Forgot to add this last time, but, if you’re not interested in crackpot nonsense, please just move along with your day. This will probably be the last post on this particular subject for a while; spoiler alert, but this approach seems to be another failure.
We last left off with a promising approach for getting the inverse square law, thwarted by the rest of our equations.
y = a * sin(θ - φ) * e^(k * (θ - φ)) * e^(-k * φ)
Let’s examine θ and φ in more detail.
In our original assumptions, θ represents the amount of time* that has passed for an imaginary graviton since it left the origin, and φ represents the amount of time* that has passed for the particle the graviton ultimately interacts with. Or, in terms of rotation; θ represents the amount of rotation necessary to arrive, and φ represents the rotation by our target particle, under the hypothesis that these two rotations “cancel out” by changing the distance in time that the imaginary graviton has to travel to arrive there. A form of Lorentz contraction, basically.
*Technically, no, it’s the number of rotations.
We’ll deal with velocity some other day; for an object in orbit, these should be identical for our use-case.
Alright. The Schwarzschild metric gives the time dilation factor as:
[(1 - ((2 * g * m) / (d * c^2))]^(1/2)
Where g is the gravitational constant, m is the mass of the origin object, d is the radial distance to the point whose time dilation factor is being considered, and c is the speed of light. Yes, there’s an issue with using the Schwarzschild metric, since the outcome of all of this should be creating that metric, with divergent results when we are outside the range of values for which the inverse square law is a good approximation. Bear with me, it should at least be reasonably accurate for what we’re doing here.
θ = ln(r / a) / k
Now, this is a change in distance, so the factor is going to apply to r, giving us:
φ = ln[r * [(1 - ((2 * g * m) / (d * c^2))]^(1/2) / a] / k
And given r = k * d / ((k^2 +1) ^ 1/2):
φ = ln[k * d / ((k^2 +1) ^ 1/2) * [(1 - ((2 * g * m) / (d * c^2))]^(1/2) / a] / k
Wow that got ugly.
…and, uh wow. Okay. So. It’s kind of hard to show my spreadsheet, but I set k=c, and fiddled with a until I got the correct value for a particular distance (that is, equal to the inverse square law), and the predicted values for gravity are off by a surprisingly predictable value; if you double the distance, the equation predicts a value twice as high as it should be. Which means that if you quadruple the distance, the predicted value is four times as high as it should be. Which in turn means - I’m getting 1/r (basically).
The issue - I think - is that the value of θ - φ is, within my margin of error, basically constant at approximately 0, the way I’m doing it here. [(1 - ((2 * g * m) / (d * c^2))]^(1/2) rapidly approaches 1 as d approaches distances relevant to the solar system. So, the only part of the equation which changes is the part of the equation that explicitly follows an inverse relationship - so I get an inverse relationship. It’s not -quite- 0, so the sine function doesn’t wipe everything out, which means that my unknown constants can “fix” things back up. But it doesn’t actually fix anything.
I can make a somewhat unprincipled* decision that relativity only applies to e^(k * (θ - φ)), and drop the -φ from the sine portion of the equation - but then, effectively, I’m just back to sin(ln(x))/x, times an additional (effective) constant, and back to the problem of trying to work out what constants might work there - and I’ve already struggled with that once before, I don’t see much value in doing it again.
* The reason I wouldn’t call this entirely unprincipled is that the exponential part of the equation is equivalent to a rotation/scaling; adjusting the rotation/scaling of the equation is exactly what I expect the relativistic behavior to do. The reason I still don’t like it is that it effectively eliminates the nice behavior where the curves physically can’t curve themselves because they’re in the same inertial frame. Also I don’t actually have a good a-priori justification for treating the two values differently, even if I can rationalize it after the fact.
The issue, in the broadest possible terms, is that, in order for this to work, the interior value of the sine needs to increase much faster than the interior value of the exponential part of the equation; in particular, for a distance r*10^m, I need the value of the interior of the sine to increase by 2pi for every increase in m of somewhere between 5 and 18, with an “ideal” somewhere around 12, to account for the hierarchy problem. Yet at the same time, I need the interior of the exponential part of the equation to barely change at all, lest it rapidly dominate the equation.
Adjusting the relativistic behavior in our somewhat unprincipled way fixes that problem, but causes other problems. Maybe there’s a solution with a -third- rotation value, but, well, I don’t know exactly what it would be, much less why it should apply in one case but not the other.
I’m not convinced this approach is entirely a bust, yet, but it’s looking awfully lot like a bust. Kind of. There might be a very weird path out, in which you have to have a particular level of mass density relative to the distances involved before larger-scale forces appear; however, this dissolves the explanatory power with regard to the Kuiper cliff, gravitational rotation curves, and dark energy, that otherwise this approach might possess - so it could conceivably work, but … only if it doesn’t predict anything interesting. Also it probably doesn’t work without some kind of unprincipled adjustment to the way I tried to deal with relativity.
Additionally, on further inspection, I strongly doubt k can be c, given its position in the sine wave; the wavelength decay is just too weak. Probably not “a”, either. So the initial premise of this approach, way back in the first post, falls apart.
Until I have another idea, which historically has taken a year or more, this will probably be it for this crackpot physics.