Gravitation



A Constructive Model of Gravitation

Raghubansh P. Singh

raghu.singh@

Abstract

The 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. Introduction

Newton (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 data

We 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 model

We 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 interaction

We 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:

[pic], (3)

where Fs is repulsive, and Gs is the static gravitational constant.

The gravitational force between m1 and m2 with momenta [pic] and [pic] respectively, as mediated by their momentum fields per Assumption (2), is given by:

[pic], (4)

where Fd is attractive or repulsive as the angle between [pic] and [pic] 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:

[pic] (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 [pic] and [pic] relative to O and have mass fields (white shade), momentum fields (darker shades), and momentum field ranges S1 and S2 respectively.

Figure 1. Masses with mass and momentum fields

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

[pic] (6)

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

[pic]; (7)

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

[pic] (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:

[pic] (9)

[pic] (10)

[pic] (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 |repulsion |

|u = b |attraction |zero |

|u > b |attraction |attraction |

4.2 Mass-energy gravitational interaction

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 interaction

Per 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 ([pic]), it holds at an arbitrary velocity ([pic]) as well.

5. Graviton

Graviton is a hypothetical quantum of energy exchanged in gravitational interactions. If b  1, thus νr  m′) changes from r = r to r = ∞.

We consider an ideal rod constituted of “atoms” of mass δm′ (> 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 [pic] in addition to the radial momentum field [pic]. Radial [pic] drifts with the mass at speed dυ and propagates radially at speed b. Transverse momentum field [pic] changes from zero to amplitude Pt and back to zero in dt; this is gravitational radiation pulse propagating outwardly at speed b.

Figure 2. Mass m and its momentum field vectors

The geometrical ratio of transverse to radial momentum fields is:

[pic], (40)

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

[pic] (41)

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

[pic] (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):

[pic] (43)

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

[pic] (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:

[pic] (45)

If [pic] and [pic] are orthogonal, the magnitude of [pic] 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:

[pic] (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).

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

[pic] (47)

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

[pic] (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:

[pic] (49)

or, [pic] (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:

[pic] (51)

[pic] (52)

[pic] (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.

13. Measurements

The model introduced three constants (Gs, Gd and σ) and one parameter ([pic]). 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 [pic]. 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 [pic] 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 predictions

Appendix 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. Remarks

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

Appendix A

Nomenclature

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

a acceleration

AU astronomical unit: 1.4959787 x 1011 m.

b speed of gravitational radiation; Eq. (5).

c speed of electromagnetic wave

e electrical charge

E energy

F Force

G classical gravitational constant

Gd 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 density

m/R point density

M mass-field strength

nt newton, unit of force.

p momentum (gravitational current)

P momentum-field strength; Eq. (41).

Q Coulomb constant

R radius

Sm momentum field range of a mass m; Eq. (1).

S12 inter-momentum field range between masses m1 and m2; Eq. (2).

[pic] velocity of masses relative to the Primordial Point; Fig. 1.

[pic] arbitrary velocity

Γ Γ = Gs (1 − u2/b2)

κ mass-energy gravitational coefficient; Eq. (13).

λ wavelength

ν frequency

σ momentum field range coefficient; Eq. (1).

τ period

Ω power

Appendix B

Estimating matter-energy gravitational coefficient κ

B-1 Estimating κ using the results from the Pound-Rebka experiment

The 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):

[pic] (B.2)

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

[pic] (B.3)

Solving (B.3) for κ, we get:

[pic] (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 mass

From (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)

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.

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 ( ................
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