measurement independence with a geomet-
ric principle: all observed correlations—including those between measurement settings and
outcomes—emerge from shared causal structure embedded in trembling spacetime. This re-
formulation explains Bell inequality violations without requiring nonlocality, retrocausality, or
indeterminism, and sets the stage for a broader reinterpretation of quantum phenomena in
terms of deterministic causal geometry.

**9.3 Comparison with Existing Interpretations**

Trembling Spacetime Relativity Theory (TSRT) introduces a novel explanatory framework in which quantum-like correlations arise from deterministic, causally structured geometry rather than from nonlocal quantum states or statistical postulates. This perspective stands in sharp contrast with several prevailing interpretations in both quantum theory and alternative approaches to entanglement. Understanding these distinctions is essential to appreciate the conceptual and physical uniqueness of TSRT.

**Contrast with Quantum Field Theory.** In conventional quantum field theory (QFT), entanglement originates from the tensor product structure of Hilbert spaces and the noncommutative nature of quantum observables [27,28]. Correlations between measurements on spacelike-separated regions are attributed to a shared quantum state that collapses nonlocally upon measurement. However, the spacetime manifold on which QFT is defined is treated as a fixed classical background, without feedback or dynamical response to quantum evolution.

By contrast, TSRT eliminates the need for ontological wavefunctions altogether. The metric of spacetime is not passive but actively fluctuates according to causal geometric laws. These fluctuations influence both particle trajectories and detector orientations through proper-time synchronized geodesics. Quantum-like correlations thus emerge not from nonlocal superpositions but from deterministic alignment of paths within a trembling causal structure. No wavefunction collapse is required, and no tension with relativity arises.

**Contrast with Semiclassical Gravity.** In semiclassical gravity, matter fields are quantized while spacetime remains classical [29, 30]. The gravitational field responds only to the expectation value of the quantum stress-energy tensor $\langle T_{\mu\nu} \rangle$, leading to a one-way coupling that cannot account for dynamic correlation formation between quantum events. Spacetime does not influence entanglement structure beyond gravitational lensing or redshift effects.

TSRT, in contrast, posits a symmetric causal interaction: the metric fluctuates dynamically and deterministically, and these fluctuations affect both matter and measurement configurations. The trembling of the metric itself becomes the source of apparent quantum behavior. This leads to entanglement correlations without resorting to semiclassical averaging, and embeds the influence of gravity into the microscopic alignment of geodesics. Thus, TSRT constitutes a more unified and relational alternative to hybrid quantum-classical frameworks.
**Contrast with Nonlocal Hidden Variable Theories.** Deterministic hidden-variable theories [5,31–34] such as Bohmian mechanics preserve realism but abandon locality. They postulate an instantaneous influence between distant entangled particles, typically mediated by a nonlocal quantum potential that lives in configuration space. While empirically consistent, these models violate relativistic causality and often require preferred reference frames.

TSRT avoids such nonlocal constructs entirely. It explains experimental violations of Bell inequalities through causal, light-cone-constrained correlations in a single spacetime manifold. The trembling metric influences both ends of the experiment through shared geodesic ancestry, but without requiring any faster-than-light signaling or global configuration variables. Unlike Bohmian mechanics, TSRT remains fully Lorentz covariant and local in the relativistic sense, while still producing deterministic outcomes.

**Summary.** TSRT offers a conceptual unification that departs from the probabilistic and nonlocal foundations of standard quantum mechanics and hidden-variable theories. It attributes entanglement to geometric causality rather than Hilbert space structure. Unlike QFT and semiclassical models, it treats spacetime as a dynamic, fluctuation-carrying field that shapes both matter evolution and detector configuration. Unlike Bohmian or superdeterministic approaches, it maintains both realism and locality without invoking unobservable nonlocality.

In doing so, TSRT preserves all empirical successes of quantum theory in known regimes, while replacing its interpretive paradoxes with a deterministic geometric mechanism. This sets the stage for a new paradigm, in which entanglement emerges not from abstract mathematical postulates, but from the physically real and testable geometry of trembling spacetime.

## 9.4 A New Paradigm: Entanglement as Geometric Alignment

The cumulative insight offered by Trembling Spacetime Relativity Theory (TSRT) is a radical but coherent reinterpretation of quantum entanglement—not as a manifestation of nonlocal quantum states, but as the observable imprint of causal geometric alignment across fluctuating spacetime.

In this paradigm, what quantum theory describes as 'entangled correlations' are understood instead as the deterministic consequence of proper-time-synchronized geodesic evolution. Particles and measurement devices follow worldlines shaped by a common trembling metric field. These fluctuations, though microscopic and Lorentz covariant, carry coherent structure across extended causal regions. The resulting alignment of detector orientations and particle trajectories is not imposed externally, but emerges from the geometry of the spacetime fabric itself.

This reformulation has deep consequences. First, it restores realism: outcomes are determined by local geometric data, not by probabilistic collapse. Second, it preserves locality: correlations arise from overlapping past light cones and the causal reach of trembling geodesics, without any superluminal influence. Third, it avoids the philosophical opacity of the wavefunction, replacing it with a physically interpretable structure—fluctuations of the metric tensor—that can, in principle, be modeled and measured.

Moreover, this framework provides a concrete path to generalizing or testing quantum theory.
Since the amplitude of trembling-induced correlations depends on parameters such as coherence length $L_c$, decay exponent $n$, and causal cutoff scale $\lambda_c$, deviations from standard quantum predictions become quantifiable rather than speculative. Long-baseline or high-curvature entanglement experiments could detect the suppression of CHSH violations, not as a breakdown of entanglement itself, but as a sign that spacetime coherence is fading over large separations.

Under this view, the Born rule is not a fundamental axiom, but an emergent statistical behavior of ensembles of trembling geodesics under symmetric conditions. Quantum superposition becomes a useful computational construct, not an ontological mystery. The 'spookiness' of entanglement dissolves, revealing a deeper layer of geometric causality beneath the probabilistic surface of quantum mechanics.

In this sense, TSRT represents not just a technical alternative to standard quantum theory, but a conceptual shift—a move from interpreting quantum phenomena as fundamentally nonlocal and indeterministic, to understanding them as emergent, causal, and geometric. It reframes the debate over Bell's theorem, not by denying its logic, but by rejecting one of its assumptions on physical, rather than metaphysical, grounds.

This new paradigm invites a reinterpretation of many open questions in foundational physics. It suggests that entanglement, coherence, and even the apparent randomness of quantum measurement may all have their roots in the same deterministic trembling of the metric that gives rise to spacetime itself.

# 10 Experimental Tests of Trembling Spacetime

Trembling Spacetime Relativity Theory (TSRT) offers concrete, falsifiable predictions that diverge from the standard quantum framework. While quantum mechanics expects CHSH inequality violations to remain invariant under ideal conditions—regardless of detector separation, gravitational environment, or experiment duration—TSRT predicts systematic modulations of entanglement correlations as functions of spatial distance, spacetime curvature, and long-timescale coherence.

These modulations arise from two foundational features of TSRT: the finite spatial and temporal coherence of the trembling metric field, and the deterministic causal alignment of detectors and particles via shared geodesic ancestry. Together, these features lead to a distinctive correlation structure that depends explicitly on spatial separation $L$, gravitational potential $\Phi(x)$, and the spectrum of long-wavelength trembling modes.

## Distance-Dependent Suppression of Entanglement

TSRT predicts that entanglement correlations diminish smoothly and deterministically with detector separation. By combining the maximal angular violation result of Equation (44) with the geometric modulation amplitude defined in Equation (38), we obtain the full TSRT prediction for the CHSH parameter as a function of detector separation:

$$S_{\text{TSRT}}(L) = 2\sqrt{2} \left( \frac{L_c}{L_c + L} \right)^n e^{-L^2/\lambda_c^2}, \qquad (57)$$
where $L_c$ is the characteristic coherence length of trembling-induced correlations, $\lambda_c$ is a causal cutoff scale beyond which coherence vanishes, and $n$ is a geometric decay exponent governed by curvature and dimensionality.

This spatial attenuation stands in sharp contrast to standard quantum mechanics, which predicts that maximal CHSH violations persist indefinitely, subject only to technical noise or decoherence. Precision Bell experiments conducted over long optical fibers, free-space links, or satellite channels thus provide a direct means of testing this prediction. A measurable falloff consistent with the TSRT formula would strongly support a geometric rather than quantum-nonlocal origin of the observed correlations.

## Gravitational Modulation of CHSH Strength

Because TSRT explicitly links correlation strength to local spacetime geometry, it predicts that entanglement strength varies with gravitational potential. Detectors placed at differing altitudes—such as sea level versus high-altitude balloon or satellite—experience different degrees of geodesic divergence and proper-time variation. As a result, CHSH violations should exhibit altitude-dependent modulation, with stronger correlations at lower curvature.

This gravitational sensitivity is absent in standard quantum mechanics and cannot be attributed to conventional relativistic time dilation. Instead, it reflects a curvature-induced suppression of causal coherence in the trembling spacetime field.

## Temporal Drift from Long-Wavelength Trembling

Beyond spatial and gravitational dependencies, TSRT predicts slow temporal variations in entanglement strength due to long-wavelength trembling modes of the metric. These low-frequency components introduce quasi-random drifts in the amplitude of CHSH correlations over long durations—even in otherwise controlled experimental conditions.

Such fluctuations are not expected in standard quantum theory, which presumes time-invariant correlation strength in stationary configurations. Their detection would require high-precision, low-drift experiments conducted over extended time intervals but are feasible with existing quantum communication platforms.

## Correlation with Detector Orientation Drift

TSRT further predicts subtle correlations between detector orientation fluctuations and environmental geophysical factors. Since trembling spacetime modulates local angles through projected curvature, detectors equipped with high-resolution angular encoders may exhibit minute alignment drifts synchronized across separated stations. These effects would exceed mechanical tolerances and noise levels expected in standard quantum scenarios and would serve as indirect evidence of trembling-induced metric alignment.

## Summary of Experimental Signatures

Together, these predicted effects—distance-dependent CHSH suppression, gravitational modulation, slow temporal drift, and angular orientation correlation—form a distinct experimental
fingerprint of trembling spacetime. Each can be tested independently and falsified with high
precision. Their collective confirmation would strongly support the TSRT hypothesis that en-
tanglement is not a fundamentally nonlocal quantum phenomenon, but a deterministic and
causally constrained imprint of metric coherence.

Should these predictions be borne out by observation, they would signal a profound shift
in our understanding of quantum correlations, recasting entanglement as a geometric effect of
fluctuating spacetime rather than a nonlocal feature of quantum theory.

11 Conclusion and Future Directions

This work proposes a fundamental reinterpretation of quantum entanglement: Bell inequality violations, long attributed to quantum nonlocality, are here understood as signatures of causal structure embedded in a trembling spacetime geometry. Within the framework of Trembling Spacetime Relativity Theory (TSRT), the breakdown of Bell's assumption of measurement independence arises not from randomness or conspiracy, but from deterministic, proper-time-synchronized fluctuations in the spacetime metric. These geometric fluctuations influence both particle trajectories and detector settings through shared causal ancestry, eliminating the need for superluminal signaling or wavefunction collapse.

TSRT offers a local, realist, and covariant account of quantum correlations. Entanglement is not interpreted as a nonlocal property of wavefunctions in Hilbert space, but as the geometric alignment of measurement axes and particle paths shaped by coherent trembling of the metric field. These metric perturbations generate statistical correlations that span spacelike separations while remaining fully constrained by causal structure.

The CHSH violations observed in Bell experiments are thus reinterpreted as determinis-
tic effects arising from trembling-induced coherence. This view bridges the conceptual divide
between quantum mechanics and general relativity: entanglement becomes a geometric phe-
nomenon, and the Born rule emerges not as a postulate, but as a statistical approximation to
geodesic ensemble dynamics in a fluctuating metric background.

Beyond conceptual insight, TSRT yields concrete, testable predictions that differ from quan-
tum mechanics in well-defined regimes. It predicts distance-dependent suppression of CHSH vio-
lations, curvature-modulated loss of coherence, and slow temporal drifts due to long-wavelength
trembling modes. These effects arise not from environmental decoherence, but from intrinsic
causal dynamics of the metric itself. Experimental pathways—including long-baseline opti-
cal tests, altitude-separated entanglement stations, satellite-based links, and precision angular
tracking—are outlined to empirically probe these deviations.

On the theoretical front, future work should refine the modeling of metric fluctuations
beyond the Gaussian approximation. This includes incorporating curvature-sensitive scaling
laws, anisotropic mode structure, and covariant formulations of geodesic ensemble dynamics
under realistic spacetime curvature. Proper-time correlation functions in curved spacetimes
and in the presence of gravitational gradients remain a promising direction for analytical and
numerical exploration.

If confirmed by experiment, the trembling spacetime hypothesis would transform our un-
derstanding of quantum correlations and resolve one of the deepest tensions in modern physics. It would unify quantum phenomena and causal structure by grounding entanglement in geometry rather than probability. In doing so, TSRT offers not just a reinterpretation of existing results, but a new physical principle: that the fabric of spacetime itself, through its microscopic trembling, encodes the correlations we observe in nature.

## 12 Acknowledgments

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 nonde-
structive 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.

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