# Is light a fluid like any other?

Jean-Paul Caltagirone

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# Is light a fluid like any other?

Jean-Paul Caltagirone

calta@ipb.fr

Bordeaux INP, University of Bordeaux, CNRS UMR-5295,
Arts et Métiers Institute of Technology, INRAE, I2M Bordeaux,
351 Cours de la Libération - Bât. A11, 33405, Talence - France

## Abstract

The wave-particle-energy triality extends the concept of wave-particle duality by associating the wave with the celerity of the medium, the velocity with that of the particle, and the energy with the product of celerity and velocity. When velocity equals celerity, the law $e = cv$ generalizes the formula of theory of relativity by removing the notion of mass. The assumptions of classical mechanics, relativity, and quantum mechanics are reexamined in light of discrete mechanics, where the conservation of acceleration plays a central role in the derivation of the law of motion. The principle of parsimony leads to the abandonment of redundant quantities in physics, for example in the equivalence between mass and energy, where energy per unit mass is the only relevant parameter; mass becomes a simple coefficient of proportionality. The units of the disjoint concepts of space and time are the only ones needed to describe all quantities in physics. The Weak Equivalence Principle (WEP) and Newton's second law are also reformulated; a paradox on a small temporal scale leads us to consider that Newton's second law is an approximation of reality. A new law of motion is presented, which resolves the paradox while remaining consistent with the results of relativity. The concepts revisiting classical mechanics and those of special and general relativity are then applied to quantum mechanics. The absence of the notion of mass allows us to glimpse a potential unification of the laws of mechanics of material media and those of particles with or without mass. A unified law of motion is thus derived, both as a wave equation and as a deterministic transport equation.

## Keywords

Weak Equivalence Principle; Wave-Particle-Energy triality; Discrete Mechanics; Helmholtz-Hodge Decomposition; Conservation of Acceleration

## 1 Introduction

Absolutely, the light is a fluid like any other. The objective here is to show that there is a physical homology between light and a fluid, that light propagates like a fluid whose propagation properties are certainly different in value, but represent strictly the same concepts. This homology leads to the derivation of a single law of motion, the same for light and fluid. The major differences in the physical properties associated with each of these two fields have led, over time, to the emergence of specific approaches such as geometric optics, quantum mechanics, and fluid mechanics, even though all these approaches can be fundamentally replaced by a single law. The unification of the laws of physics will still require significant discontinuous efforts, discarding previously introduced irreconcilable concepts in physics. This is the case with the theory of relativity and quantum mechanics, which are definitively incompatible [1]. To quantify gravity, a conceptual leap must be made, reinterpreting space-time in some way in order to create a coherent theory.

Given the differences between the concepts of quantum mechanics and those of classical or relativistic mechanics, it is unrealistic to hope for a reconciliation between these two theories. A possible unification of the laws of physics can be initiated on the basis of certain rules that may seem to contradict one or the other theory, but which preserve their intangible observations, among other things,

* apply the principle of parsimony to significantly reduce the number of variables, unknowns, physical parameters, constants, etc. The multitude of these quantities hinders the search for simple solutions and the laws that govern them. The perfect example is the presence of mass in physical equations. This concept, which dates back to the dawn of time, has been retained in all theories of mechanics, but not in electromagnetism, which shows that it must be examined carefully. Abandoning the
notion of mass leads to the elimination of a number of other quantities in physics that explicitly
or implicitly involve it;

* abandon the concept of space-time, as it is not compatible with the quantification of gravity necessary for the unification of the two theories, which are considered incompatible, each with its own supporters seeking to integrate the other into their own doctrine. In discrete mechanics, space-time does not exist, gravity is a field like any other, subject to the same rules of a unified relativistic law of motion. Space and time are disjoint concepts, the two intangible notions of physics. The results of general relativity are nevertheless found in discrete mechanics, which satisfies Lorentz invariance *a posteriori*;

* to give a more precise meaning to the concepts used in everyday language, particularly that of "speed of light", it is necessary to distinguish between the celerity of the wave and the velocity of the photon. While the assertion of special relativity strictly associates celerity and velocity, it remains, in general, an unverified postulate. Celerity is an intrinsic property of the medium, for example light in a vacuum, while velocity, a relative quantity, is associated with a particle with or without mass or with a material medium;

* refuting the exclusively random nature of quantum mechanics, which is incompatible with gravity. Light behaves both as a wave and as a stream of particles, with the wave allowing us to calculate the probability that a particle will manifest itself. The probabilistic approach is one way of addressing a physical phenomenon that involves a large number of degrees of freedom, but it is not the only one. Turbulence in fluids, for example, is treated in a complementary manner by statistical approaches to calculate a probability density, but also by deterministic means, in a direct simulation where the law of motion is supposed to represent reality on a small scale. The motion of an elementary particle, in particular the photon, will be addressed by the same physical law that governs the propagation of waves;

* if we abandon the exclusively periodic nature of waves, they can also manifest themselves through non-periodic behavior; for example, shock waves in fluid mechanics, where the initial periodic phenomenon transforms into a shock, or waves that eventually break on a beach. The equations of quantum mechanics are written in terms of a complex wave function $\psi(r, t)$ that characterizes the quantum state of a material particle. The classical concept of trajectory must be replaced by that of a state dependent on the current time. The quantum state of a particle such as an electron is characterized by the wave function, which contains all the information that can be known about the particle. This analogy with geometric optics is abandoned in favor of concepts closer to traditional mechanics, namely velocity and energy;

* let us temporarily abandon the idea of quantum entanglement, where correlations between measurements associated with two particles can only be detected once the results have been compared, which necessarily implies an exchange of classical information, in accordance with relativity. Quantum mechanics therefore respects the principle of causality. When two particles are subjected to the same field, for example gravity, they behave in exactly the same way without it being necessary to invoke entanglement, simply because the Weak Equivalence Principle is a local principle;

* revisiting the fundamental laws of physics, even if their legitimacy is not questioned. While the results of repeated observations must be considered accurate, the interpretations they give rise to may evolve over time. For example, what is the meaning of the Weak Equivalence Principle, as understood by Galileo, according to which two bodies of different natures fall at the same velocity under the effect of gravity as a local principle, or in the sense of quantum mechanics, according to which these two bodies, distant from each other, would be entangled? In an essay, Louis de Broglie [2] discussed the need to periodically subject accepted principles that are no longer questioned to thorough examination.

The application of these rules radically changes the derivation of a unified law of motion, which must
nevertheless retain the essential elements of classical mechanics or relativity, as well as those of quantum
mechanics. New abstractions are then adopted,

* the concept of the wave-particle-energy triality, where the wave, which is not necessarily periodic, is characterized by a length $\ell = ct$, the particle is defined by its velocity $v$ such that $\ell = vt$, and the energy per unit mass is defined by $\phi = cv$, the product of the celerity of light and velocity, two
disjoint quantities. The most famous formula in physics, $E = mc^2$, becomes here $\phi = E/m = c\mathbf{v}$. There is in fact no reason to systematically attribute the value of the celerity of light in the medium to the velocity of a particle. The relationship $\phi = c\mathbf{v}$ applies without restriction to a fluid or solid in motion or to a particle with or without mass, such as a photon. This energy is strictly reserved for the concept of a particle, since waves do not carry energy;

* the introduction of the concept of spin into the equations of mechanics, a property of particles that is lacking in classical mechanics. Angular momentum, the orbital spin of particles, and intrinsic spin are all related to the same concept of rotation around an axis, whether external or attached to the particle. In discrete mechanics [3], angular momentum is represented by the rotational energy per unit mass $\psi$. The intrinsic acceleration of a particle with or without mass or of a material medium is equal to the sum of two terms, the translational acceleration defined by the gradient of the scalar potential $\phi$ and the dual curl of the vector potential, $\psi$. The notion of spin is therefore naturally associated with the latter potential;

* the concept of inertia, which is always present regardless of the physical phenomenon under consideration, applies not only to turbulent fluid flows, but also to light, the propagation of electromagnetic waves with Lorentz forces, etc. Nonlinearities play a key role, to varying degrees, in all phenomena; for example, the photon has no mass but possesses inertia, which represents its only energy. Since its velocity is in principle equal to the celerity of light in a vacuum, its straight trajectory can be curved in the presence of an intense gravitational field; this corresponds to a balance between inertia and gravitational acceleration;

* the derivation of a deterministic and unique photonic equation that represents wave propagation, particle motion, and energy per unit mass. Although it is possible to apply the rules of statistical mechanics to the movements of a packet of photons and thus calculate a probability density, the movement of a single photon is deterministic, with its velocity influenced by external fields and its intrinsic inertia.

The fruitful research conducted at the end of the 19th century and the beginning of the 20th century led to the emergence of new theories that complemented the laws of modern physics developed by Galileo. The theory of special relativity developed by H. Poincaré, H. Lorentz, M. Planck, A. Einstein, and others [4, 5, 6, 7] and the theory of general relativity [8, 9, 10] greatly improved our understanding of physical phenomena. Another viewpoint that proved incompatible with the theory of relativistic gravitation, quantum mechanics, led to the emergence of the wave-particle duality principle through simple experiments such as Young's double-slit experiment [11, 12, 13]. Furthermore, J.C. Maxwell's success [14] was enshrined in his equations, unifying the laws of electrostatics and magnetism. The work of P.H. Hugoniot, E. Mach, and, more recently, C. Truesdell [15, 16, 17, 18] on mechanics followed in the footsteps of Galileo and I. Newton [19] by developing general concepts on waves, fluid mechanics, and solid mechanics. These illustrious pioneering works inspired a multitude of researchers and gave rise to new theories based on these foundations, without however managing to combine these efforts into a unified theory. General relativity and quantum mechanics, in particular, remain distinct, and Newtonian mechanics has hardly evolved in a century.

The following essay does not, of course, question these different theories, which are based primarily on indisputable observations. The aim is rather to reduce the number of physical properties, variables, and units used to describe them by applying a principle dating back to the Middle Ages, the principle of parsimony, i.e., how to model a physical phenomenon by reducing the number of quantities without altering its physical meaning. At the same time, it is necessary to clarify certain ambiguous terms that have always been used, such as the celerity of light.

## 2 Celerity and velocity in a medium or in a vacuum

Consider a material medium or vacuum in which a wave propagates at a celerity $c_l$, light or, more generally, an electromagnetic wave; these waves do not carry any energy as they propagate. The medium, including a vacuum, consists of particles with or without mass, each with its own velocity $\mathbf{v}$ and carrying energy, at least kinetic energy in the case of massless particles. Thus, triality is defined here as the coexistence of a wave, a particle (e.g., a photon), and energy quantified by the potential $\phi = c_l \mathbf{v}$. Since
the celerity of the medium is not infinite, it must be considered compressible. This is true in the case of a material medium, such as a gas, but also for light. Indeed, although the celerity of light, equal to $c_0 \approx 3 \ 10^8 \text{ ms}^{-1}$, is nearly $10^6$ times that of air, the propagation of light is not infinite and the phenomenon must be considered compressible. Contrary to what is accepted for fluid flows, it is not the Mach number $M = v/c_l$ that determines whether the flow is compressible or incompressible. Discrete mechanics introduces the factor $dt \ c_l^2 \nabla \cdot v$, where $dt$ is the time interval during which the phenomenon is observed and $\nabla \cdot v$ represents the medium's ability to be compressed. Celerity is an intrinsic property of the medium, and for a fixed value that is not infinite, the movement or propagation of waves will be more compressible as the time interval decreases. Thus, the time constant with which a phenomenon is observed will determine whether this movement is compressible or incompressible. For example, water is a rather incompressible medium at common time scales of the order of a few seconds, but this same fluid must be considered compressible at much smaller time scales $dt \ll l/c_l$ where it propagates acoustic waves. Similarly, the propagation of light can only be observed with very short time intervals of the order of magnitude of $dt \approx 10^{-17} \text{ s}$. Much larger values of $dt$ can be used for observations, but not for waves, in the case of geometric optics, for example. In short, as with a compressible fluid medium, light is a wave with velocity $c_0$, the photon is a particle with velocity $v$, and its energy is carried by the product $\phi = c_0 v$. It should be noted that energy is not included in the other two concepts and is absolutely necessary. It is the acceleration $\gamma = dv/dt$, an energy per unit mass and length, that ensures the conservation of energy for the motion of fluids or light.

At this point, in order to introduce the concept of energy, it is necessary to define the quantum of energy in a formulation where mass is absent. This quantity is Planck's constant, $h$, or $h \approx 6.626 \ 10^{-34} \text{ kg m}^2 \text{ s}^{-1}$, which must be reduced from mass. In the case of the photon, since the mass is zero, we must use the definition of mass in motion,

$$m_p = \frac{2 h \nu}{c_0^2}, \qquad (1)$$

where $m_p$ is called the relativistic mass. Here we will define a Planck constant called $\tilde{h}$, reduced by the relativistic mass,

$$\tilde{h} = \frac{h}{m_p} = \frac{c_0^2}{2 \nu}, \qquad (2)$$

whose dimension is $[\tilde{h}] = m^2 s^{-1}$ in SI units, an energy per unit mass and time. Furthermore, the energy per unit mass can be identified from the frequency of the wave or the velocity of the photon,

$$\left\{ \begin{array}{l} \phi = \tilde{h} \nu, \\ \phi = c v, \end{array} \right. \qquad (3a)$$

$$\left\{ \begin{array}{l} \phi = c v, \\ \phi = \tilde{h} \nu, \end{array} \right. \qquad (3b)$$

either the relationship between velocity and frequency,

$$v = \frac{\tilde{h}}{c} \nu. \qquad (4)$$

The concept of periodic waves is not universal; there are other types of waves, notably light waves. The classic example is that of a shock wave occurring in a gas when supersonic flow transitions to subsonic flow through a discontinuity. It is therefore necessary to abandon this concept related to the periodic nature of a wave in favor of that of the time constant $\tau$, which must also be of the same order of magnitude as the time interval $dt$ in order to understand the evolution of the phenomenon under consideration. For example, to simulate the propagation of light in a vacuum, the time interval $dt$ must be much smaller than $\tau$. The relationship derived from the previous ones, (3) and (4), leads to

$$\tau = \frac{1}{2 \nu}, \qquad (5)$$

a quantity independent of the nature of the wave, whether periodic or not.

In the case of a photon whose energy per unit mass is strictly equal to its kinetic energy $v^2/2$, equal to $c_0^2/2$ when its velocity is equal to the celerity of light in the medium, we have

$$\tilde{h} = \frac{c_0^2}{2 \nu} = c_0^2 \tau = 1, \qquad (6)$$
a reduced Planck constant $\tilde{h}$ equal to unity, which is the quantum of energy per unit mass replacing the traditional Planck constant here. It should be noted that the energy per unit mass then becomes

$$ \phi = \frac{\tilde{h}}{2\tau}, \qquad (7) $$

i.e., the kinetic energy per unit mass equal to $\phi = v^2/2$. However, there is an essential difference between the theory of relativity and discrete mechanics: the latter does not postulate that the velocity of the photon is always equal to the celerity of light, $v \neq c_0$ in the general case; the velocity of the photon may be less than or greater than $c_0$. The concept of energy quantum thus loses its status as a "grain of light" whose energy is constant and minimal insofar as the time interval may be less than the time constant $dt < \tau$.

Table 1 shows the values of the main characteristics of the photon and the electron, assuming that the photon has zero mass and energy strictly equal to its kinetic energy.

<table><caption>**Table 1.** Mass of the particle $m_p$ in kg, energy $E$ in Joule, energy per unit mass $\phi = E/m_p$ in $m^2s^{-2}$, celerity $c$ in $ms^{-1}$, reduced Planck constant $\tilde{h}$ in $m^2s^{-1}$ and order of magnitude of time lapse $\tau = \tilde{h}/c^2$ in s.</caption><thead><tr><th></th><th>mass m<sub>p</sub></th><th>energy E</th><th>&phi; = E/m</th><th>celerity c</th><th>Planck h&#771;</th><th>time lapse &tau;</th></tr></thead><tbody><tr><td>photon</td><td>0</td><td>h&nu;</td><td>9 10<sup>16</sup></td><td>3 10<sup>8</sup></td><td>1</td><td>1.11 10<sup>-17</sup></td></tr><tr><td>electron</td><td>9.11 10<sup>-31</sup></td><td>8.18 10<sup>-14</sup></td><td>9 10<sup>16</sup></td><td>3 10<sup>8</sup></td><td>1</td><td>1.11 10<sup>-17</sup></td></tr></tbody></table>

In both cases, the time constant is equal to $\tau = \tilde{h}/c_0^2$ if these particles have a velocity equal to $c_0$. The study [20] is devoted to the delay introduced in the Weak Equivalence Principle (WEP) and Newton's second law; these pillars of modern mechanics contain a paradox at very low time constants that can only be resolved if the delay $\tau$ is introduced into the formulation of these laws.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Scheme illustrating the evolution of the velocity of a particle with or without mass released at time $t = 0$ in an external acceleration field $h$ for three theories: Newtonian mechanics (NM), special relativity (SR), and discrete mechanics (DM).</figcaption></figure>

In order to clarify the role of the time constant $\tau$ in the Retarded Weak Equivalence Principle (R-WEP) [20], consider a constant acceleration $h$, for example the gravitational acceleration $h = g$. A particle with or without mass initially at rest is released into this gravitational field at time $t = 0$. The question is whether its velocity and acceleration will be consistent with Newton's second law (NM) of classical mechanics and the law of special relativity (SR). In the first case, the velocity will evolve according to the law $v = gt$ and will tend toward infinity with time. In the case of special relativity, the velocity will be determined by the Lorentz transformation, $v = gt/\sqrt{1 - \frac{g^2}{c_0^2}}$, and the velocity cannot exceed the celerity of light in a vacuum under any circumstances. Figure 1 shows the variations in velocity, and in both cases, the velocity changes to $v \approx gt$ at the beginning of the observation.

Discrete mechanics [20] invalidates the behavior of these two theories for small time constants by clarifying the origin of the paradox and introducing the delay $\tau$ that resolves it. The evolution obtained
from discrete mechanics (DM) in Figure 1 schematizes the behavior of the solution for large time constants in accordance with the theory of relativity, while resolving the paradox for small time constants.

## 3 Triality wave – particle – energy

Since the beginning of the last century, it has been accepted that light has a dual nature, that of a wave and that of a particle. The wave-particle duality postulated by A. Einstein and confirmed by L. de Broglie [12, 13, 7] is expressed by the Planck-Einstein relation:

$$E = h \nu, \qquad (8)$$

where $h = 6.626 \ 10^{-34} kg \ m^2 \ s^{-1}$ is Planck's constant. This relationship expresses the fact that energy transfer occurs in discrete steps, where $E$ is the energy quantum. In his thesis, L. de Broglie introduced the concept of momentum $p$ of a particle associated with a wavelength $\lambda$ such that $\lambda = h/p$, a relationship compatible with the definition of momentum $p = E/c_0$ and $\lambda = c_0/\nu$.

However, in the general case, momentum is simply the product of mass and velocity, $p = m \nu$. This raises the problem of the zero-mass photon, which leads to a tautology. This paradox is resolved by assuming that the photon does indeed have momentum despite its zero mass. This raises the question of the true nature of the photon's energy. This can only be related to kinetic energy $E_k = m |\nu|^2/2$, which becomes $E_k = m c_0^2/2$ if we adopt the postulate of special relativity, according to which the velocity of a photon is always equal to the celerity of light.

The proposed approach consists of removing mass from the list of quantities used to model physical phenomena while retaining the notion of equivalence between mass and energy; retaining both concepts would introduce completely unnecessary redundancy. Indeed, the Weak Equivalence Principle revisited by discrete mechanics is a local principle, independent of any reference to mass, and the notion of relativistic mass is no longer necessary. In the case of the photon, only its energy has undeniable concrete meaning. For example, Planck's constant is defined by a unit of mass equal to $\hbar = h/m_p$, where $m_p$ is the energy equivalent of the photon's mass, i.e., its relativistic mass in the theory of relativity. In this case, the equivalent mass of the photon must be redefined in terms of its energy. The concept of wave-particle duality itself is open to debate; while a wave is associated with physically well-defined quantities such as energy, wavelength, or frequency, the concept of a particle is less precise. If its mass is adopted to characterize it, we then encounter the paradox associated with the mass of the photon.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Triality wave-particle-energy; $\phi$ is the energy per unit mass, $c$ is the wave velocity, and $\nu$ is the velocity of the medium or particle.</figcaption></figure>

The proposal that emerges from these remarks is to adopt a wave-particle-energy triality, an extension of classical duality; light would be a wave of constant celerity $c_0$, the particle would be associated with a velocity $\nu$, and energy would be perfectly identified both by Planck-Einstein's law, $e = \hbar \nu$, and by the law $e = c \nu$. In turn, the energy per unit mass $\phi$ must be defined by considering a motion at velocity $\nu$ imparted to a medium or particle whose velocity is equal to $c_0$; the compression energy associated with the divergence of velocity $\nabla \cdot \nu$ leads to an energy equal to $\phi = c \nu$. When the velocity of the medium or particle is equal to the wave velocity, $\nu = c_0$, we naturally find $\phi = c_0^2$. In all other cases, the relation
$\phi = c_0 v$ remains valid, including for $v \ll c_0$ and $v > c_0$. This relation is compatible with all media or particles with or without mass. The scheme in Figure 2, whose coordinates are represented by two axes $(\phi - x)$, schematizes this wave-particle-energy triality.

This representation has the advantage of prioritizing energy over mass and describing physical quantities using only two units, length and time. The distance attributed to the wave is described by its celerity, $x = ct$, but the distance traveled by a photon with velocity $v$ during a time $t$ is equal to $x = vt$. These two distances are only identical if $v = c$, which is not always true, as shown by the redshift problem [3]. This velocity $v$ can therefore only be calculated from the photon's energy, and since its potential energy is zero because it cannot accumulate energy, it can only be the kinetic energy per unit mass $e_k = |v|^2/2$, which has undeniable physical significance. Using the definition of velocity derived from frequency, the velocity of the photon is:

$$v = \sqrt{2 \tilde{h} \nu.} \qquad (9)$$

Let $v = c_0$ be a velocity equal to the celerity of light. This observation, which is compatible with the postulate of special relativity, is only verified in the case of wave-energy equilibrium, where the energy $\phi$ is the necessary and sufficient value to set the photon in motion at velocity $c_0$; this is true for the emission of a black body at temperature $T$. If an external acceleration $h$ is applied to the photon, then its velocity can be changed to values below or above the celerity of light. The reduced Planck constant is an invariant; it cannot be changed, but if the frequency is changed, the velocity is no longer equal to $c_0$. Applying acceleration to a moving photon fundamentally changes the physical phenomenon, and the celerity of light is no longer an impassable limit; the kinetic energy of the photon can then be increased beyond its equilibrium value $|v|^2/2 > c_0^2/2$.

The return to classical mechanics highlights the physical homology between Newtonian mechanics and relativistic mechanics. To illustrate this, consider the flow of a perfect gas in a nozzle consisting of a convergent section followed by a divergent section. For a perfect fluid, in the absence of viscous effects, the generating enthalpy $h_i = h + |v|^2/2$, where $h$ is the local enthalpy, remains constant along the nozzle. The velocity of the gas in the nozzle increases until it reaches the throat, where it becomes equal to $c_l$, the celerity of sound in the medium, and then increases beyond this value if the downstream pressure satisfies the required conditions. Hugoniot's theorem [15] can be used to calculate the velocity as a function of the Mach number $M = |v|/c_l$ and the local cross-sectional area of the channel. When the downstream pressure is zero, the velocity reaches a limiting value equal to:

$$v_{lim} = \sqrt{2 \tilde{h}_i}, \qquad (10)$$

we say that gas expansion is complete. This is not just an analogy; the concepts and equations of discrete mechanics apply equally well to fluid flow and to the propagation of light. In the absence of external acceleration, the velocity of the photon is entirely defined by its initial energy and $v = c_0$. In this case, the notion of cross section no longer makes sense, and it is as if the convergent nozzle were followed by a duct of constant cross section; this is the notion of sonic blockage observed in an orifice of constant cross section separating a pressurized gas tank from the atmosphere.

The wave-particle duality does not precisely qualify the presence of these concepts in space and time. In quantum mechanics, the wave and the motion of particles cannot be observed simultaneously, because observing one obscures the other. The proposed wave-particle-energy triality provides details on their coexistence. In fluids, as in light, the observation of waves is only possible if the time constant, the time interval $dt$ of discrete mechanics, is compatible with the characteristic time, the frequency $\nu$ of a periodic wave. The time interval $dt$ must be of the order of magnitude $1/\nu$ or smaller in order to correctly capture the unstable nature of the wave, for example $dt \ll \lambda/c$ if $\lambda = c/\nu$ is the wavelength considered. It should be noted that the periodic nature of waves must be abandoned because it is generally not true, for example in the case of a shock wave in fluid mechanics or a sudden change in the refractive index of light. The particle introduces a spatial limitation; it can only be observed at very small scales compatible with those of quantum mechanics. Observation of its motion is therefore only possible if the length $l = v dt$ is compatible with the means of measurement. If $v \approx c$, it would then be possible to observe the wave and the particle simultaneously; what prevents this is the probabilistic nature of quantum mechanics, or rather the fact that a packet of photons characterized by a probability density prevails. To restore the strict duality between the photon and the wave, it is necessary to return to a deterministic aspect of its motion. Energy, on the other hand, has a much broader meaning that does not depend on space and time scales. Photons and the Sun have different energies, but they interact through gravitational acceleration, which can also be interpreted as an exchange of energy.
The deterministic nature of the motion of a fluid or light, considered both as a wave and as a particle, can be guaranteed by a single law of motion. This law must be valid for all time scales and all spatial scales. It must also possess specific properties, in particular that of completeness, representing all the physical phenomena on which it is supposed to be based. The triality symbolized by the triplet (c, v, φ) is the appropriate concept for achieving the exhaustiveness of the phenomenon in an approach based on the principle of parsimony.

# 4 Unified discrete law of motion

The completeness of a law requires that it be representative at all scales of time and space. It must also be consistent with proven theories such as those of special and general relativity established from the Lorentz transformation. For the latter, as for Newton's second law, based on the adoption of the Weak Equivalence Principle (WEP), behavior at small time scales implies an inconsistency in the form of a paradox: the energy required to achieve strict equality between proper acceleration and imposed acceleration is infinite. In order to maintain consistency with the discrete law of motion, the time delay is rewritten in the form of a ratio introducing a parameter $\tilde{h}$ [20], equivalent to a diffusion coefficient, and the wave velocity c. The equation for proper acceleration becomes,

$$ \frac{\tilde{h}}{c^2} \frac{d\gamma}{dt} + \gamma = h, \qquad (11) $$

where h represents the sum of the accelerations imposed in the form of external fields. The quantity $\tilde{h}$, the reduced Planck constant, is equal to unity. Newton's second law, initially modified by the elimination of mass, then becomes a hyperbolic differential equation (11) that eliminates the paradox at small time scales and restores the classical law at larger scales.

The law of motion of discrete mechanics [3] thus becomes itself an approximation of reality at small time scales, and it is necessary to complete it by introducing time delay at very small time scales. The hyperbolic nature of the equation is reinforced, and it reads as follows:

$$ \frac{\tilde{h}}{c^2} \frac{\partial^2 v}{\partial t^2} + \frac{\partial v}{\partial t} = -\nabla \left( \phi^o + \frac{1}{2} |v|^2 - c_l^2 dt \nabla \cdot v \right) + \nabla \otimes \left( \psi^o + \frac{1}{2} |v|^2 n - c_r^2 dt \nabla \times v \right) + h, \quad (12) $$

where the first term on the left-hand side reflects the hyperbolic behavior of the equation on very small time scales. In its form, this first term is similar to that introduced by C. Cattaneo [21] to eliminate the paradox introduced by Fourier's law at very low time constants, which presupposes instantaneous heat diffusion. For larger time constants, the behavior is consistent with that of special and general relativity and, of course, with that of the equation of motion in discrete mechanics [22].

The new law of discrete mechanics (12) therefore has legitimacy and consistency on all time and space scales when the $\tau$ delay is introduced for the case of light. At the same time, this equation becomes an alternative to the other equations of mechanics, notably the Navier-Lamé equation for solids or the Navier-Stokes equation for viscous fluid flows. However, it remains to be seen whether the $\tau$ delay is compatible with longitudinal and transverse propagation phenomena for solids and fluids. The Navier-Stokes equations also present a paradox at small time scales when a wall in contact with a viscous fluid is translated with non-zero velocity, the viscous stress is then infinite. Generally speaking, no physical phenomenon can lead to paradoxes of this type, and the legitimacy of the fundamental equations governing them must be carefully re-examined.

# 5 Conclusion

The propagation of light, or the movement of photons, is comparable to that of a fluid, which propagates acoustic waves and is also the source of motion. This is not a simple analogy, but a physical homology based on one and the same law of motion (12) that forges the same solutions for both concepts. This law of motion is just as representative of wave propagation at small time scales as it is of the motion of particles with or without mass, or of a material medium at larger time constants. It is based on the conservation of energy per unit time and length, i.e. acceleration. The wave-particle duality of
quantum mechanics is generalized here by integrating this third quantity, energy, to construct the notion of wave-particle-energy triality defined by the triplet ($c, v, \phi$).

The law of discrete motion has been widely validated by numerous reference cases for fluid and solid mechanics, heat transfer and flow in porous media. It has the property of being relativistic, but does not require immediate recourse to the Lorentz transformation, which it satisfies; here again, the test cases of special and general relativity have been successfully passed. The integration of quantum mechanical concepts - quantization - has not yet been achieved. However, the law of motion is presented as a photon equation, where the triple representation it conceals offers hope for the simulation of quantum-mechanical observations.

# Appendix - Refraction of a polarized monochromatic wave

Acoustic and light waves are of the same nature and, like the others, are described by Maxwell's equations. Other phenomena observed in other fields, such as mechanics, are not simple analogies but physical homologies described by a single law [22, 3]. The propagation of waves of very different frequencies can be described on time scales adapted to the physics of the phenomena considered, based on the equations of mechanics. For example, the propagation of ultrasonic waves in water can be modeled and simulated by the Navier-Stokes equations with time scales $dt$ of the order of a few milliseconds [23, 24, 25, 26]. These equations, like those of discrete mechanics, also allow larger time steps for steady solutions.

In the general case, radiation is transmitted, absorbed, refracted and scattered. The factors of transmission, absorption and reflection, whose sum is equal to unity, allow us to express the conservation of energy, equal to the energy of the incident radiation. These various geometrical optics mechanisms have their equivalent in mechanics, where an acoustic or tidal wave undergoes the same spatial and temporal evolutions.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Refraction at the free horizontal interface between air and water of refractive index $n = 4/3$ for an incident ray inclined at $53.13^\circ$; (a) for a direct simulation of light-wave propagation and (b) velocity field of photons refracted by the interface where $|v| = c_0$ upstream and $|v| = 3/4 c_0$ downstream of the interface.</figcaption></figure>

The example given here is that of incident monochromatic radiation of frequency $v$ and wavelength $\lambda = c/v$ on an interface separating two media of different indices, for example air and water, whose interface is flat and horizontal. The velocities of the two media are fixed by the Snell-Descartes law, $n = c_2/c_1$ where $c_1 \approx 1$ is the wave celerity in air and $c_2 = 4/3$ the wave celerity in water. When the incident ray is received by the interface at angle $\theta_1$, it is refracted at another angle into the second medium, but another part is reflected back towards the first medium without entering the second. There
is a special case where the light is not reflected and enters the second medium completely: this is the Brewster condition given by the law,

$$n_1 \sin \theta_1 = n_2 \sin \theta_2, \qquad (13)$$

and, using the expression $\sin(\theta_2) = \sin(\pi/2 - \theta_1) = \cos\theta_1$, Brewster's angle is given by the law,

$$\theta_1 = \arctan \left( \frac{n_2}{n_1} \right), \qquad (14)$$

i.e. $\theta_1 = 53.13010234^{\circ}$ and $\theta_2 = 36.86989766^{\circ}$ for the air/water pair.

Figure 3 shows the result of a direct simulation of the refraction of a longitudinal wave of frequency $\nu$ under Brewster incidence in the case of an air/water interface where $n_1/n_2 = 3/4$. This qualitative example shows the propagation of the wave in the upstream zone of the interface and its refraction by the water in the downstream zone. The discrete law of motion (12) was solved in the presence of all terms, including inertia terms, even though these are negligible under simulation conditions. When much longer time lapses are used, for example $dt \approx 1$ s, wave propagation is no longer accessible because $dt \gg \tau$ where $\tau = 1.11 \ 10^{-17}$ s is the time constant not to be exceeded for light wave simulation. Two different meshes were used for the simulations, one with $n_c = 32768$ quadrangles and the other with $n_c = 10587$ regular triangles and $n_v = 5464$ vertices.

Interpretation from a mechanical point of view involves representing photon velocities on time scales $dt$ larger than those used for wave propagation. In this case, the photon velocity remains equal to the velocity of the medium, and in the upstream zone is equal to $v_1 = c_1$, while in the downstream zone it becomes $v_2 = c_2$. Note that conservation of mass is strictly assured, since the flow rate $v_1 \ell_1 = v_2 \ell_2$ where $\ell$ is the width of each zone. In the case of an incidence equal to Brewster's, the reflected fraction is zero, which is precisely why the motion is incompressible; otherwise, part of the photon flux returns to the first medium. The flow is then divergence-free and curl-free, except at the interface where the photon flow undergoes a rotation of an angle $\theta = \theta_1 - \theta_2 = 16.23^{\circ}$. Consider the interface $\Sigma$ between the two media and regard it as a layer of length $L$ and surface area $S$ encompassing $\Sigma$. Stokes' theorem,

$$\int_S \nabla \times \mathbf{v} \, ds = \int_{\Sigma} \mathbf{v} \cdot \mathbf{t} \, dl, \qquad (15)$$

leads to a non-zero result, the motion is effectively curl-free in both media, except at the interface between the two media with different refractive indices. In this case, the velocity is equal to the celerity of light, $v_1 = c_1$ and $v_2 = c_2$, and the motion is characterized by a velocity jump $[[\mathbf{v} \cdot \mathbf{t}]] = (v_1 - v_2) \cdot \mathbf{t}$ on the interface. The interface behaves like a shock discontinuity, with light passing through the interface. In fact, Snell-Descartes' law represents the continuity of the normal velocity at the interface, and the velocity jump $[[\mathbf{v} \cdot \mathbf{t}]]$ is the second boundary condition relative to the direction tangent to the interface $\mathbf{t}$; they are written as follows,

$$\begin{cases} n_1 \sin \theta_1 = n_2 \sin \theta_2 & \text{Snell - Descartes,} \qquad (16a) \\ c_1 \sin \theta_1 - c_2 \sin \theta_2 = [[\mathbf{v} \cdot \mathbf{t}}], & (16b) \end{cases}$$

and the law of motion itself completes the conservation of energy in the domain and at the boundary conditions. The latter can be applied explicitly for each of the two domains or implicitly using Stokes' theorem (15) to define a vector potential $\psi_b = \nabla \times \mathbf{v}$ integrated into the law of motion itself. In fact, all types of boundary conditions, including surface discontinuities or shock waves, can be incorporated into the discrete law [3]. This phenomenon is similar to that observed for oblique shock waves described by Landau and Lifshitz, e.g. [27]. For shock waves in gases, the angle of refraction is fixed by the upstream Mach number. The flow of matter passes through the shockwave at an angle defined by oblique shockwave theory.

This qualitative example shows that it is possible to simulate apparently very different physical phenomena, such as steady flows or light propagation, using a single law of motion. The main parameter remains the time lapse during which these phenomena are observed.

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