Article 15: Two Derivations of Stationary Energy Theory's Equations for Gravity
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1/ Deriviation from Equations for Deflection and Splay Derived in Article 12:
We have seen earlier (in Article 7, Article 8 and Article 12) that, according to this theory, gravitational force increases with the deflection of BTTPs (α), but the “splay” of BPs (σ), due to the curvature of the Universe from being overall a 4D sphere, subtracts from the gravitational force to the extent that it is present. This means gravitational force varies with the deflection minus the splay. The deflection, itself, increases with the mass of the larger object, and decreases with distance between the objects but, as already discussed, not the square of the distance between them, and since the actual force must decrease with the square of the distance, we must divide by distance again. And since the force will also be in direct proportion to the lesser mass (m_{2}), we must multiply by that, too. We can assemble these attributes into this equation:
F_{g} = K(α – σ)m_{2}/d . . . . . . . (28)
where K is a constant, m_{2} is the lesser mass, and “d” is the distance between the objects. At smaller distances, such as for macrosized objects within our solar system, when the splay, σ, is negligible, this equation will limit to Newton’s equation for gravity, so:
Kαm_{2}/d = Gm_{1}m_{2}/d˛ . . . . . . . (29)
In Article 12 we derived Equation 22 for deflection in terms of the number of solar masses in m_{1} and the distance apart of the objects in light years. Converting this to an equation in SI units where m_{1} is in kg, distance, d, is in meters, and α is in radians, we get:
α = m_{1} × 1.485143333754 × 10^{27}/d . . . . . (30)
Substituting this in Equation 29 we get:
Km_{1}m_{2} × 1.485143333754 × 10^{27}/d˛ = Gm_{1}m_{2}/d˛
Canceling out the common terms from each side, we get:
K = G/1.485143333754 × 10^{27} . . . . . . (31)
Putting the value of G into this equation, we get
K = 6.67390 × 10^{11}/1.485143333754 × 10^{27}, or K = 4.493775 × 10^{16}.
Since c˛ = 8.98755 × 10^{16}, we can see that K = ˝c˛. Inserting this value of K = ˝c˛ into Equation 28, this theory’s basic equation for gravitational force in Newtons, with deflection and splay in radians, distance in meters, speed in meters per second, and mass in kilograms (all SI units), becomes:
F_{g} = ˝m_{2}c˛(α – σ)/d . . . . . . . (32)
The SI formulas for α and σ are equations (30) and (33) on this page, which enables this equation for gravity to be easily used in practice. The formula for α, equation (30), assumes a value for G of 6.67390 × 10^{11} and a value for the sun's mass obtained from this value of G of 1.98843 × 10^{30} kg. I have road tested this equation for gravity, and it gives the same results as Newton's equation to a high precision, where this would be expected, providing the same value of G, quoted above, is used in Newton's equation.
2/ Derivation from the Curvature of Space Indicating Gravity, Newton's Equation and the Gravitational Lensing Equation:
Although it goes further, in providing a mechanism for how matter curves space, Stationary Energy Theory is in agreement with General Relativity that matter does curve the space around it, and that the greater the curvature the stronger the gravitational force it exerts. The deflection, α, in this theory, is equal to the deflection of light as it comes in to the closest point of approach to an object, or half the total gravitational lensing of that object. As such, α is a measure of the curvature of space around an object creating a gravitational field.
This theory also goes beyond General Relativity in proposing that the "concave" curvature of space which objects with mass cause, and α is a measure of, is reduced over great distances by the "convex" curvature of space arising from the universe being, overall, a fourdimensional sphere. This "convex" curvature is called the "splay," σ, in this theory, and σ is equal to half the distance between the two objects being considered divided by the average radius of curvature of the universe in the space between them. Stationary Energy Theory thus proposes that the force of gravity is in proportion to α  σ, and can, at great distances, where σ exceeds α, become a negative force, or repulsion, instead of an attractive force. To derive an equation for gravitational force based on this does not require the mechanism proposed by this theory for how matter bends space, but depends only on the fact that space is bent. We will start by including (α  σ), the net angle of bending, as one term of the equation, and specifying 'R' as the rest of the terms, such that:
F_{g} = R(α  σ)
At close distances, such as within our solar system, where σ limits to zero:
F_{g} = Rα
Where σ = 0, this equation should give the same result as Newton's equation, so:
Rα = Gm_{1}m_{2}/d˛
So:
R = Gm_{1}m_{2}/d˛α
We have defined α as half of the gravitational lensing around an object. Since the equation for gravitational lensing is θ = 4Gm_{1}/c˛d, then
α = 2Gm_{1}/c˛d
and so:
R = (Gm_{1}m_{2}/d˛) × (c˛d/2Gm_{1})
Cancelling out the 'G's, the 'm_{1}'s and one of the 'd's, we get:
R = m_{2}c˛/2d
Substituting this back into our original equation, F_{g} = R(α  σ), we get:
F_{g} = m_{2}c˛(α  σ)/2d
Or, multiplying top and bottom by ˝, we get our Basic Equation for Gravity:
F_{g} = ˝m_{2}c˛(α – σ)/d . . . . . . . (32)
where:
α = m_{1} × 1.485143333754 × 10^{27}/d . . . . . (30)
σ = d × 1.14888 × 10^{27}/f . . . . . . (33)
'f' is a flatness factor, discussed elsewhere, which on the vast scale of the universe equals one, but is estimated to be as high as 2,500 within superclusters of galaxies, where the geometry of space is very flat.
Discussion of the Equation for Gravity Just Derived in Two Different Ways:
The above basic equation for gravity is interesting in at least two respects. Firstly, it has no need for a constant of gravitation, or any constant, for that matter (except for c, which this theory suggests may vary over large periods of time), thus it is very revealing of the nature of gravity. The lack of the need for a constant of gravitation is because the force of gravity on an object is a result of that object’s kinetic energy as it moves forward through time at the speed of light (˝m_{2}c˛). We know that force = energy/distance, and there is no reason why gravitational force should be any different. The actual gravitational force, though, is only the result of the deflected kinetic energy divided by the distance from the attracting mass, which is why it is then multiplied by the net angle of deflection (α – σ) and divided by 'd.'
Secondly, this equation shows that the force of gravity is proportional to the square of the speed of light. So if the speed of light were just over seven times its current value when the Universe was about a billion years old, as Figure 8 (in Article 7) suggests, then the force of gravity would have been about 50 times higher then than it is now. That would certainly have got that primeval matter condensed into galaxies in a hurry! It would also have acted to quickly slow the speed of expansion of the Universe, and, as a result, the speed of light, until the force of gravity became a lot weaker, and couldn’t slow it any more. The speed of expansion of the Universe would then have begun to settle around an equilibrium value, as suggested in Article 7.
Converting the Basic Equation for Gravity into a One Step Equation Including G:
Let’s continue on, though, to get an equation for gravity in terms of masses, distances and G, the recognized gravitational constant. Just as we converted Equation 22 to get Equation 30, we can also convert Equation 23 to get an equation for splay in terms of meters, with the result in radians. It is:
σ = d × 1.14888 × 10^{27}/f . . . . . . (33)
Substituting Equations 30, 31 and 33 into Equation 28 we get:
F_{g} = G((m_{1} × 1.485143333754 × 10^{27}/d) – (d × 1.14888 × 10^{27}/f))m_{2}/(d × 1.485143333754 × 10^{27})
Multiplying top and bottom of the subtracted term by d to get a common denominator:
F_{g} = G(m_{1} × 1.485143333754 × 10^{27} – d˛ × 1.14888 × 10^{27}/f)m_{2}/(d˛ × 1.485143333754 × 10^{27})
Dividing both top and bottom of the right hand side by 1.485143333754 × 10^{27}, we get:
F_{g} = Gm_{2}(m_{1} – (0.77358d˛/f))/d˛ . . . . (34)
This is a very useful equation for gravitational force as it stands, which limits to Newton’s equation when the subtracted term is relatively tiny, such as within our solar system. We could, however, take it one step further and multiply top and bottom of the subtracted term by m_{1}, so we can take out m_{1} as common factor to get:
F_{g} = Gm_{1}m_{2}(1 – (0.77358d˛/m_{1}f))/d˛ . . . . (35)
This then enables us to write the equation as in Equation 15:
F_{g} = IGm_{1}m_{2}/d˛ . . . . . . (36)
where I = 1 – (0.77358d˛/m_{1}f) . . . . . (37)
“I” is the Gravitational Intensity Factor we have previously mentioned, which we now have quantified. “f” is, of course, the “flatness factor” that we have previously talked about, which is likely to have a value of about 2,500 within our Local Supercluster of galaxies. Note that the factor of 0.77358 in the subtracted term is dependent on the radius of the Universe. This value is based on the current radius being presumed to be 46 billion light years. This factor varies inversely with the radius of the Universe, and will decrease over time.
The way in which “I” decreases slowly at first with distance ensures that “I” will always be very close to one at small fractions of the critical distance. Our sun has a critical distance of about 8.5 light years. At 0.085 light years distance, 128 times the distance of the orbit of Neptune, and 1/100 of the sun’s critical distance, “I” will only be reduced by one part in 10,000, to be I = 0.9999. At the orbit of Neptune, I = 0.9999999968. At the orbit of the Earth I = 0.999999999997. These figures show the Stationary Energy Theory is in agreement with what we know from experience — that Newton’s equation for gravitation is a very good description of gravity within our solar system.
Although “I” only differs minutely from one within the solar system, it would be interesting to see whether these minute differences could account for the tiny anomalies of planetary motion that Newton’s equation by itself can’t account for. In particular, could it account for the tiny fraction of the precession of the perihelion of Mercury’s orbit that Newton’s equation can’t account for, but general relativity can? Newton’s equation already accounts for 99.23% of the precession of Mercury’s perihelion, so it is quite conceivable that the tiny differences within the solar system, predicted by the Stationary Energy Theory, could explain the remaining 0.77%. (For a discussion of how the repulsion of nearby stars by our sun could affect the environment we live in, see Appendix E.)
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