Entropy as a Geometric Consequence of Higher Dimensions

Abstract

Entropy has traditionally been understood as a phenomenological principle, capturing time irreversibility in physical processes. In this work, we propose that entropy can emerge as a geometric property of higher-dimensional spacetime. Within a Kaluza–Klein framework featuring an additional circular dimension proportional to particle wavelength, trajectories acquire statistical multiplicity, which naturally produces a monotonic increase in entropy and offers a geometric foundation for the second law of thermodynamics. In the broader context, we note that the association between entropy and geometry is not unprecedented: Bekenstein and Hawking showed that black holes yields entropy proportional to the horizon area. Our contribution, however, is independent of that line of research and focuses on higher-dimensional spacetime. Importantly, the framework yields concrete predictions. In the arrival-time experiment of Das and Dürr, our model uniquely predicts symmetric probability distributions when the initial state is symmetric, in contrast to the non-symmetric outcomes expected from both standard quantum and Bohmian mechanics. This provides a distinctive and testable signature for hidden dimensions.

1. Introduction

Entropy has long been recognized as one of the most fundamental and elusive concepts in multiple fields. Its historical trajectory reflects the continuous attempt to reconcile microscopic dynamics with macroscopic irreversibility, determinism with the arrow of time. The story started in the nineteenth century with Clausius, who introduced entropy to quantify inefficiencies in heat engines and to capture the irreversibility inherent in the flow of energy [1], after the work of Carnot [2]. Clausius established that in every real process the entropy increases, a principle that later became the celebrated second law of thermodynamics. His formulation was phenomenological though profound, embedding in a single quantity the asymmetry of time.

Later, Boltzmann provided entropy with a statistical foundation, linking it to the microscopic world through the celebrated relation $S=k_{B}lnW,$ in which entropy measures the number of microscopic configurations consistent with a macroscopic state [3]. Irreversibility arises not as a fundamental law of mechanics, which is time-reversal invariant, but as a statistical result where systems evolve from less probable to more probable configurations. Gibbs refined this statistical perspective, extending entropy to ensembles of states and providing the modern language of statistical mechanics [4].

The twentieth century broadened the scope well beyond thermodynamics. In 1948, Shannon introduced the notion of information entropy, a measure of uncertainty and information content in communication systems [5]. His proposal mirrored the mathematical form of Boltzmann expression but carried a new interpretation where rather than measuring physical disorder, it quantified unpredictability in messages. This extension reinforced the idea that entropy is a universal property of systems governed by distributions and probabilities, either describing the movement of particles or quantifying information.

Entropy also reached the frontiers of relativity and quantum theory. In the 1970s, Bekenstein and Hawking demonstrated that black holes possess entropy proportional to the area of their event horizons [6,7]. The Bekenstein–Hawking entropy linked thermodynamics, quantum mechanics, and general relativity in a single relation, and in doing so, it suggested that entropy is deeply tied to the geometry of spacetime. This discovery inspired new perspectives: if entropy can be derived from spacetime geometry, perhaps the second law of thermodynamics is not just a statistical law but a manifestation of deeper structures of the universe.

Against this backdrop, higher-dimensional theories provide a natural setting to revisit entropy. The pioneering work of Kaluza and Klein in the 1920s sought to unify gravitation and electromagnetism by extending spacetime to five dimensions, with the extra dimension compactified into a circular structure [8,9]. This idea inspired later frameworks such as string theory, M-theory or particle physics, with proposals relying on additional dimensions to unify interactions [10,11,12,13,14,15]. While these theories have been extensively studied for their unifying potential, their implications for fundamental thermodynamic laws—especially entropy—remain relatively unexplored.

Moreover, recent studies have explored the connection between higher-dimensional theories and entropy functions, highlighting how Kaluza–Klein geometry can influence thermodynamic and informational quantities. In particular, Salti et al. [16] investigated the Kaluza–Klein nature of the entropy function in a cosmological context, providing a framework that complements the present approach, which focuses instead on the microscopic and geometric origin of entropy at the level of single-particle dynamics.

It is in this context that our contribution emerges. In a previous work [17], we introduced a framework in which an additional circular dimension, proportional to the wavelength of the particle under study, resolves the wave-particle duality in the Young double-slit experiment. This higher-dimensional geometry provided a natural explanation for interference phenomena, embedding quantum-like behavior into spacetime itself. Building on that foundation, the present study develops the idea that entropy can also be derived from the existence of an extra dimension.

The essential insight here is that when particles propagate in a spacetime with an additional dimension, their trajectories are no longer constrained to straight lines determined solely by four-dimensional dynamics. Instead, the geometry of the extra dimension induces deviations that manifest statistically as a diffusion phenomena with a multiplicity of possible outcomes [17]. These deviations generate entropy not as an imposed phenomenological principle but as a natural consequence of higher-dimensional behavior. When a particle passes through a barrier or slit, the extended-space trajectories give rise to probability distributions from which entropy appears naturally. Entropy in this framework is therefore geometric as it results from the structure of the extended spacetime.

This perspective situates entropy at the intersection of thermodynamics, quantum theory, and geometric unification. It resonates with black hole thermodynamics, where entropy is tied to horizon geometry [6,7], and with information theory, where entropy reflects uncertainty in distributions [5]. By deriving entropy within a Kaluza–Klein framework [17], our approach suggests that the arrow of time and the universality of entropy increase are not merely emergent statistical phenomena but reflections of the deeper architecture of spacetime.

Moreover, we show that this framework yields testable predictions in arrival-time measurements, particularly in the experiment proposed by Das and Dürr [18]. While standard quantum mechanics and Bohmian mechanics predict non-symmetric arrival-time distributions, our model uniquely predicts symmetric ones whenever the initial distribution is symmetric. If confirmed, such results would provide empirical evidence that thermodynamic irreversibility and the arrow of time are fundamentally linked to hidden dimensions of spacetime.

In what follows, we revisit the classical interpretations of entropy and their limitations, present the higher-dimensional framework, and derive the consequences for entropy production and the second law of thermodynamics. By anchoring entropy in geometry, this article expands its meaning from a statistical measure of disorder to a manifestation of the very structure of spacetime.

2. Historical Perspectives on Entropy

The concept of entropy emerged gradually in the nineteenth century as different minds sought to understand the efficiency and limitations of heat engines. The first milestone was achieved by Carnot, who in 1824 demonstrated that the efficiency of an idealized engine depends solely on the temperatures of the heat reservoirs between which it operates [2]. He did not use the word entropy, but his work established the principle that heat cannot be entirely converted into work, thereby laying the foundation for the second law of thermodynamics.

Clausius introduced the term entropy in 1865, providing a measure of irreversibility in thermodynamic processes [1]. He formulated the famous inequality

$$\begin{array}{c}\hfill \Delta S\ge \int {\displaystyle \frac{\delta Q}{T}},\end{array}$$

(1)

where $\delta Q$ is the infinitesimal heat exchanged with a reservoir at temperature T. For reversible processes the equality holds, while irreversible processes result in a strict increase in entropy. Clausius formulation turned Carnot insights into a general law, i.e., in any closed system, entropy tends to increase, giving time its thermodynamic arrow.

On one hand, Boltzmann transformed entropy from a phenomenological construct into a statistical one. In 1877, he proposed the celebrated relation

$$\begin{array}{c}\hfill S=k_{B}lnW,\end{array}$$

(2)

where $k_B$ is the Boltzmann constant and W is the number of microscopic states compatible with a given macroscopic configuration [3]. This interpretation explained irreversibility in terms of probability: although the microscopic laws of mechanics are reversible, macroscopic systems overwhelmingly evolve toward states with larger multiplicity.

On the other hand, Gibbs extended this statistical approach with his theory of ensembles, published in 1902 [4], when he introduced the ensemble average and defined entropy as

$$\begin{array}{c}\hfill S=-k_B\sum _i{p}_{i}ln{p}_i,\end{array}$$

(3)

where $p_i$ represents the probability of the system occupying microstate i. This form remains fundamental in modern statistical mechanics, unifying equilibrium and non-equilibrium descriptions.

Planck further emphasized the universality of entropy, particularly in his seminal work on blackbody radiation, where the quantization of energy levels implied a statistical treatment of radiation fields consistent with Boltzmann entropy [19]. He viewed entropy as a bridge between the microscopic quantum world and macroscopic thermodynamic laws.

In the twentieth century, entropy found new life in information theory, when Shannon redefined entropy as a measure of uncertainty in communication, introducing

$$\begin{array}{c}\hfill H=-\sum _i{p}_i{log}_2{p}_i,\end{array}$$

(4)

to quantify the information content of a source [5]. Despite differences in interpretation, Shannon entropy shared the same mathematical form as that of Gibbs, reinforcing the idea that entropy is a universal measure of multiplicity and unpredictability.

Later, in the realm of gravitation and cosmology, entropy was again reinterpreted. Bekenstein and Hawking showed that black holes carry entropy proportional to the area of their horizons [6,7], a discovery that linked thermodynamics, quantum theory, and spacetime geometry. This suggested that entropy is not only statistical but also geometric in nature.

Taken together, these historical developments show entropy remarkable adaptability. From Carnot engines to Clausius thermodynamics, Boltzmann statistical mechanics, Gibbs ensembles, Planck quantum theory, Shannon information, and Bekenstein–Hawking black hole thermodynamics, entropy has consistently revealed itself as a universal principle. It is within this lineage that the present work situates itself, proposing that entropy can be understood as a direct consequence of higher-dimensional geometry, specifically within a Kaluza–Klein framework [17].

Entropy and Geometry

The recognition that entropy can emerge directly from geometry has been one of the most profound developments of modern physics. The pioneering work of Bekenstein and Hawking established that black holes are thermodynamic objects, with entropy given by

$$\begin{array}{c}\hfill S_{\mathrm{BH}}={\displaystyle \frac{k_B{c}^{3}A}{4G\hslash }},\end{array}$$

(5)

where A is the area of the event horizon [6,7]. This relation not only revealed that entropy scales with area rather than volume, but also suggested that the fundamental degrees of freedom responsible for entropy may be encoded on surfaces or boundaries. Such results motivated the holographic principle and reinforced the idea that entropy is not solely a statistical construct but also an intrinsic feature of spacetime itself.

These insights create a natural bridge to higher-dimensional approaches. If entropy can arise from the geometry of a horizon, it is plausible that the geometry of additional dimensions might also generate entropy. In this sense, the proposed circular dimension turns out to be more than a unification tool as it becomes as well a source of irreversibility and randomness when projected into four dimensions.

In our earlier work [17], we demonstrated how an additional circular dimension could reproduce the interference patterns of the Young experiment. Here, we extend this framework to show that entropy naturally emerges when particles propagate in higher-dimensional spaces. Thus, entropy may be viewed as a manifestation of the hidden structure of spacetime, in direct analogy to how black hole entropy reflects the geometry of horizons.

3. Historical Perspectives on Kaluza–Klein Theory

The idea of extending spacetime beyond four dimensions originated in the early twentieth century. In 1921, Kaluza proposed a five-dimensional extension of Einstein general relativity in which the additional dimension could unify gravitation and electromagnetism [8]. In Kaluza formulation, the metric tensor of the five-dimensional spacetime contained not only the gravitational field but also terms corresponding to the electromagnetic potential. This unification was remarkable, but it lacked a clear explanation for why the extra dimension was unobservable.

Klein addressed this issue in 1926 by suggesting that the fifth dimension is compactified into a small circle of radius on the order of the Planck length [9]. The compactification made the extra dimension effectively invisible at macroscopic scales, while still permitting it to play a role in the fundamental structure of physics. Together, Kaluza and Klein laid the foundation for what is now called Kaluza–Klein theory.

Although initially met with limited attention, Kaluza–Klein ideas were revived in the mid-twentieth century as physicists sought unified frameworks. The theory inspired later approaches to unification, including supergravity, string theory, and M-theory, all of which rely on the existence of multiple compact dimensions [10,11,12,13,14,15]. In these contexts, additional dimensions serve to accommodate gauge fields, particle families, and quantum consistency conditions.

Despite its elegance, Kaluza–Klein theory faced criticism due to the lack of experimental evidence for extra dimensions and challenges in explaining the hierarchy of forces [20,21]. Nevertheless, the framework remains influential, particularly as a precursor to higher-dimensional models in modern physics.

Our contribution builds upon this historical lineage by exploring a new consequence of Kaluza–Klein extensions: the natural emergence of entropy. In our previous work [17], we demonstrated that a circular dimension proportional to the particle wavelength resolves the wave-particle duality in the Young experiment. Here, we extend the same framework to show that entropy increase is also an unavoidable feature of motion through such an extended spacetime. This perspective situates entropy at the heart of the Kaluza–Klein program, linking thermodynamics, geometry, and unification.

4. Methods

This section develops the central argument of this paper through a logical sequence of steps. First, the geometric postulate of the five-dimensional metric is presented, establishing the formal structure that extends spacetime by an additional circular dimension. Second, the projection of particle trajectories from the 5D manifold into ordinary 4D spacetime is derived, showing how the circular coordinate modulates the observable motion. Third, the resulting projection leads naturally to the probability distribution $I\left(\theta \right)$, which describes the statistical structure of the observed scattering pattern. Fourth, the corresponding informational entropy H is computed, quantifying the degree of geometric spreading induced by the extra dimension. Finally, the model predicts a measurable symmetry in the arrival-time experiment, providing a possible empirical signature of the proposed higher-dimensional geometry.

The five-dimensional spacetime can be expressed in a form that naturally separates the ordinary four-dimensional manifold from the circular coordinate. We adopt the notation $g_{AB}$ for the full metric, where capital Latin indices $A,B=0,1,2,3,5$ run over all dimensions, and Greek indices $\mu ,\nu =0,1,2,3$ refer exclusively to the four-dimensional spacetime subspace. Accordingly, the line element can be written as

$$d{s}^2={\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+{\mathsf{\Phi }}^2{\left(dy\right)}^2,$$

(6)

where ${\eta }_{\mu \nu }$ represents the Minkowski metric of the extended spacetime and $\mathsf{\Phi }$ is the scalar factor associated with the circular fifth coordinate y.

Thus, the fifth coordinate has a circular topology ($S^1$), and $f\left(y\right)$ is periodic in y and given by

$$f^A\left(y\right)=\sum _{n=-\infty }^{\infty }{\widehat{f}}^A\left(n\right)e^{(ni\pi y/\lambda )},$$

(7)

where n is an integer and i is the imaginary unit, ${\widehat{f}}^A\left(n\right)$ are Fourier coefficients and the wavelength $\lambda$ is the radius.

The projection from the five-dimensional manifold ${\mathcal{M}}^5$ to the observable four-dimensional spacetime ${\mathcal{M}}^4$ is now defined explicitly. The metric of the extended space,

$$d{s}^2={\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+{\mathsf{\Phi }}^2{\left(dy\right)}^2,$$

contains the compact circular coordinate y with topology $S^1$ and radius $R=\lambda /2\pi$. Translations along y correspond to local spatial rotations in ${\mathcal{M}}^4$. We introduce a periodic rotation matrix ${{\mathsf{\Psi }}^{\mu }}_{\nu }\left(y\right)$ satisfying [17]

$${\eta }_{\alpha \beta }={{\mathsf{\Psi }}^{\rho }}_{\alpha }\left(y\right){{\mathsf{\Psi }}^{\sigma }}_{\beta }\left(y\right){\eta }_{\rho \sigma },$$

so that infinitesimal displacements $\Delta y$ induce infinitesimal rotations of the four-dimensional tangent frame according to

$${\displaystyle \frac{d\mathsf{\Psi }}{dy}}={\displaystyle \frac{1}{R}}J\mathsf{\Psi },$$

where J is the generator of $\mathrm{SO}\left(3\right)$ rotations. The projection operator $\pi :{\mathcal{M}}^5\to {\mathcal{M}}^4$ acts by fixing y and mapping tangent vectors through this matrix,

$${\tilde{x}}^{\alpha }={{\mathsf{\Psi }}^{\alpha }}_{\rho }\left(y\right)x^{\rho },$$

so that the 4D coordinates correspond to the rotated shadow of the 5D geodesic on the hypersurface of constant y. Physically, motion along the circular dimension manifests as a rotation of the 4D spatial frame, providing the geometric basis for the modulation discussed below.

The consequence is that we should have [17],

$${\tilde{x}}^{\alpha }={\mathsf{\Psi }}_{\rho }^{\alpha }{x}^{\rho }.$$

(8)

The rotation in Equation (8) therefore does not arise from an external assumption but from the intrinsic geometry of the compact dimension: each infinitesimal shift $\Delta y$ on $S^1$ produces a local rotation $\mathsf{\Psi }\left(y\right)$ of the four-dimensional coordinates. This establishes the mathematical and physical meaning of the projection that connects the five-dimensional geodesic motion with the observable 4D trajectories.

The effect of the matrix, ${\mathsf{\Psi }}_{\rho }^{\alpha }$, in (8) is to change the spatial direction of travel of a given particle according to the shifts of y. This is an important point: as long as $\Delta y\ne 0$ we should have $\mathsf{\Psi }\ne \delta$, where $\delta$ is the identity matrix.

This is only one part of the phenomena, as we have additionally to account for the relativity effects on the paths. Indeed, if now we take into account the circular dimension effects and the Lorentz boost, $\Lambda$, we can rewrite (8) as,

$${\tilde{x}}^{\alpha }={\mathsf{\Psi }}_{\rho }^{\alpha }{\Lambda }_{\mu }^{\rho }{x}^{\mu }.$$

(9)

Equation (9) makes explicit how shifts along the compact coordinate produce measurable displacements in ordinary spacetime. The matrix $\mathsf{\Psi }\left(y\right)$ represents the geometric rotation induced by motion along the circular dimension, while $\Lambda$ is the conventional Lorentz boost acting within the four-dimensional Minkowski manifold. The boost does not originate from any new coordinate change in five dimensions; rather, it ensures that the projected coordinates are expressed in the observer inertial frame. Hence, the product ${{\mathsf{\Psi }}^{\alpha }}_{\rho }{{\Lambda }^{\rho }}_{\mu }$ defines the measurable projection operator: the rotation encodes the internal geometric effect of the compact dimension, and the Lorentz transformation relates the resulting trajectory to the laboratory frame. In this way, the link between infinitesimal displacements $\Delta y$ in the circular dimension and the corresponding observable variations $\Delta x^{\mu }$ in four dimensions becomes mathematically explicit.

Let us specify the meaning of the “extended subspace”. This subspace corresponds to the tangent bundle $\Sigma =\{x^1,x^2,x^3,y\}$ that combines the ordinary spatial coordinates with the circular coordinate y. The circular coordinate has topology $S^1$ with radius $\lambda$, so that any infinitesimal displacement $dy$ on $S^1$ can be represented as a rotation of the local spatial triad. Mathematically, this rotation is described by

$${\displaystyle \frac{d\mathsf{\Psi }}{dy}}={\displaystyle \frac{1}{\lambda }}J\mathsf{\Psi },$$

where J is the generator of $\mathrm{SO}\left(3\right)$ rotations and $\mathsf{\Psi }\left(y\right)$ the rotation matrix acting on the spatial sector of the extended manifold. Thus, motion along the circular dimension produces a continuous rotation in the extended subspace, and the projected four-dimensional coordinates transform accordingly. Thus, consider,

$$\Delta x^{\mu }=\left({{\mathsf{\Psi }}^{\mu }}_{\nu }\left(y\right)-{{\delta }^{\mu }}_{\nu }\right){{\Lambda }^{\nu }}_{\rho }{x}^{\rho },$$

(10)

which expresses this effect explicitly, showing that each infinitesimal displacement in y corresponds to a small rotation of the 4D trajectory followed by a Lorentz boost in spacetime.

Each infinitesimal displacement along the compact coordinate y corresponds to a local rotation of the four-dimensional spatial frame. Because the fifth dimension has a circular topology $S^1$ of radius $R=\lambda /2\pi$, the total rotation accumulated along a displacement y is characterized by a geometric phase $\varphi \left(y\right)=2\pi y/\lambda .$ The rotation matrix $\mathsf{\Psi }\left(y\right)$ introduced earlier therefore carries this phase as its argument, and the observable four-dimensional coordinates ${\tilde{x}}^{\mu }$ inherit this periodic dependence. Each distinct value of y defines a rotated copy of the same five-dimensional geodesic when projected onto the 4D hypersurface, differing only by the phase $\varphi \left(y\right)$.

This geometric phase provides the origin of the modulation observed in the projected trajectories. As the particle evolves, the compact coordinate y advances along the circular dimension, producing a continuous phase shift in the corresponding 4D projection. The periodic boundary condition $y\sim y+2\pi R$ ensures that the phase is cyclic and well defined. The ensemble of all such phase-shifted projections forms the interference pattern encoded in the distribution $I\left(\theta \right)$ discussed below. Hence, the phase appearing on the kinematic trajectory is purely geometric: it arises from the topology of the fifth dimension and is modulated by its periodic structure, without invoking any external field or additional dynamical assumption.

4.1. Entropy from the System of Two Chambers

To illustrate how entropy emerges within our framework, consider a simplified system of two chambers. Initially, one chamber is filled with particles while the other is empty, and the two are separated by a thin wall with a small opening. Classically, once the barrier is removed, the gas diffuses from the full chamber into the empty one until both are uniformly occupied. Entropy in this process increases, as the number of accessible microstates expands.

Now suppose instead that only a single particle passes through the slit connecting the two chambers. In four-dimensional spacetime, the particle would travel in a straight line and would simply strike the opposite wall, leaving a localized mark. However, when we extend spacetime by including an additional circular dimension, the particle motion is altered, i.e., rather than following only one path, it spreads out in a distribution of possible trajectories. The result is analogous to the single-slit diffraction experiment in optics, where a particle or wave produces an intensity distribution on a detection screen.

The scattering pattern following a slit can be viewed as a projection of a single five-dimensional trajectory into four-dimensional space. The probability density function is given by

$$I\left(\theta \right)={\sin \mathrm{c}}^2\left({\displaystyle \frac{d\pi \theta }{\lambda }}\right),$$

(11)

where $I\left(\theta \right)$ represents a normalized probability distribution. This spreading directly reflects the influence of the circular dimension: as the particle moves in the 5-th dimension, it changes the corresponding angle $\theta$, as observed in our 4-th dimensional world, which leads to the well known pattern seen the experiments for I.

Equation (11) represents the observable projection of the five-dimensional trajectory into four-dimensional space. The geometric phase $\varphi \left(y\right)=2\pi y/\lambda$ introduced earlier acts as the parameter connecting the motion along the compact coordinate to the measurable displacement in 4D. Each distinct value of $\varphi \left(y\right)$ corresponds to a rotated projection of the same deterministic 5D geodesic, and the ensemble of all such projections produces the intensity distribution $I\left(\theta \right)$ measured in the scattering pattern. The spreading of this distribution is therefore not a stochastic phenomenon but a direct geometric consequence of the periodic structure of the fifth dimension.

In this formulation, the phase $\varphi \left(y\right)$ plays the same role that an optical path difference plays in conventional interference: it modulates the amplitude of the projected trajectories as y varies over $S^1$. The observable probability density $I\left(\theta \right)$ is thus the envelope of all such phase-shifted projections, and its normalization guarantees that the distribution reflects a complete set of possible geometric configurations of the compact coordinate.

The informational entropy H associated with $I\left(\theta \right)$ quantifies the uncertainty generated by this geometric multiplicity of projections. It measures the number of distinguishable phase states available as the particle interacts with the boundary or detection plane. Consequently, entropy in this framework arises not from randomness in the particle motion but from the loss of phase information inherent to the projection of the deterministic 5D dynamics into the observable 4D domain. This connects the emergence of H directly to the geometry and periodicity of the additional dimension.

Rather, H represents an informational measure in the sense of Shannon, applicable to each discrete scattering event, and provides a geometric characterization of the loss of information associated with the projection of the five-dimensional motion into four-dimensional space. Between collisions the system evolves deterministically, and no additional information is produced; entropy increases only at the moments where the interaction imposes a boundary constraint on the trajectory, where boundary refers to the geometric constraint imposed by the measurement surface (e.g., the slit or detector plane), not a boundary in the circular coordinate itself.

As the particle continues to move through the second chamber, bouncing against walls or interacting with other particles, each scattering event introduces further multiplicity of paths. These successive contributions can be treated as independent, so that the total entropy is obtained by summing over the entropies of each stage of motion.

Non-commutativity of the transformations in (9) does not break microscopic reversibility. Each transformation remains invertible, but repeated applications lead to phase mixing among trajectories—an effect analogous to Liouvillian mixing in classical mechanics [22,23,24,25]. This cumulative process produces statistical irreversibility without violating the reversible nature of the underlying dynamics.

We can understand the intensity in (11) as the probability of the particle to reach the screen. Indeed, the function I in (11) is a probability density function (pdf), as we can easily rewrite I in the form ${\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{\infty }\mathrm{sinc}{\left(\varphi \right)}^{2}d\varphi =1$ and also, $I\ge 0$, which is another condition for the function to be a pdf. For the sake of simplicity, we will not rescale I in (11) by ${\displaystyle \frac{1}{\pi }}$ as it is not strictly necessary in the reasoning below.

The entropy, H, is defined as

$$H={\int }_{-\infty }^{\infty }Ilog\left(I\right)d\alpha ={\int }_{-\infty }^{\infty }{\mathrm{sinc}}^2\left(\alpha \right)log\left({\mathrm{sinc}}^2\left(\alpha \right)\right)d\alpha ,$$

(12)

where $\alpha =d\pi \theta /\lambda$. We show the convergence of the integral in Appendix A.1.

The integration limits in (12) represent the analytical normalization of the angular probability density and are equivalent to integrating over the finite experimental aperture. To guarantee proper normalization and positivity, the informational entropy is computed from the normalized distribution $p\left(\theta \right)=I\left(\theta \right)/\int I\left(\theta \right)d\theta$ as $H=-\int p\left(\theta \right)lnp\left(\theta \right)d\theta$, which ensures $H\ge 0$ and preserves the geometric interpretation of H as the measure of phase-space dispersion induced by the circular dimension.

In this formulation, $I\left(\alpha \right)$ denotes the normalized probability distribution of the scattering angle immediately after each interaction event. The corresponding entropy functional (12) quantifies the information produced at that moment, measuring the uncertainty introduced in the angular distribution. It does not represent the thermodynamic entropy S or the quantum von Neumann entropy, but an informational measure in the sense of Shannon [5].

As we said above, once in the second chamber, the particle follows different consecutive paths, each with a given entropy. We can regard them as mutually statistically independent and, therefore, the final entropy $H_f$ will be given by adding the contribution of each $i-th$ path, and will be given by

$$H_f=\sum _{i=1}^N{H}_i.$$

(13)

There are two important remarks to be made here. On one hand, the initial entropy, $H_1$, regards the one when the particle crosses the slit and reaches the opposite wall, while the ones aftewards $H_i,i=2,3,\dots ,N$ can be though as if the particle was moving within the chamber, either hitting other walls or other particles.

On the other hand, entropy growth arises only when considering an ensemble of scattering events, where the circular dimension introduces small phase shifts that decorrelate the projected four-dimensional trajectories. The cumulative effect of many such interactions leads to a statistical spreading of the distribution, producing an effective increase in H. Thus, the entropy discussed here characterizes the collective loss of phase correlation across repeated scattering events, rather than the dynamics of an individual particle.

Moreover, the entropy H defined in (13) measures the information associated with the multiplicity of trajectory projections. It does not represent thermodynamic entropy but an informational quantity linked to geometric degrees of freedom.

4.2. Measurement of Time: Das and Dür Experiment

An effective way to test the hypothesis of an additional circular dimension is through the experiment proposed by Das and Dürr [18] for measuring particle arrival times. In their setup, a single particle is initially confined at one end of a cylindrical trap, with a known initial probability distribution $P(x_{\mathrm{in}},t_0)$. When the trap is opened, the particle propagates freely toward detectors at the opposite end, which record the arrival-time distribution $P(x_{\mathrm{fin}},t_1)$.

In the present framework, the propagation from $(x_{\mathrm{in}},t_0)$ to $(x_{\mathrm{fin}},t_1)$ is described by the sequence of matrices introduced in (9), which combine Lorentz transformations and rotations arising from the circular dimension. The cumulative effect of these transformations defines a transition probability density

$$F(x_{\mathrm{in}},t_0\parallel x_{\mathrm{fin}},t_1)=\mathcal{T}[x_{\mathrm{in}},t_0;x_{\mathrm{fin}},t_1],$$

representing the probability that a particle initially at $x_{\mathrm{in}}$ at time $t_0$ will be detected at $x_{\mathrm{fin}}$ at time $t_1$. The measurable probability density at the detector is then obtained from the integral

$$P(x_{\mathrm{fin}},t_1)=\int F(x_{\mathrm{in}},t_0\parallel x_{\mathrm{fin}},t_1)P(x_{\mathrm{in}},t_0)d{x}_{\mathrm{in}}.$$

(14)

This expression has the structure of a convolution and establishes a direct correspondence between the theoretical model and observable quantities such as detection position and arrival-time probabilities. The transition kernel F thus encapsulates the geometric spreading introduced by the extra dimension, translating the five-dimensional dynamics into an experimentally measurable distribution.

In the present formulation, the coordinate x denotes the transverse displacement relative to the beam axis, while t represents the arrival time of the particle at the detector plane. The propagation kernel $F(x_{\mathrm{in}},t_0\parallel x_{\mathrm{fin}},t_1)$ is isotropic in the transverse direction and separable in $(x,t)$,

$$F(x_{\mathrm{in}},t_0\parallel x_{\mathrm{fin}},t_1)=f\left(\right|x_{\mathrm{fin}}-x_{\mathrm{in}}\left|\right)g(t_1-t_0),$$

so that the convolution in (14) preserves the spatial symmetry of the initial condition for each fixed $t_1$. When the initial distribution $P(x_{\mathrm{in}},t_0)$ is symmetric about the beam axis, the resulting $P(x_{\mathrm{fin}},t_1)$ remains symmetric in x. The detectors, however, record the integrated signal over x at a fixed longitudinal position, producing the arrival-time histogram

$$P\left(t_1\right)=\int P(x_{\mathrm{fin}},t_1)d{x}_{\mathrm{fin}}.$$

Because this integration acts on a function symmetric in x, the measured histogram $P\left(t_1\right)$ inherits the same symmetry. Thus, the temporal symmetry observed experimentally in Ref. [16] originates from the spatial isotropy of the propagation kernel combined with the symmetric initial condition. The result does not require an explicit time-symmetry of F, but follows directly from the geometric structure of the model and from the invariance of the transverse spatial distribution under $x\to -x$.

A notable consequence of this formulation emerges when the initial probability distribution is symmetric, that is, when there exists a point $x_0$ such that $P(x-x_0)=P(x+x_0)$ for all x. Because the convolution with F preserves symmetry for isotropic circular-dimension effects, the resulting probability $P(x_{\mathrm{fin}},t_1)$ also remains symmetric. This outcome contrasts with both standard quantum-mechanical and Bohmian analyses [18], which predict asymmetric arrival-time distributions even for symmetric initial conditions.

The statistical origin of entropy increase can be stated more precisely. Taking the probability density be obtained from the convolution (14), the corresponding informational entropy will be, $H\left(t_1\right)=-\int P(x_{\mathrm{fin}},t_1)lnP(x_{\mathrm{fin}},t_1)d{x}_{\mathrm{fin}}$. Because F introduces phase-dependent spreading, the resulting distribution $P(x_{\mathrm{fin}},t_1)$ becomes broader than $P(x_{\mathrm{in}},t_0)$, leading to $\Delta H=H\left(t_1\right)-H\left(t_0\right)>0$. This explicit formulation shows that entropy growth follows from the geometric convolution mechanism rather than from any single-particle or matrix-commutation effect.

In this context, the term distribution refers to the experimental arrival-time histograms reported in [18], which represent the probability densities of detected electron positions in the single-electron interference experiment of Tonomura et al. [26]. We use the words “histogram” and “distribution” synonymously to describe these measured probability patterns, which serve here as the empirical counterpart of the theoretical probability function $P(x_{\mathrm{fin}},t_1)$ derived in the present model.

Physically, the arrival-time symmetry predicted by this model arises because the circular dimension introduces an additional geometric degree of freedom that redistributes phases uniformly during propagation. Each transformation in the sequence of matrices corresponds to an operation coupling the circular coordinate to the extended spacetime coordinates. This mechanism ensures that while the total probability is conserved, local asymmetries average out, yielding a symmetric arrival-time profile. The Das-Dürr experiment thus provides a direct means to verify whether the microscopic propagation of particles retains geometric symmetry, offering a concrete probe of the link between higher-dimensional geometry, entropy generation, and the arrow of time.

Thus, our higher-dimensional approach provides a clear experimental signature: the observation of symmetric arrival-time distributions in such measurements would strongly support the presence of an additional circular dimension. This makes the Das-Dürr setup not only a conceptual probe of time in quantum mechanics but also a concrete test for the geometric origin of entropy and irreversibility proposed in this work.

5. Discussion and Conclusions

The central proposal of this work is conceptually simple: by introducing an additional circular dimension into spacetime, one can derive entropy growth as a direct geometric consequence. While the idea of extra dimensions dates back to the seminal work of Kaluza and Klein [8,9], most studies in this tradition have focused on unification of forces or high-energy phenomenology [15]. By contrast, few works have explored the thermodynamic implications of higher-dimensional structures, and even fewer have suggested concrete experimental tests [12,13]. In this sense, our approach provides a novel connection between extra dimensions and the second law of thermodynamics.

As demonstrated in our previous work [17], when the fifth coordinate y possesses circular topology, a displacement along this circular dimension acts as a rotation in the four-dimensional subspace. The 5D Minkowski metric admits a rotation matrix ${{\mathsf{\Psi }}^{\rho }}_{\alpha }\left(y\right)$ satisfying ${\eta }_{\alpha \beta }={{\mathsf{\Psi }}^{\rho }}_{\alpha }\left(y\right){{\mathsf{\Psi }}^{\sigma }}_{\beta }\left(y\right){\eta }_{\rho \sigma }$, so that infinitesimal shifts $\Delta y$ correspond to local rotations of the projected coordinates $x^{\mu }$. Physically, motion along the circular coordinate modulates the orientation of 4D trajectories, producing the angular variation later described by the probability distribution $I\left(\theta \right)$. This coupling between translation in the circular dimension and rotation in 4D spacetime constitutes the geometric basis for the observable phase modulation derived in Section 4.2.

Two results are particularly significant. Firstly, we demonstrated that entropy arises naturally in a two-chamber system with a narrow slit once the additional dimension is included. In the four-dimensional picture, a single particle would traverse the slit and leave a localized mark, producing no diffusion and therefore no entropy increase. However, when the circular dimension is considered, the particle spreads across a distribution of possible paths, generating entropy even at the single-particle level. As the particle undergoes further interactions within the second chamber, each event contributes additively to the total entropy.

The statistical non-commutativity of successive affine transformations combining rotations and translations ensures irreversibility, embedding the second law of thermodynamics in the geometry of extended spacetime, as shown in Appendix A.2. This statistical irreversibility is analogous to Liouvillian mixing [22,23,24,25] in classical mechanics: although Liouville theorem guarantees conservation of phase-space volume, successive non-commuting transformations continuously deform and intertwine its regions, erasing local correlations while preserving total measure. Similarly, in the present extended-space formulation, statistical non-commuting geometric transformations generate an effective phase-space mixing, providing a structural origin for entropy growth. To our knowledge, this geometric derivation of entropy growth has not been highlighted previously in the literature.

A word might be in order here regarding reversibility and the naming we gave as statistical non-commutativity rather than algebraic non-commutativity. Although each rotation and boost remains individually invertible, Appendix A.2 shows that their commutation condition,

$$[\Lambda ,\mathsf{\Psi }]=0⟺R\beta =\beta .$$

(15)

The composition of a Lorentz boost with a spatial rotation depends on the relative orientation of their axes. Commutativity between these two transformations holds whenever the boost direction is invariant under the rotation. In particular, any boost performed along an axis perpendicular to the plane of the rotation commutes with it. For non-perpendicular configurations, however, the generators of boosts and rotations satisfy non-vanishing commutation relations, leading to small but measurable geometric asymmetries in the projected 4D trajectories. This property underlies the anisotropy discussed in the following section and connects directly to the structure of the Lorentz algebra in the projected space.

Because this subset forms a measure-zero set in the continuous space of orientations, the probability of exact commutativity is practically null. Consequently, repeated applications of non-commuting transformations induce progressive phase mixing among trajectories—a process that yields statistical irreversibility without violating microscopic reversibility.

Moreover, the second law of thermodynamics is inherently statistical: entropy quantifies the logarithmic measure of accessible microstates and increases as correlations among microscopic degrees of freedom are lost. In our framework, the tendency toward entropy growth is not redefined but reinterpreted geometrically within the extended five-dimensional manifold. Analogous to Liouvillian mixing in phase-space dynamics [22,23,24,25], the informational entropy H introduced here represents a geometric measure of correlation loss induced by the topology of the circular dimension. Thus, the model offers a structural explanation for the universality of entropy increase and the emergence of temporal asymmetry.

In our framework, however, no stochastic process is assumed at any stage of the dynamics. The motion of the corpuscle in the five-dimensional manifold is entirely deterministic, governed by the geodesic structure of the extended metric. However, because the additional dimension is circular and periodic, a single 5D trajectory can give rise to multiple effective projections in ordinary 4D spacetime, each corresponding to a distinct phase of the circular coordinate. This geometric multiplicity generates an apparent dispersion in the 4D projection that mimics the statistical character of interference phenomena, even though the underlying motion remains deterministic. Thus, the probabilistic appearance of the distribution arises from the topology of the extra dimension rather than from any intrinsic randomness, distinguishing this approach from stochastic formulations.

In this framework, entropy growth is not redefined but reinterpreted geometrically within the extended five-dimensional manifold. Successive non-commuting transformations coupling the circular and extended coordinates redistribute phase relations among the projected four-dimensional trajectories, producing an effect analogous to Liouvillian mixing in classical phase-space dynamics. Thus, the informational entropy H introduced here acts as a geometric measure of correlation loss induced by the circular dimension, offering a structural explanation for the universality of entropy increase and the emergence of temporal asymmetry.

While the historical development of entropy provides essential context, its most general interpretation transcends thermodynamics. As shown by Wehrl [27], entropy represents a universal functional that measures the amount of inaccessible information in a system and remains well defined under classical, quantum, or statistical formulations. In this broader view, entropy is not restricted to energy dissipation or disorder but reflects the geometric and informational structure of phase space. Within the present model, the entropy functional H introduced in Section 4 plays this generalized role: it quantifies the information lost through the projection of deterministic motion in the five-dimensional manifold into the observable four-dimensional domain. This interpretation situates the present theory within Wehrl general framework and highlights how geometry can serve as the origin of entropy growth and temporal asymmetry.

Secondly, we showed that our framework makes specific predictions for the arrival-time experiment proposed by Das and Durr [18]. Standard quantum mechanics and Bohmian mechanics both predict non-symmetric arrival-time distributions, even when the initial state is symmetric. In contrast, our higher-dimensional model predicts that the final distribution must remain symmetric whenever the initial distribution is symmetric. This sharp distinction provides a clear and testable experimental signature for the presence of hidden dimensions. A confirmation of symmetric distributions in such an experiment would not only support the geometric origin of entropy but would also provide direct evidence for the existence of an additional circular dimension.

One of the main challenges for higher-dimensional theories, such as the original Kaluza–Klein model, is the stabilization of the additional dimension. Without a mechanism that keeps the extra dimension small, quantum or classical instabilities could lead to variations in the effective constants of nature. In the present model, the fifth coordinate is circular and its radius is proportional to the wavelength of the particle under consideration. Although this formulation ties the size of the extra dimension to a measurable quantity, it does not in itself constitute a dynamical stabilization mechanism. The stability issue can, however, be interpreted geometrically: configurations in which entropy is maximized under compactification constraints correspond to stable geometries. Entropy maximization may therefore act as a natural selection rule favoring compactified states.

Future developments may adapt stabilization mechanisms analogous to those in superstring and M-theory, where flux compactification and brane potentials provide dynamical confinement of the extra dimensions [15].

An important prediction of the model concerns the symmetry of the final probability distribution. For symmetric initial conditions $P(x_{\mathrm{in}},t_0)$, the convolution kernel $F(x_{\mathrm{in}},t_0\parallel x_{\mathrm{fin}},t_1)$ preserves this symmetry, leading to a final probability $P(x_{\mathrm{fin}},t_1)$ that remains symmetric with respect to the central axis of propagation. This behavior arises naturally from the geometric structure of the extended manifold and can be tested experimentally through arrival-time measurements or single-particle interference setups. The observation of symmetric detection statistics under controlled boundary conditions would therefore serve as an empirical validation of the proposed higher-dimensional framework.

More broadly, the analysis highlights the capacity of higher-dimensional models to illuminate properties that appear puzzling in four dimensions. The Campbell theorem [28] already demonstrated that field equations in five dimensions can reduce to four-dimensional equations with sources, suggesting that matter may be a manifestation of geometry. In an analogous spirit, our results indicate that entropy itself can be understood as a manifestation of geometry when motion is considered in a higher-dimensional spacetime.

This proposal resonates with the broader narrative that has emerged since the discovery of black hole entropy by Bekenstein and Hawking [6,7]. There, entropy was shown to be proportional to horizon area, highlighting its geometric nature. Our contribution is independent of that line of research but shares the same spirit: entropy may not simply be a statistical construct, but rather an inevitable consequence of the structure of spacetime itself.

Moreover, it is important to emphasize that the standard diffraction pattern describes the momentum-space distribution of the one slit experiment. In our model, the same form arises geometrically from the projection of a single five-dimensional trajectory onto four-dimensional spacetime, where distinct phases of the circular coordinate correspond to distinct effective momenta.

Although our proposal introduces only a single extra circular dimension, it shares with Kaluza–Klein models the generic difficulty of stabilization. In the absence of a mechanism, the radius of the circular dimension could vary, leading to inconsistencies with observed physics. In string/M-theory, stabilization is achieved through flux compactifications and potentials for moduli fields. Our phenomenological approach does not yet solve this problem, but we suggest that entropy maximization may naturally favor compactified configurations. Future work should investigate whether entropy arguments can provide a dynamical stabilization mechanism.Finally, one could raise the matter that the distribution can equivalently be viewed as arising from an ensemble of varying initial conditions. Thus, in the extended-space description, the phase of the circular coordinate effectively parameterizes this ensemble.

In conclusion, the addition of a circular dimension not only extends the unification idea of Kaluza and Klein but also sheds light on one of the most profound principles of physics: the universality of entropy increase. By deriving the second law of thermodynamics from geometry, the present work suggests that the arrow of time may ultimately be rooted in the hidden structure of spacetime. Future developments should pursue two directions. On the theoretical side, applying the framework to many-body systems and nonequilibrium dynamics could reveal new structures in entropy production. On the experimental side, the Das and Durr setup provides a clear starting point for testing the predictions of this model, and further experimental schemes may be devised to probe the geometric origin of irreversibility.

A natural extension of this framework involves formulating entropy within a density-matrix approach. In higher-dimensional theories, the circular coordinates may be viewed as hidden degrees of freedom that contribute to the statistical character of the effective four-dimensional dynamics. A rigorous treatment would describe the system by a density matrix $\rho$ defined on the full extended manifold, with the observable 4D entropy obtained by tracing over circular coordinates,

$$S=-\mathrm{Tr}(\rho ln\rho ).$$

This approach would make it possible to determine for which number of compactified dimensions entropy S is maximized, potentially providing a geometric explanation for the stability of the observed macroscopic dimensions. Such a density-matrix formulation could also bridge the present geometric model with superstring frameworks, where compactification and entanglement entropy play analogous roles.

Taken together, our results strengthen the view that entropy, far from being merely a statistical artifact, is a fundamental property of spacetime geometry. The arrow of time, as argued here, emerges not from approximation or coarse-graining but from the very architecture of higher-dimensional reality.

This work is intended as a first step toward establishing a quantitative relationship between informational entropy and the dimensional structure of spacetime.

Appendix A.1. Convergence of the Integral in (12)

The convergence of the integral in (12) can be shown by analyzing the behavior of the integrand in three critical regions. Firstly, as $\left|\alpha \right|\to \infty$, the function ${\mathrm{sinc}}^2\left(\alpha \right)$ decays quadratically, while $log\left({\mathrm{sinc}}^2\left(\alpha \right)\right)$ approaches $-log\left({\alpha }^2\right)$. The resulting product behaves therefore as $-{\displaystyle \frac{log\left({\alpha }^2\right)}{{\alpha }^2}}$, which decays rapidly and guarantees integrability at infinity. Secondly, near the zeros of the $\mathrm{sinc}$ function (i.e., $\alpha =n\pi$), the logarithmic term diverges to $-\infty$, while $-{\displaystyle \frac{log\left({\alpha }^2\right)}{{\alpha }^2}}$ approaches zero. Since the product $x^{2}log\left(x^2\right)\to 0$ as $x\to 0$, the singularities are integrable. Thirdly, at $\alpha =0$, the $\mathrm{sinc}$ function is smooth while reaching its maximum, allowing therefore the integrand to be well-behaved. Thus, the integral is convergent over $\mathbb{R}$.

Appendix A.2. Non-Commutativity Between a Lorentz Boost and a Spatial Rotation

Consider a Lorentz transformation $\Lambda \in S{O}^{+}(1,3)$ written in block form:

$$\Lambda =\left(\begin{array}{cc}\gamma & -\gamma {\mathit{\beta }}^{\mathsf{T}}\\ -\gamma \mathit{\beta }& {\mathbf{I}}_3+{\displaystyle \frac{{\gamma }^2}{\gamma +1}}\mathit{\beta }{\mathit{\beta }}^{\mathsf{T}}\end{array}\right),$$

where $\mathit{\beta }=\mathbf{v}/c$ and $\gamma ={(1-{\beta }^2)}^{-1/2}$ describe a boost along the direction of $\mathit{\beta }$.

Let the pure spatial rotation $\mathsf{\Psi }$ act only on the three spatial coordinates, whose angle of rotation is proportional to y, which is the 4-th circular additional spatial dimension. Its $4\times 4$ representation in Minkowski space will be

$$\mathsf{\Psi }=\left(\begin{array}{cc}1& 0^{\mathsf{T}}\\ 0& R\end{array}\right).$$

The composition rotation followed by the Lorentz boost is

$${\Lambda }_{\mathrm{rot}/\mathrm{boost}}=\Lambda \mathsf{\Psi }=\left(\begin{array}{cc}\gamma & -\gamma {\mathit{\beta }}^{\mathsf{T}}R\\ -\gamma \mathit{\beta }& \left(\delta +{\displaystyle \frac{{\gamma }^2}{\gamma +1}}\mathit{\beta }{\mathit{\beta }}^{\mathsf{T}}\right)R\end{array}\right),$$

where $\delta$ is the identity matrix.

In contrast, the Lorentz boost followed by rotation gives

$${\Lambda }_{\mathrm{boost}/\mathrm{rot}}=\mathsf{\Psi }\Lambda =\left(\begin{array}{cc}\gamma & -\gamma {\left(R\mathit{\beta }\right)}^{\mathsf{T}}\\ -\gamma R\mathit{\beta }& R\left(\delta +{\displaystyle \frac{{\gamma }^2}{\gamma +1}}\mathit{\beta }{\mathit{\beta }}^{\mathsf{T}}\right)\end{array}\right).$$

The two matrices are equal only when

$$R\mathit{\beta }=\mathit{\beta },$$

that is, when the boost direction is invariant under the rotation (for instance, i.e., the R matrix identity). Otherwise, ${\Lambda }_{\mathrm{rot}/\mathrm{boost}}\ne {\Lambda }_{\mathrm{boost}/\mathrm{rot}}$.

More specifically, commutativity between the Lorentz boost $\Lambda$ and the rotation $\mathsf{\Psi }$ requires

$$[\Lambda ,\mathsf{\Psi }]=0\iff R\beta =\beta .$$

This condition is satisfied whenever the boost direction $\mathit{\beta }$ is invariant under the spatial rotation, that is, when the rotation R acts about the same axis as the boost or, equivalently, when $R\mathit{\beta }=\mathit{\beta }$. In such configurations, most notably when the boost is performed along an axis perpendicular to the plane of the rotation, the two transformations commute. For all other orientations, $[\Lambda ,\mathsf{\Psi }]\ne 0$, and the non-commutativity introduces geometric asymmetries in the projected 4D trajectories. Thus, while commutativity is not restricted to trivial cases, it occurs only when the boost direction remains invariant under the rotation.

More importantly, the first, $\beta \left(\mathbf{v}\right)$ is a function of the three spatial coordinates through $\mathbf{v}$ and the second is function of a 5th independent coordinate, i.e., $\mathsf{\Psi }\left(y\right)$. Those two variables $\mathbf{v}$ and y are independent and therefore we can assume that commutativity is very unlikely to happen.

Hence, exact commutativity—and, therefore, perfect phase retracing—occurs with negligible probability. Irreversibility in this context is not algebraic but statistical, emerging from the cumulative loss of phase correlations across non-commuting transformations, in agreement with the notion of Liouvillian mixing [22,23,24,25].