Preliminary knowledge

Gravitation is the theory dealing with attraction of massive objects. Gravity is a force; it makes things move toward each other. Physics describes gravitation by using Newton's law of universal gravitation (formulated in 1687) in which the gravitational force of attraction between two objects with masses $M_1$ and $M_2$ separated by a distance $r$ has the form

\begin{align} F = - G \frac{M_1 M_2}{r^2} \end{align}

where $G = 6.674 × 10^{-11}$ N m2 kg-2 is the gravitational constant. From expression (1) we can obtain the potential energy of gravitation

\begin{align} V = - G \frac{M_1 M_2}{r} \end{align}

and the gravitational potential generated by a mass $M$ in its surrounding

\begin{align} U = - G \frac{M}{r}. \end{align}

Let us mention here the integral form of Gauss' law for gravity, which states:

\begin{align} \oint_{\partial S}\vec {g}\cdot d \vec{s} = -4 \pi GM \end{align}

where $\partial S$ is any closed surface, $d \vec {s}$ is a vector whose magnitude is the area of an infinitesimal piece of the surface $\partial S$ and whose direction is the outward-pointing surface normal; $M$ is the total mass enclosed within the surface $\partial S$. The left-hand side of equation (4) is called the flux of the gravitational field; it is always negative (or zero), and never positive (although for electricity Gauss' law allows fluxes to be either positive or negative, because the charge can be either positive or negative, while mass can only be positive).

The Gauss law (4) is interesting to us, as it introduces the gravitational field $\vec {g}$, which is a vector field that originates from the central point, i.e. the point of location of mass $M$.

Since 1916 Newton's law has been superseded by Einstein's theory of general relativity. General relativity is only required when there is a need for extreme precision, or when dealing with gravitation for very massive objects. General relativity or the general theory of relativity is the geometric theory of gravitation published by David Gilbert and Albert Einstein in 1916. The theory unifies special relativity and Newton's law of universal gravitation and describes gravity as a property of the geometry of space and time, or space-time. In particular, the curvature of space-time is directly related to the four momenta (mass-energy and linear momentum) of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of 10 differential equations.

General relativity became very popular due to the prediction of three phenomena (two of them were very new), which were confirmed experimentally in 1919-1920. Namely, general relativity predicted:

  1. The Motion of Mercury’s perihelion by an amount $\Delta \phi = 6\pi GM_{\rm Earth} /(Lc^2)$ where $M_{\rm Earth}$ is the Earth's mass, $L$ is the focal parameter, $c$ is the velocity of light;
  2. The Bending of a light ray by the sun, i.e. the following angle deviation of the ray from the direct line was derived: $\Delta \phi = 4 GM_{\rm Sun} /(r c^2)$ where $M_{\rm Sun}$ is the Sun's muss, $c$ is the velocity of light and $r$ is the radial distance from the centre of the Sun to the point where the light ray is bending;
  3. The Gravitational red shift of spectral lines $\Delta \nu = - GM \nu_0 /(r c^2)$ on the surface of the massive body whose mass is $M$, $r$ is the radius of the body, $\nu_0$ is the frequency of generated light without the presence of gravity.

General relativity predicted also gravitational time dilation and gravitational time delay, which were observed as well. However, unanswered questions remain, the most fundamental one being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity. Nevertheless, it seems this challenge cannot be resolved in principle because of principal differences in approaches to physical laws by microscopic quantum physics and phenomenological general relativity. Besides, general relativity does not look like a true physical theory but rather like an abstract mathematical theory.

In the case of general relativity, which denied the classical ether and introduced an abstract vague vacuum, we can distinguish five problems, conceptual difficulties, that do not have resolutions in the framework of relativity formalism:

  • general relativity is founded on the basic Newtonian term $- GM/r$, but cannot explain its origin;
  • a massive object can influence space-time but cannot be derived from it, because the unknown and undetermined parameter mass is entirely separated from the phenomenological notion of space-time;
  • the formalism of relativity is failing on a microscopic scale as it does not pay attention to the wave nature of matter. At distances compared to or less than the object's de Broglie wavelength $\lambda$ the formalism of general relativity has to give way to an approach based on microscopic consideration;
  • general relativity does not offer any sorts of particles/quasi-particles, which will be able to realize short-range action in the gravitational attraction of objects and hence it is a theory based on an action-at-a-distance phenomenological approach, the same as the Newtonian theory (and also quantum mechanics whose long-range action also falls within the range of its conceptual difficulties); regarding quasi-particles gravitons we can say that, based on the studies of other researchers [1] as well as experimental results [2], these are abstract mathematical objects absent in real nature;
  • light, which plays an exceptionally important role in relativity, has to be massless in theory; however, light carriers, photons, transfer momentum and energy and therefore by the principle of equivalence, photons must have non-zero mass.

Sub-microscopic consideration

1. Derivation of Newton's gravitational law for a canonical particle

The submicroscopic concept based on the constitution of physical space and submicroscopic mechanics allows a detailed theory of gravity to be derived which suggests a radically new approach to the problem of quantum gravity and allows the derivation of Newton's gravitational law from first subatomic principles. Such approach completely removes all difficulties that concern the action-at-a-distance phenomenology by introducing inertons as carriers of the interaction between massive objects.

Since any motion in the tessel-lattice generates clouds of inertons - mass excitations of the real physical space - may be considered as the actual carriers of the gravitational field as it occurs in Gauss's approach (4) to the problem of gravity.

The cloud of inertons surrounding the particle spreads out to a range $\Lambda = \lambda c/\upsilon$ from the particle center where $\lambda$ is the particle's de Broglie wavelength and $\upsilon$ and $c$ are velocities of the particle and light, respectively. Since inertons transfer fragments of the particle's mass, they also play the role of carriers of gravitational properties of the particle. First of all we should understand how inertons emitted by the particle come back to it, returning fragments of its mass as well as the velocity. The behaviour of the particle's inertons can be studied in the framework of the Lagrangian [3,4]

\begin{align} {L} = -m_0 c^2 \Big\{ \frac {T^2}{2m_0^2} {\dot m}^2 + \frac{T^2}{2\Lambda^2} {\dot {\vec \xi}}^{{\kern 2pt}2} - \frac{T}{m_0} {\dot m} \nabla {\vec \xi} \Big\}^{1/2}. \end{align}

Here $m ({\vec r}, t)$ is the current mass of the {particle-inerton cloud} system; $\vec \xi ({\vec r}, t)$ is the variable that describes a local distortion of the tessel-lattice, which can be called the rugosity or tension (see in inerton); $T$ is the time period of collisions of the particle and its inerton cloud.

The Euler-Lagrange equations for variables $m$ and $\vec \xi$ is

\begin{align} \frac{\partial}{\partial t}\frac{\partial L}{\partial {\dot q}} - \frac{\delta L}{\delta q} = 0. \end{align}

The equations for $m$ and $\vec \xi$ become

\begin{align} \frac {\partial^2 m}{\partial t^2} - \frac{m_0}{T} \nabla {\dot {\vec \xi}} =0; \end{align}
\begin{align} \frac {\partial^2 {\vec \xi}}{\partial t^2} - \frac{\Lambda^2}{m_0 T} \nabla {\dot m} =0. \end{align}

Taking the initial and boundary conditions as well as the radial symmetry into account, we can obtain the following solutions to equations (7) and (8):

\begin{align} m(r, t) = C_1 \frac{m_0}{r} \cos \frac{\pi r}{2\Lambda} \Big| \cos \frac{\pi t}{2T} \Big|, \end{align}
\begin{align} \xi(r, t) = C_2 \frac{\xi_0}{r} \sin \frac{\pi r}{2 \Lambda}(-1)^{[t/T]} \Big| \sin \frac{\pi t}{2 T} \Big|. \end{align}

These solutions exhibit the dependence $1/r$, which is typical for standing spherical waves.

The solution for mass $m$ (9) shows that at a distance $r << \Lambda$ the time-averaged distribution of mass of inertons along the radial ray which originates from the particle, becomes

\begin{align} m(r) \approx l_{\rm f} \frac{m_0}{r} \end{align}

In this region the rugosity (or tension) of space, as followed from expression (10), is: $\xi \approx 0.$

When the local deformation is distributed in space around the particle, it forms a deformation potential $\propto 1/r$ that spreads up to the distance $r= \Lambda$ from the particle's kernel-cell.

In the range covered by the deformation potential, cells of the tessel-lattice are found in the contraction state and it is this state of space which is responsible for the phenomenon of the gravitational attraction.

In terms of physics, the distribution (11) is replaced with the Newton's gravitational potential

\begin{align} U(r) = - G \frac{m_0}{r} \end{align}

where the gravitational constant $G$ plays the role of a dimensional constant. These equations, (11) and (12), are rather formal, as quantum mechanical behaviour of particles prevails at such scale. Nevertheless, this consideration allows us to understand the inner reasons responsible for the formation of Newton law of gravitation for macroscopic objects.

2. Newton law of gravitation for a macroscopic body


Fig. 1. Overlapping of inerton clouds of particles means the appearance of an excess mass in the system studied.

An object, which consists of many particles (a solid, a planet, or a star), experiences vibrations of its entities (atoms, ions, particles). Entities vibrate in the neighborhood of their equilibrium positions and/or move to new positions. These movements produce inerton clouds around the appropriate particles. Indeed, Figure 1 shows a set of particles surrounded with their inerton clouds. These clouds overlap. Note the particle together with its cloud of inertons (which exist in the real physical space) manifest themselves in conventional quantum mechanics (whose formalism has been developed in an abstract phase space) as the so-called $\psi$-wave function.

If the body studied consists of N particles, it will be characterised by N/2 modes of normal vibrations (i.e. harmonics). When particles vibrate, they clouds of inertons vibrate too and hence the same occur with overlapping zones: a set of states of overlapped zones becomes equal to N/2 as well.

What does happen in the zones of overlapped inerton clouds? In these places clouds become denser. In other words, these zones become more massive. Why? Because by definition the inerton is a carrier of mass. At the same time, by definition of mass, a gain to the mass means that the deformation of a cell at which the mass excitation (i.e. inerton) is located at the moment becomes large. That is, the cell becomes more shrunken - its volume a little bit reduced.

Overlapping of inerton clouds results in the formation of a total inerton cloud of the body [5]. Such overlapping known in nuclear physics as the mass defect, but as we can see the phenomenon is general in quantum physics - it takes place in any system where an overlapping of $\psi$-wave functions (i.e. inerton clouds) occur: the system's potential energy increases.

The spectrum of inertons of such mass defect is similar to the spectrum of phonons, as inertons immediately appear when entities move from their initial position, which is discussed in submicroscopic mechanics (we may say that a body of phonons is filled with inerton carriers). For instance, if we have a solid sphere with a radius $R_{\rm sph}$, which consists of $N_{\rm sph}$ atoms, the spectrum of acoustic waves is composed of $N_{\rm sph}/2$ waves with the wavelengths $\lambda_n = 2an$ where $a$ is the mid-distance between nearest atoms and $n = 1, 2, 3, ... , N_{\rm sph}/2$.

At the same time, inertons that accompany acoustically vibrating atoms produce also their own spectrum and the wavelengths of these collective inertonic vibrations can be estimated by expression

$\Lambda_n = 2an {\kern 2pt } c/\upsilon_{\rm sound}$.

Also note that the behaviour of these inerton oscillations obeys the law of standing spherical waves, i.e. the dependence of the front of the inerton wave must be proportional to the inverse distance from the source irradiating the wave, $1/r$. What do these waves transmit? It is obvious, local deformations of space, which gradually decrease with r by the law $1/r$; in other words, cells of space are smaller in size near the body and their size approaches its equilibrium size at the distance of $\Lambda$ from the body where the tessel-lattice is found in the degenerate state. These standing inerton waves create a deformation potential around the body.

For instance, a solid sphere with volume 1 cm3 includes around 1022 atoms; estimating the velocity of sound $\upsilon_ {\rm sound} \approx 10^3$ m/s (order of magnitude) and the distance between atoms $a=0.5$ nm, we obtain for the amplitude of the longest inerton wave: $\Lambda_{N_{\rm}/2} \sim 10^{17}$ m. Thus, up to this distance the inerton field of the solid sphere is able to propagate in the form of the standing spherical inerton wave.

To the solid sphere studied we may now apply the same consideration which has been done above for the gravity of a particle. In particular, expression (12) is also applicable for the case of an massive object.

Therefore, we were able to derive Newton's potential (12) in terms of short-range action provided by inertons, carriers of mass properties of objects. Being averaged in time, a mass field around the body studied can be considered as a stationary gravitational potential (3).

The theory presented here sheds light on the principle of equivalence, which proclaims the equivalence of gravitational and inertial masses: $m_{\rm grav} = m_{\rm inert}$. Namely, this equality, which is held in a rest-frame of the particle in question, becomes invalid in a moving reference frame. In the quantum context, this equality should be transformed to the principle of equivalence of the phases of gravitational and inertial waves, $\varphi_{\rm grav} = \varphi_{\rm inert}$. This correlation ties up the gravitational and inertial energies of the particle and also shows that the gravitational mass is completely allocated in the inertial wave that guides the particle. De Haas [6] was the first who came to this conclusion when comparing Mie’s variational principle and de Broglie's harmony of phases of a moving particle. So the matter waves consist of kernel, particle and its inerton cloud, which exchange velocity, mass and hence energy and momentum; this exchange occurs owing to the strong interaction of the particle and its inertons with the tessel-lattice and it is this interaction that causes the induction of the gravitational potential in the range of spreading of the particle's/object's inertons.

3. Correction to the Newton law of gravitation [7]

The sub-microscopic approach points out to the fact that the gravitational interaction between objects must consist of two terms: (i) the radial inerton interaction between two masses $M$ and $m$, which results in the classical Newton gravitational law (2), and (ii) the tangential inerton interaction between the masses, which is caused by the tangential component of the motion of the test mass $m$ and which is characterized by the correction:

\begin{align} \delta V = -G \frac{Mm}{r} {\kern 2pt} \frac{r^2 {\dot \phi}^2}{c^2}. \end{align}

Note that the existence of such a correction is in line with a remark by Poincaré [8] who stated that the expression for the attraction should include two components: one is parallel to the vector that joins positions of both interacting objects and the second one is parallel to the velocity of the attracted object. Thus the velocity of an object must influence the value of its gravitational potential.

By using the total expression for the gravitation

\begin{align} V= - G \frac{Mm}{r} \cdot \Big( 1+ \frac{r^2 {\dot \phi}^2}{c^2} \Big) \end{align}

we can study four problems that were investigated in the framework of general relativity, namely: 1) the motion of Mercury’s perihelion; 2) the bending of light by the sun; 3) the gravitational red shift of spectral lines; 4) the gravitational time delay effect (the Shapiro time delay effect) [10]. Expression (14) allows us to examine the three problems in the framework close to that carried out in terms of classical physics, not general relativity. Expression (14) enables the immediate and easy derivation of the same equations of motion that general relativity derived by using complicated geodesic equations. That is why having the same equations describing these three problems, we can use the same solutions pointed out in the above section Preliminary knowledge.

Therefore it does not make sense to use the complicated mathematics of general relativity to solve this or that challenge. The physics of the phenomena studied is hidden in the potential energy (14), which describes the interaction of two attracting objects.

This approach also clarifies the situation with so-called "black holes", which were introduced in physics at the end of the 1960s. The approach described above shows that a point mass $M$ at rest possesses the conventional Minkowski flat-space metric, i.e. is exactly exemplified by Newton's gravitational potential (12) and hence does not show any singularity. But this metric disturbed by a smaller mass $m$ changes to the Schwarzschild metric (or maybe another metric) in the location of the smaller mass.

In other words, a point mass does not have any peculiarity in its metric, its metric is flat/linear, although deformed in line with the radial symmetry [9]. The sub-microscopic consideration of gravity suggests no reasons to hypothesize a "black hole" solution in its modern classical sense. Only an outside source of the gravitational field is able to disturb the flat metric of a heavy central mass. Thus researchers dealing with the formalism of general relativity must be extremely careful in application of their theoretical results to the description of the surrounding.

At last, the availability of standing spherical inerton waves around massive objects, which provide short action, made it possible finally to solve the problem of so-called "dark matter" [10].


[1] A. Loinger, On black holes and graviational waves (La Goliardica Pavese, 2002); More on BH’s and GW’s. III (La Goliardica Pavese, 2007).

[2] V. Krasnoholovets and V. Byckov, Real inertons against hypothetical gravitons. Experimental proof of the existence of inertons, Indian Journal of Theoretical Physics 48, no. 1, 1-23 (2000) (also

[3] V. Krasnoholovets, Gravitation as deduced from submicroscopic quantum mechanics,

[4] V. Krasnoholovets, Reasons for the gravitational mass and the problem of quantum gravity, in Ether, Space-Time and Cosmology, Vol. 1, Eds.: M. Duffy, J. Levy and V. Krasnoholovets (PD Publications, Liverpool, 2008), pp. 419-450 (ISBN 1 873 694 10 5) (also

[5] V. Krasnoholovets, On variation in mass of entities in condensed media, Applied Physics Research 2, no. 1, 46-59 (2010), ISSN: 1916-9639; E-ISSN: 1916-9647 (direct access

[6] E. P. J. de Haas, The combination of de Broglie’s harmony of the phases and Mie’s theory of gravity results in a principle of equivalence for quantum gravity, Annales de la Fondation Louis de Broglie 29, no. 4., 707-726 (2004).

[7] V. Krasnoholovets, On microscopic interpretation of phenomena predicted by the formalism of general relativity, in: Ether Space-Time and Cosmology, Vol. 2: New Insights into a Key Physical Medium. Eds.: M. C. Duffy, J. Lévy (Apeiron, 2009), pp.417-431 (Publisher: C. Roy Keys Inc.; Apeiron. ISBN: 0973291184; 978-0973291186); in Apeiron 16, no. 3, 418-438 (2009) (direct access

[8] H. Poincaré, Sur la dynamique de l’électron, Rendiconti del Circolo matematico di Palermo 21, 129-176 (1906); also: Oeuvres, t. IX, pp. 494-550 {also in Russian translation: А. Пуанкаре, Избранные труды (H. Poincaré, Selected Transactions), ed. N. N. Bogolubov (Nauka, Moscow, 1974), vol. 3, pp. 429-486}.

[9] V. Krasnoholovets, On the gravitational time delay effect and the curvature of space, submitted and accepted.

[10] V. Krasnoholovets, Dark matter as seen from the physical point of view, Astrophysics and Space Science 335, No. 2, 619-627 (2011); (a pdf file of the paper can be download from the web site

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