Impact Models of Gravitational and Electrostatic Forces

The far-reaching gravitational force is described by a heuristic impact model with hypothetical massless entities propagating at the speed of light in vacuum transferring momentum and energy between massive bodies through interactions on a local basis. In the original publication in 2013, a spherical symmetric emission of secondary entities had been postulated. The potential energy problems in gravitationally and electrostatically bound two-body systems have been studied in the framework of this impact model of gravity and of a proposed impact model of the electrostatic force. These studies have indicated that an antiparallel emission of a secondary entity—now called graviton—with respect to the incoming one is more appropriate. This article is based on the latter choice and presents the modifications resulting from this change. The model has been applied to multiple interactions of gravitons in large mass conglomerations in several publications. They will be summarized here taking the modified interaction process into account. In addition, the speed of photons as a function of the gravitational potential is considered in this context together with the dependence of atomic clocks and the redshift on the gravitational potential.


Introduction
Newton's law of gravity gives the attraction between two spherical symmetric bodies A and B with masses M and m, respectively, for a separation distance of their centres r (large compared to the sizes of the bodies) at rest in an inertial frame of reference. The force acting between A and B is where G N ¼ 6:67408 31 ðÞ Â 10 À11 m 3 kg À1 s À2 is the constant of gravity 1 ,r is the unit vector of the radius vector r with origin at A and r ¼ |r |. The first term on the right-hand side represents the classical gravitational field of the mass M.
In close analogy, Coulomb's law yields the force of the electrostatic interaction between particles C and D with charges Q and q, respectively: where ε 0 ¼ 8:854 187 817 … Â 10 À12 Fm À1 is the electric constant in vacuum. Here charges with opposite signs lead to an attraction and with equal signs to a repulsion.
is the classical electrostatic field of a charge Q. For two electrons, e.g. the ratio of the gravitational and electrostatic forces is Eq.
(1) yields a very good approximation of the gravitational forces, unless effects treated in the general theory of relativity (GTR) [2] are of importance.
The physical processes of the gravitational and the electrostatic fields-in particular their potential energies-are still a matter of debate: Planck [3] wondered about the energy and momentum of the electromagnetic field. A critique of the classical field theory by Wheeler and Feynman [4] concluded that a theory of action at a distance, originally proposed by Schwarzschild [5], avoids the direct notion of fields. Lange [6] calls the fact "remarkable" that the motion of a closed system in response to external forces is determined by the same law as its constituents. It should be recalled here that von Laue [7] considered radiation confined in a certain volume ("Hohlraumstrahlung") and showed that the radiation contributed to the mass of the system according to Einstein's mass-energy equation (see Eq. (51)). In a discussion of energy-momentum conservation for gravitational fields, Penrose [8] finds even for isolated systems "… something a little 'miraculous' about how things all fit together, …", and Carlip [9] wrote in this context: "… after all, potential energy is a rather mysterious quantity to begin with …".
Related to the potential energy problem is the disagreement of Wolf et al. and [10] and Müller et al. [11] on whether the gravitationally redshifted frequency of an atomic clock is caused by the gravitational potential Ur ðÞ¼À G N M r (5) or by the local gravity field g ¼ ∇U.
These remarks and disputes motivated us to think about electrostatic and gravitational fields and the problems related to the potential energies.

Gravitational and electrostatic interactions
If far-reaching fields have to be avoided, gravitational and electrostatic models come to mind similar to the emission of photons from a radiation source and their absorption or scattering somewhere else-thereby transferring energy and momentum with the speed of light c 0 ¼ 299792 458 m s À1 in vacuum [12][13][14][15].
We have proposed a heuristic model of Newton's law of gravitation in [16]without far-reaching gravitational fields-involving hypothetical massless entities. Originally they had been called quadrupoles but will be called gravitons now. In subsequent studies, conducted to test the model hypothesis, it became evident that energy and momentum could not be conserved in a closed system without modifying the interaction process of the gravitons with massive bodies and massless particles, such as photons. The modification and the consequences in the context of the gravitational potential energy will be discussed in the following sections together with related topics.
The analogy between Newton's and Coulomb's laws suggests that in the latter case, an impact model might be appropriate as well-with electric dipole entities transferring momentum and energy. This has been proposed in [17]. The equations governing the behaviour of gravitons and dipoles in the next sections are very similar in line with the similarity of Newton's and Coulomb's laws.
Both concepts are required for a description of the gravitational redshift in terms of physical processes in Section 3.8.

Definitions of gravitons
Without a far-reaching gravitational field, the interactions have to be understood on a local basis with energy and momentum transfers by gravitons. This interpretation has several features in common with a theory based on gravitational shielding conceived by Nicolas Fatio de Duillier [18] at the end of the seventeenth century. A French manuscript can be found in [19], and an outline in German has been provided by Zehe [20]. Related ideas by Le Sage have been discussed in [21].
The gravitational case, in contrast to the electrostatic one, does not depend on polarized particles. Gravitons with an electric quadrupole configuration propagating with the speed of light c 0 will be postulated in the case of gravity. They are the obvious candidates as they have small interaction energies with positive and negative electric charges and, in addition, can easily be constructed with a spin of S ¼AE2, if indications to that effect are taken into account, cf. [22].
The vacuum is thought to be permeated by the gravitons that are, in the absence of near masses, isotropically distributed with (almost) no interaction among each other-even dipoles have no mean interaction energy in the classical theory (see, e.g. [23,24]). The graviton distribution is assumed to be a nearly stable, possibly slowly varying quantity in space and time. It has a constant spatial number density: Constraints on the energy spectrum of the gravitons will be considered in later sections. At this stage we define a mean energy of for a massless graviton with a momentum vector of p G .

Definitions of dipoles
A model for the electrostatic force can be obtained by introducing hypothetical electric dipoles propagating with the speed of light. The force is described by the action of dipole distributions on charged particles. The dipoles are transferring momentum and energy between charges through interactions on a local basis.
Apart from the requirement that the absolute values of the positive and negative charges must be equal, nothing is known, at this stage, about the values themselves, so charges of AE|q| will be assumed, where q might or might not be identical to the elementary charge e ¼ 1:602176 634 Â 10 À19 C (exact) [25].
The electric dipole moment is parallel or antiparallel to the velocity vector c 0 n , wheren is a unit vector pointing in a certain direction and l is the separation distance of the charges. This assumption is necessary in order to get attraction and repulsion of charges depending on their mutual polarities. In Section 2.5 it will be shown that the value |d| of the dipole moment is not critical in the context of our model. The dipoles have a mean energy where p D represents the momentum of the dipoles. As a working hypothesis, it will first be assumed that | p D | is constant remote from gravitational centres with the same value for all dipoles of an isotropic distribution. The dipole distribution is assumed to be nearly stable in space and time with a spatial number density but will be polarized near electric charges and affected by gravitational centres.

Virtual entities
As an important step, a formal way will be outlined of achieving the required momentum and energy transfers by discrete interactions. The idea is based on virtual gravitons and dipoles in analogy with other virtual particles, cf. [26][27][28].

Virtual gravitons
A particle with mass M is symmetrically emitting virtual gravitons with moments p * G and energies of T * G ≪ Mc 2 0 . The emission rate is proportional to M. The gravitons will have a certain lifetime Δt G and interact with "real" gravitons. In the literature, there are many different derivations of an energy-time relation, e.g. in [29][30][31]. Considerations of the spread of the frequencies of a limited wave-packet led Bohr [32] to an approximation for the indeterminacy of the energy that can be rewritten as with h ¼ 6:626 07015 Â 10 À34 J s (exact), the Planck constant [25]. For propagating gravitons, the equation is equivalent to the photon energy relation E ν ¼ h ν ¼ hc 0 =λ, where λ corresponds to l G , which can be considered as the wavelength of the hypothetical gravitons. Since there is experimental evidence that virtual photons (identified as evanescent electromagnetic modes) behave non-locally [33,34], the virtual gravitons might also behave non-locally. Consequently, the absorption of a real graviton could occur momentarily by a recombination with an appropriate virtual one.

Virtual dipoles
We assume that a particle with charge Q is symmetrically emitting virtual dipoles with p * D . The emission rate is proportional to its charge and the orientation such that a repulsion exists between the charge and the dipoles. The symmetric creation and destruction of virtual dipoles are sketched in Figure 1. The momentum balance is shown for the emission phase on the left and the absorption phase on the right side.
Virtual dipoles with energies of T * D ≪ m Q c 2 0 will have a certain lifetime Δt D and interact with real dipoles. The energy-lifetime relation corresponds to that of the gravitons in Eq. (11). The equation is for propagating dipoles also equivalent to the photon energy relation, with l D corresponding to λ.

Newton's law of gravity
The gravitons are absorbed by massive bodies from the background and subsequently emitted at rates determined by the mass M of the body independent of its charge: where κ G is the gravitational absorption coefficient and η G the corresponding emission coefficient.
Spatially isolated particles at rest in an inertial system will be considered first. The sum of the absorption and emission rates is set equal to the intrinsic de Broglie frequency of the particle, cf. Schrödinger's Zitterbewegung [8, [35][36][37][38][39]. Since the absorption and emission rates must be equal in Eq. (15), this gives an emission coefficient of i.e. half the intrinsic de Broglie frequency, since two virtual gravitons are involved in each absorption/emission process (cf. Figure 2). The absorption coefficient is constant, because both ρ G and η G are constant. For an electron with a mass of m e ¼ 9:109 383 56 11 ðÞ Â 10 À31 kg, the virtual graviton production rate equals its de Broglie frequency ν B G,e ¼ m e c 2 0 =h ¼ 1:235 … Â 10 20 Hz. The energy absorption rate of an atomic particle with mass M is Larger masses are thought of as conglomeration of atomic particles. The emission energy, in turn, is assumed to be reduced to per graviton, where Y (0 < Y ≪ 1) is defined as the reduction parameter. This leads to an energy emission rate of Without such an assumption, the attractive gravitational force could not be emulated, even with some kind of shadow effect as in Fatio's concept, cf. [18,19]. Interaction of gravitons with a body of mass M. A graviton arriving with a momentum p G on the left combines together with a virtual graviton with p * G ¼Àp G . The excess energy liberates a second virtual graviton with p À G on the right in a direction antiparallel to the incoming graviton. The excess energy T À G is smaller than T * G . The conceptual diagram shows gravitons with a spin S ¼AE4 Â ℏ=2 and G þ or G À orientation. It is unclear whether such a spin would have any influence on the interaction process (modified from Figure 1 of [16]).
The reduction parameter Y and its relation to the attraction are discussed below. If the energy-mass conservation [40] is applied, its consequence is that the mass of matter increases with time at the expense of the background energy of the graviton distribution.
A spherically symmetric emission of the liberated gravitons had been assumed in [16]. Further studies summarized in Sections 3.1, 3.6 and 3.8 indicated that an antiparallel emission with respect to the incoming graviton has to be assumed in order to avoid conflicts with energy and momentum conservation principles in closed systems. This important assumption can best be explained by referring to Figure 2. The interaction is based on the combination of a virtual graviton with momentum p * G and an incoming graviton with p G followed by the liberation of another virtual graviton in the opposite direction supplied with the excess energy T À G . Regardless of the processes operating in the immediate environment of a massive body, it must attract the mass of the combined real and virtual gravitons, which will be at rest in the reference frame of the body. The excess energy T À G is, therefore, reduced and so will be the liberation energy, as assumed in Eq. (18). The emission in Eq. (19) will give rise to a flux of gravitons with reduced energies in the environment of a body with mass M. Its spatial density is where the volume increase in the time interval Δt is The radial emission is part of the background in Eq. (6), which has a larger number density ρ G than ρ M r ðÞat most distances r of interest. Note that the emission of the gravitons from M does not change the number density or the total number of gravitons. For a certain r M , defined as the mass radius of M, it has to be because all gravitons of the background that come so close interact with the mass M in some way. The same arguments apply to a mass m 6 ¼ M and, in particular, to the electron mass m e . Therefore will be independent of the mass as long as the density of the background distribution is constant. The quantity σ G is a kind of surface mass density. The equation shows that σ G is determined by the electron mass radius r G,e , for which estimates will be provided in Sections 3.2 and 3.3. From Eqs. (16), (22) and (23), it follows that The flux of modified gravitons from M will interact with a particle of mass m and vice versa. The interaction rate in the static case can be found in Eqs. (15) and (20): A calculation with antiparallel emissions of the secondary gravitons shows that an interaction of a graviton with reduced momentum p À G provides Àp G 2 Y À Y 2 ÀÁ together with its unmodified counterpart from the opposite sides. The resulting imbalance will be if the quadratic terms in Y can be neglected for very small Y scenarios. The imbalance will cause an attractive force that is responsible for the gravitational pull between bodies with masses M and m. By comparing the force expression in Eq. (26) with Newton's law in Eq. (1), a relation between p G , Y, κ G and G N can be established through the constant G G : It can be seen that Y does not depend on the mass of a body. Since Eq. (18) allows stable processes over cosmological time scales only, if Y is very small, we assume in Figure 3 that Y < 10 À15 .
Note that the mass of a body and thus its intrinsic de Broglie frequency are not strictly constant in time, although the effect is only relevant for cosmological time scales (see lower panel of Figure 3). In addition, multiple interactions will occur within large mass conglomerations (see Sections 3.4-3.6) and can lead to deviations from Eqs. (1).
The graviton energy density remote from any masses will be where the last term is obtained from Eq. (27) with the help of Eqs. (16) and (22)- (24).
What will be the consequences of the mass accretion required by the modified model? With Eqs. (17), (19) and (27), it follows that the relative mass accretion rate of a particle with mass M will be which implies an exponential growth according to where M 0 ¼ Mt 0 ðÞ is the initial value at t 0 and the linear approximation is valid for small AtÀ t 0 ðÞ ¼ A Δt. The accretion rate is if the expression is evaluated in terms of recent parameters.
The gravitational quantities are displayed with these assumptions in a wide parameter range in Figure 3 (although the limits are set rather arbitrarily). The lower panel displays the time constant of the mass accretion. It indicates that a significant mass increase would be expected within the standard age of the Universe of the order of 1=H 0 (with the Hubble constant H 0 ) only for very small r G,e . Energies of E G ¼ 10 À40 to 10 À20 ðÞ J are assumed for the gravitons, as indicated in the upper and middle panels by different line styles. In the upper panel, the spatial number density of gravitons and the corresponding reduction parameter Y of Eq. (18) are plotted as functions of the electron mass radius r G,e . The range Y ≥ 1 (dark shading) is obviously completely excluded by the model. Even values greater than ≈ 10 À15 are not realistic (light shaded region), cf. paragraph following Eq. (27). The cosmic dark energy estimate 3:9 AE 0:4 ðÞ GeV m À3 ¼ 6:2 AE 0:6 ðÞ Â 10 À10 Jm À3 (see [44]) is marked in the second panel. It is well below the acceptable range (light shaded and unshaded regions in the middle panel). If, however, only the Y portion is taken into account in the dark energy estimate, the total energy density could be many orders of magnitude larger as shown for Y from 10 À10 to 10 À30 by short horizontal bars. In the lower panel, the mass accretion time constant and the time required for a relative mass increase of 1% are shown (on the right side in units of years). Indicated is also the Hubble time 1=H 0 as well as the lower limit of the electron mass radius (left triangle and dark shaded area) estimated from the Pioneer anomaly. The light shaded area takes smaller Pioneer anomalies into account (see Section 3.2). It is shown up to the vertical dotted line for the classical electron radius of 2.82 fm. The right triangle and the vertical solid line show the result in Section 3.3 based on the observed secular increase of the Sun-Earth distance [45] (modified from Figure 2 of [16]).
Fahr and Heyl [41] have suggested that a decay of the vacuum energy density creates mass in an expanding universe, and Fahr and Siewert [42] found a mass creation rate in accordance with Eq. (30).
The relative uncertainty of the present knowledge of the Rydberg constant is Sommerfeld's fine-structure constant. Since spectroscopic observations of the distant universe with redshifts up to z ≤ 0:5 are compatible with modern data, it appears to be reasonable to set 1 þ u r ðÞ M 0 ≥ Mt ðÞ> M 0 at least for the time interval Δt ≤ 1:6 Â 10 17 s. Any variation of R ∞ , caused by the linear dependence upon the electron mass, which has also been considered in [43], would then be below the detection limit for state-of-the-art methods.
From the emission rate and the lifetime of virtual gravitons in Eqs. (15) and (11), an estimate of their total number and energy at any time can thus be obtained for a body with mass M as and i.e. the mass of a particle would reside within the virtual gravitons.

Electrostatic fields and charged particles
Coulomb's law in Eq. (2) gives the attractive or repulsive electrostatic force between two charged particles at rest in an inertial system. Together with the electrostatic field in Eq. (3), it can be written as The electrostatic potential U E r ðÞof a charge Q, located at r ¼ 0, is for r > 0 The corresponding electrostatic field can thus be written as E Q r ðÞ¼À ∇U E r ðÞ .

Dipole interactions
Note that the dipoles in the background distribution, cf. Eq. (10), have no mean interaction energy, even in the classical theory (see, e.g. [24]). Whether this "background dipole radiation" and the "graviton radiation" are related to the dark matter (DM) and dark energy (DE) problems is of no concern here but could be an interesting speculation.
A charge Q absorbs and emits dipoles at a rate where η D and κ D are the corresponding (dipole) emission and absorption coefficients.
From energy conservation it follows that absorption and emission rates of dipoles in Eq. (38) of a body with charge Q must be equal. The momentum conservation can, in general, be fulfilled by isotropic absorption and emission processes.
The interaction processes assumed between a positively charged body and dipoles is sketched in Figure 4. A mass m Q of the charge Q has explicitly been mentioned, because the massless dipole charges are not assumed to absorb and emit any dipoles themselves. The conservation of momentum could hardly be fulfilled in such a process. In Section 3.8 we postulate, however, that gravitons interact with dipoles and thereby control their momentum and speed, subject to the condition that p G ≪ p D . The assumptions as outlined will lead to a distribution of the emitted dipoles in the rest frame of an isolated charge Q with a spatial density of where ΔV r is given in Eq. (21). The radial emission is part of the background ρ D , which has a larger number density than ρ D r ðÞat most distances r of interest. Note that the emission of the dipoles from Q does not change the number density ρ D in the environment of the charge but reverses the orientation of half of the dipoles affected.
The total number of dipoles will, of course, not be changed either. For a certain r Q , defined as the charge radius of Q, it has to be because all dipoles of the background that come so close interact with the charge Q in some way. The same arguments apply to a charge q 6 ¼ Q. Since ρ D cannot depend on either q or Q, the quantity must be independent of the charge and can be considered as a kind of surface charge density, cf. "Flächenladung" of an electron defined by Abraham [46], which is the same for all charged particles. The equation shows that σ Q is determined by the electron charge radius r e .
At this stage, this is a formal description awaiting further quantum electrodynamic studies in the near-field region of charges. It might, however, be instructive to provide a speculation for the dipole emission rate ΔN Q =Δt of a charge Q. The physical constants α, c 0 , h, ϵ 0 and G N can be combined to give a dipole emission coefficient as half the virtual dipole production rate and thus for a charge |e| a rate of Note that the dipole emission rate is fixed for a certain charge and does not depend on the particle mass. From Eqs. (38), (40) and (41), we get During a direct interaction, the dipole A À (arriving in Figure 4 from above on the right side) combines together with an identical virtual dipole with an opposite velocity vector. This postulate is motivated by the fact that it provides the easiest way to eliminate the charges and yield P ¼Àp D þ p * D ¼ 0, where p * D ¼ p D is the magnitude of the momentum vector of a virtual dipole, cf. Eq. (9). The momentum balance is neutral, and the excess energy T D is used to liberate a second virtual dipole B þ , which has the required orientation. The charge had emitted two virtual dipoles with a momentum of þp * D , each, and a total momentum of À p D þ p * D ÀÁ ¼ À2 p D was transferred to |Q|. The process can be described as a reflection of a dipole together with a reversal of the dipole momentum. The number of these direct interactions will be denoted by ΔN Q . The dipole of type A þ (arriving from above on the left side) can exchange its momentum in an indirect interaction only on the far side of the charge with an identical virtual dipole during its absorption (or destruction) phase (cf. Figure 1). The excess energy of T D is supplied to liberate a second virtual dipole B þ . The momentum transfer to the charge þ|Q| is zero. This process just corresponds to a double charge exchange. Designating the number of interactions of the indirect type with ΔÑ Q ,itis . Unless direct and indirect interactions are explicitly specified, both types are meant by the term "interaction". The virtual dipole emission rate in Figure 4 has to be i.e. the virtual dipole emission rate equals the sum of the real absorption and emission rates. The interaction model described results in a mean momentum transfer per interaction of p D without involving a macroscopic electrostatic field.
A quantitative evaluation gives the force acting on a test particle with charge q at a distance r from another particle with charge Q. This results from the absorption of dipoles not only from the background but also from the distribution emitted from Q according to Eq. (39) under the assumption of a constant absorption coefficient κ D in Eq. (38). The rate of interchanges between these charges then is which confirms the reciprocal relationship between q and Q. The equation, also very similar to Eq. (25), does not contain an explicit value for η D . It is important to realize that all interchange events between pairs of charged particles are either direct or indirect depending on their polarities and transfer of a momentum of AE2 p D or zero.
The external electrostatic potential of a spherically symmetric body C with charge Q is given in Eq. (37). Since the electrostatic force between the charged particles C and D is typically many orders of magnitude larger than the gravitational force, we only take the electrostatic effects into account in this section and neglect the gravitational interactions.
In order to have a well-defined configuration for our discussion, we will assume that body C with mass m C has a positive charge Q > 0 and is positioned at a distance r beneath body D (mass m D ) with either a charge þ|q| in Figure 5 or À|q| in Figure 6. Only the processes near body D are shown in detail.
The interaction rates of dipoles with bodies C and D in Eq. (47) (the same for both bodies even if |Q| 6 ¼ |q|) and the momentum transfers indicated in Figures 5 and 6, respectively, lead to a norm of the momentum change rate for bodies C and D of 13 Together with this leads, depending on the signs of the charges Q and q, to a repulsive or an attractive electrostatic force between C and D in accordance with Coulomb's law in Eq. (2). Important questions are related to the energy T D and momentum p D of the dipoles and, even more, to their energy density in space. Eqs. (9), (40), (41) and (44) together with Eq. (49) allow the energy density to be expressed by This quantity is independent of the dipole energy. It takes into account all dipoles (whether their distribution is chaotic or not). Should the energy density vary in space and/or time, the surface charge density σ Q must vary as well.
If we assume that the electron charge radius r Q in Eq. (41) equals the classical electron radius r e ¼ 2:82 fm, then an energy density of ϵ D ¼ 1:45 Â 10 29 Jm À3 (very high compared to the present cosmic dark energy estimate) follows from Eq. (50). The dipole density ρ D ¼ 7:95 Â 10 56 m À3 in Eq. (40) is also very high, leading to a dipole energy of T D ¼ 1:83 Â 10 À28 J. If, on the other hand, we identify the dipole distribution with DM with an estimated energy density of 2:48 Â 10 À10 Jm À3 and require that the dipole energy density corresponds to this value, then extreme values follow for r Q ¼ 13:9 μm, ρ D ¼ 3:28 Â 10 37 m À3 and T D ¼ 7:55 Â 10 À48 J.

Applications of impact models
The detection of gravitons and dipoles with the expected properties would, of course, be the best verification of the proposed models. Lacking this, indirect support can be found through the application of the models with a view to describe physical processes successfully for specific situations.

Gravitational potential energy
As mentioned in Section 1, the study of the potential energy problem [47] had been motivated by the remark that the potential energy is rather mysterious [9]. 2 It led to the identification of the "source region" of the potential energy for the special case of a system with two masses M E and M M subject to the condition M E ≫ M M .An attempt to generalize the study without this condition required either violations of the energy conservation principle as formulated by von Laue [49] for a closed system or a reconsideration of an assumption we made concerning the gravitational interaction process in [16]. The change necessary to comply with the energy conservation principle has been discussed in Section 2.4. A generalization of the potential energy concept for a system of two spherically symmetric bodies A and B with masses m A and m B without the above condition could then be formulated [50].
We will again exclude any further energy contributions, such as rotational or thermal energies, and make use of the fact that the external gravitational potential of a spherically symmetric body of mass M and radius r in Eq. (5) is that of a corresponding point mass at the centre.
The energy E m and the momentum p of a free particle with mass m moving with a velocity v relative to an inertial reference system are related by where p is the momentum vector [40,51]. For an entity in vacuum with no rest mass (m ¼ 0), such as a photon [15,52,53], the energy-momentum relation in Eq. (51) reduces to In [50] we assume that two spherically symmetric bodies A and B with masses m A and m B , respectively, are placed in space remote from other gravitational centres at a distance of r þ Δr reckoned from the position of A. Initially both bodies are at rest with respect to an inertial reference frame represented by the centre of gravity of both bodies. The total energy of the system then is with Eq. (51) for the rest energies and with Eq. (5) for the potential energy The evolution of the system during the approach of A and B from r þ Δr to r can be described in classical mechanics. According to Eq. (48), the attractive force between the bodies during the approach is approximately constant for r ≫ Δr > 0, resulting in accelerations of b A ¼ |K G r ðÞ |=m A and b B ¼À|K G r ðÞ |=m B , respectively. Since the duration Δt of the free fall of both bodies is the same, the approach of A and B can be formulated as Δt showing that s A m A ¼Às B m B , i.e. the centre of gravity stays at rest. Multiplication of Eq. (55) by |K G r ðÞ | gives the corresponding kinetic energy equation The kinetic energies 3 T A and T B should, of course, be the difference of the potential energy term in Eq. (54) at distances of r and r þ Δr. We find indeed for small Δr with Newton's law in Eq. (1): We may now ask the question, whether the impact model can provide an answer to the potential energy "mystery" in a closed system. Since the model implies a secular increase of mass of all bodies, it obviously violates a closed-system assumption. The increase is, however, only significant over cosmological time scales, and we can neglect its consequences in this context. A free single body will, therefore, still be considered as a closed system with constant mass. In a two-body system, both masses m A and m B will be constant in such an approximation, but now there are gravitons interacting with both masses.
The number of gravitons travelling at any instant of time from one mass to the other can be calculated from the interaction rate in Eq. (25) multiplied by the travel time r=c 0 : The same number is moving in the opposite direction. The energy deficiency of the interacting gravitons with respect to the corresponding background then is together with Eqs. (18) and (27) for each body The last term shows-with reference to Eq. (57)-that the energy deficiency ΔE G equals half the potential energy of body A at a distance r from body B and vice versa.
We now apply Eq. (59) and calculate the difference of the energy deficiencies for separations of r þ Δr and r for interacting gravitons travelling in both directions and get Consequently, the difference of the potential energies between r þ Δr and r in Eq. (57) is balanced by the difference of the total energy deficiencies.
The physical processes involved can be described as follows: 1. The number of gravitons on their way for a separation of r þ Δr is smaller than that for r, because the interaction rate depends on r À2 according to Eq. (48), whereas the travel time is proportional to r.

2.
A decrease of r þ Δr to r during the approach of A and B increases the number of gravitons with reduced energy. 3 Eqs. (51) and (52)  3. The energies liberated by energy reductions are available as potential energy and are converted into kinetic energies of bodies A and B.
4. With Eqs. (51) and (52) and the approximations in Footnote 3, it follows that the sum of the kinetic energies T A and T B , the masses A and B, plus the total energy deficiencies of the interacting gravitons can indeed be considered to be a closed system as defined by von Laue [49].

Electrostatic potential energy.
In this section we will discuss the electrostatic aspects of the potential energy. The energy density of an electrostatic field E outside of charges is given by cf., e.g. [24,54]. Applying Eq. (61) to a plane-plate capacitor with an area F,a plate separation b and charges AE|Q| on the plates, the energy stored in the field of the capacitor turns out to be With a potential difference ΔU E ¼ |E | b and a charge of Q ¼ ε 0 |E | F (increased incrementally to these values), the potential energy of Q at ΔU E is The question as to where the energy is actually stored, [54] answered by showing that both concepts implied by Eqs. (62) and (63) are equivalent. Can the impact model provide an answer for the electrostatic potential energy in a closed system, where dipoles are interacting with two charged bodies? This question we posed in [55]: the number of reversed dipoles travelling at any instant of time from a charge Q > 0toq in Figures 5 and 6 can be calculated from the interaction rate in Eq. (47) multiplied by a travel time Δt ¼ r=c 0 : The same number of dipoles is moving in the opposite direction from q to Q. From Eqs. (9), (49) and (64), we can determine the total energy of the reversed dipoles: It is equal to the absolute value of the electrostatic potential energy of a charge q at the electrostatic potential U E r ðÞin Eq. (37) of a charge Q. The evolution of the system is similar to that of the gravitational case in Section 3.1.1; however, attraction and repulsion have to be considered during the approach or separation of bodies C and D. The initial distance between C and D be r, when both bodies are assumed to be at rest, and changes to r AE Δr by the repulsive or attractive force K E r ðÞbetween the charges given by Coulomb's law in Eq. (2) The force is approximately constant for r ≫ Δr > 0 causing accelerations of b D ¼ K E r ðÞ =m D and b C ¼ÀK E r ðÞ =m C , respectively. Since the duration Δt of the motions of both bodies is the same, the separation (upper sign) or approach (lower sign) of C and D can be formulated as follows: Comparing the second term of the equation with the last one, it can be seen that s D m D ¼Às C m C , i.e. the centre of gravity stays at rest. Multiplication of Eq. (66) by K E r ðÞgives a good estimate of the corresponding kinetic energy: where v D ¼ b D Δt and v C ¼ b C Δt are the speeds of the bodies, when the distances r AE Δr between C and D are attained. The sum of the kinetic energies T C and T D must, of course, be equal to the difference of the electrostatic potential energy at distances of r and r AE Δr: The variations of the number of ΔN Q,q r ðÞdipoles in Eqs. (58) and (65) during the separation or approach of bodies C and D from r to r AE Δr are δN Q,q r, Δr ðÞ ¼ ΔN Q,q r AE Δr ðÞ À ΔN Q,q r ðÞ The number of reversed dipoles thus decreases during the separation of C and D in Figure 5. The corresponding energy variation with positive q is δE Q,q r, Δr ðÞ ¼ 2 p D c 0 δN Q,q r, Δr ðÞ ¼ À Δr |q| Q 4πε 0 r 2 < 0, cf. Eq. (65). The energy of the reversed dipoles thus decreases by the amount that fuels the kinetic energy in Eq. (67).
In the opposite case with negative q and attraction, it can be seen from Figure 6 that the increased number of reversed dipoles is actually leaving the system without momentum exchange and is lost. The momentum difference, therefore, is again negative δP Q,q r, Δr ðÞ ¼ À 2p D δN Q,q r, Δr ðÞ (71) and so is the energy of the reversed dipoles confined in the system: δE Q,q r, Δr ðÞ ¼ δP Q,q r, Δr ðÞ c 0 ¼À|q| Δr Q 4 πε 0 r 2 < 0: The electrostatically bound two-body system thus is a closed system in the sense defined by von Laue [49], slowly evolving in time during the movements of bodies C and D. The potential energy converted into kinetic energy stems from the modified dipole distributions.

Pioneer anomaly
Anomalous frequency shifts of the Doppler radio-tracking signals were detected for both Pioneer spacecraft [56]. The observations of Pioneer 10 (launched on 2 March 1972) published by the Pioneer Team will be considered during the time interval t 1 À t 0 ≈ 11:55 years ¼ 3:645 Â 10 8 s between 3 January 1987 and 22 July 1998, while the spacecraft was at heliocentric distances between r 0 ¼ 40 ua and r 1 ¼ 70:5 ua. The Pioneer team took into account all known contributions in calculating a model frequency ν mod t ðÞwhich was based on a constant clock frequency f 0 at the terrestrial control stations. Observations at times t ¼ t 0 þ Δt then indicated a nearly uniform increase of the observed frequency shift with respect to the expected one of with _ f ¼ 5:99 Â 10 À9 Hz s À1 [57]. The observations of the anomalous frequency shifts could, in principle, be interpreted as a deceleration of the heliocentric spacecraft velocity by a p ¼À 8:74 AE 1:33 ðÞ Â 10 À10 ms À2 : However, no unknown sunward-directed force could be identified [58]. Alternatively, a clock acceleration at the ground stations of could explain the anomaly. A true trajectory anomaly together with an unknown systematic spacecraft effect was considered to be the most likely interpretation by Anderson et al. [59]. Although Turyshev et al. [60] later concluded that thermal recoil forces of the spacecraft caused the anomaly of Pioneer 10, the discussion in the literature continued.
Assuming an atomic clock acceleration, a constant reference frequency f 0 for the calculation of ν mod t ðÞis not appropriate. Consequently, we modified in [61] the equation equivalent to Eq. (73) with and to Our gravitationally impact model [16] summarized in Section 2.4 leads to a secular mass increase of massive particles in Eq. (30). Consequently the Rydberg constant in Eq. (32) would increase in a linear approximation with the electron mass m e according to (80) resulting in frequency increases of atomic clocks with time. They give rise to the clock acceleration in Eq. (77), if we assume a t ¼ A. The most likely values of r G,e in Figure 3 range from 2:04 Â 10 À4 pm to 2.82 fm, the classical electron radius, corresponding with Eq. (31) to A H ¼ 2:43 Â 10 À18 s À1 ≈ H 0 , the Hubble constant, and A ≈ 1:3 Â 10 À20 s À1 . Within the uncertainty margins, the high value agrees with a t in Eq. (75) and would quantitatively account for the Pioneer frequency shift. Should the anomaly be much less pronounced, because thermal recoil forces decelerate the spacecraft, the range of r G,e in Figure 3 could accommodate smaller values of a t as well.

Sun-Earth distance increase
A secular increase of the mean Sun-Earth distance with a rate of 15 AE 4 ðÞ m per century had been reported using many planetary observations between 1971 and 2003 [45]. Neither the influence of cosmic expansion nor a time-dependent gravitational constant seems to provide an explanation [62]. As our impact model summarized in Section 2.4 leads to a secular mass increase according to Eq. (30) of all massive bodies fuelled by a decrease in energy of the background flux of gravitons, it allowed us to formulate a quantitative understanding of the effect within the parameter range of the model [63].
The value of the astronomical unit is defined by the International Astronomical Union (IAU) and the Bureau International des Poids et Mesure [64] as 1 ua ¼ 1:495978 707 Â 10 11 m (exact). The mean Sun-Earth distance r E is known with a standard uncertainty of (3 to 6) m [65][66][67].
Considering this uncertainty, the measurement of a change rate of is difficult but feasible as relative determination. A circular orbit approximation had been considered, because the mean value of r E is of interest: This follows from equating the gravitational attraction, cf. Eq. (1), and the centrifugal force with v E , the tangential orbital velocity of the Earth, where the heliocentric gravitational constant is μ ⊙ ¼ 1:327 124 400 42 Â 10 20 m 3 s À2 (IAU) and the mass of the Sun M ⊙ ¼ 1:988 42 Â 10 30 kg.
We now consider Eq. (82) not only for t 0 but also at t ¼ t 0 þ Δt assuming constant G N as well as constant v E . The latter assumption is justified by the fact that any uniformly moving particle does not experience a deceleration. It implies an increase of the momentum together with the mass accumulation of the Earth. The apparent violation of the momentum conversation principle can be resolved by considering the accompanying momentum changes of the graviton distribution. A detailed discussion of this aspect is given in Section 3 of [16]. 21 Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org /10.5772/intechopen.86744 From Eqs. (30) and (82), it follows and With the help of Eqs. (31) and (81), the electron mass radius can now be calculated. The result is close to the classical electron radius The relative accumulation rate deduced from the observations of r E finally becomes A ¼ A ua ≈ 3:2 Â 10 À20 s À1 (see Figure 3).

Secular perihelion advances in the solar system
Multiple applications of the interaction process described in Section 2.4 can produce gravitons with reduction parameters greater than Y in large mass conglomerations-within the Sun in this section. The proportionality of the linear term in the binomial theorem with the exponent n in 1 À Y ðÞ n ≈ 1 À nY for Y ≪ 1 (87) suggests that a linear superposition of the effects of multiple interactions will be a good approximation, if n is not too large. Energy reductions according to Eq. (18) are therefore not lost, as claimed by Drude [21], but they are redistributed to other emission locations within the Sun. This has two consequences: (1) the total energy reduction is still dependent on the solar mass, and (2) since emissions from matter closer to the surface of the Sun in the direction of an orbiting object is more likely to escape into space than gravitons from other locations, the effective gravitational centre should be displaced from the centre of the Sun towards that object.
Using published data on the secular perihelion advances of the inner planets Mercury, Venus, Earth and Mars of the solar system and the asteroid Icarus, we found that the effective gravitational centre is displaced from the centre of the Sun by approximately ρ ¼ 4400 m [68]. Since an analytical derivation of this value from the mass distribution of the Sun was beyond the scope of the study, future investigations need to show that the modified process with directed secondary graviton emission can quantitatively account for such a displacement.

Earth flybys
Several Earth flyby manoeuvres indicated anomalous accelerations and decelerations and led to many investigations without reaching a solution of the problem (see recent reviews by Anderson et al. [69] and Nieto and Anderson [70]). Since there is general agreement that the anomaly is only significant near perigee, we discuss here the seven passages at altitudes below 2000 km listed in Table 1 of Acedo [71]. Three of them (Galileo I, NEAR and Rosetta) we have studied in [72] assuming the gravitational impact model of Section 2.4 and multiple interactions. As in Section 3.4, the multiple interactions result in a deviation ρ of the effective gravitational centre from the geometric centre. We obtained for Galileo, NEAR and Rosetta ρ ≈ 1:3m , 3 :9 m and 0:5 m, respectively. The study had been conducted assuming a spherically symmetric emission of liberated gravitons mentioned in Section 2.4.
With the assumption of an antiparallel emission, we have repeated the analysis and found ρ ≈ 2 m for all spacecraft, provided the origin of ρ is shifted by approximately À0:6 m in the direction of the perigee of Galileo I, þ1:9 m for NEAR and À1:5 m for Rosetta. Moreover, it was possible to model the decelerations of Galileo II on 8 December 1992 with a shift of À3:4m , of Cassini on 18 August 1999 with À2:7 m and the null result for Juno on 9 October 2013 with À2m .
An origin offset of þ3:4 m opposite to the Cassini perigee could to a first approximation achieve all apparent shifts taking the geographic coordinates of the various flybys into account. A detailed study would have to consider in addition the Earth gravitational model.

Juno Jupiter flybys
Juno was inserted into an elliptical orbit around Jupiter on 4 July 2016 with an orbital period of 53.5 days. Acedo et al. [74] studied the first and the third orbit with a periapsis of "4200 km over the planet top clouds". "A significant radial component was found and this decays with the distance to the center of Jupiter as expected from an unknown physical interaction … . The anomaly shows an asymmetry among the incoming and outgoing branches of the trajectory …". The radial component is shown in their Figure 6 in the time interval t ¼À 180 to þ 180 ðÞ min around perijove for the first and third Juno flyby. The peak anomalous outward accelerations shown are in both cases: δa ¼ 7m ms À2 at t ≈ À 15 min and δa ¼ 6m ms À2 at t ≈ þ 17 min.
We applied the multiple-interaction concept of the previous Sections 3.4 and 3.5.1 in [75] and found that offsets of ρ ≈ (8 to 27) km of the gravitational from the geometric centre are required to model the acceleration in Figure 7, which is in good agreement with the observations during the Juno Jupiter flybys. The variation of ρ could be modelled by an ellipsoidal displacement of the gravitational centre offset in the direction of a flyby position near t ¼À10 min.

Rotation velocities of spiral galaxies
The rotation velocities of spiral galaxies are difficult to reconcile with the Keplerian motions, if only the gravitational effects of the visible matter are taken into account, cf. [76,77]. Dark matter had been proposed by Oort [78] and Zwicky [79] in order to understand several velocity anomalies in galaxies and clusters of galaxies. A Modification of the Newtonian Dynamics (MOND) has been introduced by Milgrom [80] that assumes a modified gravitational interaction at low acceleration levels.
The impact model of gravitation in Section 2.4 is applied to the radial acceleration of disk galaxies [81]. The flat velocity curves of NGC 7814, NGC 6503 and M 33 are obtained without the need to postulate any dark matter contribution. The concept explained below provides a physical process that relates the fit parameter of the acceleration scale defined by McGaugh et al. [82] to the mean-free path length of gravitons in the disks of galaxies. It may also provide an explanation for MOND.
McGaugh [83] has observed a fine balance between baryonic and dark mass in spiral galaxies that may point to new physics for DM or a modification of gravity. Fraternali et al. [84] have also concluded that either the baryons dominate the DM or the DM is closely coupled with the luminous component. Salucci and Turini [85] have suggested that there is a profound interconnection between the dark and the stellar components in galaxies.
The large baryonic masses in galaxies will cause multiple interactions of gravitons with matter if their propagation direction is within the disk. For each interaction the energy loss of the gravitons is assumed to be YT G (for details see Section 2.3 of [16]). The important point is that the multiple interactions occur only in the galactic plane and not for inclined directions. An interaction model is designed indicating that an amplification factor of approximately two can be achieved by six successive interactions. An amplification occurs for four or more interactions. The process works, of course, along each diameter of the disk and leads to a twodimensional distribution of reduced gravitons.
The multiple interactions do not increase the total reduction of graviton energy, because the number of interactions is determined by the (baryonic) mass of the gravitational centre according to [16]. A galaxy with enhanced gravitational acceleration in two dimensions defined by the galactic plane will, therefore, have a reduced acceleration in directions inclined to this plane.

Light deflection and Shapiro delay
The deflection of light near gravitational centres is of fundamental importance. For a beam passing close to the Sun, Soldner [86] and Einstein [87] obtained a deflection angle of 0:87 00 under the assumption that radiation would be affected in the same way as matter. Twice this value was then derived in the framework of the GTR [2] 4 and later by Schiff [88] using the equivalence principle and STR. The high value was confirmed during the total solar eclipse in 1919 for the first time [89]. This and later observations have been summarized by Mikhailov [90] and combined to a mean value of approximately 2″.
The deflection of light has also been considered in the context of the gravitational impact model summarized in Section 2.4. As a secular mass increase of matter was a consequence of this model, the question arises on how the interaction of gravitons with photons can be understood, since the photon mass is in all likelihood zero. 5 An initial attempt at solving that problem has been made in [91], where we assumed that a photon stimulates an interaction with a rate equal to its frequency ν ¼ E ν =h. It is summarized here under the assumption of an antiparallel re-emission, both for massive particles and photons.
A physical process will then be outlined that provides information on the gravitational potential U at the site of a photon emission [95]. This aspect had not been covered in our earlier paper on the gravitational redshift [96].
Interactions between massive bodies have been treated in [16] with an absorption rate of half the intrinsic de Broglie frequency of a mass, because two virtual gravitons have to be emitted for one interaction. The momentum transfer to a photon will thus be twice as high as to a massive body with a mass equivalent to E ν =c 2 0 . We then apply the momentum conservation principle to photon-graviton pairs in the same way as to photons [73] and can write after a reflection of p G We assume, applying Eq. (88) with p G ≪ p ν ¼ | p ν |, that under the influence of a gravitational centre relevant interactions occur on opposite sides of a photon with p G and p G 1 À Y ðÞ transferring a net momentum of 2 Yp G . Note, in this context, that the Doppler effect can only operate for interactions of photons with massive bodies [97,98]. Consequently, there will be no energy change of the photon, because both gravitons are reflected with constant energies under these conditions, and we can write for a pair of interactions: where p 0 ν is the photon momentum after the events. If p ν and a component of 2 Yp G are pointing in the same direction, it is c 0 < c, the speed is reduced; an antiparallel direction leads to c 0 > c. Note that this could, however, not result in 4 It is of interest in the context of this paper that Einstein employed Huygens' principle in his calculation of the deflection. 5 A zero mass of photons follows from the STR and a speed of light in vacuum c 0 constant for all frequencies. Einstein [52] used "Lichtquant" for a quantum of electromagnetic radiation; the term "photon" was introduced by Lewis [15]. With various methods the photon mass could be constrained to m ν < 10 À49 kg [92,93] or even to m ν < 6:3 Â 10 À53 kg [94].
The deflection of light by gravitational centres according to the GTR [2] and its observational detection by Dyson et al. [89] leave no doubt that a photon is deflected by a factor of two more than the expected relative to a corresponding massive particle. Since in our concept the interaction rate between photons and gravitons is twice as high as for massive particles of the same total energy, the reflection of a graviton from a photon with a momentum of 1 À Y ðÞ p G must also be antiparallel to the incoming one, i.e. a momentum of À2 Yp G will be transferred. Otherwise the correct deflection angle for photons cannot be obtained. This modified interaction process has one further important advantage: the reflected graviton can interact with the deflecting gravitational centre and transfers 2 Yp G -through the process outlined in the paragraph just before Eq. (48)-in compliance with the momentum conservation principle. In the old scheme, the violation of this principle had no observational consequences, because of the extremely large masses of relevant gravitational centres, but the adherence to both the momentum and energy conservation principles is very encouraging and clearly favours the new concept.
Basically the same arguments are relevant for the longitudinal interaction between photons and gravitons. The momentum transfer per interaction will be doubled, but the gravitational absorption coefficient will be reduced by a factor of two. Together with an increased graviton density, all quantities and results are the same as before. However, a detailed analysis shows that the momentum conservation principle is now also adhered to.

Gravitational redshift
The gravitational potential U at a distance r from a spherical body with mass M is constraint in the weak-field approximation for nonrelativistic cases by cf. [73]. A definition of a reference potential in line with this formulation is The study of the gravitational redshift, predicted for solar radiation by Einstein [109], is still an important subject in modern physics and astrophysics [95,96,[110][111][112][113][114]. This can be exemplified by two conflicting statements. Wolf et al. [10] write: "The clock frequency is sensitive to the gravitational potential U and not to the local gravity field g ¼ ∇U". Whereas it is claimed by Müller et al. [11]: "We first note that no experiment is sensitive to the absolute potential U". Support for the first alternative can be found in many publications [49,88,95,96,109,[115][116][117], but it is, indeed, not obvious how an atom can locally sense the gravitational potential U. Experiments on Earth, in space and in the Sun-Earth system, cf. [118][119][120][121][122][123], however, have quantitatively confirmed in the static weak field approximation a relative frequency shift of where ν 0 is the frequency of the radiation emitted by a certain transition at U 0 and ν is the observed frequency there, if the emission caused by the same transition had occurred at a potential U.
Since Einstein discussed the gravitational redshift and published conflicting statements regarding this effect in [2,87,109], the confusion could still not be cleared up consistently, cf., e.g. [124,125]. In most of his publications Einstein defined clocks as atomic clocks. Initially he assumed that the oscillation of an atom corresponding to a spectral line might be an intra-atomic process, the frequency of which would be determined by the atom alone. Scott [126] also felt that the equivalence principle and the notion of an ideal clock running independently of acceleration suggest that such clocks are unaffected by gravity. Einstein [2] later concluded that clocks would slow down near gravitational centres, thus causing a redshift.
The question whether the gravitational redshift is caused by the emission process (case a) or during the transmission phase (case b) is nevertheless still a matter of recent debates. Proponents are, e.g. of (a) Schiff [88], Okun et al. [116], Møller [127], Cranshaw et al. [128] and Ohanian [129], and of (b) Hay et al.
[130], Straumann [131], Randall [132] and Will [133]. It is surprising that the same team of experimenters albeit with different first authors published different views in [128,130] on the process of the Pound-Rebka-Experiment.
Pound and Snider [120] and Pound [134] pointed out that this experiment could not distinguish between the two options, because the invariance of the velocity of the radiation had not been demonstrated.
Einstein [13] emphasized that for an elementary emission process, not only the energy exchange but also the momentum transfer is of importance; see also [12,46,97]. Taking these considerations into account, we formulated a photon emission process at a gravitational potential U assuming that: 1. The atom cannot sense the potential U, in line with the original proposal by Einstein [87,109], and initially emits the same energy ΔE 0 at U < 0 and U 0 ¼ 0.
2. It also cannot directly sense the speed of light at the location with a potential U.
3. As the local speed of light is, however, cU ðÞ 6 ¼ c 0 , a photon having an energy of ΔE 0 and a momentum p 0 is not able to propagate. The necessary adjustments of the photon energy and momentum as well as the corresponding atomic quantities then lead in the interaction region to a redshift consistent with hν ¼ ΔE 0 1 þ U=c 2 0 ÀÁ and observations [96].
As outlined in Section 3.7, there is general agreement in the literature that the local speed of light is in line with Eq. (95) in Section 3.7. It has, however, to be noted that the speed cU ðÞ was obtained for a photon propagating from U 0 to U, and, therefore, the physical process which controls the speed of newly emitted photons at a gravitational potential U is not yet established.
An attempt to do that will be made by assuming an aether model. Before we suggest a specific aether model, a few statements on the aether concept in general should be mentioned. Following Michelson and Morley [135] famous experiment, Einstein [51,109] concluded that the concept of a light aether as carrier of the electric and magnetic forces is not consistent with the STR. In response to critical remarks by Wiechert [136], cf. Schröder [137] for Wiechert's support of the aether, von Laue [138] wrote that the existence of an aether is not a physical, but a philosophical problem, but later differentiated between the physical world and its mathematical formulation [139]: a four-dimensional "world" is only a valuable mathematical trick; a deeper insight, which some people want to see behind it, is not involved.
In contrast to his earlier statements, Einstein said at the end of a speech in Leiden that according to the GTR, a space without aether cannot be conceived [140] and even more detailed thus one could instead of talking about "aether" as well discuss the "physical properties of space". In theoretical physics we cannot do without aether, i.e. a continuum endowed with physical properties [141]. Michelson et al. [142] confessed at a meeting in Pasadena in the presence of H.A. Lorentz that he clings a little to the aether, and Dirac [143] wrote in a letter to Nature that there are good reasons for postulating an aether.
In [17] we proposed an impact model for the electrostatic force based on massless dipoles. The vacuum is thought to be permeated by these dipoles that are, in the absence of electromagnetic or gravitational disturbances, oriented and directed randomly propagating along their dipole axis with a speed of c 0 . There is little or no interaction among them. We suggest to identify the dipole distribution postulated in Section 2.5 with an aether. Einstein's aether mentioned above may, however, be more related to the gravitational interactions, cf. [144]. In this case, we have to consider the graviton distribution as another component of the aether.
We now assume that an individual dipole interacts with gravitons in the same way as photons in Eq. (89), i.e. according to where T D and p D refer to the energy and momentum of a dipole. The condition p D ≫ p G , cf. Eq. (88), is fulfilled in the range from Y ≈ 10 À22 to 10 À15 for all r e ≤ 2:82 fm (see Section 2.5 and Figure 3).
We can then modify Eqs. (90)-(94) by changing ν to D and find that Eqs. (95) and (98) are also valid for dipoles with a speed of c 0 for U 0 ¼ 0.
Considering that many suggestions have been made to describe photons as solitons, e.g. in [145][146][147][148][149][150], we also propose that a photon is a soliton propagating in the dipole aether with a speed of cU ðÞ , cf. Eq. (98), controlled by the dipoles moving in the direction of propagation of the photon. The dipole distribution thus determines the gravitational index of refraction, cf. Eq. (95), and consequently the speed of light cU ðÞ at the potential U. This solves the problem formulated in relation to Eq. (98) and might be relevant for other phenomena, such as gravitational lensing and the cosmological redshift, cf., e.g. [151]. Should the speculation in Section 2.5.2 be taken seriously that the dipole distribution corresponds to DM, it has to be much more evenly distributed than previously thought [152]. The light deflection would then be caused by gravitationally induced index of refraction variations.

Discussion and conclusions
With Newton's law of gravitation as starting point, the ideas presented in Section 2.4 allow an understanding of far-reaching gravitational force between massive particles as local interactions of hypothetical massless gravitons travelling with the speed of light in vacuum. The gravitational attraction leads to a general mass accretion of massive particles with time, fuelled by a decrease of the graviton energy density in space. The physical processes during the conversion of gravitational potential energy into kinetic energy have been described for two bodies with 29 Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org /10.5772/intechopen.86744 masses m A and m b , and the source of the potential energy could be identified in Section 3.1.1. In order to avoid conflicts with energy and momentum conservation, we had to modify a detail of the interaction process in Eq. (26), i.e. assume an antiparallel emission of the secondary graviton with respect to the incoming one.
Multiple interactions of gravitons leading to shifts of the effective gravitational centre of a massive body from the "centre of gravity" are treated in Sections 3.4-3.6 taking the modified concept into account. The interaction of gravitons with photons in Section 3.7 had to be modified as well, but the modification did not change the results, with the exception that now, both the energy and momentum conservation principles are fulfilled.
Our main aim in Section 3.8 was to identify a physical process that leads to a speed cU ðÞ of photons controlled by the gravitational potential U. This could be achieved by postulating an aether model with moving dipoles, in which a gravitational index of refraction n G U ðÞ ¼ c 0 =cU ðÞ regulates the emission and propagation of photons as required by energy and momentum conservation principles. The emission process thus follows Steps (1) to (3) in Section 3.8, where the local speed of light is given by the gravitational index of refraction n. In this sense, the statement that an atom cannot detect the potential U by Müller et al. [11] is correct; the local gravity field g, however, is not controlling the emission process.
A photon will be emitted by an atom with appropriate energy and momentum values, because the local speed of light requires an adjustment of the momentum. This occurs in the interaction region between the atom and its environment as outlined in Step 3.
In the framework of a recently proposed electrostatic impact model in [17], the physical processes related to the variation of the electrostatic potential energy of two charged bodies have been described, and the "source region" of the potential energy in such a system could be identified and is summarized in Section 3.1.2.
Sotiriou et al. [125] made a statement in the context of gravitational theories in "A no-progress report": "[ … ] it is not only the mathematical formalism associated with a theory that is important, but the theory must also include a set of rules to interpret physically the mathematical laws". With this goal in mind, we have presented our ideas on the gravitational and electrostatic interactions. gravitational and inertial mass. Physical Review. 1964;135:B1049-B1056