Going to step back a bit from where I currently am, to how I got to where I currently am, because it has become important to shore up some massive holes in my explanations.
A few years ago I was trying to formulate an equation; at the time, my best guess at the equation was c*sin(ln(x))/x^2, as an equation which satisfied the properties I was expecting to find:
1.) The wavefunction should be nonzero at every scale of x, for an appropriately-chosen value of c - that is, for every value of x, there should be a value c such that the function has a nonzero value, and a nonzero derivative, where “nonzero” here means “sufficiently far from zero”. (I could say arbitrarily, here, instead of sufficiently, but the meaning isn’t -quite- correct.)
Or, to phrase this another way: If you were to graph this on a piece of paper, no matter what units you place on the X axis (1, 10, 100, 1000, etc), there should be some value c for which you can see some kind of behavior going on in the graph with your eyes; the graph shouldn’t look identical to the line of the X axis itself.
2.) For any given value of c, the wavefunction should be a sufficiently close to 0 at some distance x. (That is, it should approach 0.) (This is where “arbitrarily” gets messy.)
Or, to phrase this another way: If you were to graph this on a piece of paper, no matter what units you place on the X axis (1, 10, 100, 1000, etc), there should be some value c for which you -cannot- see some kind of behavior going on in the graph with your eyes; the graph will look identical to the X and Y axes.
3.) The definite integral of the function from 0 to n should approach 0 as n approaches infinity.
Or, to phrase this another way: The graph should have a sinuisodal function, spending equal parts of its time above and below 0.
These each correspond to a symmetry I expected to find: Scalar symmetry. The last point is the conservation law that I expect corresponds to this symmetry.
In terms of normal physics - spacetime is conserved. Adding a particle to the universe does not change the “amount” of spacetime in the universe.
(On a brief semi-irrelevant aside, I expect this conservation law to “eat” all other conservation laws; it is the core thing being conserved, and everything else is mere window dressing on this conservation. Conservation of momentum is just an outcome of conservation of spacetime. But that’s where this goes from “unified field theory” to “theory of everything”, and I’m not even to the first goalpost yet.)
But, as I was developing this set of ideas, I ran into an issue: I couldn’t formulate an equation from these properties alone. I needed another property.
Around the same time I was investigating more conventional physics, trying to better understand what was going on, and I ran into a weird problem: Why do black holes have gravity?
See, gravity is a function of distance, and the effective distance from inside the black hole, to outside the black hole, is infinite. This particular problem is generally resolved by the assertion that “gravity doesn’t gravitate” - but it sure as shit -acts- like it gravitates in some other contexts; like, negative mass binding looks an awful lot like gravity gravitating, and in fact precisely mirrors the increase of volume corresponding to an increase in curvature. Also, even if gravity doesn’t gravitate, it still isn’t actually clear that a black hole, which should theoretically have collapsed beyond a kind of negative mass binding horizon at which point the mass should be negative, should in fact have any gravity, or at least not standard gravity.
I never did find a satisfactory answer to the question of why negative mass binding should stop in a black hole in conventional physics, but the line of questioning led me to consider what happens to -my- objects, as they fall into a black hole. For my objects, I conceptualize curvature as a form of rotation, so there’s no particular issue with keeping them gravitating through a collapse into a black hole - but there is, maybe, a problem with them maintaining some kind of reasonable “shape”.
So whatever my geometry is, it must be preserved through certain kinds of transformations. And as I considered that, a potential fourth property arose:
4.) The wavefunction should be self-convergent; for any possible transformation on itself, the wavefunction should be one of two possible results, the other being the null wavefunction.
If I had better understanding of what makes the bell curve special, I might have skipped a few steps here; since I do have a better understanding of what makes the bell curve special, I can in fact skip a few steps here now. So let’s rewrite that fourth property:
4.) The wavefunction should be universally convergent; for any valid transformation on any valid wavefunction, the limit of the outcome of that transformation, as it is repeatedly applied, should result in either this wavefunction or the null wave function.
So, eventually arriving at some conceptualization of this property, I set out to see what happened to the simplest possible wavefunction, the Cartesian coordinate system, as it collapses into a singularity. And in particular I was interested to see what happened to this coordinate system at the boundary - at the event horizon.
So, orienting the coordinate system just so, let us consider what happens to a coordinate system oriented such that we are measuring distance-from-the-center-of-the-forming-black-hole on one axis, and the other two axes are, if we imagine facing outward from this point, an arbitrarily-chosen up, down, left, and right, with “forward” representing movement away from the center of the forming black hole, and “backward” representing movement towards it (and past it).
As spacetime distorts, our Cartesian coordinate system distorts as well; our four orthogonal arms (up, down, left, right) begin to bend backwards a bit, for example, and forward becomes increasingly scrunched up. The curve backwards increases as the event horizon continues forming, bending around the event horizon to point backwards - and then, as it finishes forming, wrapping around it. A straight line in our once-Cartesian coordinate system going up will eventually return to where we departed from. Light can’t escape - forward doesn’t exist either. But - backwards stops being meaningful, too. What happened to forwards and backwards? Well, a photon heading straight out away from the black hole is trapped in the same place; any motion “forwards” arrives at the same location. Just as our orthogonal arms were forced into closed loops - so too has forward, in some strange sense, been forced to loop back in on itself. Since the only point of closure for it is “backwards” - then “forwards” and “backwards” have -also- become a closed loop.
Okay, so we have this weird kind of sphere, which set me off on a long string of research into n-spheres, which ultimately wasn’t terribly helpful.
What did end up being helpful was when I asked a different question: Is the distance you can travel in “up” meaningful? That is, is the surface of a black hole actually a -surface-, or is it kind of a single point?
Lots of physics assumes it is a surface. I’m less certain, because *forwards doesn’t connect to anything* - or at least anything besides itself. The whole “surface” analogy is based on a kind of presumption that it is a sphere with a meaningful connection to the general topology of the universe. If “forwards” doesn’t connect to anything besides itself, it’s not really a surface of a sphere, it is kind of its own mathematical object.
What kind of object? Well, if we consider “forward/backward” as a closed dimension of 0 measure, that’s kind of just a point. Do up/down and left/right have meaningful dimensionality? How far “up” do you have to travel to arrive back at the point you started? And the answer is kind of weird: In one sense, it’s infinite; you’ve kind of swapped time and space, and the infinite future has to be represented here. But, if we consider this construct to be the surface of a sphere, we know the radius is 0, by looking at forward/backward. I decided that “0” was at least something I could proceed with, but I didn’t forget that infinity, and it will come back later.
So, suppose it’s a point; our three dimensions have collapsed into zero dimensions. Now, add just a -little- bit more mass to our black hole. Something is happening to our point; it’s being pulled more, in some fashion, in something like a direction. But it has nowhere to go; it’s already a point. And yet something is happening, in the same “direction” as the direction which collapsed it into a point.
I could visualize it, sort of, as spacetime turning “upside down and backwards”, or “inside out and backwards”. But this is kind of impossible to explain to somebody on its own.
So, I tried to find some kind of mathematical concept that models this.
There’s a geometric notion called suspension - this, very very roughly, corresponds to adding a dimension. There is an inverse operation called a desuspension - this, very very roughly, corresponds to subtracting a dimension. Here is exactly the operation I was hoping for.
…but the only information I could find on desuspension left it as an open question to the reader what, exactly, happens when you try to desuspend a point. But alright, we have this operation, which subtracts a dimension - and, considering the problem, what we’re looking for is something like a “negative dimension”. The only information I could find on “negative dimensions” were some references to them being used in the field of linguistics, with a somewhat cryptic comment about some unusual properties they have when you zoom in on the origin, suggesting that in some sense negative dimensions get bigger as you approach the origin, as opposed to smaller.
I didn’t know the definition of fractal dimensionality at the time, so this was, well, a little mysterious. For those who are unfamiliar, fractal dimensionality deals with non-positive-integer dimensions, and extends the concept of dimensionality to deal with these objects; the basic idea is that a fraction of a dimension changes the way the measure changes as you perform certain kinds of operations on it.
So, if you cut a line into pieces of equal size, say, one-half, then measure, the distance, is halved. If you cut the -length- of a square in half, however, what you’re really doing is cutting the square into four smaller squares, so the area is 1/4th. With a cube, cutting the “length” in half nets you 8 smaller cubes - the volume of each of which is 1/8th the original cube.
So, for this particular kind of partitioning, where you cut a thing into smaller versions of itself with half the extent, for a number of dimensions n, the measure of the object after this operation is 2^(-n).
A negative dimension is one in which performing this operation gets you a larger object than you started with. In negative three dimensions, if you cut a negative cube into eight constituent pieces, they’re each eight times larger than the cube you started with.
This actually pretty neatly describes what I mean by a topology that is “inside out”. The missing piece to explain is “backwards”; imagine, for a moment, a mirror. The left side of your face, if you are looking into the mirror, is the right side of the face of your reflection. This is the correct orientation - so, when I say “backwards”, imagine a mirror whose left matches your left.
This “backwards” behavior becomes necessary later.
Alright, so, something strange happens to our point, and we end up with a negative dimension after it is pulled inside out and backwards. But let’s consider the point’s perspective, for a moment, because, in true relativistic fashion, from its perspective, it’s still oriented normally. So, what does a negative dimension look like from a point’s perspective?
Well, if positive dimensions are outside the point - negative dimensions really only have one direction to go. Inside the point.
So, zoom in on our point at 0. It’s still a point. Zoom in further. It’s still a point. Crank that zoom to infinity - and there we go. A negative dimension points -inward- from 0, to an infinity, instead of -outward-, to infinity.
Now, here’s a neat little thing: We don’t know which dimension we just took the negative of! There are an infinite number of possible negative dimensions! However, negative dimensions, because they point inward, are -convergent-. They all start and end at the same two points.
We can arrange these to visualize them however we like; we can even flatten them onto a plane. Let’s make them into a circle, and think about this circle in terms of its radius - a line connecting the origin to a point on the boundary. (Remember that the entire boundary is “0”, however, we’re just doing this to visualize it.)
So, since we are “inside” 0, 0 is the boundary of the circle. If we are to assign an infinity to our negative dimension, it must lay at the origin of our disk. Now, notice that this implies something about the distribution of numbers, specifically integers - they cannot be evenly distributed on the radius line of this circle while preserving certain important properties, like sequentiality. It isn’t a circle, but if we visualize it as one anyways, there is no way to evenly distribute integers on the radius line while preserving properties we care about. To demonstrate this, place any finite number anywhere on the radial line; there must be a finite number of numbers from that point to 0, but an infinite number of numbers from that point to infinity.
But - I’m not done yet. See, we can organize those negative dimensions any way we like, and once we notice that the point density gets really nuts as you approach the origin (infinity), maybe we should organize those negative dimensions in a way that does something about it, so they behave in a more orderly fashion. See, “density” screams “curvature” at me. So, is there a way we can curve those dimensions so that we get something more sane? Is there a curve that traverses the sphere in such a manner that the point density remains a little more constant?
I -think- the answer is “yes”, and I -think- the answer is - a logarithmic spiral. It is some kind of spiral, at least, and a logarithmic spiral seems to fit. These spirals still form a complete circle, just rotate the object all the way around the boundary, and you’ll see that we still have complete coverage.
Note that, returning to the radius perspective, because that is still a sensible object to think about - we have a different way of counting distance in this circle, which is equally valid. We have the unit distance, which goes from 0 at the edges of our circle, to infinity at the origin. But we also have the negative-dimensional-measure - which goes from 0 at the origin, to infinity at the edges of our circle.
So, we have three ways of counting distance. Radial distance, radial measure, and the arc length of some kind of spiral.
I’m now going to skip ahead in real-world time a lot - see, I got stuck here, for a long time. I knew this object I found was important, but I had no idea -what- it was. Until I drew a picture of it for somebody (hello Pato!), and they went - “This looks exactly like the Riemann sphere”. (I leave out a lot of irrelevant nonsense that Pato patiently sat through.)
To which my (internal) reaction was “What? It’s a disk. How could it possibly look like a sphere?” And I spent a few minutes screwing around before I actually looked that object up.
And when I looked up what a Riemann sphere was, uh, yes. I had drawn a Riemann Sphere. I actually put the wrong coordinates in my picture - I put “infinity/2” at the midway point, not because it went there, but because I knew the distribution of points was uneven, and was trying to convey that idea in MS Paint.
See, a Riemann Sphere is constructed by taking two copies of the complex plane, inverting one of them, and sort of “gluing” them together. The inverted copy of the complex plane corresponds to the behavior of the negative dimension, and the non-inverted copy corresponds to the behavior of the measure.
If I had noticed that I had 0’s on the boundary of my object, and done the natural thing and glued them together - I would have the Riemann Sphere.
Or if, when I was thinking about all the infinite negative dimensions connecting 0 to infinity, maybe it would have made more sense to think about how I could organize those negative dimensions if I left 0 and infinity as single points, instead of smearing 0 across the boundary of a circle.
But a Riemann Sphere is generally constructed by an arbitrary fiber bundle.
This object is constructed by a bundle of negative dimensions, whose inherent properties lead to a very particular interpretation: Whatever the spherical equivalent of that spiral is, it is something meaningful.
I tried to figure out what the spiral was.
Letting ω be the rotation about the axis passing through the poles (so longitude), and θ be the orthogonal rotation (latitude). Suppose there exists a function f(ω) = θ. Letting ζ = tan(θ/2) * e^(-i*ω), and m(ζ) = tan(θ/2) - where m(ζ) is the "magnitude" or measure - what must f(ω) be, such that: integral [2π * (n), 2π * (n+1)] m(ζ) dω = integral [2π * (n+1), 2π * (n+2)] m(ζ) dω For all integers n >= 0.
I couldn’t solve that quickly. But then I gave up on a unique solution.
If I just assume that f(ω) is maximally convenient for integration - if it is e^(-i*w) / 2 - I get an equation which I think satisfies my constraint; assuming I did all the math correctly, 2/i * ln(cos(e^(-iω)/2) + c. And the cosine in there is always, for positive integer values of n, 1/2, so my equality is kind of trivially true.
So - a solution is possible for such a spherical helix. Getting something meaningful out of it, however, is going to require tensor mathematics. Because that “inside out and backwards” part is going to come into play now.
See, the negative dimensions happily lived in their little point all along. They weren’t created by the black hole, they’re their own mathematical objects living their own mathematical lives. The black hole made us think about them, but it made us think about them in the context of something happening to them, something “pulling one out”, so to speak.
So - all that structure I described as living on the -inside- of the circle? Well, the circle has been pulled inside out. All that now lives -outside- the circle. It still preserves its mathematical properties, but now these properties exist outside of the point. So “inside out” is fine.
“Backwards”, however, may take some explaining. And may help to explain why I also sometimes described it as “upside down and backwards”. Because, if this description of black holes is correct - and the point I described is, in an important sense, the black hole itself - then the interior and the exterior are the same space; the interior space of the black hole has been pulled out by its own forces and ejected; those negative dimensions that lived inside it happily until they were violently ripped out.
Which means something very interesting is going on at the event horizon - because it is the point at which an arrowing pointing into the black hole turns around and points outward again.
If we imagine the event horizon as a ribbon of space, it has been turned around on itself. The easiest way to do this is to just turn it upside down - but this will reverse the chirality of any object that passes through it, which, if you’re familiar with some deeper physics, is a Very Bad Thing. In order to preserve chirality, the ribbon has to not only turn back over on itself, but also twist once - like if you cut a Mobius Strip in half and pressed both cuts flat against each other. If an object -could- pass through this loop intact, it would arrive back out going the opposite direction, entirely unchanged save for that it is now facing the opposite direction.
Do we have something that satisfies our four properties?
Well … I have no idea! We have something, 2/i * ln(cos(e^(-iω)/2) + c, that could kind of be thought of as a “wave function”. But it’s really more of a coordinate system, or at least one of three dimensions of a three dimensional coordinate system. Given that each curve can be clockwise or counterclockwise, we can construct a second dimension from the second curve that passes through a given point, and a third dimension from either the latitude or longitude, depending on how the angles of those curves line up. (You can get three lines that are ninety degrees apart on the surface of a sphere). So there’s the three dimensions we live in and observe every day. And the bizarre relationship between time and space is reflected in the bizarre relationship between distance from the poles and measure.
My suspicion, or perhaps hope, is that all of physics will just drop out of the tensor equation expressing this coordinate system. But there are lots of bits and bobs I haven’t talked about here that do not seem like they will drop out nicely - there’s almost certainly a lot of work to be done porting this into quantum mechanics.
A particle should be considered (insofar as we imagine it a particle - it’s really more the whole spiral coordinate system, where what we think of as the particle is more like a particular coordinate in time, which the particle isn’t “at”, because it’s everywhere, but which we can kind of think of being an equivalent to its position … yeah this gets messy) as falling into the origin of this spiral construct; its future is “inward”. If it is falling in one of the two spirals described above, the rotation about the spiral is, from certain perspectives, “spin”. Whether or not it is a proper spinor, however, I’m uncertain of. This tight loop, again from certain perspectives, will also, I don’t think coincidentally, resemble movement about a closed dimension.
So I don’t think it is impossible! Indeed, it looks incredibly possible to express quantum mechanical behavior in this framework.
(I have another hint: The n-body problem for particles needs to be expressed in terms of maximum-entropy attractor points in the configuration space. For particles, their relative time is so much faster than ours that, if there are any maximum-entropy attractor points, they will, from our perspective, immediately fall into them.)
At this point I have to learn tensors to continue any further - this is a hard stop for me, I cannot proceed without them - so, well, I guess this is it on the crackpot physics for now.