A Constructive Model of Gravitation

Raghubansh P. Singhraghu.singh@verizon.net

AbstractThe paper presents a physical model in which mass fields and momentum fields mediate gravitational interactions.The model addresses: Gravitational interaction between masses, between mass and energy, and between photons; Gravity’s effect on

spectral lines, time periods of atomic clocks, and lengths of material rods; Gravitational radiation; Mercury’s orbital precession rate; and the Pioneer effect. Of particular importance, it calculates gravitational radiation power emissions from the moon, the planets of the sun, and the binary pulsars PSR B1913+16. It reflects upon time.

The model rediscovers the initial predictions of general relativity. It makes new predictions: Gravitational interactions can be attractive, repulsive, or zero; Photons gravitationally interact albeit negligibly; the Classical gravitational constant is not a constant; and Accelerating masses generate gravitational radiation, which has a four-lobed quadrupole pattern and propagates at less than the speed of light.

The paper puts forward specific suggestions for measuring the model’s constants, including the speed of gravitational radiation, and for detecting gravitational radiation.

Keywords: Gravitation; Gravitational charge; Gravitational current; Gravitational field; Mass field; Momentum field; Gravitational interaction; Gravitational force; Gravitational constant; Gravitational radiation; Gravitational wave; Time.

Terms. Entity’ stands for an item of matter, antimatter, or energy. ‘Charge’ without a qualifier means electrical charge.

1. IntroductionNewton (c. 1686) discovers the law that gravitational attraction between two bodies is proportional directly to the

product of their masses and inversely to the square of their separation distance.Einstein1 (c. 1915) publishes the general theory of relativity, according to which gravitation is due to the curvature

which matter creates in the field of space-time geometry. The field of space-time geometry is the gravitational field.Milne2 (c. 1935) holds that “geometry can be selected primarily by the nature of underlying phenomenon and the

convenience of representing and analyzing that phenomenon; and transformations of coordinates alone are but translations of language and have not necessarily much to do with phenomena.”

Biswas3 (1994) explains the first three predictions of general relativity by introducing a second-rank symmetric tensor into special relativity as a potential rather than a metric.

The strong, the weak, and electromagnetic interactions are mediated by the strong, the weak, and electromagnetic fields associated with the color, the weak, and electrical charges of matter and antimatter. These fundamental charges and related currents respectively are the static and dynamic properties of matter and antimatter. The strong and the weak interactions are mediated at microscopic levels; electromagnetic interactions occur at microscopic through macroscopic levels; and gravitational interactions are known to be effective at macroscopic levels. (At microscopic levels, fields are quanta; at macroscopic levels, fields are effectively continuous with values at each space-time point.)

Along the lines stated above, gravitational interaction will be addressed in three parts, of which this paper is the first:

- Part 1: Classical gravitation theory.  Gravitational interaction between entities will be formulated in terms of their pertinent static and dynamic properties and associated fields at the macroscopic level.

Mass, as gravitational charge, is a static property of matter and antimatter. Momentum, as gravitational current, is a dynamic property of entities.

In this part: non-gravitational and other extraneous effects will be ignored; special relativity will be postponed; gauge symmetries will not be required; and quantum theory will be invoked only elementarily.

- Part 2: Classical gravitational field theory. Lorentz and gauge transformations will be considered.

- Part 3: Quantum gravitational field theory. Quantum mechanics will be considered.

(Appendix A has the nomenclature. In mathematical expressions, the order of operations is as follows: multiplication, division, addition, and subtraction.)

2. Assumptions:We make two assumptions:

(1) Matter (or antimatter) has an envelope of intrinsic mass field.

(2) An entity in motion has an envelope of momentum field.

The range of mass field is infinity. The effective range of momentum field is proportional to the momentum.

3. Physical dataWe will use the following data in calculations later:

(a) Speed of light (c): 2.998 x 108 m/s;(b) Classical gravitational constant (G): 6.672 x 10–11 nt-m2/kg2;(c) Sun’s mass: 1.989 x 1030 kg;(d) Sun’s radius: 6.963 x 108 m;(e) Earth’s mass: 5.976 x 1024 kg;(f) Earth’s radius: 6.378 x 106 m;(g) Farthest Kuiper Belt bodies from the sun: ~ 103 AU; 4

(h) Diameter of the Milky Way galaxy: ~ 105 ly; 4

(i) Ratio of electrical to gravitational force: 1040. 5

4. The gravitation modelWe define momentum field range Sm of a mass m with momentum p:

Sm = σ p , (1)

where σ is momentum field range coefficient. A mass with momentum of 1 kg-m/s has momentum field range of σ meter.The inter-momentum field range S12 between masses m1 and m2 is thus:

S12 = S1 + S2 (2)

4.1 Mass-mass gravitational interactionWe consider two masses m1 and m2 separated by distance r.The gravitational force between m1 and m2 , as mediated by their mass fields per Assumption (1), is given by:

,

(3)

where Fs is repulsive, and Gs is the static gravitational constant.The gravitational force between m1 and m2 with momenta and respectively, as mediated by their momentum

fields per Assumption (2), is given by:

,

(4)

where Fd is attractive or repulsive as the angle between and is acute or obtuse, and Gd is the dynamic gravitational constant.

The dimension of Gs /Gd is of the square of speed. Denoting this speed by b, which would be the speed of mass-momentum wave (gravitational radiation), we have:

(5)

Fig. 1 shows an arbitrary sector of the universal sphere with center at the Primordial Point O, the “point” of the Big Bang. The Primordial Point is the sole space-time reference point for all entities in the universe. Masses m1 and m2 are at distances r1 and r2 from O. The angle between r1 and r2 at O is α. The masses have velocity vectors and relative to O and have mass fields (white shade), momentum fields (darker shades), and momentum field ranges S1 and S2 respectively.

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m1 m2rS1

S2

u1u2

r1

r2

Figure 1. Masses with mass and momentum fields

At r ≤ S12, momentum fields are effective. Eq. (4) then becomes:

(6)

By definition p = m u, so Eq. (6) becomes:

;

(7)

At r > S12, mass fields are predominant. From (3), (5), and (7), the resultant force between m1 and m2 is given as:

(8)

Eqs. (7) and (8) are similar to Newton’s law of gravitation but in the form only. The term in the bracket of (7), the so-called universal gravitational constant, is not universally constant in space and in time, as it varies with ( u1u2) and with cosα.

We simplify. The age of the universe is large (about 14 BY). So, in Fig. 1, as r1 → ∞, r2 → ∞, and r ≤ S12, angle α is acute and small. Setting u1 = u = u2 in (7) and (8), we re-express (6), (7), and (8) respectively as follows:

(9)

(10)

(11)

Eq. (10) holds in the regions of bound systems; Eq. (11) holds in the regions beyond momentum field ranges. In  (10), as r ≤ S12, the Cavendish experiment measures the classical gravitational constant G ≡ Gd u2. Both G and Γ = Gs (1 − u2/b2) vary with u2; however, on the human space-time scale, as u is virtually constant, they are virtually constant.

Based on (10) and (11), Table I has the signs of gravitational interaction, which is attractive in the inner regions (r ≤ S12) regardless of the value of u, and repulsive, attractive, or zero in the outer regions (r > S12) depending on the magnitude of u/b.

Table I. Interaction signs with respect to u/b and S12

u/b r ≤ S12 r > S12

u < b attraction repulsionu = b attraction zerou > b attraction attraction

4.2 Mass-energy gravitational interaction

3

O

α

An energy quantum (E, c) is at distance r from a mass (m, u). Per Assumptions (1) and (2), the energy quantum has no mass field but has momentum field – by its momentum (p = E /c). The attractive force between the mass and the energy quantum is mediated by their momentum fields and, from (9), is given by:

FmE  =  κ mE/r2 , (12)

where κ is mass-energy gravitational coefficient, which varies with u:

κ = Gd u/c = G/uc (13)

4.3 Photon-photon gravitational interactionPer Assumptions (1) and (2), photons have no mass fields but have momentum fields – by their momenta ( p = E /c).

Photons gravitationally interact via their momentum fields. Eq. (9) applies. Absolute, maximum gravitational force between the electromagnetic waves (λ) is given by substituting p = h/λ and r = λ/2 in (9):

Fλλ ≈ 4 Gd h2 / λ4, (14)

where h is Planck constant. Even for extremely energetic electromagnetic waves (λ ≈ 10–18 m), maximum gravitational force between them ≈ 10–25 nt. That is, there exists gravitational force between electromagnetic waves, but it is virtually zero.

4.4 Antimatter gravitation. Antimatter has the same mass as its counterpart matter but equal and opposite value of some other property. Negative mass is considered to be nonexistent. The model applies to matter-antimatter and antimatter-antimatter gravitational interactions – regardless of the sign of mass.

4.5 Even though the model is developed for masses with primordial velocity ( ), it holds at an arbitrary velocity ( ) as well.

5. GravitonGraviton is a hypothetical quantum of energy exchanged in gravitational interactions. If b < c, gravitons would have

non-zero mass. In the third part of the model, we will address the quantization of mass-momentum (gravitational) field.Gravitational force is lopsidedly feeble relative to other fundamental forces: (g : w : e : s ≈ 1 : 1030 : 1040 : 1042). That is,

the time taken for the emission or absorption of gravitons would be lopsidedly longer: ( g : w : e : s ≈ 1033 : 103 : 10–12 : 10–

23 sec). (In comparison, a proton’s lifetime is estimated to be about 1038 secs and the universe’s age to be about 1018 secs.)

6. Vibrating particle in gravitational fieldWe derive the change in frequency (ν) of vibration of a particle as its position relative to a mass (m, u) changes. The

mass is a sphere of radius R at r = 0. The particle’s energy E is proportional to ν. Gravitational attraction between the mass and the particle is mediated by their momentum fields as given by (12).

6.1 Vibrating particle outside the massAs the particle is moved from r = r ≥ R to r = (r + x), the change in its energy is given by:

(15)

Carrying out the integration, we have:

(16)

The term in the bracket > 1, thus νr < νx. The particle vibrates at lower frequency closer to the mass.Using (16), we estimate κ. The details are in Appendix B, which shows:

κ = 1.552 x 10–27 nt-s2/kg2 (present-day) (17)

As x → ∞, Eq. (16) reduces to:

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(18)

If the vibrating particle serves as an emitter of light, its wavelength at the surface of the sun and at infinity, from (18), (17), and data 3(c, d), are related by:

λR = (1 + 4.433 x 10–6) λ∞

(19)

That is, spectral lines produced at the sun are redshifted by about 4.433  x 10–6 of their wavelengths compared to those produced at infinity.7 Wavelengths as emitted are longer closer to the mass.

If the vibrating particle serves as an atomic clock, its time period at the surface of the earth and at infinity, from (18), (17), and data 3(e, f), are related by:

τR = (1 + 1.454 x 10–9) τ∞

(20)

That is, as the atomic clock is moved from infinity to the earth, its time period is relatively dilated by about 1.454  x 10–9. Time runs slower closer to the mass.

6.2 Vibrating particle inside the massAs the particle is moved from r = 0 to r = r ≤ R, the change in its energy is given by:

,

(21)

where mr is the mass contained within r = r ≤ R.

Carrying out the integration, we have:

(22)

The term in the bracket < 1 but > 0, thus ν0 < νr. The particle vibrates at lower frequency closer to the center of the mass.Light wavelength at the surface and at the center of the sun, from (22), (17), and data 3(c,  d), are related by:

λR = (1 – 2.22 x 10–6) λ0

(23)

That is, spectral lines produced at the sun’s center are redshifted by about 2.22  x 10–6 of their wavelengths compared to those produced at its surface.7 Wavelengths as emitted are longer closer to the center of the mass.

An atomic clock’s time period at the surface and at the center of the earth, from (22), (17), and data 3(e,  f), are related by:

τR = (1 – 7.27 x 10–10) τ0

(24)

That is, as the atomic clock is moved from the earth’s surface to the center, its time period is relatively dilated by about 7.27 x 10–10. Time runs slower closer to the center of the mass.

6.3 Rod in gravitational fieldWe address the change in the length of a rod of mass m′ as its position relative to another mass m (>> m′) changes from

r = r to r = ∞.We consider an ideal rod constituted of “atoms” of mass δm′ (<< m′) and charge e spaced equally by d. Such an atom,

under the electrostatic forces of its neighboring atoms, undergoes oscillations with frequency ν as given by:

ν = w / d n,

(25)

where w and n are parameters. (A one-dimensional rod is treated as an example in Appendix C.)

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The gravitational field of mass m affects the oscillations of δm′ according to (18). Substituting (25) in (18), we have:

dr = (1 + κ m/r)1/n d∞

(26)

As dr > d∞ , the rod is longer closer to the mass. As the material rod is moved from infinity to the earth’s surface, it is relatively elongated by about 1.0 x 10–9.

If the gravitational field is not uniform over the rod, the spacings are affected non-uniformly; so, the rod is deformed.

6.4 Vibrating particle and point-dense massThe factor m/r in (18), (22), and (26) is meaningful as range density (m/r) or point density (m/R).A mass of infinitely high point-density may be indicated by m/R → ∞. An example of such a mass is a black hole.From (18), as m/R → ∞, τR /τ∞ → ∞. Time period at the surface tends to infinity; time virtually stops.From (18), as m/R → ∞, λR /λ∞ → ∞; λR → ∞, νR → 0, νR λR = c.  At the surface, light travels at c with nearly flat

waveform.From (26), as m/R → ∞, dR /d∞ → ∞. At the surface, rod flattens to the point where it disintegrates.

6.5 Vibrating particle and no massFor a vanishing mass, its point density m/R → 0/0, which is an indeterminate. From (18), as m/R → 0/0,

(τR – τ∞)/τ∞ → 0/0. One plausible interpretation would be τR → τ∞ → 0; that is, time periods may cease to exist in the absence of mass.

6.6  The effect of other fundamental fields, particularly electromagnetic, on time and length is not known.

7. Estimation of the constants and parametersWe estimate Gs, Gd, b, σ, Γ, and u theoretically. Coefficient κ was estimated in (17).

(1) With reference to the sun, Eqs. (13) and (17) and data 3(a, b) yield:

u = 1.433 x 108 m/s (present-day) (27)

Gd = 3.249 x 10–27 nt-s2/kg2

(28)

(2) Datum 3(g) will be considered as the sun’s momentum field range. From (1), (27), and data 3(c,  g), we have:

σ = 5.26 x 10–25 s/kg (29)

A 1 kg point-mass with a speed of 1 m/s has an effective momentum field range of the order 10–24 m.From (1), (27), (29), and datum 3(h), the black hole at the center of the Milky Way galaxy has m ≈ 6.3 x 1036 kg.

(3) We consider electrical force (Fe) and gravitational force (Fg) between two particles of charge e (1.602 x 10–19 coul) and mass m (4.0 x 10–29 kg), intermediate between a proton and an electron.

The speed (b) of gravitational radiation is expressed below.

b2 = (Q /Gd) (e/m)2 (Fg /Fe), (30)

where Q is the Coulomb constant = 8.988 x 109 nt-m2/coul2. Appendix D has the details.From (30), (5), (28) and datum 3(i), we calculate:

b = 6.661 x 107 m/s (31)

Gs = 1.442 x 10–11 nt-m2/kg2 (32)

Γ = Gs (1 − u2/b2) = (-) 5.231 x 10–11 nt-m2/kg2 (present-day) (33)

u/b = 2.15 (present-day) (34)

u/c = 0.478 (present-day) (35)

b/c = 0.2222 (36)

The present-day primordial speed (u) of the masses in the universe would be about 48% of the speed (c) of light. The speed (b) of gravitational radiation would be about 22.22% of the speed (c) of light.

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(4) Estimations of Gs, Gd, b, u, κ, and Γ are independent of σ; estimation of σ is dependent on u. Suggestions for measuring b, u, and σ are presented later.

8. Gravitational deflection of light at a massFrom (12), the gravitational deflection θ of electromagnetic waves with impact parameter d at a mass m is given by:

θ = 2 tan–1(κ m/d) (37)

For small deflections, θ ≈ 2Gm/duc; from (35), as present-day u ≈ c/2, θ ≈ 4Gm/dc2. Appendix E has the details.For a ray of light grazing the sun, we have, from (37), (17), and data 3(c, d), the deflection angle θ ≈ 1.83 arc-secs.(The 1919 expedition determined the deflection to be 1.7 arc-secs, the 1929 expedition 2.2 arc-secs, and later

measurements from 1.5 to 3 arc-secs.8, 9)

9. Escape radius for light near a massTo escape a mass m, light must be outside a critical impact parameter (escape radius) Re , which, from (37), is given by :

Re  =  κ m  =  Gm /uc (38)

From (35), as present-day u ≈ c/2, Re ≈ 2Gm/c2. From (38), (17), and (29), as if they were points, the black hole at the center of the Milky Way, the sun, and the earth would have Re ≈ 1010 m, 3 km, and 1 cm respectively.

10. Mercury’s orbital precession rateAn observer measures Mercury’s orbital precession rate to be 5600 arc-secs/century. After filtering out the observer-

planet relative motions, a 575 arc-secs/century is left over, which can be accounted for in flat space-time geometry:- Price and Rush 

10 and others show that classical-mechanical contribution is about 532 arc-secs/century; and - Biswas3 shows that a Lorentz covariant modification of the Newtonian potential contributes about 43 arc-secs/century.

A second-rank symmetric tensor is introduced into special relativity – as a potential rather than a metric.

11. The Pioneer effectThe masses in the universe do not have exactly the same primordial velocity ; that is, G and Γ are not constant across

space and over time. Even an infinitesimal change in changes G and Γ, which leads to the perturbation of trajectories and orbits. A change in may be caused by a local force, such as of gravitational, electromagnetic, or other origin.

From (10), for a mass at r from another mass m, variations du, dG, and da are given below:

da  =  (m /r2) dG  =  (2m Gd /r2) u du ;   r ≤ S12

(39)

We now use (39) to address the so-called Pioneer effect. The Pioneer-10 and Pioneer-11 spacecraft, after they passed about 20 AU on their trajectories, were observed to have a small, additional acceleration a = (8.74 + 1.33) x 10–10 m/s2

toward the sun. From (39), a 0.00296% increase in u leads to 0.00593% increase in G at their sites resulting in their additional acceleration of 8.74 x 10–10 m/s2 toward the sun.

12. Gravitational radiation from accelerating massWe derive gravitational Larmor’s formula. We follow Thomson’s treatment of electromagnetic radiation from

accelerating charges.11 We apply the formula to explain gravitational radiation from bodies in circular and elliptical orbits.Fig. 2 shows a mass m with momentum at t = 0, which accelerates at for dt and gains speed dυ. The mass continues

with dυ for t = t >> dt. Changes propagate at a finite speed (b), so, later at (dt + t), momentum field component 0-A-B turns into t-C-A-B. Consequently, there is a transverse momentum field in addition to the radial momentum field . Radial

drifts with the mass at speed dυ and propagates radially at speed b. Transverse momentum field changes from zero to amplitude Pt and back to zero in dt; this is gravitational radiation pulse propagating outwardly at speed b.

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tr = b t. Δ = b dt. l = dυ t sin φ.

φdυ t

Pr

Pt

lm

0

Δr

A B

Cυ, a, p

Figure 2. Mass m and its momentum field vectors

The geometrical ratio of transverse to radial momentum fields is:

,

(40)

where the radial momentum field Pr is given by (4) as follows:

(41)

Substituting (41) in (40) and using t = r/b and dυ/dt = a, we get:

(42)

Radial Pr varies with 1/r2, transverse Pt with 1/r. That is, transverse Pt survives over radial Pr at great distances.We define gravitational Poynting intensity Ig (watt/m2):

(43)

Substituting (42) in (43), we get gravitational radiation intensity Igt due to transverse Pt :

(44)

Gravitational radiation intensity Igt falls off as 1/r2 and its angular variation is shown in Fig. 3 (a), which is a four-lobed quadrupole pattern. Fig. 3 (a), in turn, shows that gravitational radiation accelerates a mass at 450 to its propagation direction.

In contrast, the angular variation of electromagnetic radiation pulse intensity is shown in Fig.  3 (b), which, in turn, shows that electromagnetic radiation accelerates a charge at 900 to its propagation direction.

Gravitational radiation power (Ω) emitted is given by integrating (44) over all directions:

(45)

If and are orthogonal, the magnitude of stays constant, and its direction changes but only uniformly. Thus, a mass in a circular orbit does not emit gravitational radiation, because no momentum-field pulse develops. A mass in motion in an eccentric orbit or an arbitrary trajectory gravitationally radiates. In contrast, a charge even in a circular orbit emits electromagnetic radiation.

Carrying out the integration in (45), we get gravitational Larmor’s formula:

(46)

That is, a mass with motive power of p a ≈ 1025 watts emits gravitational radiation power of about one watt. In contrast, a charge emits electromagnetic radiation power proportional to (e a)2.

We address gravitational radiation from masses in elliptical orbits. Fig.  3 (c) shows mass m2 in an elliptical orbit around mass m1. The elliptical orbit is given by semimajor axis A and eccentricity ε. Appendix F has the details on Eqs. (47) - (53).

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From (46), gravitational radiation power Ω emitted from a point (r, θ) on the orbit is given below:

(47)

Gravitational radiation power emission is maximum (Ωmax) at a pair of angle θ, as given by:

(48)

The variation of Ω with θ is shown in Fig. 3 (d), which shows the emission of gravitational radiation power in a pair of pulses with peaks Ωmax at θ+ and θ– in an orbital period.

Figure 3.  (a) Variation with φ in the intensity I of gravitational radiation. (b) Variation with φ in the intensity I of electromagnetic radiation. (c) Mass m2 in an elliptical orbit around mass m1. (d) Variation of gravitational radiation power emission Ω with θ.

Gravitational radiation energy (Eo) emitted in one orbital period (τ) is given by integrating Ω(θ) in (47) as follows:

(49)

or,

(50)

Emission of gravitational radiation reduces the kinetic energy of the orbiting mass, and, so, shrinks its orbit. Changes ΔA, Δε, and Δτ due to loss of gravitational radiation energy Eo per orbital period τ are given below:

(51)

(52)

(53)

12.1 Gravitational radiation powers from selected bodies, including the moon and PSR B1913+16, are given in Appendix G. Appendix G-1 raises the possibility of detecting gravitational radiation – emanating from the moon.

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(d)(b) (c)(a)

Arθ

m1

m2

r = A(1 – ε2)/(1 + ε cosθ) θ

0

Ω(θ)

π 2π0 θ–

Ωmax

θ+

a

φm

a

φe

I

Ig

13. MeasurementsThe model introduced three constants (Gs, Gd and σ) and one parameter ( ). The validity of the model acutely depends

on G, c, b, u, and σ. We presume G and c remain constant during measurements. We now suggest measurements for b, u, and σ.

(1) Appendix H has the outlines for measuring the speed (b) of gravitational radiation.

(2a) Measurements of the gravitational deflection θ of light at an impact parameter d from a mass m yield the magnitude of velocity . From (37) and (13), u = G m/d c tan(θ/2).

(2b) We are unable to make a suggestion for determining the direction of of the sun and of the solar system.

(3) Measurements of the range Sm of the gravitational attraction of a mass m yield the value of σ. From (1), σ = Sm /m u, where u is the present-day primordial speed from (2a) above.

(4) Constants Gd and Gs may be calculated as follows: Gd = G/u2; Gs = b2 Gd .

(5) The rest may be calculated as follows: κ = Gd u/c; Γ = Gs (1 − u2/b2).

14. Results and predictionsAppendix I lists the results and predictions of this model and compares them with those of general relativity.To estimate σ, we considered the outer Kuiper Belt instead of the Oort Cloud, because the latter’s spread is not clearly

known. Regardless, the results from the model listed in Table I-1, except the mass of the Milky Way’s black hole, are not affected by the value of σ.

The model ignores non-gravitational and other extraneous effects, as such effects are difficult to account for and quantify – especially in astronomical observations.

15. RemarksFaraday introduced the concept of field in physics. Classical physics introduced gravitational field and electromagnetic

field. Modern physics introduced the strong nuclear field and the weak nuclear field. General relativity introduced space-time geometry field for gravity. This model introduces mass-momentum field for gravity.

With the introduction of mass-momentum field for gravity, the four fundamental interactions may have a common underlying theme: Matter particles have the fundamental charges and currents, which have the fields and field quanta (force particles), which, in turn, mediate the fundamental interactions between the matter particles. (The fundamental charges are: color; weak; electrical; and mass.)

Electromagnetic and gravitational interactions are similar in some aspects. They are repulsive and attractive; and they are mediated by their respective electric-magnetic and mass-momentum fields associated respectively with the mass and charge properties of interacting entities. Charge field extends out to infinity, so does mass field.

Electromagnetic and gravitational interactions are not so similar in other aspects. Charge is either positive or negative, but mass is known to be only positive. A charge in motion has its magnetic field extending out to infinity, but a mass in motion has its momentum field which is limited in both range and direction. A one-kilogram mass at a speed of one meter a second has its momentum field extending out to about 10–24 meter. That is, it would be challenging to detect that momentum field of a mass which acquires motion anew within the laboratory. (This stated momentum field should not be confused with the pervasive momentum fields due to motions of the bodies relative to the Primordial Point and to each other in the universe.)

The model finds that gravitational “constants” G and Γ are not constant, because is not constant. That is, gravitational forces have been evolving across the space and over the time since the Big Bang and would be doing so into the future.

Special relativity was postponed, primarily because we were not sure as to whether b is less than, equal to, or greater than c. If b were less than c, gravitons would be bosons with non-zero mass. In the next part of the model, we will consider Lorentz transformations and also explore whether transformations in (x, y, z, ibt) coordinate systems will be needed as well.

Gauge symmetry was not required, because we were not sure what kind of it to look for as mass-momentum field does not seem to be inspiringly similar to the well-studied electric-magnetic field. A theory should demonstrate some symmetry, but sometimes it may not. In the next parts of the model, we will try to extract suitable gauge symmetries.

Imposing Lorentz covariance and gauge covariance should improve accuracy with experiments but will turn the equations unnecessarily mathematically complex and the paper too lengthy at the cost of the basic physical insight. Discovering the origin of gravitation is of the foremost necessity. (According to the Standard Model of Elementary Particles, matter particles exchanging force particles originate the strong, the weak, and electromagnetic interactions.)

Finally, we reflect upon time. The model finds that gravitational field affects the speed of time: Time periods may not exist in the absence of mass; Time periods are infinitesimal at an infinitesimal mass; Time periods are longer closer to mass; and Time periods are infinitely long at a mass of infinitely high point-density. The effect of any of the other

10

fundamental fields on time periods is not known. We advance the following hypothesis for validation: The genesis of time lies in interactions.

11

Appendix A

Nomenclature

Symbols a and α, symbols k and κ, and symbols P and p may look almost indistinguishable.

a accelerationAU astronomical unit: 1.4959787 x 1011 m.b speed of gravitational radiation; Eq. (5).c speed of electromagnetic wavee electrical chargeE energyF ForceG classical gravitational constantGd dynamic gravitational constant; Eq. (4).Gs static gravitational constant; Eq. (3).h Planck constant: 6.626 x 10–34 nt-m-s.Ig gravitational Poynting intensity; Eq. (43).ly light-year: 9.4605 x 1015 m.m mass (gravitational charge)m/r range densitym/R point densityM mass-field strengthnt newton, unit of force.p momentum (gravitational current)P momentum-field strength; Eq. (41).Q Coulomb constantR radiusSm momentum field range of a mass m; Eq. (1).S12 inter-momentum field range between masses m1 and m2; Eq. (2).

velocity of masses relative to the Primordial Point; Fig. 1.arbitrary velocity

Γ Γ = Gs (1 − u2/b2)κ mass-energy gravitational coefficient; Eq. (13).λ wavelengthν frequencyσ momentum field range coefficient; Eq. (1).τ periodΩ power

12

Appendix B

Estimating matter-energy gravitational coefficient κ

B-1 Estimating κ using the results from the Pound-Rebka experimentThe Pound-Rebka experiment finds that spectral lines produced at the earth’s surface (r = R) are redshifted by

Δ = 5.13 x 10–15 compared to those produced at height x = 22.5 m.6, 7 That is,

λR = (1 + Δ) λx , (B.1)

From (16), we have (using ν λ = c):

(B.2)

Comparing (B.1) with (B.2), we get:

(B.3)

Solving (B.3) for κ, we get:

(B.4)

Substituting x = 22.5 m, Δ = 5.13 x 10–15, and data 3(e, f) in (B.4), we get:

κ = 1.552 x 10–27 nt-s2/kg2

(B.5)

The value of κ in (B.5) is copied in (17), which is then used to estimate u and Gd in (27) and (28) respectively.

B-2 Estimating κ using the magnitude of deflection of light by massFrom (37), the gravitational deflection θ of electromagnetic waves with impact parameter d at a mass m is given by:

θ = 2 tan–1(κ m/d) (B.6)

The 1929 expedition determined that light was deflected at the sun by 2.2 arc-secs. 8 Substituting θ = 2.2 arc-secs and data 3(c, d) in (B.6), we get:

κ = 1.867 x 10–27 nt-s2/kg2 (present-day) (B.7)

Eqs. (13), (B.7), and (B.3), expression G ≡ Gd u2, and data 3(a, b) yield:

u = 1.191 x 108 m/s (present-day) (B.8)

Gd = 4.704 x 10–27 nt-s2/kg2

(B.9)

13

Δ = 6.17 x 10–15

(B.10)

That is, according to the model, if θ = 2.2 arc-secs, spectral lines produced at the earth’s surface would be redshifted by 6.17 x 10–15 compared to those produced at a height of x = 22.5 m; if θ = 1.83 arc-secs, this relative redshift would be by 5.13 x 10–15, which is the observed magnitude.

B-3 We opted (B.5) over (B.7) for coefficient κ, as terrestrial experiments would be relatively more accurate and reliable than astronomical observations.

14

Appendix C

One-dimensional rigid rod

We intend to arrive at a mathematically simple relationship between the frequency (ν) of oscillations of the “particles” which constitute a rod and the spacings (d) between them.

Fig. C-1 shows a one-dimensional rod of mass m constituted of “particles” of mass δm (<< m) and charge e spaced equally by d. Such a particle at O, under the electrostatic forces of its neighboring particles, oscillates between points i and j with frequency ν and displacement ε (< d).

Figure C-1.  A particle (δm, e) at O under the electrostatic forces of the neighboring particles at x and –x.

Electrostatic force on δm when at point i is given by the following, where Q is the Coulomb’s constant:

(C.1)

Similarly, the electrostatic force on δm when at point j is given by:

(C.2)

From (C.1) and (C.2), particle δm has acceleration a directed toward the neutral point O, as given by:

(C.3)

The standard equation for a simple harmonic motion is: . Comparing this with (C.3), we get:

(C.4)

That is, ν2 is inversely proportional to d 3. (This is remarkably similar to Kepler’s third law of orbital motions!)

Appendix D

Comparative strengths of static gravitational and electrical forces

We consider two particles, separated by distance r, each of mass m and charge e.Static gravitational force between the particles, from (3) and (5), is given by:

Fs = Gd b2 m2

/ r2 ,

(D.1)

where Gd is the dynamic gravitational constant and b is the speed of gravitational radiation.Static electrical force between the particles is given by:

Fe = Q e2 / r2

(D.2)

15

(δm, e) δm, e (δm, e)

dd

εi jOε–x x

where Q is the Coulomb constant.From (D.1) and (D.2), we get:

Fe /Fg = (Q/Gd) (e/m)2 (1/b2) (D.3)

16

Appendix E

Deflection of light ray by mass

Fig. E-1 (a) shows a light ray (ν) at impact parameter d from mass m. The light ray traces a hyperbola and has initial momentum and final momentum . The gravitational force between the light ray and the mass is mediated by their momentum fields. Fig. E-1 (b) shows the vectorial relationships among the momenta.

Figure E-1. (a) Deflection of a light ray (ν) by a mass m. (b) Momentum vectors of the light ray

Change (Δp) in momentum (p) of the light ray, from Fig. E-1 (b), is given by:

Δp = 2 p sin(θ/2) (E.1)

where p is the magnitude of initial and final momenta.Angular momentum (L) of the light ray of energy Eν is conserved; that is:

(E.2)

or,

(E.3)

By definition, a force ( ) is given by:

, (E.4)

where, in this case, F will be the gravitational force between mass (m) and light ray (Eν), as given by (12) and (13):

(E.5)

From (E.4), using (E.3) and (E.5), we get:

(E.6)

Equating (E.6) to (E.1) and using Ev = p c and G = Gd u2, we get:

tan(θ/2) = Gd m u /c d = G m /d u c (E.7)

Using κ = Gd u/c = G/uc from (13) in (E.7), we get:

17

θαα

m

νd

φd

pi

pfO

A

rF

θ θ/2pi

pf Δp

A

O

(a) (b)

Δp

α = (π – θ)/2 Δp = 2 p sin(θ/2)

θ = 2 tan-1(κ m /d) (E.8)

For small deflections, θ ≈ 2Gm/duc. From (35), the present-day u ≈ c/2; that is, θ ≈ 4Gm/d c2.

E.1 From (35), the present-day u ≈ c/2. From (38), we get escape radius for light Re = 2Gm/c2.

18

Appendix F

Gravitational radiation power emission from mass in elliptical orbit

Fig. F-1 shows a mass m2 in an elliptical orbit around another mass m1. We derive gravitational radiation power emission from the orbiting mass. The reduced-mass frame will be used. The reduced mass at m2 is μ = m1 m2 / (m1 + m2), and the compensated mass at m1 is (m1 + m2). Gravitational masses are not reduced or compensated.

Figure F-1. A mass m2 in elliptical orbit around another mass m1.

From the geometry of the ellipse, we have:

r = A (1 – ε2) / (1 + ε cosθ) (F.1)

(F.2)

(F.3)

Angular momentum vector ( ), by definition, is given below, where is momentum vector:

(F.4)

or,    sin2α = L2 / r2

p2

(F.5)

The angular momentum (L) of m2 in an elliptical orbit is a constant of motion as given by:

(F.6)

From (46), the gravitational Larmor’s formula, we have:

,

(F.7)

where acceleration a = G m1 / r2.With (F.5), (F.6), and (F.7), we get from physics:

19

O: centerA: semimajor axisε: eccentricity

rθ

α

A

m2 υ, p

m1

υ: velocityp = μ υ: momentum

O

(F.8)

Equating (F.8) to (F.3) and using (F.1) we get gravitational radiation power emission from a point (r, θ) on the orbit:

(F.9)

Using ∂Ω/∂θ = 0 in (F.9), we get maximum gravitational radiation power (Ωmax) from points at angles (θ):

,

(F.10)

that is, power emission is maximum when the orbiting mass is θ+ or θ– away from the periastron.Total gravitational radiation energy (Eo) emission in one orbital period (τ) is given by integrating (F.9) as follows:

(F.11)

Carrying out the following integration is straightforward:

(F.12)

From (F.11), (F.9), and (F.12), Eo is given by:

(F.13)

From (F.5), (F.6), and (F.1), the speed (υ) of m2 at a point (r, θ) on the ellipse is given by:

(F.14)

Kinetic energy (K) of m2 per orbital period is given below, where υ2(θ) comes from (F.14):

(F.15)

Emission of gravitational radiation power (Ω) reduces kinetic energy (K). Changes ΔA and ΔL in one orbital period (τ) are:

(F.16)

20

,

(F.17)

where Eo is gravitational radiation energy emitted in one orbital period (τ) as given in (F.13).Substituting (F.6) and (F.15) in (F.16) and (F.17), we get changes ΔA and Δε per orbital period:

(F.18)

(F.19)

Using Kepler’s third law, which applies to binary systems, we have:

(F.20)

From (F.20), first-order change in the orbital period (Δτ) per orbital period (τ) is given by:

,

(F.21)

where ΔA is given in (F.18).

21

Appendix G

Gravitational radiation emissions from selected binary systems

We use (47) – (53) to estimate the magnitudes of gravitational radiation from Moon, the planets, and PSR  B1913+16. We recall that as the eccentricity of an orbit goes to zero, so does the emission of gravitational radiation power.

G-1 Gravitational radiation from MoonWe will use the following data:4

(a) Mass of Earth (m1): 5.976 x 1024 kg;(b) Mass of Moon (m2): 7.348 x 1022 kg;(c) Semimajor axis of the lunar orbit (A): 3.844 x 108 m;(d) Eccentricity of the lunar orbit (ε): 0.055;(e) Lunar orbital period (τ): 27.322 days.From (47), the variation of Ω with θ is plotted in Fig. 3(d), which shows the emission of gravitational radiation power in

a pair of pulses at peaks Ωmax = 2.306 watts at θ = + 830.8 away from the perihelion at approximately 14.6 days apart and then at 12.7 days apart per orbital period of 27.3 days.

From (50), gravitational radiation energy emitted in an orbital period is Eo = 2.702 x 106 joules.From (51), the semimajor axis is decreasing at 4.338 x 10–15 m an orbital period. From (53), the orbital period is

decreasing at 4.0 x 10–17 sec in an orbital period.Even though the moon has low orbital eccentricity, its mass is only about 0.0123 times that of the earth. That is, the

moon barely manages to emit gravitational radiation power at peaks of low but appreciable 2,306 watts when it is at + 830.8 away from its perihelion. The power emission falls off and would be of the order of 10 –17 watts at the earth, which is too weak to be detectable. However, this power of about 2,300 watts could be detectable by a detector in an artificial satellite at a calculated distance from the moon.

G-2 Gravitational radiation from MarsWe will use the following data:4

(a) Mass of Sun (m1): 1.989 x 1030 kg;(b) Mass of Mars (m2): 6.5736 x 1023 kg;(c) Semimajor axis of the Martian orbit (A): 2.278 x 1011 m;(d) Eccentricity of the Martian orbit (ε): 0.093374;(e) Martian orbital period (τ): 5.94 x 107 secs.From (47), the variation of Ω with θ is plotted in Fig. 3(d), which shows the emission of gravitational radiation power in

a pair of pulses at peaks Ωmax = 0.45 watts at θ = + 790.75 away from the perihelion.

G-3 Gravitational radiation from other planets of the sunAs in the case of Mars, the planets of the sun have low orbital eccentricities and small masses compared to the mass of

the sun. That is, the planets emit no appreciable gravitational radiation.

G-4 Gravitational radiation from binary pulsars PSR B1913+16The neutron stars are orbiting each other around their center of mass. Their physical and orbital data are: 12

(a) Mass of the first star (m1): 2.866 x 1030 kg;(b) Mass of the second star (m2): 2.759 x 1030 kg;(c) Semimajor axis of each orbit (A): 1.95 x 109 m;(d) Eccentricity of each orbit (ε): 0.617131;(e) Orbital period (τ): 2.791 x 104 secs; 7.752 hours.(f) Distance from Earth: 21,000 light years.From (47), the variation of Ω with θ is plotted in Fig. 3(d), which shows the emission of gravitational radiation power in

a pair of pulses at peaks Ωmax = 1.674 x 1026 watts at θ = + 530.85 away from the periastron at approximately 6.58 hours apart and then at 1.17 hours apart per orbital period of 7.752 hours.

From (50), gravitational radiation energy emitted in one orbital period is Eo = 1.645 x 1030 joules.From (51), due to the emission of gravitational radiation, the semimajor axis is decreasing at 3.71  mm per orbital period;

from (52), eccentricity is decreasing at 9.71 x 10–13 per orbital period; and from (53), orbital period is decreasing at 7.987 x 10–8 sec per orbital period.

The pulsars are too far away for a detector on the earth to detect even the peak gravitational radiation power (Ω max) – unless, of course, the detector is placed in an orbit close enough to the pulsars, but this is not possible either.

22

Appendix H

Determining the speed of gravitational radiation from binary systems

Fig. H-1 shows mass m2 in elliptical orbit about mass m1. The ellipse is given by semimajor axis A and eccentricity ε.Fig. 3(d) shows variation with θ of gravitational radiation power Ω from m2. Maximum gravitational radiation power

Ωmax is emitted, when m2 is either at (r+, θ+) or (r−, θ−). Angle θ+ is given by (F.10) in Appendix F.

Figure H-1. Mass m1 around mass m1 in elliptical orbit.

We outline the steps to help carry out the measurement:

(1) Determine the distance of mass m2 from the detector: d.

(2) Detect by electromagnetic signals as mass m2 arrives at point (r+, θ+) and note the time (tc).

(3) Then, note the time (tg) when peak gravitational power Ωmax from (r+, θ+) is detected. The peak power emission could have come from one of the previous orbital periods. (We presume that no entity can exceed the speed of light.)

(4) If the electromagnetic signal comes from a specific orbital period, and the peak gravitational power signal comes from a prior nth orbital period (τ), we have:

tg – tc = (n τ + d/b) – d/c (H.1)

or,       

(H.2)

(5) Make adjustments for the Doppler effects.

(6) Repeat steps 1 – 5 for the next peak gravitational power Ωmax from point (r– , θ–).

In Appendix G, we estimated peak gravitational radiation power from the moon and from PSR B1913+16. We now apply (H.2) to the two astronomical binary systems:

(1) PSR B1913+16. This binary system is 21,000 light-years away; and each star has an orbital period of 2.791  x 104

secs. That is, there are minimum 2.0 x 107 orbital periods during the time light travels from the stars to the detector. Moreover, extraneous agents between the stars and the detector would affect gravitational radiation and electromagnetic radiation differently. Therefore, it is not possible to ascertain n. That is, Eq. (H.2) is not applicable here.

(2) Earth-Moon. The moon is 1.2813 light-seconds away; its orbital period is 27.322 days. That is, the moon has barely moved along its orbit (≈ 10–7 of a period) in the time light travels to the detector. Extraneous agents between the moon and the detector would affect gravitational radiation and electromagnetic radiation differently but insignificantly. That is, Eq. (H.2) is applicable here as n = 0. Eq. (H.2) then becomes:

(H.3)

23

θ+

θ–

r

θ– = - θ+

m1

m2

r+

r–A

m2

m2

According to Appendix G-1, the detector should be placed in an artificial satellite around the moon.

24

Appendix I

Results and predictions

Table I-1. Results and predictions of this model and of general relativity

Topic This Model General RelativityRedshift and time-period dilation at Sun 4.43 x 10–6 2.12 x 10–6

Redshift and time-period dilation at Earth 1.45 x 10–9 0.7 x 10–9

Deflection of light at Sun(observed: 1.5 to 3 arc-secs)9 1.83 arc-secs 1.75 arc-secs

Elongation of a rod at Sun 2.96 x 10–6

Elongation of a rod at Earth 9.70 x 10–10

Time at a black hole virtually stops virtually stopsWaveform of light at a black hole nearly flatRod at a black hole elongates to disintegrationMass of the Milky Way galaxy’s Black Hole 3.2 x 106 suns 4 x 106 sunsEscape radius for light near the Milky Way galaxy’s Black Hole 1010 m 1.2 x 1010 m

Escape radius for light at Sun-as-a-point 3 km 3 kmEscape radius for light at Earth-as-a-point 1 cm 1 cmMercury’s orbital precession rate(observed: 575 arc-secs a century)

Classical plus relativistic mechanics:532 + 43 + ... arc-secs/century.10, 3 532 + 43 + … arc-secs/century.

Photon-photon gravitational interaction exists but virtually zeroMatter-antimatter and antimatter-antimatter gravitational interactions

applies(regardless of the sign of mass)

Gravitational interaction attraction, repulsion, or zero attraction

Gravitational constant: - G is not constant- Gs and Gd are constant

- G is constant

Gravitational radiation:- Emission:- Speed:- Pattern:- Accelerates a mass:

- by accelerating mass- b ≈ 6.661 x 107 m/s- quadrupole, Fig. 3(a).- at 450 to propagation direction

- by accelerating mass- b = c = 2.998 x 108 m/s.13

- quadrupole and higher multipoles- at 900 to propagation direction13

Binary pulsars PSR B1913+16:- Gravitational radiation energy emission:- Decrease rate in semimajor axis:- Decrease rate in eccentricity:- Decrease rate in orbital period:(observed: 6.745 x 10–8 sec /orbital period)12

- 1.645 x 1030 joules /orbital period- 3.71 mm /orbital period- 9.707 x 10–13 /orbital period- 7.987 x 10–8 sec /orbital period

- 2.015 x 1029 joules /orbital period- 3.5 mm /orbital period

- 6.705 x 10–8 sec /orbital period

The model agrees on the older predictions of general relativity directionally and well within an order of magnitude – despite lacking accurate magnitudes of b, Gd, κ, and σ. (Mercury’s orbital precession rate is well explained by classical mechanics and special relativity.)

The 1919 expedition reported that light rays were deflected inward by 1.7 arc-secs as they passed grazing the sun. Later measurements found the deflection to be from 1.5 to 3 arc-secs. At least one-fifth of observed deflections could be due to the non-gravitational effects (electromagnetic, thermal, etc.) in the solar atmosphere alone. 9

The model and general relativity differ on gravitational radiation. One key difference may be illustrated by the action of gravitational radiation as it propagates through a cloud of gas, say, along the z axis. According to the model, the gas cloud takes an “hour-glass” shape, whose axis is the z axis. According to general relativity: in a half cycle, while the gas cloud is expanding along the x axis, it is also contracting along the y axis; in the next half cycle, the pair of motions reverses.

More than one theory may explain several gravitational phenomena, but one and only one theory shall explain the physics of gravitational radiation (that is, its emission, propagation, structure, speed, and polarization). There are indirect evidences of the existence of gravitational radiation, such as from the shrinking of the orbits of PSR B1913+16, but they do not shed light on its physics. So far, no gravitational radiation has been detected.14

The model makes new predictions: Gravitational interaction can be attractive, repulsive, or zero; the Classical gravitational constant is not a constant; Photons gravitationally interact albeit negligibly; and Accelerating masses generate gravitational radiation which has a four-lobed quadrupole pattern and propagates at less than the speed of light.

25

AcknowledgementThe manuscript was posted on three websites. We received many comments – many negative, several positive. We list

the salient critical comments followed by our responses to the commentators. Several commentators used the word ‘contradict’ loosely. We used the comments to clarify ambiguities and improve the model. We thank the websites and the commentators.

Comment 1. The model reinstitutes Newton’s action-at-distance force.In the model, mass-momentum fields mediate gravitational force. The field here and now depends on the field in the

immediate neighborhood and past; on the field here and now depends the field in the immediate neighborhood and future. That is, the force is communicated from one point to the next by the field at a finite speed.

Mediation-by-fields obviates action-at-distance.

Comment 2. There is no evidence of momentum field. What is momentum field anyway?Momentum field range coefficient σ ≈ 10–24 s/kg. That is, detecting the presence of the momentum field of a mass-in-

motion in laboratories would be very challenging. (This momentum field should not be confused with the pervasive momentum fields due to motions of the masses relative to the Primordial Point and to each other.)

We cannot answer what is momentum field – in the same sense that we cannot answer what is magnetic field.

Comment 3. Is mass field the Higgs field?We do not know. (The origin of mass is not relevant at the macroscopic levels. Quantum theory will be included in Part

3.)

Comment 4. Are mass fields and momentum fields dark matter and dark energy?We do not know. (The universe is filled with mass fields, momentum fields, and other property fields and their sources.)

Comment 5.  I will not read your paper until your equations are gauge invariant.Gauge symmetry was not extracted, because we were not sure what kind of it to look for as mass-momentum field does

not seem to be inspiringly similar to the well-studied electric-magnetic field. A theory should demonstrate some symmetry, but sometimes it may not. Existence of gauge symmetries does not make a theory more (or less) physical. In the next part of the model, as we begin developing gravitational field theories, we will try to extract suitable gauge symmetries.

Comment 6. The model contradicts special relativity.You did not mention as to where and how the model contradicts special relativity.This model begins in classical physics. Classical mechanics is an approximation to special-relativistic mechanics.

Approximation is not contradiction.Imposing Lorentz covariance will, of course, improve accuracy with experiments but will turn the equations

unnecessarily mathematically complex and the paper too lengthy at the cost of initial physical insight in this first, classical part. Special relativity will be considered in the next part – so will be gauge transformations.

If b were less than c, gravitons would be bosons with non-zero mass. If b < c, covariance of gravitation equations under transformations in also (x, y, z, ibt) coordinates should be explored.

Comment 7. The model contradicts general relativity.You did not mention as to where and how the model contradicts general relativity. The model and general relativity are

structurally independent of each other.The model makes no predictions directionally contradictory to general relativity; small numerical differences are not

contradictions. As any theory will always remain a theory, an older theory should not be invoked to falsify a proposed theory; only an indisputable experiment may falsify a theory.

The model and general relativity agree that gravitational radiation exists but differ on other aspects of it. These differences are not contradictions, as no gravitational radiation has been detected anyway from any source far or near.

Comment 8. The model is speculative and outside the mainstream physics.The evolutionary history of physis is replete with out-of-mainstream theories of their times. The model is not any more

speculative than such post-modern theories as string theories, hidden dimensions, supersymmetry, quantum gravity, et al. Despite innumerable journal papers and doctoral theses, those theories seem to be beyond the reach of experiments. The possibility exists for confirming or falsifying one or more of the new predictions of this model.

Physics has acquired this enormous knowledge on gravitation; however, its understanding of gravitational radiation is far from being satisfactory. Deciphering gravitation is essential to the successful journey of the species beyond the solar system and beyond the Milky Way. Ascertaining the existence of gravitational radiation is not enough. Physics must wait for experimental data on the physics of gravitational radiation (emission, propagation, structure, speed, and polarization) in order to discover the workings of gravitation that Nature is still hiding.

26

References1. A. Einstein, The Meaning of Relativity, Princeton, 1955.2. E. A. Milne, Relativity Gravitation and World-Structure, Oxford, 1935.3. T. Biswas, Special Relativistic Newtonian Gravity, Foundations of Physics, v24n4, 1994.4. I. Ridpath, Ed., Oxford Dictionary of Astronomy, Oxford, 2003.5. A. Isaacs, Ed., Oxford Dictionary of Physics, Oxford, 2003.6. R. V. Pound and G. A. Rebka, Jr., Apparent Weight of Photons, Phys. Rev. Letts., 4 (337), 1960.7. This interpretation follows Einstein’s as given in Ref. 1 (92).8. E. Freundlich, H. v. Klüber, and A. v. Brunn; Zs. f. Astrophys., 3, 171, 1931.9. R. Adler, M. Bazin and M. Schiffer, Introduction to General Relativity (129, 193), McGraw-Hill, 1965.10. M. P. Price and W. F. Rush, Nonrelativistic contribution to Mercury’s perihelion precession, Am. J. Phys., v47n6, 1979.11. M. S. Longair, High Energy Astrophysics, v. 1, Cambridge, 1981.12. Data on the PSR B1913+16 system, www.johnstonsarchive.net/relativity/binpulsar.html, 6/25/2013.13. P. G. Bergmann, The Riddle of Gravitation (136), Dover, 1992.14. B. P. Abbott, et al., Search for Gravitational Waves from Low Mass Compact Binary Coalescence in 186 Days of LIGO’s

Fifth Science Run, ligo-p0900009-v11, arXiv:0905.3710v3 [gr-qc] 5 Oct 2009.

April 16, 2014

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