I'm playing with dozens of subtle variations on the same basic equation. The trouble is, while I have a pretty good idea what I want the equation to do, getting it to behave that way is pushing the limits on my conceptual understanding of the mathematical relationships involved.
I expect the force equation to behave approximately like f(x) = sin(ln(x))/x^2. It might also be something like (cos(ln(x)s) - sin(ln(x)))/x^2, which is close enough.
Force is, in these terms, proportional to the rate of change of spacetime density. So I'm looking at the derivative of spacetime density in one of those equations.
The total change in spacetime must be 0. So the integral of force, if I am thinking correctly, from 0 to infinity must be 0; alternatively, the absolute value of the integral between any two finite points must be less than some value N, where N might be proportional to the mass of the object (or should be 1, for massless equations, such as we are playing with here). This is implied by conservation of spacetime.
More, because I think the "true" structure of the wave is a sine wave, which is self-interfering, we should be able to construct a recursive structure for the equation in place of ln(x), which is a placeholder I chose because it looks right, rather than any good reason to actually expect it to be right. The fact that my best guess at the "real" equation for distance, f(x) = x + sin(f(x))/x^2, looks very similar when graphed, is really neat. It's not great evidence to anyone else, any more than that the equation bears some resemblance to the graph of the nuclear forces, because they have different priors than I did.
To explain the priors thing: Somebody will win the lottery, somebody winning the lottery isn't surprising. However, it is quite surprising to the person it happens to. Likewise, on a population scale, somebody coming across equations that look a particular way isn't surprising. But it is quite surprising when it happens to you.
So I know the evidence doesn't carry very much weight for anyone else. But personally it is pretty neat.
Anyways, on to what I think the equations (might) be:
f(d) = d + integral(sin(f(d))/d^2) from 0 to d
Where d is Minkowski distance
Proper distance: f(d)
Density Equation: Derivative(f(d))
Or: 1 + sin(f(d)) / d^2
(Maybe)
Which means the force equation will be something like:
Sin(f(d1))/d^2 - sin(f(d2))/d^2
Yeah, that's a lot like a second derivative. But we aren't concerned with instantaneous change in density, I don't think. (Maybe we are, however, in which case my equations are wrong by a factor of d, because the observed behavior of gravity is that it drops off with distance squared, as we expect from a two dimensional surface area expanding over a three dimensional space. This is where my conceptual understanding runs into a wall; I have trouble conceptualizing the derivative of density, so I have trouble figuring out what the equations need to be. I -think- the total force exerted will be the integral of the derivative from d1 to d2, because that's the total change in density, so I think the equations are correct, I'm just not certain of that.)
But, as mentioned, I suspect the recursive function might simplify to sin(ln(x)).
For a simple idea of what this looks like, run this in a graphing calculator: sin(x+sin(x)/x^2)/x^2
Then run sin(x+sin(sin(x+sin(sin(x+sin(sin(x+sin(x)/x^2)/x^2)/x^2)/x^2)/x^2)/x^2)/x^2)/x^2
This looks like a slower-frequency-decaying version of: sin(ln(x))/x^2
So sin(ln(x))/x^2 might be the equation. It's certainly easier to work with than a recursive function, so I'll treat it as correct-enough for now. The major issue with it as the grand unified field equation is that it might not decay fast enough; that's hard to evaluate, given that the input parameter is Minkowski distance, which we can't measure directly.
Now, I've previously commented on "zeroes" being points of interest. However, the zeroes we are interested in are not the zeroes in the density equation, but rather the zeroes in the derivative of the density equation. What matters isn't whether the change in density compared to flat space is positive or negative, but whether the rate of change is positive or negative. This is important because it changes which part of the curve we are interested in; in particular, it means that gravity gets stronger quite slowly at first, and the change is negligible compared to the x^2 term we are dividing by. It is only as we get relatively far away that gravity starts increasingly quickly enough to be noticeable, quickly enough to start producing the flat rotation curve we observe in galaxies.
Which is to say, we don't need dark matter.
The same behavior exists with regard to the next repulsive phase out; the cosmological constant phase starts off weak, and accelerates in strength as you move away. Thus, we don't need dark energy, either.
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Returning to this question, I have been playing with different equations. And si(x) definitely has a decay rate that is perfect for our expectations.
More, I realized I probably don't need to simplify the recursive equation - we can get the proper distance just by measuring. How was I planning on getting the Minkowski distance, anyways?
So I moved onto units. Now, if I'm right, there should be a decaying heirarchical relationship between all the different point masses, and their size. So, in theory, I should be able to get a dimensionless ratio by dividing mass by the mass of a fundamental particle, and distance by it's radius.
In practice, we don't know the radius of any fundamental particles. I tried the reduced Compton Wavelength for electron width, and used electron mass.
New problems! The numbers are enormous. Also, substituting the way I expect to substitute doesn't work. Using F(x) results in a raw distance term inside my sinuisodal function that ends up destroying the wavelength decay. Turning into into a dimensionless term by dividing by the reduced Compton Wavelength just makes things worse.
mR = origin mass / particle mass (I/e, planet mass over electron mass - yes, these numbers are huge)
dR = distance from origin / particle width (I/e, radius of Earth divided by, maybe, reduced Compton Wavelength?)
dW = particle width (reduced Compton Wavelength)
So currently like:
playing with something like
F=(dW/(c^2*mE)) * mR* sin(mR*(si(dR) - si(1)))/x^2
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A few new thoughts...
I think the basic idea there may be correct, but we can simplify it a bit.
Relativity has simpler units, specifically, we can test against.
eG = 1.866*10^-26 m/kg
We might get that value here:
eG = wF/mF * sin ( si ( c * m1 * d / (mF*wF)))
Where wF is the width of the fundamental particle, mF is the mass of the fundamental particle, m1 is the mass of the origin object, d is the distance from the origin object to the point in space under consideration, and c is a potential conversion to radians.
The thing is, there is a term inside that sin function I am dropping. I am dropping it because I think it may be negligible for our purposes, even though in practical terms it dominates the value of the function. Specifically, we should have x+si(), rather than just si().
Additionally, that m1/mF might need to be outside the si(), rather than inside it, as a multiplier. That is the case in the last version I added here, but having thought about it more, I realize I just don't know.
And one more uncertainty: si() may need to be sin(x)/x - why? Because we may be interested in the rate of change of the distance, rather than the distance itself.
I'd know the answers with more accuracy if I could figure out how to actually simplify the original recursive function; as-is, there is a lot of guesswork involved.
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