We compare <i>Newton</i>’s force law of universal gravitation with a corrected simple approach based on <i>Bhandari</i>’s recently presented work, where the gravitation constant <i>G</i> is maintained. A reciprocity relation exists between both alternative gravity formulas with respect to the distances between mass centers. We conclude a one-to-one mapping of the two gravitational formulas. We don’t need <i>Einstein</i>’s construct of spacetime bending by matter.
Newton’s well-known law of universal gravitation, acting along the line intersecting two “point” bodies of masses m1 respectively m2 with a distance of R, takes the form of [
F N = G ⋅ m 1 ⋅ m 2 R 2 (1)
where G = 6.67430 ( 15 ) × 10 − 11 ( m 3 kg ⋅ s 2 ) is the gravitation constant [
Later, Einstein postulated in his general theory of relativity that gravitation is curvature of spacetime caused by massive bodies [
G μ v = R μ v − 1 2 R ˜ g μ v + Λ g μ v = 8 π G c 4 T μ v = κ ⋅ T μ v (2)
where G μ v represents the Einstein tensor, R μ v is the Ricci tensor, T μ v is the energy-momentum tensor, g μ v is a metric tensor, R ˜ is the curvature scalar (don’t confuse with the distance R), c is the speed of light, and Λ is the cosmological constant that is attributed to large scale dynamics of the cosmos.
Already in 1953, Sciama presented a reliable theory about the origin of gravity (inertia) as inductive effect of distant matter. The gravitational field of a moving universe was calculated by means of Maxwell-type field equations [
Recently, Bhandari and Bhandari set out to demonstrate that an external energy source may power our universe and that gravity may be due to cancellation of energy lines in the shadow regions of mass objects, creating an energy vacuum that causes the gravitational force between the masses [
The reader may also compare this effect with the Casimir effect as the dominant interaction between nano-scale objects, where the omnipresent quantum electromagnetic vacuum energy in parts is displaced between, for instance, two perfectly conducting parallel plates [
The many attempts to expand Einstein’s approach are beyond the intention of this contribution. For instance, a post-quantum theory of classical gravity discussed the need of quantization of gravity [
The present author has just commented on the Bhandari approach, explained in the Introduction Chapter, and developed a simple formula that describes the gravitational force quite accurately by maintaining the well-known G constant [
V 2 = 2 3 π r 2 3 ( 1 − 1 − ( r 1 R ) 2 ) (3)
V 1 = 2 3 π r 1 3 ( 1 − 1 − ( r 2 R ) 2 ) (4)
Dividing by the volume of each sphere gives us simpler relative volume expression V ′ i :
V ′ 2 = 1 2 ( 1 − 1 − ( r 1 R ) 2 ) = 1 2 ( 1 − cos ( δ 1 ) ) (5)
V ′ 1 = 1 2 ( 1 − 1 − ( r 2 R ) 2 ) = 1 2 ( 1 − cos ( δ 2 ) ) (6)
where δ i are the respective half-cone angles.
The shadow is not a cone but a spherical cutout, a cone capped by a spherical section. The cone base radius a is equal for both bodies, because:
a = r 1 ⋅ r 2 R (7)
The diameter d of the cut circle, where the two shadow cones meet (
d ≈ 2 ⋅ r 1 ⋅ r 2 r 1 + r 2 (8)
giving a circular area of:
A ≈ π ⋅ ( r 1 ⋅ r 2 r 1 + r 2 ) 2 = π ⋅ ( a ⋅ R r 1 + r 2 ) 2 (9)
We can also write down the interesting reciprocity relation:
2 d = r − 1 ≈ r 1 − 1 + r 2 − 1 (10)
which resembles the relation R 12 − 1 = R 1 − 1 + R 2 − 1 for parallel connection of Ohmic resistors.
Indeed, when replacing charge by mass, we jump from electrodynamics to gravity.
The gravitational force law then results in:
F = 4 ⋅ G ⋅ M 1 ⋅ M 2 R 2 ( 1 − 1 − ( r 2 R ) 2 ) ( 1 − 1 − ( r 1 R ) 2 ) (11)
For r i ≪ R ,
1 − ( r i R ) 2 ≈ 1 − 1 2 ( r i R ) 2 (12)
F ≈ 4 ⋅ G ⋅ M 1 ⋅ M 2 r 1 2 r 2 2 4 R 2 = 16 ⋅ G ⋅ M 1 ⋅ M 2 a 2 (13)
The case R → 2 r i + δ in Equation (12) will be considered in a separate contribution.
When finally denote the force by ExS (means external energy source) and obtain the following formula, where M1 respectively M2 are now masses of the shadow spherical cutouts, and r1 respectively r2 are the radii of these spherical bodies (
F E x S = 16 ⋅ G ⋅ M 1 ⋅ M 2 ⋅ R 2 r 1 2 ⋅ r 2 2 (14a)
F E x S = 16 ⋅ G ⋅ M 1 ⋅ M 2 ⋅ a − 2 (14b)
Remarkably, the squared distance between the centers of the masses is reciprocal in relation (14) compared to the Newtonian gravity formula:
F N = G ⋅ m 1 ⋅ m 2 R 2 , F E x S = 16 ⋅ G ⋅ M 1 ⋅ M 2 ⋅ R 2 r 1 2 ⋅ r 2 2 (15)
In the Newtonian formula, the gravitational force FN is inversely proportional to the second power of distance of the mass centers, whereas for FExS we estimated the second power of distance between the mass centers in the nominator.
Reciprocity relations are frequently observed in physics. Most important is the reciprocal duality between particles and waves [
Examples demonstrating the accuracy of our gravity Formula (14) are given in Ref. [
r J u p = ( 3 m 4 π D ) 1 3 = 6.99125 × 10 7 ( m ) (16)
In the same way, we get for the Sun r S u n = 6.96025 × 10 8 ( m ) .
We calculated gravitational forces between the Sun and the following planets:
Earth F E x S = 3.5448 × 10 22 ( N ) , F N = 3.5424 × 10 22 ( N ) , F E x S F N = 1.0007 (17a)
Mars F E x S = 1.7953 × 10 21 ( N ) , F N = 1.7934 × 10 21 ( N ) , F E x S F N = 1.0011 (17b)
Jupiter F E x S = 4.5719 × 10 23 ( N ) , F N = 4.5659 × 10 23 ( N ) , F E x S F N = 1.0013 (17c)
The gravitational force between Earth and Moon is calculated to be:
F E x S = 1.979 × 10 20 ( N ) , F N = 1.981 × 10 20 ( N ) , F E x S F N = 0.9990 (18)
As a resume, we conclude a one-to-one mapping of the two gravitational formulas. However, relation (14) is only an approximation [
We can in Equation (14b) transform the masses in rest energies and get the following equation:
F E x S = 16 G c 4 ( N − 1 ) ⋅ M 1 c 2 a ( N ) ⋅ M 2 c 2 a ( N ) (19)
where 4 G c 4 = κ ′ = 3.305088 × 10 − 44 ( N − 1 ) . Using the maximum force being [
F max = c 4 4 G = 1 κ ′ (20)
we finally get:
F E x S = 4 F max ( N − 1 ) ⋅ M 1 c 2 a ( N ) ⋅ M 2 c 2 a ( N ) (21)
The reader may compare the factor κ = 8 π G c 4 in Einstein’s field equations with the similar factor κ ′ = 4 G c 4 in our relation (19):
κ ′ = κ 2 π (22)
The new approach is of course a simplified one and does not consider variations in the G “constant”. For instance, periodic variations of the gravitational constant and the length of day (LOD) were recently attributed by Guynn to the influence of Jupiter’s orbit and alignment relative to the galaxy [
We should all work hard now to bring together the ideas of Guynn [
The energy lines or an energy field penetrating our world may be similar to the Higgs field. Similarities between the Higgs boson as composite particle with no spin and superconductivity may pave the way to understand dark constituents of our universe [
m H ≈ 133.3959128 ( m p + m e ) = 125.2298 GeV / c 2 (23)
where
133.3959128 = α 1 φ 2 (24)
and
α 1 = φ 2 m H i m p + m e (25)
α 1 = 50.9527 ∘ is a new magic angle [
10 43 π 20 = D L N 20 = 133.3959128 (26)
A “thought-provoking concept” as potential explanation for dark matter rooted in information physics was recently suggested by Menin [
According to Sciama, the Newtonian gravity equation arises from the gravitational effect of a rotating universe in agreement with Mach’s principle. However, based on the concept of Bhandari, gravity may arise from displacement of an energy field in the shadow of massive objects. An alternative gravity formula maps one-to-one Newton’s results, but indicates reciprocity to Newton’s formula with respect to the squared distance between the mass centers. Einstein’s construct of spacetime bending is unnecessary.
The author declares no conflicts of interest regarding the publication of this paper.
Otto, H.H. (2024) Reciprocity Relation between Alternative Gravity Formulas. Journal of Applied Mathematics and Physics, 12, 1432-1440. https://doi.org/10.4236/jamp.2024.124088
Volume of a cone V c o n e = 1 3 π a 2 ( r − h ) (27)
Volume of a sphere section V s e c = π 6 ⋅ h ( 3 a 2 + h 2 ) (28)
Volume of a sphere cutout V c u t o u t = 2 3 π r 2 h (29)
h = r − r 2 − a 2 (30)
V 2 = 2 3 π r 2 2 ( r 2 − r 2 2 − r 1 2 R 2 r 2 2 ) (31)
V 2 = 2 3 π r 2 3 ( 1 − 1 − ( r 1 R ) 2 ) = 2 3 π r 2 3 ( 1 − cos ( δ 1 ) ) (32)