3.7 Light deflection and Shapiro delay

The deflection of light near gravitational centres is of fundamental importance. For a beam passing close to the Sun, Soldner [86] and Einstein [87] obtained a deflection angle of 0:87<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 combined to a mean value of approximately 2″.

The deflection of light has also been considered in the context of the gravitational impact model summarized in Section 2.4. As a secular mass increase of matter was a consequence of this model, the question arises on how the interaction of gravitons with photons can be understood, since the photon mass is in all likelihood zero.<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 ν ¼ Eν=h. It is summarized here under the assumption of an antiparallel re-emission, both for massive particles and photons.

A physical process will then be outlined that provides information on the gravitational potential U at the site of a photon emission [95]. This aspect had not been covered in our earlier paper on the gravitational redshift [96].

Interactions between massive bodies have been treated in [16] with an absorption rate of half the intrinsic de Broglie frequency of a mass, because two virtual gravitons have to be emitted for one interaction. The momentum transfer to a photon will thus be twice as high as to a massive body with a mass equivalent to Eν=c<sup>2</sup> 0.

We then apply the momentum conservation principle to photon-graviton pairs in the same way as to photons [73] and can write after a reflection of p<sup>G</sup>

$$\mathbf{p}\_{\nu} + \mathbf{p}\_{\mathrm{G}} = \mathbf{p}\_{\nu} + 2\,\mathbf{p}\_{\mathrm{G}} - \mathbf{p}\_{\mathrm{G}} = \mathbf{p}\_{\nu}^{\*} - \mathbf{p}\_{\mathrm{G}} \tag{88}$$

with ∣pG∣ ¼ p<sup>G</sup> ¼ TG=c0.

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 write for a pair of interactions:

$$E\_{\nu} = |\mathbf{p}\_{\nu}|\boldsymbol{\varepsilon} = |\mathbf{p}\_{\nu} + 2\boldsymbol{Y}\mathbf{p}\_{\mathrm{G}}|\boldsymbol{\varepsilon}' = |\mathbf{p}\_{\nu}|\boldsymbol{\varepsilon}' = E\_{\nu}',\tag{89}$$

where p<sup>0</sup> <sup>ν</sup> is the photon momentum after the events. If p<sup>ν</sup> and a component of 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

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

<sup>E</sup> (solid bar) and the radial distance r of Juno from the centre (dash-dot curve) (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

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 two-

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

stellar components in galaxies.

Planetology - Future Explorations

Figure 7.

of [75]).

88

radius of Jupiter RJ

dimensional distribution of reduced gravitons.

reduced acceleration in directions inclined to this plane.

<sup>4</sup> It is of interest in the context of this paper that Einstein employed Huygens' principle in his calculation of the deflection.

<sup>5</sup> 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>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].

c<sup>0</sup> >c0, because c ¼ c<sup>0</sup> can only be attained in a region with an isotropic distribution of gravitons with a momentum of pG, i.e. with a gravitational potential U<sup>0</sup> ¼ 0.

The momentum p<sup>ν</sup> of a photon radially approaching a gravitational centre will be treated in line with Eq. (6) of [17] for massive bodies, however, with twice the rate of interaction. Since we know from observations that the deflection of light near the Sun is very small, the momentum variation caused by the weak and static gravitational interaction is also very small. The momentum change rate of the photon can then be approximated by

$$\frac{\delta \mathcal{p}\_{\nu}}{\Delta t} \approx 2 \, G\_{\text{N}} M\_{\odot} \, \frac{\hat{r}}{r^{2}} \frac{p\_{\nu}}{c\_{0}},\tag{90}$$

predicted and subsequently observed Shapiro delay [102–107] but also indirectly by

observational detection by Dyson et al. [89] leave no doubt that a photon is deflected by a factor of two more than the expected relative to a corresponding massive particle. Since in our concept the interaction rate between photons and gravitons is twice as high as for massive particles of the same total energy, the reflection of a graviton from a photon with a momentum of 1ð Þ � Y p<sup>G</sup> must also be antiparallel to the incoming one, i.e. a momentum of �2Y p<sup>G</sup> will be transferred. Otherwise the correct deflection angle for photons cannot be obtained. This modified interaction process has one further important advantage: the reflected graviton can interact with the deflecting gravitational centre and transfers 2Y pG—through the process outlined in the paragraph just before Eq. (48)—in compliance with the momentum conservation principle. In the old scheme, the violation of this principle had no observational consequences, because of the extremely large masses of relevant gravitational centres, but the adherence to both the momentum and energy conservation principles is very encouraging and clearly favours the new concept. Basically the same arguments are relevant for the longitudinal interaction between photons and gravitons. The momentum transfer per interaction will be doubled, but the gravitational absorption coefficient will be reduced by a factor of two. Together with an increased graviton density, all quantities and results are the same as before. However, a detailed analysis shows that the momentum conserva-

The deflection of light by gravitational centres according to the GTR [2] and its

The gravitational potential U at a distance r from a spherical body with mass M

¼ � <sup>G</sup>N<sup>M</sup> c2 0 r

cf. [73]. A definition of a reference potential in line with this formulation is

[109], is still an important subject in modern physics and astrophysics [95, 96, 110–114]. This can be exemplified by two conflicting statements. Wolf et al. [10] write: "The clock frequency is sensitive to the gravitational potential U and not to the local gravity field g ¼ ∇U". Whereas it is claimed by Müller et al. [11]: "We first

Support for the first alternative can be found in many publications [49, 88, 95, 96, 109, 115–117], but it is, indeed, not obvious how an atom can locally sense the gravitational potential U. Experiments on Earth, in space and in the Sun-Earth system, cf. [118–123], however, have quantitatively confirmed in the static weak

> <sup>≈</sup> <sup>Δ</sup><sup>U</sup> c2 0

where ν<sup>0</sup> is the frequency of the radiation emitted by a certain transition at U<sup>0</sup> and ν is the observed frequency there, if the emission caused by the same transition

<sup>¼</sup> <sup>U</sup> � <sup>U</sup><sup>0</sup> c2 0

The study of the gravitational redshift, predicted for solar radiation by Einstein

≤ 0 (96)

, (97)

is constraint in the weak-field approximation for nonrelativistic cases by

U c2 0

�1 ≪

note that no experiment is sensitive to the absolute potential U".

field approximation a relative frequency shift of

had occurred at a potential U.

91

ν � ν<sup>0</sup> ν0

<sup>¼</sup> <sup>Δ</sup><sup>ν</sup> ν0

the deflection of light [89].

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

tion principle is now also adhered to.

3.8 Gravitational redshift

U<sup>∞</sup> ¼ 0 for r ¼ ∞.

where r ¼ ∣r∣ is the distance of the photon from the centre, and the position vector of the photon is rr^with a unit vector r^. The small deflection angle also allows an approximation of the actual path by a straight line along an x axis: x≈c<sup>0</sup> t. The normalized momentum variation along the trajectory then is

$$\frac{\delta p\_{\nu}}{p\_{\nu}} \cos \theta \approx \frac{2G\_{\text{N}}M\_{\odot}}{c\_{0}} \frac{\varkappa}{r^{3}} \Delta t. \tag{91}$$

The corresponding component perpendicular to the trajectory is

$$\frac{\delta p\_{\nu}}{p\_{\nu}}\sin\theta \approx \frac{2G\_{\text{N}}M\_{\odot}}{c\_{0}}\frac{R}{r^{3}}\Delta t,\tag{92}$$

where R is the impact parameter of the trajectory. Integration of Eq. (91) over t from �∞ to x=c<sup>0</sup> (for details see [17]) yields

$$\frac{\left[\Delta p\_{\nu}(r)\right]\_{\text{x}}}{p\_{\nu}} \approx \frac{2\,\text{G}\_{\text{N}}\,\text{M}\_{\odot}}{c\_{0}^{2}r} = \frac{2\,\text{G}\_{\text{N}}\,\text{M}\_{\odot}}{c\_{0}^{2}\sqrt{\text{R}^{2} + \text{x}^{2}}}\,\text{.}\tag{93}$$

If we apply Eq. (89) to a photon approaching the Sun along the x axis starting from infinity with E<sup>ν</sup> ¼ p<sup>ν</sup> c0, and considering that the y component in Eq. (91) is much smaller than the x component in Eq. (92) for x ≫ R, the photon speed c rð Þ as a function of r can be determined from

$$p\_{\nu}c\_{0} \approx \left\{ p\_{\nu} + \left[ \Delta p\_{\nu}(r) \right]\_{\ge} \right\} c(r). \tag{94}$$

Division by p<sup>ν</sup> then gives with Eq. (93)

$$\frac{1}{[n\_{\rm G}(r)]\_{\rm x}} = \frac{c(r)}{c\_0} \approx \mathbf{1} - \frac{2\,\mathrm{G\_{\rm N}}\,\mathrm{M\_{\odot}}}{c\_0^2 \,\mathrm{r}} = \mathbf{1} + \frac{2\,\mathrm{U}(r)}{c\_0^2} \tag{95}$$

as a good approximation of the inverse gravitational index of refraction along the x axis. The same index has been obtained albeit with different arguments, e.g. in [99, 100]. The resulting speed of light is in agreement with evaluations by Schiff [88], for a radial propagation<sup>6</sup> in a central gravitational field, and Okun [101] —calculated on the basis of the standard Schwarzschild metric. A decrease of the speed of light near the Sun, consistent with Eq. (95), is not only supported by the

<sup>6</sup> Einstein [108] states explicitly that the speed at a certain location is not dependent on the direction of the propagation.

c<sup>0</sup> >c0, because c ¼ c<sup>0</sup> can only be attained in a region with an isotropic distribution of gravitons with a momentum of pG, i.e. with a gravitational potential U<sup>0</sup> ¼ 0. The momentum p<sup>ν</sup> of a photon radially approaching a gravitational centre will be treated in line with Eq. (6) of [17] for massive bodies, however, with twice the rate of interaction. Since we know from observations that the deflection of light near the Sun is very small, the momentum variation caused by the weak and static gravitational interaction is also very small. The momentum change rate of the

δp<sup>ν</sup>

normalized momentum variation along the trajectory then is

δp<sup>ν</sup> pν

δp<sup>ν</sup> pν

<sup>Δ</sup>pνð Þ<sup>r</sup> � �

pν

x

<sup>¼</sup> c rð Þ c0

from �∞ to x=c<sup>0</sup> (for details see [17]) yields

function of r can be determined from

6

90

the propagation.

Division by p<sup>ν</sup> then gives with Eq. (93)

1 ½ � nGð Þr <sup>x</sup>

The corresponding component perpendicular to the trajectory is

<sup>Δ</sup><sup>t</sup> <sup>≈</sup> <sup>2</sup>G<sup>N</sup> <sup>M</sup><sup>⊙</sup>

where r ¼ ∣r∣ is the distance of the photon from the centre, and the position vector of the photon is rr^with a unit vector r^. The small deflection angle also allows an approximation of the actual path by a straight line along an x axis: x≈c<sup>0</sup> t. The

> cos <sup>ϑ</sup> <sup>≈</sup> <sup>2</sup>GNM<sup>⊙</sup> c0

sin <sup>ϑ</sup><sup>≈</sup> <sup>2</sup>G<sup>N</sup> <sup>M</sup><sup>⊙</sup> c0

<sup>≈</sup> <sup>2</sup>GNM<sup>⊙</sup> c2

<sup>p</sup><sup>ν</sup> <sup>c</sup><sup>0</sup> <sup>≈</sup> <sup>p</sup><sup>ν</sup> <sup>þ</sup> <sup>Δ</sup>pνð Þ<sup>r</sup> � �

where R is the impact parameter of the trajectory. Integration of Eq. (91) over t

If we apply Eq. (89) to a photon approaching the Sun along the x axis starting from infinity with E<sup>ν</sup> ¼ p<sup>ν</sup> c0, and considering that the y component in Eq. (91) is much smaller than the x component in Eq. (92) for x ≫ R, the photon speed c rð Þ as a

n o

<sup>≈</sup><sup>1</sup> � <sup>2</sup>GNM<sup>⊙</sup> c2

as a good approximation of the inverse gravitational index of refraction along the x axis. The same index has been obtained albeit with different arguments, e.g. in [99, 100]. The resulting speed of light is in agreement with evaluations by Schiff [88], for a radial propagation<sup>6</sup> in a central gravitational field, and Okun [101] —calculated on the basis of the standard Schwarzschild metric. A decrease of the speed of light near the Sun, consistent with Eq. (95), is not only supported by the

Einstein [108] states explicitly that the speed at a certain location is not dependent on the direction of

r^ r2 pν c0

x

R

<sup>0</sup> <sup>r</sup> <sup>¼</sup> <sup>2</sup>GNM<sup>⊙</sup> c2 0

x

<sup>0</sup> <sup>r</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>R</sup><sup>2</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup>

, (90)

<sup>r</sup><sup>3</sup> <sup>Δ</sup>t: (91)

<sup>r</sup><sup>3</sup> <sup>Δ</sup>t, (92)

<sup>p</sup> : (93)

c rð Þ: (94)

(95)

2U rð Þ c2 0

photon can then be approximated by

Planetology - Future Explorations

predicted and subsequently observed Shapiro delay [102–107] but also indirectly by the deflection of light [89].

The deflection of light by gravitational centres according to the GTR [2] and its observational detection by Dyson et al. [89] leave no doubt that a photon is deflected by a factor of two more than the expected relative to a corresponding massive particle. Since in our concept the interaction rate between photons and gravitons is twice as high as for massive particles of the same total energy, the reflection of a graviton from a photon with a momentum of 1ð Þ � Y p<sup>G</sup> must also be antiparallel to the incoming one, i.e. a momentum of �2Y p<sup>G</sup> will be transferred. Otherwise the correct deflection angle for photons cannot be obtained. This modified interaction process has one further important advantage: the reflected graviton can interact with the deflecting gravitational centre and transfers 2Y pG—through the process outlined in the paragraph just before Eq. (48)—in compliance with the momentum conservation principle. In the old scheme, the violation of this principle had no observational consequences, because of the extremely large masses of relevant gravitational centres, but the adherence to both the momentum and energy conservation principles is very encouraging and clearly favours the new concept.

Basically the same arguments are relevant for the longitudinal interaction between photons and gravitons. The momentum transfer per interaction will be doubled, but the gravitational absorption coefficient will be reduced by a factor of two. Together with an increased graviton density, all quantities and results are the same as before. However, a detailed analysis shows that the momentum conservation principle is now also adhered to.

#### 3.8 Gravitational redshift

The gravitational potential U at a distance r from a spherical body with mass M is constraint in the weak-field approximation for nonrelativistic cases by

$$-1 \ll \frac{U}{c\_0^2} = -\frac{G\_\mathrm{N}M}{c\_0^2 \, r} \le 0 \tag{96}$$

cf. [73]. A definition of a reference potential in line with this formulation is U<sup>∞</sup> ¼ 0 for r ¼ ∞.

The study of the gravitational redshift, predicted for solar radiation by Einstein [109], is still an important subject in modern physics and astrophysics [95, 96, 110–114]. This can be exemplified by two conflicting statements. Wolf et al. [10] write: "The clock frequency is sensitive to the gravitational potential U and not to the local gravity field g ¼ ∇U". Whereas it is claimed by Müller et al. [11]: "We first note that no experiment is sensitive to the absolute potential U".

Support for the first alternative can be found in many publications [49, 88, 95, 96, 109, 115–117], but it is, indeed, not obvious how an atom can locally sense the gravitational potential U. Experiments on Earth, in space and in the Sun-Earth system, cf. [118–123], however, have quantitatively confirmed in the static weak field approximation a relative frequency shift of

$$\frac{\nu - \nu\_0}{\nu\_0} = \frac{\Delta \nu}{\nu\_0} \approx \frac{\Delta U}{c\_0^2} = \frac{U - U\_0}{c\_0^2},\tag{97}$$

where ν<sup>0</sup> is the frequency of the radiation emitted by a certain transition at U<sup>0</sup> and ν is the observed frequency there, if the emission caused by the same transition had occurred at a potential U.

Since Einstein discussed the gravitational redshift and published conflicting statements regarding this effect in [2, 87, 109], the confusion could still not be cleared up consistently, cf., e.g. [124, 125]. In most of his publications Einstein defined clocks as atomic clocks. Initially he assumed that the oscillation of an atom corresponding to a spectral line might be an intra-atomic process, the frequency of which would be determined by the atom alone. Scott [126] also felt that the equivalence principle and the notion of an ideal clock running independently of acceleration suggest that such clocks are unaffected by gravity. Einstein [2] later concluded that clocks would slow down near gravitational centres, thus causing a redshift.

von Laue [138] wrote that the existence of an aether is not a physical, but a philosophical problem, but later differentiated between the physical world and its mathematical formulation [139]: a four-dimensional "world" is only a valuable mathematical trick; a deeper insight, which some people want to see behind it, is

In contrast to his earlier statements, Einstein said at the end of a speech in Leiden that according to the GTR, a space without aether cannot be conceived [140] and even more detailed thus one could instead of talking about "aether" as well discuss the "physical properties of space". In theoretical physics we cannot do without aether, i.e. a continuum endowed with physical properties [141]. Michelson et al. [142] confessed at a meeting in Pasadena in the presence of H.A. Lorentz that he clings a little to the aether, and Dirac [143] wrote in a letter to Nature that there

In [17] we proposed an impact model for the electrostatic force based on massless dipoles. The vacuum is thought to be permeated by these dipoles that are, in the absence of electromagnetic or gravitational disturbances, oriented and directed randomly propagating along their dipole axis with a speed of c0. There is little or no interaction among them. We suggest to identify the dipole distribution postulated in Section 2.5 with an aether. Einstein's aether mentioned above may, however, be more related to the gravitational interactions, cf. [144]. In this case, we have to

We now assume that an individual dipole interacts with gravitons in the same

where T<sup>D</sup> and p<sup>D</sup> refer to the energy and momentum of a dipole. The condition

We can then modify Eqs. (90)–(94) by changing ν to D and find that Eqs. (95)

Considering that many suggestions have been made to describe photons as solitons, e.g. in [145–150], we also propose that a photon is a soliton propagating in the dipole aether with a speed of c Uð Þ, cf. Eq. (98), controlled by the dipoles moving in the direction of propagation of the photon. The dipole distribution thus determines the gravitational index of refraction, cf. Eq. (95), and consequently the speed of light c Uð Þ at the potential U. This solves the problem formulated in relation to Eq. (98) and might be relevant for other phenomena, such as gravitational lensing and the cosmological redshift, cf., e.g. [151]. Should the speculation in Section 2.5.2 be taken seriously that the dipole distribution corresponds to DM, it has to be much more evenly distributed than previously thought [152]. The light deflection would

<sup>p</sup><sup>D</sup> <sup>≫</sup> <sup>p</sup>G, cf. Eq. (88), is fulfilled in the range from <sup>Y</sup> <sup>≈</sup> <sup>10</sup>�<sup>22</sup> to 10�<sup>15</sup> for all

then be caused by gravitationally induced index of refraction variations.

With Newton's law of gravitation as starting point, the ideas presented in Section 2.4 allow an understanding of far-reaching gravitational force between massive particles as local interactions of hypothetical massless gravitons travelling with the speed of light in vacuum. The gravitational attraction leads to a general mass accretion of massive particles with time, fuelled by a decrease of the graviton energy density in space. The physical processes during the conversion of gravitational potential energy into kinetic energy have been described for two bodies with

<sup>0</sup> ¼ ∣p<sup>0</sup> <sup>D</sup>∣c <sup>0</sup> ¼ T<sup>0</sup>

D, (99)

consider the graviton distribution as another component of the aether.

T<sup>D</sup> ¼ ∣pD∣c ¼ ∣p<sup>D</sup> þ 2Y pG∣c

and (98) are also valid for dipoles with a speed of c<sup>0</sup> for U<sup>0</sup> ¼ 0.

not involved.

are good reasons for postulating an aether.

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

way as photons in Eq. (89), i.e. according to

r<sup>e</sup> ≤ 2:82 fm (see Section 2.5 and Figure 3).

4. Discussion and conclusions

93

The question whether the gravitational redshift is caused by the emission process (case a) or during the transmission phase (case b) is nevertheless still a matter of recent debates. Proponents are, e.g. of (a) Schiff [88], Okun et al. [116], Møller [127], Cranshaw et al. [128] and Ohanian [129], and of (b) Hay et al. [130], Straumann [131], Randall [132] and Will [133]. It is surprising that the same team of experimenters albeit with different first authors published different views in [128, 130] on the process of the Pound-Rebka-Experiment.

Pound and Snider [120] and Pound [134] pointed out that this experiment could not distinguish between the two options, because the invariance of the velocity of the radiation had not been demonstrated.

Einstein [13] emphasized that for an elementary emission process, not only the energy exchange but also the momentum transfer is of importance; see also [12, 46, 97]. Taking these considerations into account, we formulated a photon emission process at a gravitational potential U assuming that:


As outlined in Section 3.7, there is general agreement in the literature that the local speed of light is

$$c(U) \approx c\_0 \left( 1 + \frac{2U}{c\_0^2} \right) \tag{98}$$

in line with Eq. (95) in Section 3.7. It has, however, to be noted that the speed c Uð Þ was obtained for a photon propagating from U<sup>0</sup> to U, and, therefore, the physical process which controls the speed of newly emitted photons at a gravitational potential U is not yet established.

An attempt to do that will be made by assuming an aether model. Before we suggest a specific aether model, a few statements on the aether concept in general should be mentioned. Following Michelson and Morley [135] famous experiment, Einstein [51, 109] concluded that the concept of a light aether as carrier of the electric and magnetic forces is not consistent with the STR. In response to critical remarks by Wiechert [136], cf. Schröder [137] for Wiechert's support of the aether, Impact Models of Gravitational and Electrostatic Forces DOI: http://dx.doi.org/10.5772/intechopen.86744

Since Einstein discussed the gravitational redshift and published conflicting statements regarding this effect in [2, 87, 109], the confusion could still not be cleared up consistently, cf., e.g. [124, 125]. In most of his publications Einstein defined clocks as atomic clocks. Initially he assumed that the oscillation of an atom corresponding to a spectral line might be an intra-atomic process, the frequency of which would be determined by the atom alone. Scott [126] also felt that the equivalence principle and the notion of an ideal clock running independently of acceleration suggest that such clocks are unaffected by gravity. Einstein [2] later concluded that clocks would slow down near gravitational centres, thus causing a redshift. The question whether the gravitational redshift is caused by the emission process (case a) or during the transmission phase (case b) is nevertheless still a matter of recent debates. Proponents are, e.g. of (a) Schiff [88], Okun et al. [116], Møller [127], Cranshaw et al. [128] and Ohanian [129], and of (b) Hay et al. [130],

Straumann [131], Randall [132] and Will [133]. It is surprising that the same team of experimenters albeit with different first authors published different views in

Pound and Snider [120] and Pound [134] pointed out that this experiment could not distinguish between the two options, because the invariance of the velocity of

Einstein [13] emphasized that for an elementary emission process, not only the energy exchange but also the momentum transfer is of importance; see also [12, 46, 97]. Taking these considerations into account, we formulated a photon emission process

1.The atom cannot sense the potential U, in line with the original proposal by Einstein [87, 109], and initially emits the same energy ΔE<sup>0</sup> at U <0 and

2. It also cannot directly sense the speed of light at the location with a potential U.

3.As the local speed of light is, however, c Uð Þ 6¼ c0, a photon having an energy of ΔE<sup>0</sup> and a momentum p<sup>0</sup> is not able to propagate. The necessary adjustments of the photon energy and momentum as well as the corresponding atomic quantities then lead in the interaction region to a redshift consistent with

As outlined in Section 3.7, there is general agreement in the literature that the

2U c2 0

(98)

c Uð Þ≈ c<sup>0</sup> 1 þ

in line with Eq. (95) in Section 3.7. It has, however, to be noted that the speed c Uð Þ was obtained for a photon propagating from U<sup>0</sup> to U, and, therefore, the physical process which controls the speed of newly emitted photons at a gravita-

An attempt to do that will be made by assuming an aether model. Before we suggest a specific aether model, a few statements on the aether concept in general should be mentioned. Following Michelson and Morley [135] famous experiment, Einstein [51, 109] concluded that the concept of a light aether as carrier of the electric and magnetic forces is not consistent with the STR. In response to critical remarks by Wiechert [136], cf. Schröder [137] for Wiechert's support of the aether,

[128, 130] on the process of the Pound-Rebka-Experiment.

the radiation had not been demonstrated.

Planetology - Future Explorations

at a gravitational potential U assuming that:

The initial momentum thus is p<sup>0</sup> ¼ ΔE0=c0.

0

tional potential U is not yet established.

and observations [96].

U<sup>0</sup> ¼ 0.

<sup>h</sup><sup>ν</sup> <sup>¼</sup> <sup>Δ</sup>E<sup>0</sup> <sup>1</sup> <sup>þ</sup> <sup>U</sup>=c<sup>2</sup>

local speed of light is

92

von Laue [138] wrote that the existence of an aether is not a physical, but a philosophical problem, but later differentiated between the physical world and its mathematical formulation [139]: a four-dimensional "world" is only a valuable mathematical trick; a deeper insight, which some people want to see behind it, is not involved.

In contrast to his earlier statements, Einstein said at the end of a speech in Leiden that according to the GTR, a space without aether cannot be conceived [140] and even more detailed thus one could instead of talking about "aether" as well discuss the "physical properties of space". In theoretical physics we cannot do without aether, i.e. a continuum endowed with physical properties [141]. Michelson et al. [142] confessed at a meeting in Pasadena in the presence of H.A. Lorentz that he clings a little to the aether, and Dirac [143] wrote in a letter to Nature that there are good reasons for postulating an aether.

In [17] we proposed an impact model for the electrostatic force based on massless dipoles. The vacuum is thought to be permeated by these dipoles that are, in the absence of electromagnetic or gravitational disturbances, oriented and directed randomly propagating along their dipole axis with a speed of c0. There is little or no interaction among them. We suggest to identify the dipole distribution postulated in Section 2.5 with an aether. Einstein's aether mentioned above may, however, be more related to the gravitational interactions, cf. [144]. In this case, we have to consider the graviton distribution as another component of the aether.

We now assume that an individual dipole interacts with gravitons in the same way as photons in Eq. (89), i.e. according to

$$T\_{\rm D} = |\mathbf{p\_{D}}|\boldsymbol{\varepsilon} = |\mathbf{p\_{D}} + 2\,\mathrm{Y}\,\mathbf{p\_{G}}|\boldsymbol{\varepsilon}' = |\mathbf{p\_{D}'}|\boldsymbol{\varepsilon}' = T\_{\rm D}',\tag{99}$$

where T<sup>D</sup> and p<sup>D</sup> refer to the energy and momentum of a dipole. The condition <sup>p</sup><sup>D</sup> <sup>≫</sup> <sup>p</sup>G, cf. Eq. (88), is fulfilled in the range from <sup>Y</sup> <sup>≈</sup> <sup>10</sup>�<sup>22</sup> to 10�<sup>15</sup> for all r<sup>e</sup> ≤ 2:82 fm (see Section 2.5 and Figure 3).

We can then modify Eqs. (90)–(94) by changing ν to D and find that Eqs. (95) and (98) are also valid for dipoles with a speed of c<sup>0</sup> for U<sup>0</sup> ¼ 0.

Considering that many suggestions have been made to describe photons as solitons, e.g. in [145–150], we also propose that a photon is a soliton propagating in the dipole aether with a speed of c Uð Þ, cf. Eq. (98), controlled by the dipoles moving in the direction of propagation of the photon. The dipole distribution thus determines the gravitational index of refraction, cf. Eq. (95), and consequently the speed of light c Uð Þ at the potential U. This solves the problem formulated in relation to Eq. (98) and might be relevant for other phenomena, such as gravitational lensing and the cosmological redshift, cf., e.g. [151]. Should the speculation in Section 2.5.2 be taken seriously that the dipole distribution corresponds to DM, it has to be much more evenly distributed than previously thought [152]. The light deflection would then be caused by gravitationally induced index of refraction variations.

#### 4. Discussion and conclusions

With Newton's law of gravitation as starting point, the ideas presented in Section 2.4 allow an understanding of far-reaching gravitational force between massive particles as local interactions of hypothetical massless gravitons travelling with the speed of light in vacuum. The gravitational attraction leads to a general mass accretion of massive particles with time, fuelled by a decrease of the graviton energy density in space. The physical processes during the conversion of gravitational potential energy into kinetic energy have been described for two bodies with

#### Planetology - Future Explorations

masses m<sup>A</sup> and mb, and the source of the potential energy could be identified in Section 3.1.1. In order to avoid conflicts with energy and momentum conservation, we had to modify a detail of the interaction process in Eq. (26), i.e. assume an antiparallel emission of the secondary graviton with respect to the incoming one.

Multiple interactions of gravitons leading to shifts of the effective gravitational centre of a massive body from the "centre of gravity" are treated in Sections 3.4–3.6 taking the modified concept into account. The interaction of gravitons with photons in Section 3.7 had to be modified as well, but the modification did not change the results, with the exception that now, both the energy and momentum conservation principles are fulfilled.

Our main aim in Section 3.8 was to identify a physical process that leads to a speed c Uð Þ of photons controlled by the gravitational potential U. This could be achieved by postulating an aether model with moving dipoles, in which a gravitational index of refraction nGð Þ¼ U c0=c Uð Þ regulates the emission and propagation of photons as required by energy and momentum conservation principles. The emission process thus follows Steps (1) to (3) in Section 3.8, where the local speed of light is given by the gravitational index of refraction n. In this sense, the statement that an atom cannot detect the potential U by Müller et al. [11] is correct; the local gravity field g, however, is not controlling the emission process.

A photon will be emitted by an atom with appropriate energy and momentum values, because the local speed of light requires an adjustment of the momentum. This occurs in the interaction region between the atom and its environment as outlined in Step 3.

In the framework of a recently proposed electrostatic impact model in [17], the physical processes related to the variation of the electrostatic potential energy of two charged bodies have been described, and the "source region" of the potential energy in such a system could be identified and is summarized in Section 3.1.2.

Sotiriou et al. [125] made a statement in the context of gravitational theories in "A no-progress report": "[ … ] it is not only the mathematical formalism associated with a theory that is important, but the theory must also include a set of rules to interpret physically the mathematical laws". With this goal in mind, we have presented our ideas on the gravitational and electrostatic interactions.

### Acknowledgements

This research has made extensive use of the Smithsonian Astrophysical Observatory (SAO)/National Aeronautics and Space Administration Astrophysics Data System (NASA/ADS). Administrative support has been provided by the Max-Planck-Institute for Solar System Research and the Indian Institute of Technology (Banaras Hindu University).

Author details

95

Klaus Wilhelm1† and Bhola N. Dwivedi<sup>2</sup>

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

\*Address all correspondence to: bnd.app@iitbhu.ac.in

University), Varanasi, India

† These authors contributed equally.

provided the original work is properly cited.

\*†

1 Max-Planck-Institut für Sonnensystemforschung (MPS), Göttingen, Germany

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 Department of Physics, Indian Institute of Technology (Banaras Hindu

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

masses m<sup>A</sup> and mb, and the source of the potential energy could be identified in Section 3.1.1. In order to avoid conflicts with energy and momentum conservation, we had to modify a detail of the interaction process in Eq. (26), i.e. assume an antiparallel emission of the secondary graviton with respect to the incoming one. Multiple interactions of gravitons leading to shifts of the effective gravitational centre of a massive body from the "centre of gravity" are treated in Sections 3.4–3.6 taking the modified concept into account. The interaction of gravitons with photons in Section 3.7 had to be modified as well, but the modification did not change the results, with the exception that now, both the energy and momentum conservation

Our main aim in Section 3.8 was to identify a physical process that leads to a speed c Uð Þ of photons controlled by the gravitational potential U. This could be achieved by postulating an aether model with moving dipoles, in which a gravitational index of refraction nGð Þ¼ U c0=c Uð Þ regulates the emission and propagation of photons as required by energy and momentum conservation principles. The emission process thus follows Steps (1) to (3) in Section 3.8, where the local speed of light is given by the gravitational index of refraction n. In this sense, the statement that an atom cannot detect the potential U by Müller et al. [11] is correct; the

A photon will be emitted by an atom with appropriate energy and momentum values, because the local speed of light requires an adjustment of the momentum. This occurs in the interaction region between the atom and its environment as

In the framework of a recently proposed electrostatic impact model in [17], the physical processes related to the variation of the electrostatic potential energy of two charged bodies have been described, and the "source region" of the potential energy in such a system could be identified and is summarized in Section 3.1.2. Sotiriou et al. [125] made a statement in the context of gravitational theories in "A no-progress report": "[ … ] it is not only the mathematical formalism associated with a theory that is important, but the theory must also include a set of rules to interpret physically the mathematical laws". With this goal in mind, we have presented our ideas on the gravitational and electrostatic interactions.

This research has made extensive use of the Smithsonian Astrophysical Observatory (SAO)/National Aeronautics and Space Administration Astrophysics Data System (NASA/ADS). Administrative support has been provided by the Max-Planck-Institute for Solar System Research and the Indian Institute of Technology

local gravity field g, however, is not controlling the emission process.

principles are fulfilled.

Planetology - Future Explorations

outlined in Step 3.

Acknowledgements

(Banaras Hindu University).

94
