Article 19: Refining the Equations for Gravitational Red Shift and Gravity
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By plugging the SI equation for “α” (Equation 30 from Article 15) into the equation for gravitational red shift (Equation 18 from Article 8), we can get an equation for Z in terms of the mass of a body in kg, and its radius in meters (r):
Z = sin(7.4243 x 10^{28}m/r) . . . . . . (39)
According to this equation, Z reaches a value of one, where no light could escape an object, when that object has a m/r ratio of 739,860 times that of the sun. This compares with m/r ratios of 474,006 and 176,628 times that of the sun for the two equations general relativity provides. As was just pointed out, though, if a massive object is spinning very fast, its Gravitational Intensity Factor, “I” could be much greater than one, as the negative splay would add to “α” rather than subtract from it. This would allow a body to reach Z = 1, and become a black hole, at a much lower m/r ratio. This theory’s general equation for gravitational red shift, would then be:
Z = sin(7.4243 x 10^{28}Im/r) . . . . . . (40)
For I = 4.19 this equation would give the same m/r ratio for Z = 1 as general relativity’s “long” equation.
The factor of 7.4243 x 10^{28} present in Equations 39 and 40 just happens to equal to G/c². This means we can rewrite Equation 40 as:
Z = sin(IGm/c²r) . . . . . . . (41)
At the small angles seen when this equation is applied to the sun, and even much larger stars, the sine of the angle in radians limits, to a high degree of accuracy, to the angle itself, and unless the body is rotating at very high speed, “I”, at the short distances involved, will limit to one. So in these usual situations, the equation becomes the recognizable “short” equation of general relativity (which, apparently, Newton’s theory of gravity can also derive) of:
Z = Gm/c²r . . . . . . . . (42)
Since Z = sin(½α), in these usual situations, Z = ½α, and ½α = Gm/c²r, so:
α = 2Gm/c²r . . . . . . . . (43)
Since the deflection of light when it passes a body is, as we determined earlier, 2α, then the equation for gravitational lensing limits, in these usual situations, to the wellknown:
θ = 4Gm/c²r . . . . . . . . (44)
Interestingly, all these equations have the factor G/c² in them. Since we determined earlier that the force of gravity, and hence G, vary with c², and c varies with the rate of the expansion of the Universe, which appears to have been varying over time, neither G nor c are actually universal constants that have been constant at all times since the Big Bang. Since G varies with c², though, G/c² would be a truly universal constant, that would have always kept the same value since the Big Bang, and will always keep it in the future. It would make sense to use this as a universal constant. It would be (to a higher precision than we have up to now been using):
U = G/c² = 7.42471382 x 10^{28}. Then we could say that α = 2Um/r, θ = 4Um/r and Z = Um/r. This would make it clear that these quantities do not vary over time, even as c varies. Uc² could be used to replace G in equations for gravitational force, making it clear that the force of gravitation varies with the square of the speed of light. Equation 34, the equation for gravitational force in this theory, would become:
F_{g} = Um_{2}c²(m_{1} – (0.77373d²/f))/d² . . . . . (45)
Since, as previously mentioned, the factor of 0.77373 in the equation varies inversely with the radius of the Universe, it might make sense to introduce a radius of the Universe quotient “q”, where
q = radius of Universe/35.59158 billion light years. This would mean q = 1.2924405 at the moment (1/0.77373), when the radius of the Universe is 46 billion light years. Using this, the general equation for gravitational force would become:
F_{g} = Um_{2}c²(m_{1} – d²/qf)/d² . . . . . . (46)
This shows quite clearly just what factors gravitational force depends on, with the use of just one truly universal constant, U.
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