Abstract

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.

Keywords: gravitation, electrostatics, potential energies, gravitational redshift and anomalies, secular mass increase

### 1. 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

$$\mathbf{K}\_{\rm G}(r) = -\frac{G\_{\rm N}M\hat{r}}{r^2}m,\tag{1}$$

where <sup>G</sup><sup>N</sup> <sup>¼</sup> <sup>6</sup>:67408 31 ð Þ� <sup>10</sup>�<sup>11</sup> <sup>m</sup><sup>3</sup> kg�<sup>1</sup> <sup>s</sup>�<sup>2</sup> is the constant of gravity<sup>1</sup> , 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.

<sup>1</sup> This value and those of other constants (except h, Planck's constant, and e, charge of electron; cf. page 4 and SI, 9th edition 2019) are taken from CODATA 2014 [1].

In close analogy, Coulomb's law yields the force of the electrostatic interaction between particles C and D with charges Q and q, respectively:

$$\mathbf{K}\_{\rm E}(r) = \frac{Q\,\hat{r}}{4\,\pi r^2 \,\varepsilon\_0} \,\,\mathrm{q},\tag{2}$$

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

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

Both concepts are required for a description of the gravitational redshift in terms

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<sup>0</sup> 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

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:

<sup>ρ</sup><sup>G</sup> <sup>¼</sup> <sup>Δ</sup>N<sup>G</sup>

Constraints on the energy spectrum of the gravitons will be considered in later

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.

<sup>Δ</sup><sup>V</sup> : (6)

T<sup>G</sup> ¼ ∣pG∣c<sup>0</sup> ¼ p<sup>G</sup> c<sup>0</sup> (7)

similar in line with the similarity of Newton's and Coulomb's laws.

S ¼ �2, if indications to that effect are taken into account, cf. [22].

sections. At this stage we define a mean energy of

2.2 Definitions of dipoles

67

for a massless graviton with a momentum vector of pG.

with related topics.

of physical processes in Section 3.8.

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

2.1 Definitions of gravitons

where <sup>ε</sup><sup>0</sup> <sup>¼</sup> <sup>8</sup>:854187 817 … � <sup>10</sup>�<sup>12</sup> F m�<sup>1</sup> is the electric constant in vacuum. Here charges with opposite signs lead to an attraction and with equal signs to a repulsion.

$$E\_Q(r) = \frac{Q\,\hat{r}}{4\,\pi r^2 \,\varepsilon\_0} \tag{3}$$

is the classical electrostatic field of a charge Q. For two electrons, e.g. the ratio of the gravitational and electrostatic forces is

$$R\_{\rm G}^{\rm E} = \frac{|\mathbf{K}\_{\rm E}(r)|}{|\mathbf{K}\_{\rm G}(r)|} = 4.16574 \times 10^{42} \,\text{.}\tag{4}$$

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

$$U(r) = -\frac{G\_\text{N}\mathbf{M}}{r} \tag{5}$$

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.

#### 2. 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 <sup>c</sup><sup>0</sup> <sup>¼</sup> 299792458 m s�<sup>1</sup> in vacuum [12–15].

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

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.

#### 2.1 Definitions of gravitons

In close analogy, Coulomb's law yields the force of the electrostatic interaction

where <sup>ε</sup><sup>0</sup> <sup>¼</sup> <sup>8</sup>:854187 817 … � <sup>10</sup>�<sup>12</sup> F m�<sup>1</sup> is the electric constant in vacuum. Here charges with opposite signs lead to an attraction and with equal signs to a

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

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

U rð Þ¼� <sup>G</sup>N<sup>M</sup>

These remarks and disputes motivated us to think about electrostatic and grav-

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 <sup>c</sup><sup>0</sup> <sup>¼</sup> 299792458 m s�<sup>1</sup> in vacuum [12–15].

r

Q r^ 4π r<sup>2</sup> ε<sup>0</sup>

> Q r^ 4π r<sup>2</sup> ε<sup>0</sup>

q, (2)

<sup>¼</sup> <sup>4</sup>:<sup>16574</sup> � <sup>10</sup><sup>42</sup>: (4)

(3)

(5)

between particles C and D with charges Q and q, respectively:

is the classical electrostatic field of a charge Q.

RE

energy is a rather mysterious quantity to begin with … ".

itational fields and the problems related to the potential energies.

atomic clock is caused by the gravitational potential

2. Gravitational and electrostatic interactions

or by the local gravity field g ¼ ∇U.

66

<sup>G</sup> <sup>¼</sup> <sup>∣</sup>KEð Þ<sup>r</sup> <sup>∣</sup> ∣KGð Þr ∣

repulsion.

Planetology - Future Explorations

KEð Þ¼ r

E<sup>Q</sup> ð Þ¼ r

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<sup>0</sup> 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 ¼ �2, 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:

$$
\rho\_{\rm G} = \frac{\Delta N\_{\rm G}}{\Delta V}.\tag{6}
$$

Constraints on the energy spectrum of the gravitons will be considered in later sections. At this stage we define a mean energy of

$$T\_{\mathbf{G}} = |\mathbf{p}\_{\mathbf{G}}| c\_0 = p\_{\mathbf{G}} c\_0 \tag{7}$$

for a massless graviton with a momentum vector of pG.

#### 2.2 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.

#### Planetology - Future Explorations

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 �∣q∣ will be assumed, where q might or might not be identical to the elementary charge <sup>e</sup> <sup>¼</sup> <sup>1</sup>:<sup>602176634</sup> � <sup>10</sup>�<sup>19</sup> C (exact) [25].

The electric dipole moment is

$$
\pm \; d = |q| \; l = |q| l \; \hat{n} \tag{8}
$$

parallel or antiparallel to the velocity vector c<sup>0</sup> n^, where n^ 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

$$T\_{\rm D} = |\mathbf{p\_{D}}|c\_{0} = p\_{\rm D}c\_{0},\tag{9}$$

where p<sup>D</sup> represents the momentum of the dipoles. As a working hypothesis, it will first be assumed that ∣p<sup>D</sup> ∣ 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

$$
\rho\_{\rm D} = \frac{\Delta N\_{\rm D}}{\Delta V},
\tag{10}
$$

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.

Conceptional presentation of the creation and destruction phases of virtual dipole pairs by a charge and the corresponding momentum vectors of the virtual dipoles (long arrows). The dipoles are assumed to have a spin of

We assume that a particle with charge Q is symmetrically emitting virtual

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

<sup>D</sup> ≪ mQ c<sup>2</sup>

<sup>T</sup><sup>D</sup> <sup>¼</sup> <sup>h</sup> <sup>c</sup><sup>0</sup> lD

is for propagating dipoles also equivalent to the photon energy relation, with l<sup>D</sup>

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

<sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>ρ</sup><sup>G</sup> <sup>κ</sup>G<sup>M</sup> <sup>¼</sup> <sup>η</sup>GM, (15)

T <sup>∗</sup> <sup>D</sup> <sup>≈</sup> <sup>h</sup> Δt<sup>D</sup>

corresponds to that of the gravitons in Eq. (11). The equation

ΔNM

<sup>D</sup> . The emission rate is proportional to its charge and the orientation

<sup>0</sup> will have a certain lifetime Δt<sup>D</sup> and

(13)

(14)

2.3.2 Virtual dipoles

corresponding to λ.

charge:

69

2.4 Newton's law of gravity

Virtual dipoles with energies of T <sup>∗</sup>

S ¼ �2 � ℏ=2 (short arrows) (Figure 2 of [17]).

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

interact with real dipoles. The energy-lifetime relation

dipoles with p <sup>∗</sup>

right side.

Figure 1.

but will be polarized near electric charges and affected by gravitational centres.

#### 2.3 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. [26–28].

#### 2.3.1 Virtual gravitons

A particle with mass M is symmetrically emitting virtual gravitons with moments p <sup>∗</sup> <sup>G</sup> and energies of T <sup>∗</sup> <sup>G</sup> ≪ M c<sup>2</sup> 0. The emission rate is proportional to M. The gravitons will have a certain lifetime Δt<sup>G</sup> and interact with "real" gravitons. In the literature, there are many different derivations of an energy-time relation, e.g. in [29–31]. 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

$$T\_{\rm G}^{\*} \approx \frac{h}{\Delta t\_{\rm G}}\tag{11}$$

with <sup>h</sup> <sup>¼</sup> <sup>6</sup>:<sup>62607015</sup> � <sup>10</sup>�<sup>34</sup> J s (exact), the Planck constant [25]. For propagating gravitons, the equation

$$T\_{\rm G} = h \frac{c\_0}{l\_{\rm G}} \tag{12}$$

is equivalent to the photon energy relation E<sup>ν</sup> ¼ hν ¼ h c0=λ, where λ corresponds to lG, which can be considered as the wavelength of the hypothetical

### Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

#### Figure 1.

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 �∣q∣ will be assumed, where q might or might not be identical to the

parallel or antiparallel to the velocity vector c<sup>0</sup> n^, where n^ 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

where p<sup>D</sup> represents the momentum of the dipoles. As a working hypothesis, it will first be assumed that ∣p<sup>D</sup> ∣ 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

<sup>ρ</sup><sup>D</sup> <sup>¼</sup> <sup>Δ</sup>N<sup>D</sup>

but will be polarized near electric charges and affected by gravitational centres.

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. [26–28].

A particle with mass M is symmetrically emitting virtual gravitons with

T <sup>∗</sup> <sup>G</sup> <sup>≈</sup> <sup>h</sup> Δt<sup>G</sup>

with <sup>h</sup> <sup>¼</sup> <sup>6</sup>:<sup>62607015</sup> � <sup>10</sup>�<sup>34</sup> J s (exact), the Planck constant [25]. For propa-

<sup>T</sup><sup>G</sup> <sup>¼</sup> <sup>h</sup> <sup>c</sup><sup>0</sup> lG

is equivalent to the photon energy relation E<sup>ν</sup> ¼ hν ¼ h c0=λ, where λ corresponds to lG, which can be considered as the wavelength of the hypothetical

The gravitons will have a certain lifetime Δt<sup>G</sup> and interact with "real" gravitons. In the literature, there are many different derivations of an energy-time relation, e.g. in [29–31]. 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

<sup>G</sup> ≪ M c<sup>2</sup>

� d ¼ ∣q∣ l ¼ ∣q∣l n^ (8)

T<sup>D</sup> ¼ ∣p<sup>D</sup> ∣c<sup>0</sup> ¼ p<sup>D</sup> c0, (9)

<sup>Δ</sup><sup>V</sup> , (10)

0. The emission rate is proportional to M.

(11)

(12)

elementary charge <sup>e</sup> <sup>¼</sup> <sup>1</sup>:<sup>602176634</sup> � <sup>10</sup>�<sup>19</sup> C (exact) [25].

The electric dipole moment is

Planetology - Future Explorations

have a mean energy

2.3 Virtual entities

2.3.1 Virtual gravitons

gating gravitons, the equation

<sup>G</sup> and energies of T <sup>∗</sup>

moments p <sup>∗</sup>

rewritten as

68

Conceptional presentation of the creation and destruction phases of virtual dipole pairs by a charge and the corresponding momentum vectors of the virtual dipoles (long arrows). The dipoles are assumed to have a spin of S ¼ �2 � ℏ=2 (short arrows) (Figure 2 of [17]).

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.

#### 2.3.2 Virtual dipoles

We assume that a particle with charge Q is symmetrically emitting virtual dipoles with p <sup>∗</sup> <sup>D</sup> . 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 <sup>∗</sup> <sup>D</sup> ≪ mQ c<sup>2</sup> <sup>0</sup> will have a certain lifetime Δt<sup>D</sup> and interact with real dipoles. The energy-lifetime relation

$$T\_{\rm D}^{\*} \approx \frac{h}{\Delta t\_{\rm D}}\tag{13}$$

corresponds to that of the gravitons in Eq. (11). The equation

$$T\_{\rm D} = h \frac{c\_0}{l\_{\rm D}} \tag{14}$$

is for propagating dipoles also equivalent to the photon energy relation, with l<sup>D</sup> corresponding to λ.

#### 2.4 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:

$$\frac{\Delta N\_M}{\Delta t} = \rho\_\text{G} \,\kappa\_\text{G} M = \eta\_\text{G} M,\tag{15}$$

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

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

<sup>G</sup> and an incoming graviton with p<sup>G</sup> followed by the liberation

G. Regardless of the processes operating in the immediate environment of

1 4π r<sup>2</sup> c<sup>0</sup>

¼ η<sup>G</sup>

M 4π r<sup>2</sup> c<sup>0</sup>

<sup>2</sup> c<sup>0</sup> Δt: (21)

, (22)

<sup>G</sup> is,

(23)

, (20)

of another virtual graviton in the opposite direction supplied with the excess

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

> <sup>¼</sup> <sup>Δ</sup>NM Δt

ΔVr ¼ 4π r

gravitons. For a certain rM, defined as the mass radius of M, it has to be

<sup>ρ</sup><sup>G</sup> <sup>¼</sup> <sup>Δ</sup>NM ΔVr 

<sup>σ</sup><sup>G</sup> <sup>¼</sup> <sup>m</sup> 4π r<sup>2</sup> m

The radial emission is part of the background in Eq. (6), which has a larger number density ρ<sup>G</sup> than ρMð Þr at 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

rM

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 6¼ M and, in parti-

> <sup>¼</sup> <sup>M</sup> 4π r<sup>2</sup> M

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)

will be independent of the mass as long as the density of the background distribution is constant. The quantity σ<sup>G</sup> is a kind of surface mass density. The equation shows that σ<sup>G</sup> is determined by the electron mass radius rG,e, for which estimates will be provided in Sections 3.2 and 3.3. From Eqs. (16), (22) and (23), it

¼ ηG c0

M 4π r<sup>2</sup> M

<sup>¼</sup> <sup>m</sup><sup>e</sup> 4π r<sup>2</sup> G,e

κ<sup>G</sup> σ<sup>G</sup> ¼ c0: (24)

in the environment of a body with mass M. Its spatial density is

ΔNM ΔVr

where the volume increase in the time interval Δt is

ρMð Þ¼ r

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

cular, to the electron mass me.

Therefore

follows that

and (20):

71

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�

distribution.

momentum p <sup>∗</sup>

energy T�

#### Figure 2.

Interaction of gravitons with a body of mass M. A graviton arriving with a momentum p<sup>G</sup> on the left combines together with a virtual graviton with p <sup>∗</sup> <sup>G</sup> ¼ �pG. The excess energy liberates a second virtual graviton with p� G on the right in a direction antiparallel to the incoming graviton. The excess energy T� <sup>G</sup> is smaller than T <sup>∗</sup> <sup>G</sup> . The conceptual diagram shows gravitons with a spin S ¼ �4 � ℏ=2 and G<sup>þ</sup> or G� orientation. It is unclear whether such a spin would have any influence on the interaction process (modified from Figure 1 of [16]).

where κ<sup>G</sup> is the gravitational absorption coefficient and η<sup>G</sup> 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, 35–39]. Since the absorption and emission rates must be equal in Eq. (15), this gives an emission coefficient of

$$
\eta\_{\rm G} = \kappa\_{\rm G} \rho\_{\rm G} = \frac{1}{2} \frac{c\_0^2}{h} = 6.782 \times 10^{49} \text{ s}^{-1} \text{ kg}^{-1}, \tag{16}
$$

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<sup>G</sup> and η<sup>G</sup> are constant. For an electron with a mass of <sup>m</sup><sup>e</sup> <sup>¼</sup> <sup>9</sup>:10938356 11 ð Þ� <sup>10</sup>�<sup>31</sup> kg, the virtual graviton production rate equals its de Broglie frequency ν<sup>B</sup> G,e <sup>¼</sup> <sup>m</sup><sup>e</sup> <sup>c</sup><sup>2</sup> <sup>0</sup>=<sup>h</sup> <sup>¼</sup> <sup>1</sup>:<sup>235</sup> … � <sup>10</sup><sup>20</sup> Hz.

The energy absorption rate of an atomic particle with mass M is

$$\frac{\Delta N\_{\rm M}}{\Delta t} \, T\_{\rm G}^{\rm ab} = \kappa\_{\rm G} \rho\_{\rm G} \, MT\_{\rm G}.\tag{17}$$

Larger masses are thought of as conglomeration of atomic particles. The emission energy, in turn, is assumed to be reduced to

$$T\_{\rm G}^{\rm em} = (\mathbf{1} - Y) T\_{\rm G} \tag{18}$$

per graviton, where Y (0<Y ≪ 1) is defined as the reduction parameter. This leads to an energy emission rate of

$$\frac{\Delta N\_{\rm M}}{\Delta t} \, T\_{\rm G}^{\rm em} = -\eta\_{\rm G} M(1 - Y) \, T\_{\rm G}.\tag{19}$$

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]. Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

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 <sup>∗</sup> <sup>G</sup> and an incoming graviton with p<sup>G</sup> followed by the liberation of another virtual graviton in the opposite direction supplied with the excess energy T� 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� <sup>G</sup> 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

$$\rho\_M(r) = \frac{\Delta N\_M}{\Delta V\_r} = \frac{\Delta N\_M}{\Delta t} \frac{1}{4\pi r^2 c\_0} = \eta\_G \frac{M}{4\pi r^2 c\_0},\tag{20}$$

where the volume increase in the time interval Δt is

$$
\Delta V\_r = 4 \,\text{\pi}r^2 c\_0 \Delta t. \tag{21}
$$

The radial emission is part of the background in Eq. (6), which has a larger number density ρ<sup>G</sup> than ρMð Þr at 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 rM, defined as the mass radius of M, it has to be

$$
\rho\_{\rm G} = \left[\frac{\Delta N\_{\rm M}}{\Delta V\_{r}}\right]\_{r\_{\rm M}} = \frac{\eta\_{\rm G}}{c\_{0}} \frac{M}{4\pi r\_{\rm M}^{2}}\,,\tag{22}
$$

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 6¼ M and, in particular, to the electron mass me.

Therefore

where κ<sup>G</sup> is the gravitational absorption coefficient and η<sup>G</sup> the corresponding

Interaction of gravitons with a body of mass M. A graviton arriving with a momentum p<sup>G</sup> on the left combines

conceptual diagram shows gravitons with a spin S ¼ �4 � ℏ=2 and G<sup>þ</sup> or G� orientation. It is unclear whether

such a spin would have any influence on the interaction process (modified from Figure 1 of [16]).

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, 35–39]. Since the absorption and emission rates must be equal in Eq. (15), this gives an emission

<sup>h</sup> <sup>¼</sup> <sup>6</sup>:<sup>782</sup> � <sup>1049</sup> <sup>s</sup>

<sup>0</sup>=<sup>h</sup> <sup>¼</sup> <sup>1</sup>:<sup>235</sup> … � <sup>10</sup><sup>20</sup> Hz.

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<sup>G</sup> and η<sup>G</sup> are constant. For an electron with a mass of <sup>m</sup><sup>e</sup> <sup>¼</sup> <sup>9</sup>:10938356 11 ð Þ� <sup>10</sup>�<sup>31</sup> kg, the virtual graviton production rate equals its

�<sup>1</sup> kg�<sup>1</sup>

<sup>G</sup> ¼ �pG. The excess energy liberates a second virtual graviton with p�

<sup>G</sup> ¼ κ<sup>G</sup> ρGMTG: (17)

<sup>G</sup> ¼ ð Þ 1 � Y T<sup>G</sup> (18)

<sup>G</sup> ¼ �ηGMð Þ 1 � Y TG: (19)

, (16)

<sup>G</sup> is smaller than T <sup>∗</sup>

G

<sup>G</sup> . The

<sup>η</sup><sup>G</sup> <sup>¼</sup> <sup>κ</sup><sup>G</sup> <sup>ρ</sup><sup>G</sup> <sup>¼</sup> <sup>1</sup>

G,e <sup>¼</sup> <sup>m</sup><sup>e</sup> <sup>c</sup><sup>2</sup>

This leads to an energy emission rate of

2 c2 0

on the right in a direction antiparallel to the incoming graviton. The excess energy T�

The energy absorption rate of an atomic particle with mass M is

T<sup>a</sup><sup>b</sup>

Larger masses are thought of as conglomeration of atomic particles.

per graviton, where Y (0<Y ≪ 1) is defined as the reduction parameter.

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].

Tem

Tem

ΔNM Δt

The emission energy, in turn, is assumed to be reduced to

ΔNM Δt

emission coefficient.

together with a virtual graviton with p <sup>∗</sup>

Planetology - Future Explorations

de Broglie frequency ν<sup>B</sup>

coefficient of

70

Figure 2.

$$
\sigma\_{\rm G} = \frac{m}{4\,\pi r\_m^2} = \frac{M}{4\,\pi r\_M^2} = \frac{m\_{\rm e}}{4\,\pi r\_{\rm G,e}^2} \tag{23}
$$

will be independent of the mass as long as the density of the background distribution is constant. The quantity σ<sup>G</sup> is a kind of surface mass density. The equation shows that σ<sup>G</sup> is determined by the electron mass radius rG,e, for which estimates will be provided in Sections 3.2 and 3.3. From Eqs. (16), (22) and (23), it follows that

$$
\kappa\_{\mathcal{G}} \sigma\_{\mathcal{G}} = \mathcal{c}\_0. \tag{24}
$$

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):

Planetology - Future Explorations

$$\frac{\Delta N\_{M,m}(r)}{\Delta t} = \kappa\_{\rm G} m \frac{\Delta N\_M}{\Delta V\_r} = \frac{\kappa\_{\rm G} \eta\_{\rm G}}{c\_0} \frac{Mm}{4\pi r^2} = \frac{\kappa\_{\rm G} c\_0}{2h} \frac{mM}{4\pi r^2} = \kappa\_{\rm G} M \frac{\Delta N\_m}{\Delta V\_r} = \frac{\Delta N\_{m,M}(r)}{\Delta t}.\tag{25}$$

A calculation with antiparallel emissions of the secondary gravitons shows that an interaction of a graviton with reduced momentum p� <sup>G</sup> provides �p<sup>G</sup> <sup>2</sup><sup>Y</sup> � <sup>Y</sup><sup>2</sup> together with its unmodified counterpart from the opposite sides. The resulting imbalance will be

$$\frac{\Delta P\_{M,m}(r)}{\Delta t} \approx -2 \,\mathrm{p\_G} \, Y \, \frac{\Delta N\_{M,m}(r)}{\Delta t} = - \,\mathrm{p\_G} \, Y \, \kappa\_\mathrm{G} \, \frac{c\_0}{h} \frac{Mm}{4 \,\pi r^2},\tag{26}$$

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 pG, Y, κ<sup>G</sup> and G<sup>N</sup> can be established through the constant GG:

$$\mathbf{G}\_{\rm G} = p\_{\rm G} \, Y \kappa\_{\rm G} \approx 4 \,\pi \,\mathbf{G}\_{\rm N} \frac{h}{\varepsilon\_0} = \mathbf{1.853\ldots} \times \mathbf{10}^{-51} \,\,\mathbf{m}^4 \,\,\mathbf{s}^{-2}.\tag{27}$$

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�15.

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

$$
\epsilon\_{\rm G} = T\_{\rm G} \rho\_{\rm G} = \frac{2\pi G\_{\rm N}}{Y} \sigma\_{\rm G}^2,\tag{28}
$$

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

$$A = \frac{1}{M} \frac{\Delta \mathbf{M}}{\Delta t} = \frac{2\pi G\_{\rm N}}{c\_0} \sigma\_{\rm G} = \frac{2\pi G\_{\rm N}}{c\_0} \frac{m\_{\rm e}}{4\pi r\_{\rm G,e}^2},\tag{29}$$

which implies an exponential growth according to

$$M(t) = M\_0 \exp\left[A\left(t - t\_0\right)\right] \approx M\_0 \left(1 + A\,\Delta t\right),\tag{30}$$

where M<sup>0</sup> ¼ M tð Þ<sup>0</sup> is the initial value at t<sup>0</sup> and the linear approximation is valid for small A tð Þ¼ � t<sup>0</sup> AΔt. The accretion rate is

$$A = \frac{1.014 \times 10^{-49}}{r\_{\text{G,e}}^2} \text{ m}^2 \text{ s}^{-1},\tag{31}$$

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<sup>0</sup> (with the Hubble constant H0) only for very

the observed secular increase of the Sun-Earth distance [45] (modified from Figure 2 of [16]).

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

Energies of EG <sup>¼</sup> <sup>10</sup>�<sup>40</sup> to 10�<sup>20</sup> ð Þ <sup>J</sup> 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 rG,e. The range Y ≥1 (dark shading) is obviously completely excluded by the model. Even values greater than ≈10�<sup>15</sup> are not realistic (light shaded region), cf. paragraph following Eq. (27). The cosmic dark energy estimate ð Þ <sup>3</sup>:<sup>9</sup> � <sup>0</sup>:<sup>4</sup> GeV <sup>m</sup>�<sup>3</sup> <sup>¼</sup> ð Þ� <sup>6</sup>:<sup>2</sup> � <sup>0</sup>:<sup>6</sup> <sup>10</sup>�<sup>10</sup> J m�<sup>3</sup> (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�<sup>10</sup> to 10�<sup>30</sup> 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=H0 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

small rG,<sup>e</sup>.

73

Figure 3.

if the expression is evaluated in terms of recent parameters.

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

#### Figure 3.

ΔNM,<sup>m</sup>ð Þr

<sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>κ</sup><sup>G</sup> <sup>m</sup>

Planetology - Future Explorations

imbalance will be

ΔNM ΔVr

ΔP<sup>M</sup>,mð Þr

established through the constant GG:

assume in Figure 3 that Y <10�15.

of a particle with mass M will be

<sup>A</sup> <sup>¼</sup> <sup>1</sup> M

for small A tð Þ¼ � t<sup>0</sup> AΔt. The accretion rate is

ΔM

which implies an exponential growth according to

<sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>2</sup>πG<sup>N</sup> c0

<sup>A</sup> <sup>¼</sup> <sup>1</sup>:<sup>014</sup> � <sup>10</sup>�<sup>49</sup> r2 G,e

if the expression is evaluated in terms of recent parameters.

from Eqs. (1).

(22)–(24).

72

<sup>¼</sup> <sup>κ</sup><sup>G</sup> <sup>η</sup><sup>G</sup> c0

an interaction of a graviton with reduced momentum p�

<sup>Δ</sup><sup>t</sup> <sup>≈</sup> � <sup>2</sup> <sup>p</sup><sup>G</sup> <sup>Y</sup>

G<sup>G</sup> ¼ p<sup>G</sup> Y κ<sup>G</sup> ≈4πG<sup>N</sup>

Mm <sup>4</sup><sup>π</sup> <sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>κ</sup><sup>G</sup> <sup>c</sup><sup>0</sup> 2h

mM <sup>4</sup><sup>π</sup> <sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>κ</sup>G<sup>M</sup>

<sup>Δ</sup><sup>t</sup> ¼ �p<sup>G</sup> <sup>Y</sup> <sup>κ</sup><sup>G</sup>

<sup>¼</sup> <sup>1</sup>:<sup>853</sup> … � <sup>10</sup>�<sup>51</sup> m4 <sup>s</sup>

A calculation with antiparallel emissions of the secondary gravitons shows that

together with its unmodified counterpart from the opposite sides. The resulting

if the quadratic terms in Y can be neglected for very small Y scenarios.

h c0

The graviton energy density remote from any masses will be

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

<sup>ϵ</sup><sup>G</sup> <sup>¼</sup> <sup>T</sup><sup>G</sup> <sup>ρ</sup><sup>G</sup> <sup>¼</sup> <sup>2</sup>πG<sup>N</sup>

where the last term is obtained from Eq. (27) with the help of Eqs. (16) and

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

where M<sup>0</sup> ¼ M tð Þ<sup>0</sup> is the initial value at t<sup>0</sup> and the linear approximation is valid

<sup>Y</sup> <sup>σ</sup><sup>2</sup>

<sup>σ</sup><sup>G</sup> <sup>¼</sup> <sup>2</sup>πG<sup>N</sup> c0

M tðÞ¼ M<sup>0</sup> exp ½ � A tð Þ � t<sup>0</sup> ≈ M<sup>0</sup> ð Þ 1 þ AΔt , (30)

m<sup>2</sup> s �1

m<sup>e</sup> 4π r<sup>2</sup> G,e

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

ΔNM,mð Þr

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 pG, Y, κ<sup>G</sup> and G<sup>N</sup> can be

ΔNm ΔVr

> c0 h

Mm

�2

G, (28)

, (29)

, (31)

<sup>¼</sup> <sup>Δ</sup>Nm,<sup>M</sup>ð Þ<sup>r</sup>

<sup>G</sup> provides �p<sup>G</sup> <sup>2</sup><sup>Y</sup> � <sup>Y</sup><sup>2</sup>

<sup>4</sup><sup>π</sup> <sup>r</sup><sup>2</sup> , (26)

: (27)

<sup>Δ</sup><sup>t</sup> : (25)

Energies of EG <sup>¼</sup> <sup>10</sup>�<sup>40</sup> to 10�<sup>20</sup> ð Þ <sup>J</sup> 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 rG,e. The range Y ≥1 (dark shading) is obviously completely excluded by the model. Even values greater than ≈10�<sup>15</sup> are not realistic (light shaded region), cf. paragraph following Eq. (27). The cosmic dark energy estimate ð Þ <sup>3</sup>:<sup>9</sup> � <sup>0</sup>:<sup>4</sup> GeV <sup>m</sup>�<sup>3</sup> <sup>¼</sup> ð Þ� <sup>6</sup>:<sup>2</sup> � <sup>0</sup>:<sup>6</sup> <sup>10</sup>�<sup>10</sup> J m�<sup>3</sup> (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�<sup>10</sup> to 10�<sup>30</sup> 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=H0 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]).

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<sup>0</sup> (with the Hubble constant H0) only for very small rG,<sup>e</sup>.

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

$$R\_{\infty} = \frac{a^2 m\_e c\_0}{2h} = 10\,\text{973}\,\text{731.568508\text{ m}}^{-1}\tag{32}$$

"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

where η<sup>D</sup> and κ<sup>D</sup> are the corresponding (dipole) emission and absorption coeffi-

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

The interaction processes assumed between a positively charged body and dipoles is sketched in Figure 4. A mass mQ 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

Virtual dipoles emitted by a charge þ∣Q∣ (with mass mQ ) interact with real dipoles, A<sup>þ</sup> and A� arriving with a momentum �pD, each. On the left, a dipole A<sup>þ</sup> combines in the lower dashed-dotted box with virtual dipole A<sup>þ</sup> in its destruction phase and liberates dipole Bþ. No momentum will be transferred to the central charge with

D

<sup>D</sup> . The other types of interaction—called direct interaction, in contrast to the indirect one on the left also require two virtual dipoles, one of them combines in its creation phase with dipole A� (in the upper box with dashed-dotted boundaries), and the other one is liberated by the excess energy of the annihilation. The

¼ �2pD. No spin reversal has been assumed in both cases.

<sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>κ</sup><sup>D</sup> <sup>ρ</sup><sup>D</sup> <sup>∣</sup>Q<sup>∣</sup> <sup>¼</sup> <sup>η</sup><sup>D</sup> <sup>∣</sup>Q∣, (38)

interesting speculation.

cients.

processes.

that p<sup>G</sup> ≪ pD.

Figure 4.

pD <sup>¼</sup> <sup>p</sup> <sup>∗</sup>

75

central charge received a momentum of � pD <sup>þ</sup> <sup>p</sup> <sup>∗</sup>

A charge Q absorbs and emits dipoles at a rate

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

ΔNQ

is <sup>u</sup><sup>r</sup> <sup>≈</sup>5:<sup>9</sup> � <sup>10</sup>�12, where

$$a = \frac{e^2}{2\epsilon\_0 c\_0 h} = 7.297\,352\,5664(17) \times 10^{-3} \tag{33}$$

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 1ð Þ þ u<sup>r</sup> M<sup>0</sup> ≥ M tð Þ> M<sup>0</sup> at least for the time interval <sup>Δ</sup><sup>t</sup> <sup>≤</sup>1:<sup>6</sup> � <sup>10</sup><sup>17</sup> s. Any variation of <sup>R</sup>∞, 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

$$\mathcal{N}\_{\rm G}^{\rm tot} = \Delta t\_{\rm G} \frac{\mathcal{M}c\_0^2}{h} \tag{34}$$

and

$$T\_{\rm G}^{\rm tot} = N\_{\rm G}^{\rm tot} T\_{\rm G}^\* \approx Mc\_0^2,\tag{35}$$

i.e. the mass of a particle would reside within the virtual gravitons.

#### 2.5 Coulomb's law

#### 2.5.1 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

$$\mathbf{K}\_{\rm E}(r) = \mathbf{E}\_{\rm Q}(r)q. \tag{36}$$

The electrostatic potential UEð Þr of a charge Q, located at r ¼ 0, is for r>0

$$U\_{\rm E}(r) = \frac{Q}{4\pi r \varepsilon\_0}.\tag{37}$$

The corresponding electrostatic field can thus be written as E<sup>Q</sup> ð Þ¼� r ∇UEð Þr .

#### 2.5.2 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

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

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 1ð Þ þ u<sup>r</sup> M<sup>0</sup> ≥ M tð Þ> M<sup>0</sup> at least for the time interval <sup>Δ</sup><sup>t</sup> <sup>≤</sup>1:<sup>6</sup> � <sup>10</sup><sup>17</sup> s. Any variation of <sup>R</sup>∞, caused by the linear dependence upon the electron mass, which has also been considered in [43], would then be below the

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

<sup>G</sup> T <sup>∗</sup>

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

The electrostatic potential UEð Þr of a charge Q, located at r ¼ 0, is for r>0

The corresponding electrostatic field can thus be written as E<sup>Q</sup> ð Þ¼� r ∇UEð Þr .

Note that the dipoles in the background distribution, cf. Eq. (10), have no mean

Q 4π rε<sup>0</sup>

UEð Þ¼ r

interaction energy, even in the classical theory (see, e.g. [24]). Whether this

M c<sup>2</sup> 0

<sup>G</sup> ≈ M c<sup>2</sup>

<sup>h</sup> (34)

0, (35)

: (37)

KEð Þ¼ r E<sup>Q</sup> ð Þr q: (36)

Ntot <sup>G</sup> ¼ Δt<sup>G</sup>

i.e. the mass of a particle would reside within the virtual gravitons.

Ttot <sup>G</sup> <sup>¼</sup> <sup>N</sup>tot

<sup>2</sup><sup>h</sup> <sup>¼</sup> 10 973731:568508 m�<sup>1</sup> (32)

<sup>2</sup>ε<sup>0</sup> <sup>c</sup><sup>0</sup> <sup>h</sup> <sup>¼</sup> <sup>7</sup>:297 3525664 17 ð Þ� <sup>10</sup>�<sup>3</sup> (33)

creation rate in accordance with Eq. (30).

is <sup>u</sup><sup>r</sup> <sup>≈</sup>5:<sup>9</sup> � <sup>10</sup>�12, where

Planetology - Future Explorations

body with mass M as

2.5 Coulomb's law

2.5.2 Dipole interactions

74

and

<sup>R</sup><sup>∞</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup> <sup>m</sup><sup>e</sup> <sup>c</sup><sup>0</sup>

<sup>α</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup>

detection limit for state-of-the-art methods.

2.5.1 Electrostatic fields and charged particles

electrostatic field in Eq. (3), it can be written as

"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

$$\frac{\Delta N\_{\text{Q}}}{\Delta t} = \kappa\_{\text{D}} \rho\_{\text{D}} |Q| = \eta\_{\text{D}} |Q|,\tag{38}$$

where η<sup>D</sup> and κ<sup>D</sup> 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 mQ 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<sup>G</sup> ≪ pD.

#### Figure 4.

Virtual dipoles emitted by a charge þ∣Q∣ (with mass mQ ) interact with real dipoles, A<sup>þ</sup> and A� arriving with a momentum �pD, each. On the left, a dipole A<sup>þ</sup> combines in the lower dashed-dotted box with virtual dipole A<sup>þ</sup> in its destruction phase and liberates dipole Bþ. No momentum will be transferred to the central charge with pD <sup>¼</sup> <sup>p</sup> <sup>∗</sup> <sup>D</sup> . The other types of interaction—called direct interaction, in contrast to the indirect one on the left also require two virtual dipoles, one of them combines in its creation phase with dipole A� (in the upper box with dashed-dotted boundaries), and the other one is liberated by the excess energy of the annihilation. The central charge received a momentum of � pD <sup>þ</sup> <sup>p</sup> <sup>∗</sup> D ¼ �2pD. No spin reversal has been assumed in both cases.

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

$$\rho\_Q(r) = \frac{\Delta N\_Q}{\Delta V\_r} = \frac{1}{4\pi r^2 c\_0} \frac{\Delta N\_Q}{\Delta t} = \eta\_D \frac{|Q|}{4\pi r^2 c\_0},\tag{39}$$

dipole Bþ, which has the required orientation. The charge had emitted two virtual

�2p<sup>D</sup> 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<sup>þ</sup> (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<sup>D</sup> is supplied to liberate a second virtual dipole Bþ. The momentum transfer to the charge þ∣Q∣ is zero. This process just corresponds to a double charge exchange. Designating the number of interac-

<sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>Δ</sup>N^<sup>Q</sup> <sup>þ</sup> <sup>Δ</sup>N~<sup>Q</sup>

with <sup>Δ</sup>N~<sup>Q</sup> <sup>¼</sup> <sup>Δ</sup>N^<sup>Q</sup> <sup>¼</sup> <sup>Δ</sup>NQ <sup>=</sup>2. Unless direct and indirect interactions are explicitly specified, both types are meant by the term "interaction". The virtual dipole

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<sup>D</sup> without involving a macroscopic electrostatic field.

∣Q∣ ∣q∣

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

In order to have a well-defined configuration for our discussion, we will assume that body C with mass m<sup>C</sup> has a positive charge Q >0 and is positioned at a distance

r beneath body D (mass mD) with either a charge þ∣q∣ in Figure 5 or �∣q∣ in

The interaction rates of dipoles with bodies C and D in Eq. (47) (the same for both bodies even if ∣Q∣ 6¼ ∣q∣) and the momentum transfers indicated in Figures 5 and 6, respectively, lead to a norm of the momentum change rate for

Figure 6. Only the processes near body D are shown in detail.

<sup>4</sup><sup>π</sup> <sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>κ</sup><sup>D</sup> <sup>∣</sup>Q<sup>∣</sup>

which confirms the reciprocal relationship between q and Q. The equation, also very similar to Eq. (25), does not contain an explicit value for ηD. 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

ΔNq ΔVr

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<sup>D</sup> in

ΔNQ

ΔN <sup>∗</sup> Q <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>2</sup> <sup>Δ</sup>NQ

Eq. (38). The rate of interchanges between these charges then is

<sup>¼</sup> <sup>κ</sup><sup>D</sup> <sup>η</sup><sup>D</sup> c0

ΔNQ ΔVr

<sup>D</sup> , each, and a total momentum of � <sup>p</sup><sup>D</sup> <sup>þ</sup> <sup>p</sup> <sup>∗</sup>

<sup>Δ</sup><sup>t</sup> (45)

<sup>Δ</sup><sup>t</sup> , (46)

<sup>¼</sup> <sup>Δ</sup>Nq,<sup>Q</sup> ð Þ<sup>r</sup>

<sup>Δ</sup><sup>t</sup> , (47)

D <sup>¼</sup>

dipoles with a momentum of <sup>þ</sup><sup>p</sup> <sup>∗</sup>

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

tions of the indirect type with ΔN~<sup>Q</sup> , it is

emission rate in Figure 4 has to be

ΔNQ,<sup>q</sup>ð Þr

�2p<sup>D</sup> or zero.

bodies C and D of

77

<sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>κ</sup><sup>D</sup> <sup>∣</sup>q<sup>∣</sup>

the gravitational interactions.

where ΔVr is given in Eq. (21). The radial emission is part of the background ρD, which has a larger number density than ρDð Þr at most distances r of interest. Note that the emission of the dipoles from Q does not change the number density ρ<sup>D</sup> 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 rQ , defined as the charge radius of Q, it has to be

$$
\rho\_{\rm D} = \left[\frac{\Delta N\_{\rm Q}}{\Delta V\_r}\right]\_{r\_{\rm Q}} = \frac{\eta\_{\rm D}}{c\_0} \frac{|Q|}{4\pi r\_{\rm Q}^2},\tag{40}
$$

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 6¼ Q. Since ρ<sup>D</sup> cannot depend on either q or Q, the quantity

$$
\sigma\_{\mathcal{Q}} = \frac{|Q|}{4\,\pi r\_{\mathcal{Q}}^2} = \frac{|q|}{4\,\pi r\_q^2} = \frac{|\mathcal{e}|}{4\,\pi r\_{\mathcal{e}}^2} \tag{41}
$$

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<sup>Q</sup> is determined by the electron charge radius re.

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 ΔNQ =Δt of a charge Q. The physical constants α, c0, h, ϵ<sup>0</sup> and G<sup>N</sup> can be combined to give a dipole emission coefficient

$$\eta\_{\rm D} = \frac{1}{2} \frac{a^2 c\_0^2}{h \sqrt{\epsilon\_0 \, G\_{\rm N}}} = 1.486 \times 10^{56} \text{ s}^{-1} \text{ C}^{-1} \tag{42}$$

as half the virtual dipole production rate and thus for a charge ∣e∣ a rate of

$$\frac{\Delta N\_{\epsilon}}{\Delta t} = \eta\_{\text{D}}|\epsilon| = 2.380 \times 10^{37} \text{ s}^{-1}.\tag{43}$$

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

$$
\kappa\_{\mathbb{D}} \sigma\_{\mathbb{Q}} = \mathfrak{c}\_{\mathbb{D}}.\tag{44}
$$

During a direct interaction, the dipole A� (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 <sup>P</sup> ¼ �p<sup>D</sup> <sup>þ</sup> <sup>p</sup> <sup>∗</sup> <sup>D</sup> <sup>¼</sup> 0, where <sup>p</sup> <sup>∗</sup> <sup>D</sup> ¼ p<sup>D</sup> is the magnitude of the momentum vector of a virtual dipole, cf. Eq. (9). The momentum balance is neutral, and the excess energy T<sup>D</sup> is used to liberate a second virtual

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

The assumptions as outlined will lead to a distribution of the emitted dipoles in

where ΔVr is given in Eq. (21). The radial emission is part of the background ρD, which has a larger number density than ρDð Þr at most distances r of interest. Note that the emission of the dipoles from Q does not change the number density ρ<sup>D</sup> in the environment of the charge but reverses the orientation of half of the dipoles

The total number of dipoles will, of course, not be changed either. For a certain

<sup>¼</sup> <sup>η</sup><sup>D</sup> c0

∣Q∣ 4π r<sup>2</sup> Q

<sup>¼</sup> <sup>∣</sup>e<sup>∣</sup> 4π r<sup>2</sup> e

rQ

Q in some way. The same arguments apply to a charge q 6¼ Q. Since ρ<sup>D</sup> cannot

because all dipoles of the background that come so close interact with the charge

<sup>¼</sup> <sup>∣</sup>q<sup>∣</sup> 4π r<sup>2</sup> q

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<sup>Q</sup> is determined by

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 ΔNQ =Δt of a charge Q. The physical constants α, c0, h, ϵ<sup>0</sup> and G<sup>N</sup> can be combined to give a dipole emission

<sup>p</sup> <sup>¼</sup> <sup>1</sup>:<sup>486</sup> � <sup>10</sup><sup>56</sup> <sup>s</sup>

as half the virtual dipole production rate and thus for a charge ∣e∣ a rate of

<sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>η</sup><sup>D</sup> <sup>∣</sup>e<sup>∣</sup> <sup>¼</sup> <sup>2</sup>:<sup>380</sup> � <sup>10</sup><sup>37</sup> <sup>s</sup>

Note that the dipole emission rate is fixed for a certain charge and does not

During a direct interaction, the dipole A� (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

magnitude of the momentum vector of a virtual dipole, cf. Eq. (9). The momentum balance is neutral, and the excess energy T<sup>D</sup> is used to liberate a second virtual

ΔNQ <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>η</sup><sup>D</sup>

∣Q∣ 4π r<sup>2</sup> c<sup>0</sup>

, (39)

, (40)

�<sup>1</sup> C�<sup>1</sup> (42)

: (43)

<sup>D</sup> ¼ p<sup>D</sup> is the

�1

κ<sup>D</sup> σ<sup>Q</sup> ¼ c0: (44)

<sup>D</sup> <sup>¼</sup> 0, where <sup>p</sup> <sup>∗</sup>

(41)

the rest frame of an isolated charge Q with a spatial density of

ΔNQ ΔVr

<sup>ρ</sup><sup>D</sup> <sup>¼</sup> <sup>Δ</sup>NQ ΔVr � �

<sup>σ</sup><sup>Q</sup> <sup>¼</sup> <sup>∣</sup>Q<sup>∣</sup> 4π r<sup>2</sup> Q

α<sup>2</sup> c<sup>2</sup> 0 h ffiffiffiffiffiffiffiffiffiffiffi ϵ<sup>0</sup> G<sup>N</sup>

depend on the particle mass. From Eqs. (38), (40) and (41), we get

<sup>¼</sup> <sup>1</sup> 4π r<sup>2</sup> c<sup>0</sup>

ρ<sup>Q</sup> ð Þ¼ r

Planetology - Future Explorations

rQ , defined as the charge radius of Q, it has to be

<sup>η</sup><sup>D</sup> <sup>¼</sup> <sup>1</sup> 2

ΔNe

way to eliminate the charges and yield <sup>P</sup> ¼ �p<sup>D</sup> <sup>þ</sup> <sup>p</sup> <sup>∗</sup>

depend on either q or Q, the quantity

the electron charge radius re.

coefficient

76

affected.

dipole Bþ, which has the required orientation. The charge had emitted two virtual dipoles with a momentum of <sup>þ</sup><sup>p</sup> <sup>∗</sup> <sup>D</sup> , each, and a total momentum of � <sup>p</sup><sup>D</sup> <sup>þ</sup> <sup>p</sup> <sup>∗</sup> D <sup>¼</sup> �2p<sup>D</sup> 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<sup>þ</sup> (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<sup>D</sup> is supplied to liberate a second virtual dipole Bþ. 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 ΔN~<sup>Q</sup> , it is

$$\frac{\Delta \mathbf{N}\_Q}{\Delta t} = \frac{\Delta \hat{\mathbf{N}}\_Q + \Delta \hat{\mathbf{N}}\_Q}{\Delta t} \tag{45}$$

with <sup>Δ</sup>N~<sup>Q</sup> <sup>¼</sup> <sup>Δ</sup>N^<sup>Q</sup> <sup>¼</sup> <sup>Δ</sup>NQ <sup>=</sup>2. 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

$$\frac{\Delta N\_Q^\*}{\Delta t} = 2 \frac{\Delta N\_Q}{\Delta t},\tag{46}$$

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<sup>D</sup> 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<sup>D</sup> in Eq. (38). The rate of interchanges between these charges then is

$$\frac{\Delta N\_{Q,q}(r)}{\Delta t} = \kappa\_\mathcal{D} |q| \frac{\Delta N\_Q}{\Delta V\_r} = \frac{\kappa\_\mathcal{D} \eta\_\mathcal{D}}{c\_0} \frac{|Q| |q|}{4\pi r^2} = \kappa\_\mathcal{D} |Q| \frac{\Delta N\_q}{\Delta V\_r} = \frac{\Delta N\_{q,Q}(r)}{\Delta t},\tag{47}$$

which confirms the reciprocal relationship between q and Q. The equation, also very similar to Eq. (25), does not contain an explicit value for ηD. 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 �2p<sup>D</sup> 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<sup>C</sup> has a positive charge Q >0 and is positioned at a distance r beneath body D (mass mD) with either a charge þ∣q∣ in Figure 5 or �∣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∣ 6¼ ∣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

#### Figure 5.

Body C with charge Q >0 and mass mC is positioned in this configuration beneath body D with charge þ∣q∣ and mass mD leading to an electrostatic repulsion of the bodies. This results from the reversal of dipoles by the charge Q followed by direct interactions with the charge þ∣q∣ as defined on the right-hand side of Figure 4. Two reversals are schematically indicated in columns I and III. The dipoles arriving in columns II and IV from below have the same polarity as if they would be part of the background distribution and do not contribute to the momentum transfer, because of a compensation by dipoles arriving from above. The net momentum transfer caused by the two interacting reversed dipoles thus is 4pD, i.e. 2p<sup>D</sup> per dipole (modified from Figure 3 of [17]).

$$\left|\frac{\Delta P\_{\rm Q,q}(r)}{\Delta t}\right| = 2p\_{\rm D}\frac{\Delta N\_{\rm Q,q}(r)}{\Delta t} = 2p\_{\rm D}\frac{\kappa\_{\rm D}\eta\_{\rm D}}{c\_{0}}\frac{Q|q|}{4\pi r^{2}}.\tag{48}$$

Together with

$$p\_{\rm D} \frac{\kappa\_{\rm D} \eta\_{\rm D}}{c\_{0}} = \frac{T\_{\rm D} \kappa\_{\rm D} \eta\_{\rm D}}{c\_{0}^{2}} = \frac{1}{2\varepsilon\_{0}}\tag{49}$$

Eq. (50). The dipole density <sup>ρ</sup><sup>D</sup> <sup>¼</sup> <sup>7</sup>:<sup>95</sup> � <sup>10</sup><sup>56</sup> <sup>m</sup>�<sup>3</sup> in Eq. (40) is also very high, leading to a dipole energy of <sup>T</sup><sup>D</sup> <sup>¼</sup> <sup>1</sup>:<sup>83</sup> � <sup>10</sup>�<sup>28</sup> J. If, on the other hand, we identify the dipole distribution with DM with an estimated energy density of 2:<sup>48</sup> � <sup>10</sup>�<sup>10</sup> J m�<sup>3</sup> and require that the dipole energy density corresponds to this value, then extreme values follow for rQ <sup>¼</sup> <sup>13</sup>:9μm, <sup>ρ</sup><sup>D</sup> <sup>¼</sup> <sup>3</sup>:<sup>28</sup> � 1037 <sup>m</sup>�<sup>3</sup> and <sup>T</sup><sup>D</sup> <sup>¼</sup> <sup>7</sup>:<sup>55</sup> � <sup>10</sup>�<sup>48</sup> J.

Body C is again positioned beneath body D. The charge of D is now �∣q∣, however, leading to an electrostatic attraction of the bodies. The attraction results from the reversal of dipoles by the charge Q >0 followed by indirect interactions with charge �∣q∣ as defined on the left-hand side of Figure 4. Two events without momentum transfer are schematically indicated in columns II and IV. The dipoles arriving in columns I and III from below have the same polarity as if they would be part of the background distribution. The same is true for all dipoles arriving from above. The net momentum transfer caused by the two reversed dipoles thus is �4pD,

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

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].<sup>2</sup>

In this context, it is of interest that Brillouin [48] discussed this problem in relation to the electrostatic

3. Applications of impact models

i.e. �2p<sup>D</sup> per dipole (modified from Figure 4 of [17]).

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

3.1 Potential energies

2

79

potential energy.

Figure 6.

3.1.1 Gravitational potential energy

physical processes successfully for specific situations.

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<sup>D</sup> and momentum p<sup>D</sup> 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

$$
\epsilon\_{\rm D} = T\_{\rm D} \rho\_{\rm D} = \frac{\sigma\_{\rm D}^2}{2\varepsilon\_0}.\tag{50}
$$

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<sup>Q</sup> must vary as well.

If we assume that the electron charge radius rQ in Eq. (41) equals the classical electron radius <sup>r</sup><sup>e</sup> <sup>¼</sup> <sup>2</sup>:82 fm, then an energy density of <sup>ϵ</sup><sup>D</sup> <sup>¼</sup> <sup>1</sup>:<sup>45</sup> � <sup>10</sup><sup>29</sup> J m�<sup>3</sup> (very high compared to the present cosmic dark energy estimate) follows from

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

#### Figure 6.

ΔPQ,qð Þr Δt

 <sup>¼</sup> <sup>2</sup>p<sup>D</sup>

pD

κ<sup>D</sup> η<sup>D</sup> c0

ΔNQ,qð Þr

Body C with charge Q >0 and mass mC is positioned in this configuration beneath body D with charge þ∣q∣ and mass mD leading to an electrostatic repulsion of the bodies. This results from the reversal of dipoles by the charge Q followed by direct interactions with the charge þ∣q∣ as defined on the right-hand side of Figure 4. Two reversals are schematically indicated in columns I and III. The dipoles arriving in columns II and IV from below have the same polarity as if they would be part of the background distribution and do not contribute to the momentum transfer, because of a compensation by dipoles arriving from above. The net momentum transfer caused by the two interacting reversed dipoles thus is 4pD, i.e. 2p<sup>D</sup> per dipole (modified from Figure 3 of [17]).

> <sup>¼</sup> <sup>T</sup><sup>D</sup> <sup>κ</sup><sup>D</sup> <sup>η</sup><sup>D</sup> c2 0

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

Important questions are related to the energy T<sup>D</sup> and momentum p<sup>D</sup> of the dipoles and, even more, to their energy density in space. Eqs. (9), (40), (41) and

<sup>ϵ</sup><sup>D</sup> <sup>¼</sup> <sup>T</sup><sup>D</sup> <sup>ρ</sup><sup>D</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup>

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<sup>Q</sup> must vary as well.

If we assume that the electron charge radius rQ in Eq. (41) equals the classical electron radius <sup>r</sup><sup>e</sup> <sup>¼</sup> <sup>2</sup>:82 fm, then an energy density of <sup>ϵ</sup><sup>D</sup> <sup>¼</sup> <sup>1</sup>:<sup>45</sup> � <sup>10</sup><sup>29</sup> J m�<sup>3</sup> (very high compared to the present cosmic dark energy estimate) follows from

(44) together with Eq. (49) allow the energy density to be expressed by

<sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>2</sup>p<sup>D</sup>

<sup>¼</sup> <sup>1</sup> 2ε<sup>0</sup>

D 2ε<sup>0</sup> κ<sup>D</sup> η<sup>D</sup> c0

Q ∣q∣

<sup>4</sup><sup>π</sup> <sup>r</sup><sup>2</sup> : (48)

: (50)

(49)

 

Planetology - Future Explorations

Together with

Eq. (2).

78

Figure 5.

Body C is again positioned beneath body D. The charge of D is now �∣q∣, however, leading to an electrostatic attraction of the bodies. The attraction results from the reversal of dipoles by the charge Q >0 followed by indirect interactions with charge �∣q∣ as defined on the left-hand side of Figure 4. Two events without momentum transfer are schematically indicated in columns II and IV. The dipoles arriving in columns I and III from below have the same polarity as if they would be part of the background distribution. The same is true for all dipoles arriving from above. The net momentum transfer caused by the two reversed dipoles thus is �4pD, i.e. �2p<sup>D</sup> per dipole (modified from Figure 4 of [17]).

Eq. (50). The dipole density <sup>ρ</sup><sup>D</sup> <sup>¼</sup> <sup>7</sup>:<sup>95</sup> � <sup>10</sup><sup>56</sup> <sup>m</sup>�<sup>3</sup> in Eq. (40) is also very high, leading to a dipole energy of <sup>T</sup><sup>D</sup> <sup>¼</sup> <sup>1</sup>:<sup>83</sup> � <sup>10</sup>�<sup>28</sup> J. If, on the other hand, we identify the dipole distribution with DM with an estimated energy density of 2:<sup>48</sup> � <sup>10</sup>�<sup>10</sup> J m�<sup>3</sup> and require that the dipole energy density corresponds to this value, then extreme values follow for rQ <sup>¼</sup> <sup>13</sup>:9μm, <sup>ρ</sup><sup>D</sup> <sup>¼</sup> <sup>3</sup>:<sup>28</sup> � 1037 <sup>m</sup>�<sup>3</sup> and <sup>T</sup><sup>D</sup> <sup>¼</sup> <sup>7</sup>:<sup>55</sup> � <sup>10</sup>�<sup>48</sup> J.

### 3. 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.

#### 3.1 Potential energies

#### 3.1.1 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].<sup>2</sup>

<sup>2</sup> In this context, it is of interest that Brillouin [48] discussed this problem in relation to the electrostatic potential energy.

It led to the identification of the "source region" of the potential energy for the special case of a system with two masses M<sup>E</sup> and M<sup>M</sup> subject to the condition M<sup>E</sup> ≫ MM. 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<sup>A</sup> and m<sup>B</sup> 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 Em 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

$$E\_m^2 - \mathbf{p}^2 c\_0^2 = m^2 c\_0^4,\tag{51}$$

where p is the momentum vector

$$p = \nu \frac{E\_m}{c\_0^2} \tag{52}$$

The kinetic energies<sup>3</sup> T<sup>A</sup> and T<sup>B</sup> 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

m<sup>A</sup> m<sup>B</sup>

We may now ask the question, whether the impact model can provide an answer

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

> κG 8π h

> > c0 8π h

The same number is moving in the opposite direction. The energy deficiency of the interacting gravitons with respect to the corresponding background then is

The last term shows—with reference to Eq. (57)—that the energy deficiency ΔE<sup>G</sup>

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

Consequently, the difference of the potential energies between r þ Δr and r in

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�<sup>2</sup> according to Eq. (48),

2.A decrease of r þ Δr to r during the approach of A and B increases the number

<sup>0</sup> [c] and <sup>γ</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

energy of a massive body: T ¼ E � E<sup>0</sup> ¼ E<sup>0</sup> ð Þ γ � 1 . The evaluations for T<sup>A</sup> and T<sup>B</sup> agree in very good

1 � v<sup>2</sup>=c<sup>2</sup> 0

p yield the relativistic kinetic

m<sup>A</sup> m<sup>B</sup>

m<sup>A</sup> m<sup>B</sup>

<sup>r</sup> ¼ � <sup>G</sup><sup>N</sup>

1 r � <sup>1</sup> r þ Δr

2

ΔNmA,m<sup>B</sup> ð Þ¼ r

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<sup>A</sup> and m<sup>B</sup> will be constant in such an approximation, but now there

<sup>r</sup><sup>2</sup> <sup>Δ</sup><sup>r</sup> <sup>¼</sup> <sup>∣</sup>KGð Þ<sup>r</sup> <sup>∣</sup>Δr: (57)

<sup>r</sup> : (58)

m<sup>A</sup> m<sup>B</sup>

� �: (60)

<sup>r</sup> : (59)

small Δr with Newton's law in Eq. (1):

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

G<sup>N</sup> m<sup>A</sup> m<sup>B</sup>

are gravitons interacting with both masses.

together with Eqs. (18) and (27) for each body

c0 8π h m<sup>A</sup> m<sup>B</sup>

<sup>r</sup> ¼ �G<sup>G</sup>

equals half the potential energy of body A at a distance r from body B and

2f g ΔEGð Þ� r þ Δr ΔEGð Þr ¼ G<sup>N</sup> m<sup>A</sup> m<sup>B</sup>

Eq. (57) is balanced by the difference of the total energy deficiencies. The physical processes involved can be described as follows:

whereas the travel time is proportional to r.

of gravitons with reduced energy.

Eqs. (51) and (52) together with <sup>E</sup><sup>0</sup> <sup>¼</sup> m c<sup>2</sup>

approximation with Eq. (56) for small v<sup>A</sup> and vB.

ΔEGð Þ¼� r p<sup>G</sup> Y κ<sup>G</sup>

time r=c0:

vice versa.

and get

3

81

1 r � <sup>1</sup> r þ Δr � � <sup>≈</sup> <sup>G</sup><sup>N</sup>

[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

$$E\_{\nu} = p\_{\nu} c\_0. \tag{53}$$

In [50] we assume that two spherically symmetric bodies A and B with masses m<sup>A</sup> and mB, 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

$$E\_{\rm S} = (m\_{\rm A} + m\_{\rm B})c\_0^2 - G\_{\rm N} \frac{m\_{\rm A} m\_{\rm B}}{r + \Delta r}. \tag{54}$$

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<sup>A</sup> ¼ ∣KGð Þr ∣=m<sup>A</sup> and b<sup>B</sup> ¼ �∣KGð Þr ∣=mB, 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

$$
\Delta r = \mathfrak{s}\_{\mathcal{A}} - \mathfrak{s}\_{\mathcal{B}} = \frac{1}{2} \left( b\_{\mathcal{A}} - b\_{\mathcal{B}} \right) \left( \Delta t \right)^{2} = \frac{1}{2} \left( \frac{1}{m\_{\mathcal{A}}} + \frac{1}{m\_{\mathcal{B}}} \right) \left| K\_{\mathcal{G}}(r) \right| \left( \Delta t \right)^{2}, \tag{55}
$$

showing that s<sup>A</sup> m<sup>A</sup> ¼ �s<sup>B</sup> mB, i.e. the centre of gravity stays at rest. Multiplication of Eq. (55) by ∣KGð Þr ∣ gives the corresponding kinetic energy equation

$$|K\_{\rm G}(r)|\Delta r = \frac{1}{2} \left( \frac{K\_{\rm G}^2(r) \left(\Delta t\right)^2}{m\_{\rm A}} + \frac{K\_{\rm G}^2(r) \left(\Delta t\right)^2}{m\_{\rm B}} \right) = \frac{1}{2} m\_{\rm A} v\_{\rm A}^2 + \frac{1}{2} m\_{\rm B} v\_{\rm B}^2 = T\_{\rm A} + T\_{\rm B}. \tag{56}$$

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

It led to the identification of the "source region" of the potential energy for the special case of a system with two masses M<sup>E</sup> and M<sup>M</sup> subject to the condition M<sup>E</sup> ≫ MM. 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<sup>A</sup> and m<sup>B</sup> 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

The energy Em and the momentum p of a free particle with mass m moving with

Em c2 0

p ¼ v

In [50] we assume that two spherically symmetric bodies A and B with masses m<sup>A</sup> and mB, 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

[40, 51]. For an entity in vacuum with no rest mass (m ¼ 0), such as a photon

2 <sup>0</sup> � G<sup>N</sup>

The evolution of the system during the approach of A and B from r þ Δr to r can

2

showing that s<sup>A</sup> m<sup>A</sup> ¼ �s<sup>B</sup> mB, i.e. the centre of gravity stays at rest. Multiplica-

<sup>G</sup>ð Þ<sup>r</sup> ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup> m<sup>B</sup>

tion of Eq. (55) by ∣KGð Þr ∣ gives the corresponding kinetic energy equation

1 m<sup>A</sup> þ 1 m<sup>B</sup>

� �

¼ 1 <sup>2</sup> <sup>m</sup><sup>A</sup> <sup>v</sup><sup>2</sup> <sup>A</sup> þ 1 <sup>2</sup> <sup>m</sup><sup>B</sup> <sup>v</sup><sup>2</sup>

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<sup>A</sup> ¼ ∣KGð Þr ∣=m<sup>A</sup> and b<sup>B</sup> ¼ �∣KGð Þr ∣=mB, respectively. Since the duration Δt of the free fall of both bodies is the same, the approach of A

0, (51)

: (54)

, (55)

<sup>B</sup> ¼ T<sup>A</sup> þ TB:

(56)

<sup>∣</sup>KGð Þ<sup>r</sup> <sup>∣</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup>

E<sup>ν</sup> ¼ p<sup>ν</sup> c0: (53)

m<sup>A</sup> m<sup>B</sup> r þ Δr (52)

a velocity v relative to an inertial reference system are related by

E2 <sup>m</sup> � <sup>p</sup><sup>2</sup> <sup>c</sup> 2 <sup>0</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup> <sup>c</sup> 4

[15, 52, 53], the energy-momentum relation in Eq. (51) reduces to

E<sup>S</sup> ¼ ð Þ m<sup>A</sup> þ m<sup>B</sup> c

<sup>2</sup> ð Þ <sup>b</sup><sup>A</sup> � <sup>b</sup><sup>B</sup> ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup> <sup>¼</sup> <sup>1</sup>

þ K2

!

rest energies and with Eq. (5) for the potential energy

and B can be formulated as

<sup>∣</sup>KGð Þ<sup>r</sup> <sup>∣</sup>Δ<sup>r</sup> <sup>¼</sup> <sup>1</sup>

80

<sup>Δ</sup><sup>r</sup> <sup>¼</sup> <sup>s</sup><sup>A</sup> � <sup>s</sup><sup>B</sup> <sup>¼</sup> <sup>1</sup>

2

K2

<sup>G</sup>ð Þ<sup>r</sup> ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup> m<sup>A</sup>

corresponding point mass at the centre.

Planetology - Future Explorations

where p is the momentum vector

The kinetic energies<sup>3</sup> T<sup>A</sup> and T<sup>B</sup> 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):

$$\,\_\text{G}\mathbf{G}\_\text{N}\,m\_\text{A}\,m\_\text{B}\left(\frac{1}{r}-\frac{1}{r+\Delta r}\right)\approx\,\text{G}\_\text{N}\,\frac{m\_\text{A}m\_\text{B}}{r^2}\,\Delta r=|\mathbf{K}\_\text{G}(r)|\,\Delta r.\tag{57}$$

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<sup>A</sup> and m<sup>B</sup> 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=c0:

$$
\Delta N\_{m\_{\rm A}, m\_{\rm B}}(r) = \frac{\kappa\_{\rm G}}{8\pi h} \frac{m\_{\rm A} m\_{\rm B}}{r} . \tag{58}
$$

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

$$
\Delta E\_{\rm G}(r) = -p\_{\rm G} Y \kappa\_{\rm G} \frac{c\_0}{8\pi h} \frac{m\_{\rm A} m\_{\rm B}}{r} = -\rm G\_{\rm G} \frac{c\_0}{8\pi h} \frac{m\_{\rm A} m\_{\rm B}}{r} = -\frac{G\_{\rm N}}{2} \frac{m\_{\rm A} m\_{\rm B}}{r} . \tag{59}
$$

The last term shows—with reference to Eq. (57)—that the energy deficiency ΔE<sup>G</sup> 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

$$2\left\{\Delta E\_{\rm G}(r+\Delta r)-\Delta E\_{\rm G}(r)\right\}=G\_{\rm N}m\_{\rm A}m\_{\rm B}\left(\frac{1}{r}-\frac{1}{r+\Delta r}\right).\tag{60}$$

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:


<sup>3</sup> Eqs. (51) and (52) together with <sup>E</sup><sup>0</sup> <sup>¼</sup> m c<sup>2</sup> <sup>0</sup> [c] and <sup>γ</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v<sup>2</sup>=c<sup>2</sup> 0 p yield the relativistic kinetic energy of a massive body: T ¼ E � E<sup>0</sup> ¼ E<sup>0</sup> ð Þ γ � 1 . The evaluations for T<sup>A</sup> and T<sup>B</sup> agree in very good approximation with Eq. (56) for small v<sup>A</sup> and vB.


#### 3.1.2 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

$$w = \frac{\varepsilon\_0}{2} \text{ E}^2,\tag{61}$$

motions of both bodies is the same, the separation (upper sign) or approach (lower

2

1 2 K2 <sup>E</sup>ð Þr m<sup>A</sup>

<sup>≈</sup> � <sup>Δ</sup><sup>r</sup> Q q

Comparing the second term of the equation with the last one, it can be seen that s<sup>D</sup> m<sup>D</sup> ¼ �s<sup>C</sup> mC, i.e. the centre of gravity stays at rest. Multiplication of Eq. (66) by

ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup> <sup>þ</sup>

where v<sup>D</sup> ¼ b<sup>D</sup> Δt and v<sup>C</sup> ¼ b<sup>C</sup> Δt are the speeds of the bodies, when the distances r � Δr between C and D are attained. The sum of the kinetic energies T<sup>C</sup> and T<sup>D</sup> must, of course, be equal to the difference of the electrostatic potential energy at

The variations of the number of ΔNQ,<sup>q</sup>ð Þr dipoles in Eqs. (58) and (65) during

� 1 r ≈ ∓Δ<sup>r</sup>

The number of reversed dipoles thus decreases during the separation of C and D

cf. Eq. (65). The energy of the reversed dipoles thus decreases by the amount

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

The electrostatically bound two-body system thus is a closed system in the sense

δPQ,<sup>q</sup>ð Þ¼� r, Δr 2p<sup>D</sup> δNQ,<sup>q</sup>ð Þ r, Δr (71)

Q

1 r � Δr

<sup>δ</sup>EQ,<sup>q</sup>ð Þ¼ <sup>r</sup>, <sup>Δ</sup><sup>r</sup> <sup>2</sup>p<sup>D</sup> <sup>c</sup><sup>0</sup> <sup>δ</sup>NQ,<sup>q</sup>ð Þ¼� <sup>r</sup>, <sup>Δ</sup><sup>r</sup> <sup>Δ</sup>r∣q<sup>∣</sup> <sup>Q</sup>

and so is the energy of the reversed dipoles confined in the system:

δEQ,<sup>q</sup>ð Þ¼ r, Δr δPQ,<sup>q</sup>ð Þ r, Δr c<sup>0</sup> ¼ �∣q∣Δr

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

1 r � <sup>1</sup> r � Δr

1 m<sup>C</sup> þ 1 m<sup>D</sup> <sup>K</sup>Eð Þ<sup>r</sup> ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup>

ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup> <sup>¼</sup> <sup>1</sup>

η<sup>D</sup> κ<sup>D</sup> c2 0

Q ∣q∣

<sup>2</sup> <sup>m</sup><sup>D</sup> <sup>v</sup><sup>2</sup> D þ 1 <sup>2</sup> <sup>m</sup><sup>C</sup> <sup>v</sup><sup>2</sup> C

<sup>4</sup>π ε<sup>0</sup> <sup>r</sup><sup>2</sup> <sup>&</sup>gt;0: (68)

<sup>4</sup><sup>π</sup> <sup>r</sup><sup>2</sup> : (69)

<sup>4</sup>π ε<sup>0</sup> <sup>r</sup><sup>2</sup> <sup>&</sup>lt;0, (70)

<sup>4</sup>π ε<sup>0</sup> <sup>r</sup><sup>2</sup> <sup>&</sup>lt;0: (72)

: (66)

(67)

<sup>2</sup> ð Þ <sup>b</sup><sup>D</sup> � <sup>b</sup><sup>C</sup> ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup> <sup>¼</sup> <sup>1</sup>

sign) of C and D can be formulated as follows:

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

1

<sup>4</sup>π ε<sup>0</sup> <sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>

¼ T<sup>D</sup> þ T<sup>C</sup> >0,

f g U rð Þ� U rð Þ � <sup>Δ</sup><sup>r</sup> <sup>q</sup> <sup>¼</sup> Q q

that fuels the kinetic energy in Eq. (67).

modified dipole distributions.

negative

83

KEð Þr gives a good estimate of the corresponding kinetic energy:

2 K2 <sup>E</sup>ð Þr m<sup>B</sup>

4π ε<sup>0</sup>

the separation or approach of bodies C and D from r to r � Δr are

Q ∣q∣ 4π

in Figure 5. The corresponding energy variation with positive q is

δNQ,<sup>q</sup>ð Þ¼ r, Δr ΔNQ,<sup>q</sup>ð Þ� r � Δr ΔNQ,<sup>q</sup>ð Þr

<sup>¼</sup> <sup>η</sup><sup>D</sup> <sup>κ</sup><sup>D</sup> c2 0

�Δr ¼ �ð Þ¼� s<sup>D</sup> � s<sup>C</sup>

�ΔrKEð Þ¼� <sup>r</sup> <sup>Δ</sup><sup>r</sup> Q q

distances of r and r � Δr:

cf., e.g. [24, 54]. Applying Eq. (61) to a plane-plate capacitor with an area F, a plate separation b and charges �∣Q∣ on the plates, the energy stored in the field of the capacitor turns out to be

$$W = \frac{\varepsilon\_0}{2} \operatorname{E}^2 F b = \frac{\varepsilon\_0}{2} \operatorname{E}^2 V. \tag{62}$$

With a potential difference ΔU<sup>E</sup> ¼ ∣E∣b and a charge of Q ¼ ε<sup>0</sup> ∣E∣F (increased incrementally to these values), the potential energy of Q at ΔU<sup>E</sup> is

$$W = \frac{1}{2} Q \Delta U\_{\rm E}.\tag{63}$$

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 >0 to q in Figures 5 and 6 can be calculated from the interaction rate in Eq. (47) multiplied by a travel time Δt ¼ r=c0:

$$
\Delta N\_{Q,q}(r) = \frac{\kappa\_\mathcal{D} \eta\_\mathcal{D}}{c\_0^2} \frac{Q|q|}{4\pi r}. \tag{64}
$$

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:

$$
\Delta E\_{Q,q}(r) = 2\,\Delta N\_{Q,q}(r)p\_{\rm D}c\_0 = \frac{Q|q|}{4\,\pi\varepsilon\_0 r}.\tag{65}
$$

It is equal to the absolute value of the electrostatic potential energy of a charge q at the electrostatic potential UEð Þr in 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 � Δr by the repulsive or attractive force KEð Þr between the charges given by Coulomb's law in Eq. (2) The force is approximately constant for r ≫ Δr>0 causing accelerations of b<sup>D</sup> ¼ KEð Þr =m<sup>D</sup> and b<sup>C</sup> ¼ �KEð Þr =mC, respectively. Since the duration Δt of the

3.The energies liberated by energy reductions are available as potential energy

4.With Eqs. (51) and (52) and the approximations in Footnote 3, it follows that the sum of the kinetic energies T<sup>A</sup> and TB, the masses A and B, plus the total energy deficiencies of the interacting gravitons can indeed be considered to be

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 �∣Q∣ on the plates, the energy stored in the field of

<sup>E</sup><sup>2</sup> F b <sup>¼</sup> <sup>ε</sup><sup>0</sup>

With a potential difference ΔU<sup>E</sup> ¼ ∣E∣b and a charge of Q ¼ ε<sup>0</sup> ∣E∣F (increased

The question as to where the energy is actually stored, [54] answered by show-

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 >0 to q in Figures 5 and 6 can be calculated from the

> κ<sup>D</sup> η<sup>D</sup> c2 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

<sup>Δ</sup>EQ,<sup>q</sup>ð Þ¼ <sup>r</sup> <sup>2</sup>ΔNQ,<sup>q</sup>ð Þ<sup>r</sup> <sup>p</sup><sup>D</sup> <sup>c</sup><sup>0</sup> <sup>¼</sup> <sup>Q</sup> <sup>∣</sup>q<sup>∣</sup>

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 � Δr by the repulsive or attractive force KEð Þr between the charges given by Coulomb's law in Eq. (2) The force is approximately constant for r ≫ Δr>0 causing accelerations of b<sup>D</sup> ¼ KEð Þr =m<sup>D</sup> and b<sup>C</sup> ¼ �KEð Þr =mC, respectively. Since the duration Δt of the

It is equal to the absolute value of the electrostatic potential energy of a charge q

Q ∣q∣

4π ε<sup>0</sup> r

2

, (61)

E<sup>2</sup>V: (62)

<sup>4</sup><sup>π</sup> <sup>r</sup> : (64)

: (65)

<sup>2</sup> <sup>Q</sup> <sup>Δ</sup>UE: (63)

<sup>w</sup> <sup>¼</sup> <sup>ε</sup><sup>0</sup> 2 E2

<sup>W</sup> <sup>¼</sup> <sup>ε</sup><sup>0</sup> 2

incrementally to these values), the potential energy of Q at ΔU<sup>E</sup> is

ing that both concepts implied by Eqs. (62) and (63) are equivalent.

interaction rate in Eq. (47) multiplied by a travel time Δt ¼ r=c0:

at the electrostatic potential UEð Þr in Eq. (37) of a charge Q.

ΔNQ,<sup>q</sup>ð Þ¼ r

<sup>W</sup> <sup>¼</sup> <sup>1</sup>

and are converted into kinetic energies of bodies A and B.

a closed system as defined by von Laue [49].

3.1.2 Electrostatic potential energy.

Planetology - Future Explorations

the capacitor turns out to be

dipoles:

82

motions of both bodies is the same, the separation (upper sign) or approach (lower sign) of C and D can be formulated as follows:

$$\pm \Delta r = \pm (\mathfrak{s}\_{\rm D} - \mathfrak{s}\_{\rm C}) = \pm \frac{1}{2} (b\_{\rm D} - b\_{\rm C}) \left(\Delta t\right)^{2} = \frac{1}{2} \left(\frac{1}{m\_{\rm C}} + \frac{1}{m\_{\rm D}}\right) K\_{\rm E}(r) \left(\Delta t\right)^{2}.\tag{66}$$

Comparing the second term of the equation with the last one, it can be seen that s<sup>D</sup> m<sup>D</sup> ¼ �s<sup>C</sup> mC, i.e. the centre of gravity stays at rest. Multiplication of Eq. (66) by KEð Þr gives a good estimate of the corresponding kinetic energy:

$$\begin{split} \pm \Delta r K\_{\rm E}(r) &= \pm \Delta r \frac{Qq}{4\pi\varepsilon\_{0}r^{2}} = \frac{1}{2} \frac{K\_{\rm E}^{2}(r)}{m\_{\rm B}} \left(\Delta t\right)^{2} + \frac{1}{2} \frac{K\_{\rm E}^{2}(r)}{m\_{\rm A}} \left(\Delta t\right)^{2} = \frac{1}{2} m\_{\rm D} v\_{\rm D}^{2} + \frac{1}{2} m\_{\rm C} v\_{\rm C}^{2} \\ &= T\_{\rm D} + T\_{\rm C} > 0, \end{split} \tag{67}$$

where v<sup>D</sup> ¼ b<sup>D</sup> Δt and v<sup>C</sup> ¼ b<sup>C</sup> Δt are the speeds of the bodies, when the distances r � Δr between C and D are attained. The sum of the kinetic energies T<sup>C</sup> and T<sup>D</sup> must, of course, be equal to the difference of the electrostatic potential energy at distances of r and r � Δr:

$$\{U(r) - U(r \pm \Delta r)\}q = \frac{Qq}{4\pi\varepsilon\_0} \left(\frac{1}{r} - \frac{1}{r \pm \Delta r}\right) \approx \pm \Delta r \frac{Qq}{4\pi\varepsilon\_0 r^2} > 0.\tag{68}$$

The variations of the number of ΔNQ,<sup>q</sup>ð Þr dipoles in Eqs. (58) and (65) during the separation or approach of bodies C and D from r to r � Δr are

$$
\begin{split}
\delta \mathsf{N}\_{Q,q}(r,\Delta r) &= \Delta \mathsf{N}\_{Q,q}(r \pm \Delta r) - \Delta \mathsf{N}\_{Q,q}(r) \\
&= \frac{\eta\_{\mathsf{D}} \kappa\_{\mathsf{D}}}{c\_{0}^{2}} \frac{\mathsf{Q}|q|}{4\pi} \left[\frac{\mathsf{1}}{r \pm \Delta r} - \frac{\mathsf{1}}{r}\right] \approx \mp \Delta r \frac{\eta\_{\mathsf{D}} \kappa\_{\mathsf{D}}}{c\_{0}^{2}} \frac{\mathsf{Q}|q|}{4\pi r^{2}}.\end{split} \tag{69}
$$

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

$$
\delta E\_{\mathcal{Q}\mathcal{A}}(r, \Delta r) = 2p\_{\mathcal{D}}c\_0 \delta \mathcal{N}\_{\mathcal{Q}\mathcal{A}}(r, \Delta r) = -\Delta r|q| \frac{\mathcal{Q}}{4\pi\varepsilon\_0 r^2} < 0,\tag{70}
$$

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

$$
\delta P\_{Q, \emptyset}(r, \Delta r) = -2p\_{\text{D}} \delta \mathcal{N}\_{Q, \emptyset}(r, \Delta r) \tag{71}
$$

and so is the energy of the reversed dipoles confined in the system:

$$
\delta E\_{Q,q}(r,\Delta r) = \delta P\_{Q,q}(r,\Delta r)c\_0 = -|q|\Delta r \frac{Q}{4\pi\varepsilon\_0 r^2} < 0. \tag{72}
$$

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.

#### 3.2 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 <sup>t</sup><sup>1</sup> � <sup>t</sup><sup>0</sup> <sup>≈</sup>11:55years <sup>¼</sup> <sup>3</sup>:<sup>645</sup> � <sup>10</sup><sup>8</sup> s between 3 January 1987 and 22 July 1998, while the spacecraft was at heliocentric distances between r<sup>0</sup> ¼ 40 ua and r<sup>1</sup> ¼ 70:5 ua. The Pioneer team took into account all known contributions in calculating a model frequency νmodð Þt which was based on a constant clock frequency f <sup>0</sup> at the terrestrial control stations. Observations at times t ¼ t<sup>0</sup> þ Δt then indicated a nearly uniform increase of the observed frequency shift with respect to the expected one of

$$
\nu\_{\rm obs}(t) - \nu\_{\rm mod}(t) = 2\dot{\hat{f}}\,\Delta t\tag{73}
$$

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

resulting in frequency increases of atomic clocks with time. They give rise to the clock acceleration in Eq. (77), if we assume at ¼ A. The most likely values of rG,e in Figure 3 range from 2:<sup>04</sup> � <sup>10</sup>�<sup>4</sup> pm to 2.82 fm, the classical electron radius,

corresponding with Eq. (31) to <sup>A</sup><sup>H</sup> <sup>¼</sup> <sup>2</sup>:<sup>43</sup> � <sup>10</sup>�<sup>18</sup> <sup>s</sup>�<sup>1</sup> <sup>≈</sup> <sup>H</sup>0, the Hubble constant, and <sup>A</sup> <sup>≈</sup>1:<sup>3</sup> � <sup>10</sup>�<sup>20</sup> <sup>s</sup>�1. Within the uncertainty margins, the high value agrees with at 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 rG,<sup>e</sup> in Figure 3 could accommodate smaller values of at as well.

A secular increase of the mean Sun-Earth distance with a rate of 15 ð Þ � 4 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 gravi-

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 understand-

Astronomical Union (IAU) and the Bureau International des Poids et Mesure [64] as 1 ua <sup>¼</sup> <sup>1</sup>:<sup>495978707</sup> � <sup>10</sup><sup>11</sup> m (exact). The mean Sun-Earth distance <sup>r</sup><sup>E</sup> is known

is difficult but feasible as relative determination. A circular orbit approximation

<sup>¼</sup> <sup>μ</sup><sup>⊙</sup> v2 E

<sup>2</sup><sup>h</sup> <sup>m</sup><sup>e</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>A</sup>Δ<sup>t</sup> (80)

<sup>¼</sup> ð Þ <sup>4</sup>:<sup>8</sup> � <sup>1</sup>:<sup>3</sup> nm s�<sup>1</sup> (81)

: (82)

R <sup>∗</sup>

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

tational constant seems to provide an explanation [62].

with a standard uncertainty of (3 to 6) m [65–67].

and the mass of the Sun <sup>M</sup><sup>⊙</sup> <sup>¼</sup> <sup>1</sup>:<sup>98842</sup> � <sup>10</sup><sup>30</sup> kg.

85

detailed discussion of this aspect is given in Section 3 of [16].

Δr<sup>E</sup>

ing of the effect within the parameter range of the model [63].

<sup>Δ</sup><sup>t</sup> <sup>¼</sup> ð Þ <sup>15</sup> � <sup>4</sup> <sup>m</sup> <sup>3</sup>:<sup>156</sup> � <sup>10</sup><sup>9</sup> <sup>s</sup>

had been considered, because the mean value of r<sup>E</sup> is of interest:

The value of the astronomical unit is defined by the International

Considering this uncertainty, the measurement of a change rate of

<sup>r</sup><sup>E</sup> <sup>¼</sup> <sup>G</sup>NM<sup>⊙</sup> v2 E

This follows from equating the gravitational attraction, cf. Eq. (1), and the centrifugal force with vE, the tangential orbital velocity of the Earth, where the heliocentric gravitational constant is <sup>μ</sup><sup>⊙</sup> <sup>¼</sup> <sup>1</sup>:327 12440042 � <sup>10</sup><sup>20</sup> m3 <sup>s</sup>�<sup>2</sup> (IAU)

We now consider Eq. (82) not only for t<sup>0</sup> but also at t ¼ t<sup>0</sup> þ Δt assuming constant G<sup>N</sup> as well as constant vE. 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

<sup>∞</sup> ðÞ¼ <sup>t</sup> <sup>α</sup><sup>2</sup> <sup>c</sup><sup>0</sup>

mass m<sup>e</sup> according to

3.3 Sun-Earth distance increase

with \_ <sup>f</sup> <sup>¼</sup> <sup>5</sup>:<sup>99</sup> � <sup>10</sup>�<sup>9</sup> Hz s�<sup>1</sup> [57].

The observations of the anomalous frequency shifts could, in principle, be interpreted as a deceleration of the heliocentric spacecraft velocity by

$$a\_{\mathbf{p}} = -(8.74 \pm 1.33) \times 10^{-10} \text{ m s}^{-2}.\tag{74}$$

However, no unknown sunward-directed force could be identified [58]. Alternatively, a clock acceleration at the ground stations of

$$a\_l = \frac{a\_p}{c\_0} = (2.92 \pm 0.44) \times 10^{-18} \text{ s}^{-1} \tag{75}$$

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 <sup>0</sup> for the calculation of νmodð Þt is not appropriate. Consequently, we modified in [61] the equation

$$
\left[\nu\_{\rm obs}(t) - f\_{\rm o}\right] - \left[\nu\_{\rm mod}(t) - f\_{\rm o}\right] = 2\dot{f}\,\Delta t,\tag{76}
$$

equivalent to Eq. (73) with

$$f(t) = f\_0 + \dot{f}\,\Delta t = f\_0 \left(1 + \frac{\dot{f}}{f\_0}\,\Delta t\right) = f\_0 \left(1 + a\_t\,\Delta t\right) \tag{77}$$

and

$$
\nu\_{\rm mod}^{\*}(t) = \nu\_{\rm mod}(t) + \mathcal{Q}\dot{f}\,\Delta t\tag{78}
$$

to

$$\begin{aligned} \left[\nu\_{\text{obs}}(t) - \left(f\_{\text{0}} + \dot{f}\,\Delta t\right)\right] - \left[\nu\_{\text{mod}}^{\*}(t) - \left(f\_{\text{0}} + \dot{f}\,\Delta t\right)\right] \\ = \left[\nu\_{\text{obs}}(t) - f(t)\right] - \left[\nu\_{\text{mod}}^{\*}(t) - f(t)\right] = \mathbf{0}. \end{aligned} \tag{79}$$

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

3.2 Pioneer anomaly

Planetology - Future Explorations

expected one of

with \_

the literature continued.

equivalent to Eq. (73) with

equation

and

to

84

<sup>f</sup> <sup>¼</sup> <sup>5</sup>:<sup>99</sup> � <sup>10</sup>�<sup>9</sup> Hz s�<sup>1</sup> [57].

natively, a clock acceleration at the ground stations of

at <sup>¼</sup> <sup>a</sup><sup>p</sup> c0

νobsðÞ�t f <sup>0</sup>

ν ∗

<sup>ν</sup>obsðÞ�<sup>t</sup> <sup>f</sup> <sup>0</sup> <sup>þ</sup> \_

h i � �

f tðÞ¼ <sup>f</sup> <sup>0</sup> <sup>þ</sup> \_

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 <sup>t</sup><sup>1</sup> � <sup>t</sup><sup>0</sup> <sup>≈</sup>11:55years <sup>¼</sup> <sup>3</sup>:<sup>645</sup> � <sup>10</sup><sup>8</sup> s between 3 January 1987 and 22 July 1998, while the spacecraft was at heliocentric distances between r<sup>0</sup> ¼ 40 ua and r<sup>1</sup> ¼ 70:5 ua. The Pioneer team took into account all known contributions in calculating a model frequency νmodð Þt which was based on a constant clock frequency f <sup>0</sup> at the terrestrial control stations. Observations at times t ¼ t<sup>0</sup> þ Δt then indicated a

nearly uniform increase of the observed frequency shift with respect to the

<sup>ν</sup>obsðÞ�<sup>t</sup> <sup>ν</sup>modðÞ¼ <sup>t</sup> <sup>2</sup> \_

The observations of the anomalous frequency shifts could, in principle, be

<sup>a</sup><sup>p</sup> ¼ �ð Þ� <sup>8</sup>:<sup>74</sup> � <sup>1</sup>:<sup>33</sup> <sup>10</sup>�<sup>10</sup> m s�<sup>2</sup>

However, no unknown sunward-directed force could be identified [58]. Alter-

<sup>¼</sup> ð Þ� <sup>2</sup>:<sup>92</sup> � <sup>0</sup>:<sup>44</sup> <sup>10</sup>�<sup>18</sup> <sup>s</sup>

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

Assuming an atomic clock acceleration, a constant reference frequency f <sup>0</sup> for the

� � <sup>¼</sup> <sup>2</sup> \_

f f 0 Δt !

modðÞ¼ <sup>t</sup> <sup>ν</sup>modð Þþ<sup>t</sup> <sup>2</sup> \_

� <sup>ν</sup> <sup>∗</sup>

modðÞ�<sup>t</sup> <sup>f</sup> <sup>0</sup> <sup>þ</sup> \_

h i � �

calculation of νmodð Þt is not appropriate. Consequently, we modified in [61] the

� � � <sup>ν</sup>modðÞ�<sup>t</sup> <sup>f</sup> <sup>0</sup>

<sup>f</sup> <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>f</sup> <sup>0</sup> <sup>1</sup> <sup>þ</sup> \_

f Δt

<sup>¼</sup> <sup>ν</sup>obs <sup>½</sup> ðÞ�<sup>t</sup> f tð Þ� � <sup>ν</sup> <sup>∗</sup>

interpreted as a deceleration of the heliocentric spacecraft velocity by

f Δt (73)

: (74)

�<sup>1</sup> (75)

f Δt, (76)

¼ f <sup>0</sup> ð Þ 1 þ at Δt (77)

f Δt (78)

f Δt

modð Þ�<sup>t</sup> f tð Þ � � <sup>¼</sup> <sup>0</sup>: (79)

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<sup>e</sup> according to

$$R\_{\infty}^{\,\*}(t) = \frac{a^2 c\_0}{2h} m\_{\text{e}} \left( \mathbf{1} + A \,\Delta t \right) \tag{80}$$

resulting in frequency increases of atomic clocks with time. They give rise to the clock acceleration in Eq. (77), if we assume at ¼ A. The most likely values of rG,e in Figure 3 range from 2:<sup>04</sup> � <sup>10</sup>�<sup>4</sup> pm to 2.82 fm, the classical electron radius, corresponding with Eq. (31) to <sup>A</sup><sup>H</sup> <sup>¼</sup> <sup>2</sup>:<sup>43</sup> � <sup>10</sup>�<sup>18</sup> <sup>s</sup>�<sup>1</sup> <sup>≈</sup> <sup>H</sup>0, the Hubble constant, and <sup>A</sup> <sup>≈</sup>1:<sup>3</sup> � <sup>10</sup>�<sup>20</sup> <sup>s</sup>�1. Within the uncertainty margins, the high value agrees with at 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 rG,<sup>e</sup> in Figure 3 could accommodate smaller values of at as well.

#### 3.3 Sun-Earth distance increase

A secular increase of the mean Sun-Earth distance with a rate of 15 ð Þ � 4 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 <sup>¼</sup> <sup>1</sup>:<sup>495978707</sup> � <sup>10</sup><sup>11</sup> m (exact). The mean Sun-Earth distance <sup>r</sup><sup>E</sup> is known with a standard uncertainty of (3 to 6) m [65–67].

Considering this uncertainty, the measurement of a change rate of

$$\frac{\Delta r\_{\rm E}}{\Delta t} = \frac{(15 \pm 4)}{3.156 \times 10^9} \frac{\text{m}}{\text{s}} = (4.8 \pm 1.3) \text{ nm} \text{ s}^{-1} \tag{81}$$

is difficult but feasible as relative determination. A circular orbit approximation had been considered, because the mean value of r<sup>E</sup> is of interest:

$$
\sigma\_{\rm E} = \frac{G\_{\rm N} M\_{\odot}}{\upsilon\_{\rm E}^2} = \frac{\mu\_{\odot}}{\upsilon\_{\rm E}^2}. \tag{82}
$$

This follows from equating the gravitational attraction, cf. Eq. (1), and the centrifugal force with vE, the tangential orbital velocity of the Earth, where the heliocentric gravitational constant is <sup>μ</sup><sup>⊙</sup> <sup>¼</sup> <sup>1</sup>:327 12440042 � <sup>10</sup><sup>20</sup> m3 <sup>s</sup>�<sup>2</sup> (IAU) and the mass of the Sun <sup>M</sup><sup>⊙</sup> <sup>¼</sup> <sup>1</sup>:<sup>98842</sup> � <sup>10</sup><sup>30</sup> kg.

We now consider Eq. (82) not only for t<sup>0</sup> but also at t ¼ t<sup>0</sup> þ Δt assuming constant G<sup>N</sup> as well as constant vE. 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].

From Eqs. (30) and (82), it follows

$$r\_{\rm E}(t) = r\_{\rm E} + \Delta r\_{\rm E} \approx \frac{G\_{\rm N}}{v\_{\rm E}^2} M\_{\odot} \left( 1 + A \,\Delta t \right) \tag{83}$$

and

$$\frac{\Delta r\_{\rm E}}{\Delta t} \approx r\_{\rm E} A. \tag{84}$$

(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≈1:3 m, 3:9 m and 0:5 m, respectively. The study had been conducted assuming a spherically symmetric emission of liberated gravitons mentioned in

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

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 �0:6 m in the direction of the perigee of Galileo I, þ1:9 m for NEAR and �1:5 m for Rosetta. Moreover, it was possible to model the decelerations of

Galileo II on 8 December 1992 with a shift of �3:4 m, of Cassini on 18 August 1999

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 ¼ �ð Þ 180 to þ 180 min around perijove for the first and third Juno flyby. The peak anomalous outward accelerations shown are in both cases: <sup>δ</sup><sup>a</sup> <sup>¼</sup> 7 mm s�<sup>2</sup> at <sup>t</sup><sup>≈</sup> � 15 min and <sup>δ</sup><sup>a</sup> <sup>¼</sup> 6 mm s�<sup>2</sup>

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

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

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

offset in the direction of a flyby position near t ¼ �10 min.

3.6 Rotation velocities of spiral galaxies

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

with �2:7 m and the null result for Juno on 9 October 2013 with �2 m.

Section 2.4.

Earth gravitational model.

3.5.2 Juno Jupiter flybys

at t ≈ þ 17 min.

levels.

87

With the help of Eqs. (31) and (81), the electron mass radius can now be calculated. The result is

$$r\_{\rm G, \ell} = \left( r\_{\rm E} \frac{\Delta t}{\Delta r\_{\rm E}} \, \text{1.014} \times 10^{-49} \text{ m}^2 \text{ s}^{-1} \right)^{-1/2} = \left( \text{1.8}\_{-0.2}^{+0.4} \right) \text{ fm}, \tag{85}$$

close to the classical electron radius

$$r\_{\text{e}} = a^2 a\_0 = 2.82 \text{ fm.} \tag{86}$$

The relative accumulation rate deduced from the observations of r<sup>E</sup> finally becomes <sup>A</sup> <sup>¼</sup> <sup>A</sup>u<sup>a</sup> <sup>≈</sup>3:<sup>2</sup> � <sup>10</sup>�<sup>20</sup> <sup>s</sup>�<sup>1</sup> (see Figure 3).

#### 3.4 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

$$(\mathbf{1} - \mathbf{Y})^n \approx \mathbf{1} - n\mathbf{Y} \quad \text{for} \quad \mathbf{Y} \ll \mathbf{1} \tag{87}$$

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.

#### 3.5 Planetary flyby anomalies

#### 3.5.1 Earth flybys

Several Earth flyby manoeuvres indicated anomalous accelerations and decelerations and led to many investigations without reaching a solution of the problem

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

(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≈1:3 m, 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 �0:6 m in the direction of the perigee of Galileo I, þ1:9 m for NEAR and �1:5 m for Rosetta. Moreover, it was possible to model the decelerations of Galileo II on 8 December 1992 with a shift of �3:4 m, of Cassini on 18 August 1999 with �2:7 m and the null result for Juno on 9 October 2013 with �2 m.

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.

#### 3.5.2 Juno Jupiter flybys

From Eqs. (30) and (82), it follows

Planetology - Future Explorations

Δt Δr<sup>E</sup>

close to the classical electron radius

becomes <sup>A</sup> <sup>¼</sup> <sup>A</sup>u<sup>a</sup> <sup>≈</sup>3:<sup>2</sup> � <sup>10</sup>�<sup>20</sup> <sup>s</sup>�<sup>1</sup> (see Figure 3).

in the binomial theorem with the exponent n in

3.5 Planetary flyby anomalies

3.5.1 Earth flybys

86

3.4 Secular perihelion advances in the solar system

and

calculated. The result is

rG,<sup>e</sup> ¼ r<sup>E</sup>

<sup>r</sup>EðÞ¼ <sup>t</sup> <sup>r</sup><sup>E</sup> <sup>þ</sup> <sup>Δ</sup>r<sup>E</sup> <sup>≈</sup> <sup>G</sup><sup>N</sup>

<sup>1</sup>:<sup>014</sup> � <sup>10</sup>�<sup>49</sup> <sup>m</sup><sup>2</sup> <sup>s</sup>

�1=<sup>2</sup>

Δr<sup>E</sup>

With the help of Eqs. (31) and (81), the electron mass radius can now be

The relative accumulation rate deduced from the observations of r<sup>E</sup> finally

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

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

Several Earth flyby manoeuvres indicated anomalous accelerations and decelerations and led to many investigations without reaching a solution of the problem

graviton emission can quantitatively account for such a displacement.

v2 E

�1

M<sup>⊙</sup> ð Þ 1 þ AΔt (83)

fm, (85)

<sup>Δ</sup><sup>t</sup> <sup>≈</sup>r<sup>E</sup> <sup>A</sup>: (84)

<sup>¼</sup> <sup>1</sup>:8þ0:<sup>4</sup> �0:<sup>2</sup>

<sup>r</sup><sup>e</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup> <sup>a</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>:82 fm: (86)

ð Þ <sup>1</sup> � <sup>Y</sup> <sup>n</sup> <sup>≈</sup><sup>1</sup> � nY for <sup>Y</sup> <sup>≪</sup> <sup>1</sup> (87)

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 ¼ �ð Þ 180 to þ 180 min around perijove for the first and third Juno flyby. The peak anomalous outward accelerations shown are in both cases: <sup>δ</sup><sup>a</sup> <sup>¼</sup> 7 mm s�<sup>2</sup> at <sup>t</sup><sup>≈</sup> � 15 min and <sup>δ</sup><sup>a</sup> <sup>¼</sup> 6 mm s�<sup>2</sup> 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 ¼ �10 min.

#### 3.6 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

3.7 Light deflection and Shapiro delay

Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

bined to a mean value of approximately 2″.

Eν=c<sup>2</sup> 0.

with ∣pG∣ ¼ p<sup>G</sup> ¼ TG=c0.

write for a pair of interactions:

<sup>m</sup><sup>ν</sup> <sup>&</sup>lt;10�<sup>49</sup> kg [92, 93] or even to <sup>m</sup><sup>ν</sup> <sup>&</sup>lt;6:<sup>3</sup> � <sup>10</sup>�<sup>53</sup> kg [94].

where p<sup>0</sup>

of the deflection.

4

5

89

re-emission, both for massive particles and photons.

covered in our earlier paper on the gravitational redshift [96].

The deflection of light near gravitational centres is of fundamental importance.

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.<sup>5</sup> 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

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

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

We then apply the momentum conservation principle to photon-graviton pairs

We assume, applying Eq. (88) with p<sup>G</sup> ≪ p<sup>ν</sup> ¼ ∣p<sup>ν</sup> ∣, that under the influence of a gravitational centre relevant interactions occur on opposite sides of a photon with p<sup>G</sup> and p<sup>G</sup> ð Þ 1 � Y transferring a net momentum of 2Y pG. 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

<sup>0</sup> ¼ ∣pν∣c

<sup>ν</sup> is the photon momentum after the events. If p<sup>ν</sup> and a component of

It is of interest in the context of this paper that Einstein employed Huygens' principle in his calculation

A zero mass of photons follows from the STR and a speed of light in vacuum c<sup>0</sup> 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

<sup>0</sup> ¼ E<sup>0</sup>

<sup>ν</sup> � p<sup>G</sup> (88)

<sup>ν</sup>, (89)

ν ¼ Eν=h. It is summarized here under the assumption of an antiparallel

in the same way as to photons [73] and can write after a reflection of p<sup>G</sup>

E<sup>ν</sup> ¼ ∣pν∣c ¼ ∣p<sup>ν</sup> þ 2YpG∣c

2Yp<sup>G</sup> are pointing in the same direction, it is c<sup>0</sup> <c, the speed is reduced; an antiparallel direction leads to c<sup>0</sup> >c. Note that this could, however, not result in

<sup>p</sup><sup>ν</sup> <sup>þ</sup> <sup>p</sup><sup>G</sup> <sup>¼</sup> <sup>p</sup><sup>ν</sup> <sup>þ</sup> <sup>2</sup> <sup>p</sup><sup>G</sup> � <sup>p</sup><sup>G</sup> <sup>¼</sup> <sup>p</sup> <sup>∗</sup>

For a beam passing close to the Sun, Soldner [86] and Einstein [87] obtained a deflection angle of 0:87<sup>00</sup> 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]<sup>4</sup> 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 com-

Figure 7.

Anomalous radial outward acceleration δa experienced by Juno near the perijove at time t ¼ 0 (solid curve with diamond signs). It is composed of δaU calculated from the adjusted potential and δaM calculated from the adjusted centrifugal energy (see effective potential energy equation 14 of [73]). A multi-interaction process has been assumed within the mass 1.89858 � 1027 kg of Jupiter. It causes an offset <sup>ρ</sup> of the effective pivotal point of the gravitational attraction from the geometric centre of Jupiter (dotted curve). Also shown are the equatorial radius of Jupiter RJ <sup>E</sup> (solid bar) and the radial distance r of Juno from the centre (dash-dot curve) (Figure 7 of [75]).

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 Y T<sup>G</sup> (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.
