Previous articles concerning EMST [7, 8] were based on the assumption, that the value of v _4D is constant, or more precisely, that v _4D = c, where c is the speed of light in a vacuum. For the theory of gravitation, it is necessary to abandon this assumption. Let us now assume that the speed v _4D is locally and temporally variable in the stationary coordinate system S. Each point in space E₄-B (x, y, z, w) at coordinate time t can be assigned a value v _4D = f(x, y, z, w, t), where f is a continuous differentiable function. The quantity v _4D is a scalar field in space E₄-B.

The rate of flow of coordinate time t is directly related to the value of v _4D . This clearly arises from the metric (Equation 6). The time δt required to move a particle-wave by given values δx, δy, δz, δw is the greater, the lower the value of v _4D is.

The same effect also affects the flow of proper time, and thus also the running of clocks. It follows from relation (Equation 7) that the value of δτ increases as v _4D decreases. Since we always measure proper time by the number of repetitions of a regularly recurring event (e.g., the oscillation of a quartz crystal, the frequency of radiation coming from the electron shell of cesium, etc.), there will be fewer repetitions when δτ is larger, and the clock will show a smaller increase in time. This phenomenon affects all clocks regardless of their design – the fact was first discussed by A. Einstein in [9]. In the terminology of Einstein’s theory of relativity, this is a “universal effect” independent of the design of the clock.

Clocks affected only by universal effects, i.e., not affected by the specific design of the particular clock, will be referred to as “ideal”. All further considerations will be based on the use of ideal clocks for time measurements.

As an example of the ideal clock, we can use the so-called particle clock. It is a particle-wave moving cyclically in dimension w. One cycle of the particle-wave is equal to one “tick” of the clock. The time between ticks is given by the length of the particle-wave’s path and the speed of its motion. It is clear that the speed of the clock’s “ticking” is directly proportional to the value of v _4D (for an invariant path), (Figure 1).

To consider the motion of clock in E₃, we must replace v _4D in relation (Equation 7) by v _w ( v w = v 4 D 2 − v 3 D 2 , see Equation 1). The relation then takes the form δ τ = δ w / v w . Alternatively, we can consider the prolongation of the particle-wave trajectory within a single cycle. In both cases, the result will be the same – prolongation of the duration of a single tick means slower “ticking” rate.

The slowing down of the motion of particle-waves in E₄-B and the slowing down of the clocks have the same origin and act against each other. If, in some part of space, the value of v _4D is lower than c by, say, 1%, the particle-wave will travel the given path in objectively longer time (an increase of 1%), but the lower rate of the clock (a decrease of 1%) will completely compensate the effect. Therefore, the change in v _4D speed will not be noticeable locally. The ratio of the path and the measured time will remain equal to c.

The change in speed v _4D cannot be detected locally by any known method. It is the speed at which the fundamental building blocks of matter (i.e., the particle-waves) move, and therefore it affects all physical processes. It affects the speed of quarks, photons, and leptons, it determines the speed of electrons in the electron shells of atoms and the frequencies of their emitted photons. Since the time component of all ongoing processes is affected, as well as all related physical quantities (frequency, speed, acceleration, force, etc.), it is not possible to detect the change in v _4D by their measurement and comparison.

However, the change in v _4D can be detected at greater distances by comparison of the clock rates of a pair of ideal clocks 1 and 2. If these are located in areas with different v _4D , one clock will run slower than the other. The mutual ratio is given by C l o c k R a t e 1 C l o c k R a t e 2 = v 4 D , 1 v 4 D , 2 (8) where ClockRate is the number of “ticks” per selected time interval. By comparison of the mutual tick rates of remote clocks, regional variations in v _4D can be determined. If an extensive network of ideal, mutually non-moving clocks were available, deviations in ClockRate would indicate regional variations in the value of v _4D , (Figure 2).

In the following, we will assume that material bodies create a zone with a spatially variable value of v _4D around themselves – we will call this zone the gravitational field.

Let us consider a particle-wave, such as a photon, in the gravitational field. We will determine its frequency f using a clock (at rest) located in its immediate vicinity. If the photon moves freely downward in the gravitational field, its potential energy E _p will decrease, and its kinetic energy E _k will increase. According to the law of conservation of energy, its total energy E remains constant. The following generally applies to every particle-wave: E = E 0 + E p + E k = c o n s t (9)

The quantity E ₀ is the rest energy of a particle-wave, which is zero (negligible) in the case of a photon. It should be noted that EMST does not distinguish between massive and massless particles and assigns potential energy to both. Its origin will be explained in Sect. 2.3.

Since the total energy E = h f remains unchanged, f also remains unchanged. The frequency of a free-falling particle-wave does not change. However, its wavelength decreases together with decrease of v _4D (λ _4D = v _4D /f). The decrease in λ _4D is accompanied by an increase in momentum p 4 D = h / λ 4 D , see (Equation 4), and an increase in mass m = p 4 D / v 4 D = h f / v 4 D 2 (10)

Although the frequency of a particle-wave does not change during free fall, the dependence of ClockRate on the local value of v _4D causes the frequency determined by different reference clocks is different. Phenomena such as gravitational redshift and blueshift do not reflect a change in the energy of the particle-wave, but rather a change in the rate (ClockRate) of the reference clocks. For instance, a clock in interstellar space runs faster than an imaginary clock on the surface of a massive star. In other words, time on the surface of the star flows more slowly, and all physical processes, including photon emission, are also slower.

The dependence of the particle-wave frequency on the value of v _4D at the location of the reference clock is not surprising. It expresses the relative nature of potential energy. It is always related to a specific reference level. The EMST shows that the reference level is given by the local value of v _4D . A change in the reference level corresponds to the relocation of the reference clock to a location with a different v _4D . The result is a change in the frequency of the observed particle-wave, and thus its energy.

Let us consider a space and a pair of identical particle-waves A and B (e.g., electrons) arbitrarily distributed in it, as well as a pair of identical clocks 1 and 2. The particle-waves and clocks are at rest (v _3D = 0). The particle-waves would have the same energy if they were located in areas of the same v _4D , but the value of v _4D is different in the given locations. The following applies: v 4 D , B < v 4 D , A , v 4 D , 2 < v 4 D , 1 (Figure 3).

Let us denote the frequency of particle-wave A relative to clock 1 as f _A,1 , and relative to clock 2 as f _A,2 . The following applies: E A , 2 E A , 1 = h   f A , 2 h   f A , 1 = C l o c k R a t e 1 C l o c k R a t e 2 = v 4 D , 1 v 4 D , 2

It is clear from the formula that the frequency, and therefore also the energy of the particle-wave is greater relative to the slower clock 2.

A similar situation arises when comparing the frequencies of particle-waves A and B. If we relate them to clock 1, we get f A , 1 f B , 1 = v 4 D , A v 4 D , B (11) i.e., the particle-wave in the region of greater v _4D has a higher frequency and thus also higher energy: E A , 1 E B , 1 = h   f A , 1 h   f B , 1 = v 4 D , A v 4 D , B (12)

When using clock 2, we obtain different frequencies, but their ratio will be the same. This shows that identical particle-waves (e.g., free electrons at rest) have different energies depending on the local value of v _4D .

Particle-waves A and B have different frequencies, but their wavelength λ 4 D = v 4 D / f is the same, see (Equation 11). This corresponds to the same value of the momentum p _4D and all of its components. It should be noted that p _4D is a vector in E₄ with components p _x , p _y , p _z , and p _w , which correspond to the projections of the wavelength λ _4D in the direction of the coordinate axes. For example, p _x = h/λ _x , see [8].

The wavelengths of identical particle-waves do not depend on the local value of v _4D . This fact can be used for distance measurement. The length of rigid measuring rods is determined by the distance between individual atoms in the crystal lattice. It can be reasonably assumed that these are fixed multiples of the wavelengths of electrons in the electron shells of atoms. That is, the dimensions of the rod do not depend on the local v _4D . The results of length measurements using rods will match the results of length measurements using light if local clocks are used for the transit time determination.

Using relations (Equation 10) and (Equation 12), we can write: m A m B = h   f A v 4 D , A 2 h   f B v 4 D , B 2 = v 4 D , B v 4 D , A (13)

Particle-wave B has a higher mass than particle-wave A despite having lower energy.

If the distribution of the v _4D is constant in time, it can be used to define a potential field in space – the gravity potential V. This term includes both gravitational potential (caused by the gravitational effects of matter) and inertial potential (caused by the acceleration of the reference frame); if both are present, it is their sum.

The gravity potential creates a scalar field in space E₃ (the compactified fourth dimension can be neglected when describing gravity). The function V = V(x, y, z) is continuous and differentiable.

The gravity potential expresses the potential energy of a body of unit mass relative to a selected reference level ( V = E p / m ). The potential decreases downward and increases upward. If the reference level of the potential is located outside the gravity field, the potential anywhere inside the field is negative.

Let us consider the potential of particle-wave A relative to clock 1 from the previous case. The energy of any particle-wave is given by E = m   v 4 D 2 = p 4 D v 4 D . The momentum p _4D is independent of the v _4D , so the infinitesimal change of E with v _4D can be written as d E = p 4 D   d v 4 D = m   v 4 D   d v 4 D . The value dE represents the change in the potential energy of the particle-wave, i.e., dE = dE _p . Therefore the change in potential is given by d V = d E p / m = v 4 D   d v 4 D .

The potential of particle-wave A relative to clock 1 is V A , 1 = ∫ 1 A v 4 D   d v 4 D = v 4 D 2 2 1 A + C = v 4 D , A 2 2 − v 4 D , 1 2 2 + C = − 1 2 v 4 D , 1 2 1 − v 4 D , A 2 v 4 D , 1 2 + C

The integration constant C is determined by the choice of value of the potential at the location of the reference clock. The logical choice is V ₁ = 0, which leads to C = 0.

The gravity potential (of particle-wave) at point A relative to reference clock 1 located outside the gravity field ( v 4 D , 1 = c ) is V A = − 1 2 c 2 1 − v 4 D , A 2 c 2 (14)

This defines the functional dependence of V on the v _4D .

The inverse relationship gives the ratio of v _4D to c, v 4 D , A c = 1 + 2 V A c 2 (15) which can be modified for the mutual ratio of clock rates C l o c k R a t e A C l o c k R a t e 1 = 1 + 2 V A c 2 (16)

It can also be written in the form known from the GTR d τ = d t 1 + 2 V A c 2 (17) which gives the increment of proper time τ at a point with gravity potential V _A relative to the independently flowing coordinate time t outside the gravity field. It can be seen that the EMST result is identical to the GTR result.

Regardless of the location of the reference clock, v _4D = c always applies in its vicinity. Equations 14–17 are therefore valid not only for reference clocks located outside the field, but for any location of the clock.

In classical mechanics, the gradient of potential is acceleration g r a d   V = ∂ V ∂ x , ∂ V ∂ y , ∂ V ∂ z = − a x , − a y , − a z = − a → (18)

In the EMST, any accelerated motion is a consequence of the variable value of v_4D. The above-mentioned relationship between acceleration and potential must, therefore, be verified and correctly interpreted. It will be shown that the vector a → does not represent kinematic acceleration in some situations. I will refer to it as the “vector a → ” (instead of acceleration a → in cases where confusion might arise.

The relationship between the potential V and the velocity v _4D was derived above (Equation 15). It yields d v 4 D d V = d c 1 + 2 V c 2 d V = c 2 1 1 + 2 V c 2 2 c 2 = 1 v 4 D (19) and so g r a d   v 4 D = ∂ v 4 D ∂ x , ∂ v 4 D ∂ y , ∂ v 4 D ∂ z = ∂ v 4 D ∂ V g r a d   V = 1 v 4 D g r a d   V = − a → v 4 D

The dependence of vector a → on the velocity v _4D is given by the formula a → = − v 4 D   g r a d   v 4 D (20)

For the analysis of particle-wave motion, it is convenient to decompose vector a → into two components – a component parallel to the direction of particle-wave motion a ∥ → and a component perpendicular to it a ⊥ → . At the same time, we introduce an auxiliary coordinate system with its origin at the center of the particle-wave and axes ξ and η (see Figure 4a).

In the coordinate system ξ, η, the vector a → can be written as a → = a a ∥ , a ⊥ , where a ∥ = − v 4 D ∂ v 4 D ∂ ξ , a ⊥ = − v 4 D ∂ v 4 D ∂ η (21)

The relationship of these components to the vector a → is given by formulas: a ∥ = a → cos   ε , a ⊥ = a →   sin   ε (22)

The angle ε between the 4D direction of particle-wave motion and the direction of vector a → can be calculated using formula: cos ⁡ ε = sin ⁡ α ⁡ cos ⁡ γ (23)

Angle γ is the analogue to ε in ordinary space E₃, while angle α is the inclination of the 4D direction of particle-wave motion relative to axis w (Figure 4b). It holds that sin α = v_3D/v_4D.

It might seem that the value a ∥ affects the speed of the particle-wave and the value a ⊥ affects its direction, but only the second part is true. The component a ∥ has no effect on the particle-wave speed. This is because the particle-wave is an undulation of the transmission medium and its speed is determined solely by the properties of this medium, specifically the value of v _4D – see (Equation 15). It is obvious that, with decreasing value of V (downward movement), the value of v _4D also decreases. The quantity v _4D is a 4D speed, which is not directly observable for most particle-waves, so we do not register the phenomenon. The v _4D is identical to the commonly observed 3D speed only for photons (and other particle-waves with zero rest mass). It can be claimed that a photon falling into a gravitational well slows down. However, this paradox is not new. The decrease in the speed of light in a gravitational field (relative to a reference clock located outside the field) is generally known and accepted within GTR.

Regardless of the decrease in speed, the 4D momentum and mass of the particle-wave increase as it falls into the gravitational well (see Sect. 2.2 above).

The transverse component a ⊥ → is a consequence of the change of v _4D in the transverse direction. The particle-wave has non-zero dimensions and its individual parts move at different speeds - depending on the local value of v _4D (Figure 5). One side of the wave moves faster than the other, causing wavefronts to twist. The direction of the wave bends towards the area of lower v _4D . The situation is analogous to atmospheric refraction of light, where different speeds of light (different refractive indices) cause its path to curve. This is a manifestation of Fermat’s principle, which states that waves do not propagate along the shortest path, but along the path with the shortest time.

The radius of bending can be determined by the following consideration: The ratio of the path on the outer side of the arc to the path of the center of the particle-wave is equal to the ratio of local 4D speeds: l + d l / l = v 4 D + d v 4 D / v 4 D .

The ratio of distances can be replaced by the ratio of radii r + d r / r , where the increment of radius dr is equal to the negative increment of dη: l + d l / l = r + d r / r = r − d η / r .

It can be written − d η / r = d v 4 D / v 4 D , hence 1 r = − d v 4 D d η 1 v 4 D (24)

The radius depends on the rate of decrease in v _4D (i.e., on d v 4 D / d η ). By combining relations (Equation 24) and (Equation 21), it can be written r = v 4 D 2 a ⊥   ⇒   a ⊥ = v 4 D 2 r (25)

The relationship (Equation 25) shows that the transverse component of the vector a → makes the particle-wave move in a circle with radius r. Centripetal acceleration of the same magnitude would have the same effect. The variable value of v _4D curves the trajectory of the particle-wave the same way as it would be curved by the acceleration a ⊥ caused by some acting force (inertial, gravitational, etc.).

The deviation from the straight line is equal to η = ξ ² /2r, which, after substituting r from (Equation 25) and ξ = v _4D t, leads to η = v 4 D 2 t 2 a ⊥ 2 v 4 D 2 = 1 2 a ⊥ t 2 .

It can be seen that the quantity a ⊥ is the actual kinematic acceleration with which the particle-wave moves in space E₃.

In non-relativistic cases, the difference between a ⊥ → and a   → can be neglected, since particle-waves moving at non-relativistic 3D speeds (v _3D << v _4D ) perform their 4D motion primarily in the w dimension. The direction of their motion is therefore practically perpendicular to any acceleration vector a → = a x , a y , a z . In these cases, the last formula can be rewritten into the familiar form η = 1 2 a   t 2 (26)

It expresses the change in position of the particle-wave in E₃ under the influence of acceleration a = − g r a d   V . The Equation 26 applies to all non-relativistically moving particles of matter, as well as composite bodies. The change in the state of motion of a body is independent of its mass, 3D velocity, direction of motion, or composition. And, as will be shown later, it does not depend on the way in which the gravity field was created.

In relativistic cases, the kinematic acceleration a ⊥ is given by Equations 22, 23. The term sin α in Equation 23 acts equivalently as a relativistic mass increase in GTR.

The derivation of gravitational acceleration given above applies to any particle-wave (photon, neutrino, electron, muon, …) moving in any direction and at any (permissible) 3D velocity. It should be noted that the accelerated motion of particle-waves in the gravity field is not the result of acceleration or force acting at a distance, but rather the result of different values of v _4D in the immediate vicinity of the particle-wave. Acceleration or force is only a useful mathematical aid, not the real cause of the change in the particle’s state of motion. Moreover, the usefulness of this aid is limited – see the component a ∥ mentioned above.

Once it has been clarified that the cause of the curvature of particle-waves trajectories is the variable value of v _4D , the question inevitably arises: What is the cause of this variability? It is useful to follow Einstein’s approach here and focus on the similarity between gravitational and inertial forces.

Let us consider space E₄-B filled with a homogeneous isotropic medium in which all particle-waves move at the same 4D speed, namely, speed c. In this space, there is a flat rotating disk that rotates at an angular velocity ω relative to the stationary coordinate system S (x, y, z). The coordinate system S′ (x', y', z') is rigidly connected to the disk. The axis of rotation is the common axis z ≡ z'.

Any point on the disk moves relative to S at a speed v s = r s   ω , where r _s is the distance from the z-axis, r s = x 2 + y 2 . The non-zero value of v _s causes the speed of particle-waves relative to the rotating system S′ to be lower than relative to the system S. This refers to the average two-way speed. The formula v 4 D ′ = c 2 − v s 2 holds. It obviously holds for cases where v s ⊥ v 4 D ′ , i.e., in cases where the particle-wave motion is perpendicular to v _s . The validity for other directions of particle-wave motion is not obvious, but it also holds for them. A detailed justification can be found in [7].

The formula v 4 D ′ = c 2 − ω 2 r s 2 shows that in the rotating coordinate system, the velocity v _4D ’ is variable. It can be speculated that this variability causes the curvature of the trajectories of free particle-waves, and such curvature is identical to the effect of centrifugal acceleration a c = ω 2 r s .

Derivation of the above formula for v 4 D ′ with respect to radius r _s gives d v 4 D ′ d r s = − 2 ω 2 r s 2 c 2 − ω 2 r s 2 = − ω 2 r s v 4 D ′

By substituting into (Equation 21) and considering d v 4 D ′ d r s ≡ d v 4 D d η (these are identical variables with different notation), we obtain the formula a ⊥ ′ = v 4 D ′   ω 2 r s v 4 D ′ = ω 2 r s

Clearly, a c = a ⊥ ′ applies, i.e., the acceleration a c caused by the rotation of the disk is equal to the acceleration a ⊥ ′ of a particle-wave caused by the local change in v _4D '.

It is important to note that the change in v _4D ’ with the spatial position, and thus also the acceleration a ⊥ ′ , are phenomena tied to the rotating system S'. From the perspective of the stationary system S, the v _4D is the same everywhere (v _4D = c) and particle-waves move without acceleration (inertial motion). That is, the spatial variability of v 4 D ′ is a consequence of the motion of the coordinate system S′ relative to S.

The acceleration a ⊥ ′ can also be determined in another way, namely, by means of the function V' = f(x', y', z') describing the distribution of potential in the system S′ and subsequently by the relationship between acceleration and gradient of V’. For a rotating disk V ′ = − 1 2 ω 2 r s 2 applies. In the cylindrical coordinates (r, φ, z), the parameters of the acceleration vector can be determined from the potential V': a → a r , a φ , a z = − g r a d   V ′ = − ∂ V ′ ∂ r , 1 r ∂ V ′ ∂ φ , ∂ V ′ ∂ z = ω 2 r s , 0 , 0 i.e., the acceleration has a radial direction and is equal to a c . We have arrived at the same result as above, but the given procedure is usually easier.

The advantage of utilizing potential can be demonstrated in another typical situation – linear accelerated motion. Consider a system S′ moving with acceleration relative to the stationary system S. Assume constant acceleration a _S in the direction of the common axis x ≡ x'.

In this case, it is difficult to directly determine the change in v _4D ’ with position ( d v 4 D ′ / d x ′ ). This is because of v _4D ’s dependence on time and the simultaneous impossibility of mutual synchronization of remote clocks. Their time shift changes over time. However, it is possible to determine the ratio of the v _4D ’ flow at different points of S′ by comparison of the clocks (ClockRate) located there (Equation 8) or to calculate it using the gravity potential – see (Equation 15). The potential in this case is given by V ′ = a s   x ', i.e., ∂ V ′ / ∂ x ′ = a s . Using (Equation 19), we obtain ∂ v 4 D ′ / ∂ x ′ = a s / v 4 D ′ . The known change in v _4D ’ with x' allows us to calculate the curvature of the path of a free particle-wave (Equation 24). Its acceleration relative to S′ can be determined using (Equation 20) or (Equation 18): a → a x , a y , a z = − g r a d   V ′ = − ∂ V ′ ∂ x , ∂ V ′ ∂ y , ∂ V ′ ∂ z = − a s , 0 , 0

The acceleration a → with which the free particle-wave moves relative to S′ is, except for the sign, the same as the acceleration of S′ relative to S.

It has been shown that, in non-inertial systems, the speed v _4D is a function of position. This applies to both linear accelerated as well as rotational motion. Since any motion of coordinate systems relative to each other can be understood as a combination of translations and rotations, it can be argued that v _4D is a function of position in all non-inertial systems. Its change with position subsequently affects the direction of motion of particle-waves.

It is important to note that the acceleration of particle-waves caused by the variable value of v _4D is exactly opposite to the acceleration of S′ relative to S. The inertia of matter is not a strange opposition of matter change its state of motion, but a natural property of waves (i.e., particle-waves) to move along the trajectory with the shortest time (Fermat’s principle).

Since it has been shown that the spatially variable v _4D is the cause of inertial acceleration, it can be expected to be the cause of gravitational acceleration as well. Logically, the question arises: What is the cause of the variability of v _4D in the case of gravitation?

Based on the analogy between inertia and gravitation, one possibility is that the cause of the variable v _4D is some motion of the reference frame relative to the stationary frame. However, the specific character of gravitation (e.g., the existence of a central gravitational field) does not allow us to find an adequate type of motion within the E₃ or E₄-B space (for global Euclidean systems). Considering a hypothetical fifth spatial dimension, a suitable type of motion can be found – it is the rotation of E₄-B relative to the stationary space E₅, where areas of lower potential are further from the axis of rotation (larger radius r _V ) than areas of higher potential. For the speed v _v in the additional 5^th dimension, v V = r V   Ω applies. As a result, the speed v _4D is lower in areas of lower potential, since v 4 D ′ = c 2 − v v 2 applies. This model can be optimized to correspond to actual gravity, but the question is whether it corresponds to reality in general. It adds another large, yet undetected spatial dimension and, at the same time, assumes the existence of a yet undetected angular velocity Ω , at which the entire universe (space E₄-B) rotates. However, the main disadvantage of this model is that it ignores the fundamental importance of mass as a source of gravitation.

Since Newton’s time, it has been known that matter is the source of gravitation. The cause of local variability of v _4D should therefore be the presence of matter, i.e., of particle-waves. These are packets of energy moving through space in the form of waves. Each particle-wave is, by its very nature, a source of motion of the transmission medium. The motion of the medium is cyclic – after short time, the transmission medium returns to its initial position – it does not shift anywhere. However, its average speed is non-zero.

The average speed of the transmission medium through which an undulation passes depends on the amplitude, frequency, waves shape, and type of undulation (sinusoidal, helical, etc.). Of these quantities, only the frequency is known with certainty (E = h f). Therefore, a direct calculation of the average speed is not possible without further assumptions.

However, a different approach can be used: It is known that the energy of any wave motion consists of the elastic energy of the transmission medium and its kinetic energy. For example, in the case of waves on a string, the ratio of these energies is 1:1, i.e., half of the energy is kinetic and half is elastic [10, page 34]. For other types of wave motion, this ratio may be different. For our purposes, it suffices to assume that the wave motion of all particle-waves has the same character, and so the ratio of elastic and kinetic energy is the same. In that case, the ratio of the kinetic energies of particle-waves A and B will be equal to the ratio of their total energies. Since kinetic energy is proportional to the square of the velocity of the transmission medium, E k = 1 / 2 m v m 2 , the ratio of the mean square velocities v _m of the transmission medium as a result of the wave motion of particle-waves with energies E _A and E _B will be v m , A v m , B = E A E B

Based on v 4 D = c 2 − v m 2 and (Equation 14) it holds V A = − 1 2 v m , A 2 (27)

i.e., the ratio of the potentials is proportional to the ratio of the energies of the particle-waves V A V B = E A E B

The formula shows that particle-waves with n times greater energy create a gravitational field with n times greater potential.

From the known property of gravitational potential (Newton’s relation V = – GM/r) and relation (Equation 27), it follows that v m = 2 G M r (28)

i.e., v _m decreases with the growing distance r from the center of the particle-wave. It is a natural property of waves that they are not sharply outlined. Instead, they oscillate the surrounding medium, propagating not only forward, but also, to a certain extent, sideways. This also applies to particle-waves.

The motion of the transmission medium caused by a particle-wave is its motion relative to the stationary coordinate system S. This motion affects the local value of v _4D ( v 4 D = c 2 − v m 2 ), slows down other particle-waves, bends their trajectories, and simultaneously slows down the running of clocks.

The wave motion of the transmission medium caused by a single isolated particle-wave is understandably very small. The corresponding value of the mean square velocity can be determined using Equation 28. An electron at a distance of 1 m with a mass of m _e = 9.1 x 10 ⁻³¹ kg causes the transmission medium to move at a mean velocity of v _m = 1.1 x 10 ⁻²⁰ m/s. This corresponds to a wave amplitude not exceeding 10 ^–40 m for a frequency of f = m e c 2 h = 1.2   x   10 20 Hz. For more massive particles, the mean square velocity is higher, while the amplitude of the carrier medium is lower. For a proton with a mass of m _p = 1.67 x 10 ⁻²⁷ kg, the values are as follows: v _m = 4.7 x 10 ⁻¹⁹ m/s, f = 2.3   x   10 23 Hz, and the amplitude does not exceed 10 ^–42 m. However, there is a large number of particle-waves in space, and all the waves combine with each other. A superposition of waves occurs. This results in wave superposition, in which the velocities, as well as the actual displacements, add together.

The maximum displacement that the transmission medium can achieve as a result of Earth’s gravitation can be determined by the following simplified reasoning:

The mean square velocity v _m for the Earth’s mass of 6 × 10 ²⁴ kg is equal to 2.8 × 10 ⁷ m/s. The frequencies of individual particle-waves range from 10 ²⁰ Hz (electron) to 10 ²³ Hz (proton and other baryons). Using the lowest of the frequencies mentioned, 10 ²⁰ Hz, the displacement is approximately 10 ^–13 m. This is an upper (maximum) estimate. The displacements are, therefore, very small, which explains the inability to detect the wave motion (i.e., deformations) of the transmission medium in our surroundings.

The question arises: In which dimension the transmission medium oscillates? Based on the existence of polarization phenomena (polarization of photons, electrons, or neutrons), it can be assumed that particle-waves have the nature of transverse waves. However, this in itself does not fully answer the question of the direction of the transmission medium’s displacements. It could be a wave motion in some fifth dimension, i.e., in such a way that E₄-B is a hyperplane (membrane) in space E₅. However, there are no indications for such an arrangement. It is more likely that the displacements of the transmission medium lie within E₄-B, i.e., they have a very general direction in four-dimensional space. The existence of the intrinsic (spin) angular momentum of particles supports this arrangement. The wave motion of the transmission medium thus remotely resembles the propagation of the transverse waves through rigid bodies, such as S-waves through the Earth.

In wave superposition, the instantaneous displacements of the transmission medium are added together at each point in space. The same applies to the addition of velocities. The resulting velocity of the transmission medium depends on the magnitude and direction of the individual velocities. Let us now try to find a general relationship for calculating the mean square velocity v _m of the transmission medium caused by the wave motion of a large number of particle-waves. All these particle-waves act simultaneously at the same point in space.

The composition of the mean square velocities of the transmission medium requires, among other things, consideration of the nature of its oscillation. This may be oscillation in a straight line, in a plane, or in 3D space, in all cases perpendicular to the direction of motion of the particle-wave in E₄-B. The specific variant depends on the polarization of the particle-wave. In any case, the mean square velocity can be decomposed into components in the direction of the coordinate axes. The components represent the mean square velocities in the individual dimensions – x, y, z, w: v m 2 = v m x 2 + v m y 2 + v m z 2 + v m w 2

In this way, we can decompose the mean square velocities of all acting particle-waves, and then sum their effects by components. In the summation, the contributions of individual particle-waves can be considered statistically uncorrelated (the mean square velocities corresponding to individual particle-waves are independent of each other, and the same applies to their components in the direction of the coordinate axes). This means that the mean square velocities can be added in the same way as other statistically independent quantities expressed by their mean square value. This is analogous to the propagation of standard deviations in statistical analysis. The x-components v m x , 1 , v m x , 2 , …, v m x , n corresponding to individual particle-waves 1 to n are added according to the formula v m x 2 = ∑ i = 1 n v m x , i 2 . The same applies for components in other axes. The formula for the resulting mean square velocity can be derived as follows: v m 2 = ∑ i = 1 n v m x , i 2 + ∑ i = 1 n v m y , i 2 + ∑ i = 1 n v m z , i 2 + ∑ i = 1 n v m w , i 2 = ∑ i = 1 n v m x , i 2 + v m y , i 2 + v m z , i 2 + v m w , n 2 = ∑ i = 1 n v m , i 2

It is, therefore, a simple sum of the squares of the mean square velocities. The direction of oscillations has no effect on the result.

The resulting mean square velocity does not depend on the direction of motion, the polarization of individual particles-waves, or the number of dimensions in which the transmission medium oscillates. The formula also applies to the case of space E₄-B as a membrane (hyperplane) in multidimensional space. The only limitations are given by the statistical nature of the formula, which assumes a large number of simultaneously acting particle-waves and their mutual uncorrelatedness (independence).

Using (Equation 27), the formula can be rewritten as V = ∑ i = 1 n V i (29)

The Equation 29 expresses the composition of the gravitational potentials of individual particle-waves. It is their simple sum at any point in space. This formula is identical to the formula valid in Newtonian gravitation, and simultaneously in the Newtonian limit of GTR.

Waves propagate in the transmission medium at speed v _4D , which determines the speed of gravitation. When calculating the potential, it is therefore necessary to take into account the time delay – the time it takes for the waves to reach the given point in space.

The standard metric of space E₄ has the form δ s 2 = δ x 2 + δ y 2 + δ z 2 + δ w 2 (30) where δs is the measure of the separation (distance) between two adjacent points in space. The separation is given in a measure of length (e.g., meters). However, the metric of space E₄-B is given by a slightly different relation (Equation 6), where the spatial separation δs is replaced by the time separation δt. It applies δ s = v 4 D   δ t (31)

The relationship (Equation 31) reflects the fact that length measurements are always related to the flow of time. In a situation where all matter takes the form of waves, it is not possible to define a unit of length that is independent of time. This is also reflected in the modern definition of the meter in the SI system. It includes not only time but also the speed of light, [11].

If the value of the speed of light was the same throughout the universe, there would be no fundamental difference between metrics (Equation 6) and (Equation 30). Both would be Euclidean. In the EMST, however, the speed of light (more precisely, the speed of particle-wave motion v _4D ) is variable. Due to the presence of matter, the ratio between δs and δt changes and the Euclidean metric (Equation 6) ceases to be Euclidean. A logical contradiction arises – if the metric (Equation 30) is used, space E₄-B is Euclidean, but if (Equation 6) is used, it is not Euclidean. From a geometric point of view, the correct metric is (Equation 30), as it works with units of length and ignores the speed v _4D . From a physical point of view, however, we prefer the metric (Equation 6), because there is no such thing as a universal unit of length independent of the v _4D . This applies primarily to the measurement of lengths using the transit time of light. However, it also applies partially to “rigid measuring rods.” Our beliefs about them are misleading. These measuring rods are made up of waves, and their dimensions are derived from the wavelengths of the particle-waves that make them up. And as we know, wavelength depends on the v _4D and the frequency – that is, on the flow of time. So, we cannot a priori assume that they’re independent of the v _4D .

The fact that the presence of matter affects the motion of particle-waves (changing both their speed and trajectory) gives the impression that space is curved by the influence of matter. The shortest line connecting two points (realized, for example, by the path of photons) is not identical to the shortest line according to Euclidean metrics (Equation 30), see Figure 6. Gravitational effect influences transit times, wavelengths, and directions. It creates a perfect illusion of a curved space. However, if we consider the local values of v _4D and rectify distances, wavelengths, and directions for their influence, i.e., if we switch to purely geometric units free from the influence of the variable v _4D , we find that space is still Euclidean and the metric (Equation 30) applies without any exception.

In the following text, three types of lengths will be distinguished:

Euclidean lengths s_E are purely geometric lengths independent of the v_4D value. They are expressed in a standard length unit, which is the “meter” as defined by the SI, realized in an area outside the influence of the gravitational field. Here, I refer to the dependence of v _4D and the flow of time on the gravitational potential, see (Equations 15–17). The formula δs _E = c δt applies, where t represents coordinate time unaffected (not slowed) by the effect of gravitation.

Local lengths s_L: The Euclidean meter, according to the SI definition, is, coincidentally, identical to the meter realized locally within a gravitational field. The requirement is that the gravitational potential of the reference clock and along the measured length are the same. As mentioned above, the rate of the clock depends on v _4D – see (Equation 8). For the same number of clock ticks (for the same amount of time), light travels the same distance. The wavelengths of particle-waves (electrons, locally emitted photons, etc.) will also be the same, i.e., length measuring tools will also have the same dimensions. Local lengths are therefore identical to Euclidean lengths, but unlike them, they are physically realizable in a gravitational field. However, their definition does not allow the measurement of lengths between points with different gravitational potential.

Corrected lengths s _C : To determine lengths passing through areas with different v _4D , the SI meter definition can be applied, except that the reference clock is not located at the same potential level as the measured length. It can be located outside the gravitational field (V = 0, v _4D = c) or anywhere within it (V < 0, v _4D < c). A different potential level at the location of the reference clock will affect the resulting lengths – these will be different from Euclidean ones. A variable value of v _4D in the area of the measured length, combined with a constant value of v _4D at the location of the reference clock, will cause the length of the meter to be also variable. Corrected lengths in the area of variable V (i.e., variable v _4D ) do not create a Euclidean system.

The increment of the corrected length δ s C is given by δ s C = v 4 D δ t = 1 + 2 V c 2 c   δ t = k   δ s E , where k = 1 + 2 V c 2 (32)

The values of v _4D and V are related to the potential level of the reference clock. If this clock is located outside the gravitational field, then V ≤ 0 , v 4 D ≤ c applies throughout the entire space. Length increments δ s C inside the gravitational field (where v 4 D < c ) are smaller than δ s E . Any distance measured using increments δ s C will therefore be greater than the same distance measured using δ s E (i.e., by SI meter). The formula s C = 1 k s E applies.

The finding that space is Euclidean, even in areas affected by gravitation, is useful in deriving the metric of the central gravitational field. This term refers to the gravitational field generated by a point object or a body with a spherically symmetric distribution of mass. Central gravitational fields are of great importance in physics, because they describe the gravitational influence of planets and stars on bodies in their vicinity.

For the derivation of the metric of the central gravitational field, we start with the Euclidean metric of four-dimensional space. Subsequently, we implement the influence of the variable v _4D into it – i.e., we move from a purely geometric model to a physical model.

The E₄ metric (as well as E₄-B metric) in spherical coordinates (supplemented by the fourth coordinate w) is given by the relation δ s E 2 = δ r E 2 + r E 2   δ θ 2 + sin 2 ⁡ θ   δ ϕ 2 + δ w E 2

Here, r E , θ and ϕ are polar coordinates in E₃, δ r E , δ θ and δ ϕ are their increments, and δ w E is the increment in the fourth coordinate. All length quantities are Euclidean.

For the transition to the physical model, it is necessary to include the influence of the variable v _4D – move to Corrected lengths. For the derivation, a mathematical model involving a pair of concentric circles C and C+ will be used. The centers of the circles are located in the central body and their radii are r E and r E + = r E + δ r E , δ r E →   0 . The reference level of gravitational potential for corrected lengths is the potential on circle C ( V R = − G M / r E ), i.e., the length of this circle expressed in Corrected lengths will be the same as in Euclidean lengths. The potential on circle C+ is V = − G M / r E + δ r E . The circumferences of the circles in Corrected lengths relative to the reference potential V _R are

O C = O E = 2 π r E and O C + = k R / k   2 π r E + δ r E , where k is given by (Equation 32) and k R = 1 + 2 V R c 2 = 1 − 2 G M r E   c 2 = 1 − R r E (33)

Since k ≠ k _R , the circumference of circle O _C+ differs from the Euclidean O E + = 2 π r E + δ r E .

The value of k can be expressed as a multiple of k _R , assuming δ r E ≪ r E . k = 1 − R r E + δ r E ≈ 1 − R r E 1 − δ r E r E = 1 − R r E + R r E 2   δ r E = k R 2 + R r E 2   δ r E = k R 1 + R k R 2   r E 2   δ r E ≈ k R 1 + R 2   k R 2   r E 2   δ r E ≈ k R 1 1 − R 2   k R 2   r E 2   δ r E

Subsequently, the increase in the radius in Corrected lengths can be compared with the same increase in Euclidean lengths: δ r C = r C + − r C = O C + − O C 2 π = k R k r E + δ r E − r E = 1 − R 2   k R 2   r E 2   δ r E r E + δ r E − r E = r E + δ r E − R 2   k R 2 r E   δ r E − R 2   k R 2   r E 2   δ r E 2 − r E ≈ δ r E   1 − R 2   k R 2 r E = δ r E   1 − R 2   r E − R = k mod   δ r E where k mod = 1 − R 2   r E − R (34)

In the transition from Euclidean lengths to Corrected lengths, only the increment of the radius changes ( δ r C = k mod   δ r E ). The circumference of the circle O _C and thus also the derived radius r C = O C / 2 π remain unchanged ( r C = r E ). The same applies to the increment in dimension w, which lies in the same potential as circle C ( δ w C = δ w E ) We can write δ s E 2 = δ r C 2 k mod 2 + r C 2   δ θ 2 + sin 2 ⁡ θ   δ ϕ 2 + δ w C 2 .

The value δ w C is, from a physical point of view, the increment of proper time determined by local clock δ w C = c   δ τ .

The increment δ s E represents the distance between points in space E₄-B. Its expression using the time of stationary reference clocks located outside the gravitational field leads to the formula δ s E = v 4 D δt – see (Equation 31). Considering the ratio between v _4D and c (Equation 15) and the relationships (Equation 33), we obtain the metric of space in the area of the central gravitational field c 2 δ t 2 k R 2 = δ r C 2 k mod 2 + r C 2   δ θ 2 + sin 2 ⁡ θ   δ ϕ 2 + c 2 δ τ 2 (35)

Coordinates and their increments are always real numbers in the EMST and can be both positive and negative. On the other hand, time increments δt and δτ are always non-negative (they are related to the increments of the path or the number of clock ticks – see [7]). The coefficients k _R and k _mod are given by Equations 33, 34), where r C = r E . The quantity R = 2 G M / c 2 is known from the GTR as the Schwarzschild radius.

For R ≪ r C , the difference between k _R and k _mod is negligible. We can write k mod = 1 − R 2   r C − R ≈ 1 − R r C = k R (36)

The metric (Equation 35) can be rewritten for R ≪ r C in the form known from GTR 1 − R r C c 2 δ t 2 = δ r c 2 1 − R r C + r C 2   δ θ 2 + sin 2 ⁡ θ   δ ϕ 2 + c 2 δ τ 2 i.e., in the form of the Schwarzschild metric. The meaning of the individual quantities is the same as in GTR. This also applies to the radius r C , which is the radial coordinate calculated from the circumference of a circle. The corrected lengths form the same set of coordinates that was used by Schwarzschild in his solution.

As mentioned above, in the presented theory of gravitation, the principle source of gravitational attraction is the energy of matter, not its mass. The energy-mass ratio (which is constant in the GTR) is variable in the EMST – it depends on the variable value of v _4D , E = m   v 4 D 2 , see (Equation 5). Therefore, it is not possible to interchange mass and energy in the formulas for gravitational potential or acceleration. As the potential decreases (moving down into the gravitational well), the energy of the particle-waves decreases, but their mass increases.

It is necessary to keep in mind the following when assessing the influence of gravitational potential on physical quantities: -

The dimensions of objects are independent of potential (Local lengths are equal to Euclidean lengths,

The above is reflected in the implementation of the fundamental units of these quantities (meter, second, kilogram).

In general, physical quantities containing only length quantities (distance, area, volume, etc.) and quantities containing mass and time in the opposite power (momentum [kg m s ^-1 ], angular momentum [kg m ² s ^-1 ], etc.) are not dependent on gravitational potential. For example, the value of Planck’s constant h [kg m ² s ^-1 ] is also independent of V.

The formulas describing the effects of gravitation are similar to Newton’s, but they are based on particle-wave energy as the principal source of gravitation. Energy refers to the total energy of a particle-wave ( E = h   f = E 0 + E p + E k ), see Equations 2, 9. The two most fundamental gravitation formulas take the form V = − ϰ ¯ E r C (37) a = − ϰ ¯ E r C 2 (38) i.e., the product GM is substituted by the product of ϰ ¯ E .

It can be seen that the mass of the central body M is replaced by its energy E, and Newton’s gravitational constant G is replaced by a new constant ϰ ¯ . For weak gravitational fields, the value of ϰ ¯ can be derived from G [kg ^-1 m ³ s ^-2 ], ϰ ¯ = G / c 2 = 7 , 425 × 10 − 28 [kg ^-1 m]. It is important to note that neither G nor ϰ ¯ are constants in the true sense of the word. Both are functions of the gravitational potential V, and therefore also of v _4D . For reference clocks “1” and “2” in regions with different potentials (see example above), ϰ ¯ 2 / ϰ ¯ 1 = v 4 D , 2 / v 4 D , 1 and G 2 / G 1 = v 4 D , 2 3 / v 4 D , 1 3 . The constant ϰ ¯ is less dependent on the value of the gravitational potential than G.

The gravitational potential of composite bodies is the sum of the gravitational potentials of individual particle-waves. The total value can be determined by simple addition – see relation (Equation 29). At this stage, the different energies of identical particle-waves (e.g., free electrons at rest) in places with different potentials become significant. Particle-waves deeper in the gravitational well have lower energy than identical particle-waves higher up – see Sect. 2.6.