1. Introduction
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 [1] :
(1)
where
is the gravitation constant [2] .
Later, Einstein postulated in his general theory of relativity that gravitation is curvature of spacetime caused by massive bodies [3] . With methods of differential geometry, he derived in 1915 his field equations:
(2)
where
represents the Einstein tensor,
is the Ricci tensor,
is the energy-momentum tensor,
is a metric tensor,
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 [4] . Newtonian Equation (1) arises from the gravitational effect of a rotating universe in agreement with Mach’s principle. He suggested for general motions that inertial forces have to be derived from a tensor potential.
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 [5] . The reader should study this new approach in the original publication.
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 [6] . The inverse quartic dependence with distance is different from the inverse quadratic dependence of gravity in Newton’s approach respectively electromagnetic forces. Also, forces between interacting electric dipoles respectively magnetic dipoles show an inverse quartic dependence with distance.
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 [7] .
2. An Alternative Gravity Formula
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 [8] . With the aid of Figure 1 and Figure 2, we can estimate the shadow regions of massive objects verifying the red volume of the shadow region as (see Appendix):
(3)
(4)
Dividing by the volume of each sphere gives us simpler relative volume expression
:
(5)
(6)
where
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:
(7)
The diameter d of the cut circle, where the two shadow cones meet (Figure 1), can be approximated by:
(8)
giving a circular area of:
(9)
We can also write down the interesting reciprocity relation:
(10)
which resembles the relation
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:
(11)
For
,
(12)
Figure 1. Illustration of the spherical cutoff shadow between two spherical bodies of radii r1 respectively r2 with a separation of R. Red: Projected shadow volumes as spherical cutouts. In the grey region, energy lines are devoid.
(13)
The case
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 (Figure 1 and Figure 2):
(14a)
(14b)
Remarkably, the squared distance between the centers of the masses is reciprocal in relation (14) compared to the Newtonian gravity formula:
,
(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 [9] . The reciprocity relation between Sommerfeld’s structure constant [10] and Guynn’s galactic velocity is another prominent example [11] [12] [13] . In mathematics, the golden mean is a famous example:
, where
,
.
Examples demonstrating the accuracy of our gravity Formula (14) are given in Ref. [8] and below. For sake of compatibility, we used successfully weighted radii in case of eccentric planets such as Jupiter, which should be also included in Wikipedia.
(16)
Figure 2. Explanation of the variables used. Red: The projected cone with a base radius a, completed by a spherical cap of height h to give the spherical cutout.
In the same way, we get for the Sun
.
We calculated gravitational forces between the Sun and the following planets:
Earth
(17a)
Mars
(17b)
Jupiter
(17c)
The gravitational force between Earth and Moon is calculated to be:
(18)
As a resume, we conclude a one-to-one mapping of the two gravitational formulas. However, relation (14) is only an approximation [8] , and if we consider the external energy source approach as true, then vice versa the Newtonian gravity relation must be considered as a quite good approximation.
We can in Equation (14b) transform the masses in rest energies and get the following equation:
(19)
where
. Using the maximum force being [14] :
(20)
we finally get:
(21)
The reader may compare the factor
in Einstein’s field equations with the similar factor
in our relation (19):
(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 [15] , based on measurements of Anderson et al. [16] . Jupiter is by far the most massive planet in the solar system. It is always engineers who come up with such excellent ideas. The late astronomer Johannes Kepler (*1571, †1630) would be very pleased about Gynn’s finding.
3. Outlook
We should all work hard now to bring together the ideas of Guynn [11] [15] , Bhandari and Bhandari [5] , Suleiman [17] , El Naschie [18] , Pellis [19] , Markoulakis [20] and some others, including ideas of the present author [12] [13] [21] , into a common picture of new physics and reality. We will understand, why obscure dark matter, which is strongly coupled to moving baryonic matter, may be explained by the speed dependent “viscous” drag exerted on moving objects by the repressed otherwise invisible (superluminal) construct of energy lines from an external energy source, similar to the recently successfully verified effect of gravitomagnetism as kinetic effect caused by mass “currents” (charge is replaced by mass) on gravity [22] [23] .
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 [24] . A simple formula for the mass
of the Higgs boson was recently derived by the present author [24] :
(23)
where
(24)
and
(25)
is a new magic angle [25] and
is the golden mean,
is the proton mass respectively
the electron mass. The number in relation (16) is related to Dirac’s large number (DLN) [26] [27] :
(26)
A “thought-provoking concept” as potential explanation for dark matter rooted in information physics was recently suggested by Menin [28] .
4. Conclusion
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.
Appendix
Volume of a cone
(27)
Volume of a sphere section
(28)
Volume of a sphere cutout
(29)
(30)
(31)
(32)