A Crackpot Physics

A

Adirian

Registered Member
Beginning with what I initially thought was a not-that-interesting observation:

Assuming motion through time, Lorentz Contraction can be treated as the cause of motion, rather than the interaction of relativity and motion.

Specifically, given that gravity is contracted, a slight asymmetry in the contraction in the vector of motion (that is, if the forward vector is slightly more contracted), motion through time will give rise to motion through space.

In terms of macroscopic physics, this means acceleration is a wave transformation of an object's gravitational field, and leads naturally to a speed of light limitation (as this is the point when the Lorentz Contraction, viewed as a wave, is a singularity).

Considering the effect on a given object's gravitational field as it falls into a gravitational field, an asymmetric contraction arises, from an outside perspective; because gravitational fields are a change in distance, from an outside perspective treated as occupying a flat space, the leading (downward) portion of the gravitational field of the falling object traverses more distance than the trailing (upward) portion relative to the observer than the observed distance, which itself is traversing more distance. As the object falls, even without considering Lorentz Contraction from a velocity perspective, an outside observer would observe a slightly asymmetric contraction increasing as the object falls. In order to arrive at Lorentz Contraction, it is only necessary that some portion of this transformation persists, and is the phenomenon we refer to as motion.

In this formulation, Lorentz Contraction is "real" in a particular sense that the observed contraction is a result of increased curvature in the vector of motion, which corresponds to greater surface area relative to a flat space. In much the same way that the interior space of a gravity well will exceed the exterior dimensions, the observed contraction arises because contracted gravitational field gives rise to greater interior dimensions than the exterior measure, from an outside observer's perspective. (The reverse holds as well, as relativity remains untouched.)

This is, I believe, technically incorrect with respect to conventional considerations of relativity, but I have been unable to find a case where they aren't equivalent. (It does require differentiating between time dilation caused by velocity from time dilation caused by acceleration, however.)
 
Adirian said:
Beginning with what I initially thought was a not-that-interesting observation:

Assuming motion through time, Lorentz Contraction can be treated as the cause of motion, rather than the interaction of relativity and motion.

Specifically, given that gravity is contracted, a slight asymmetry in the contraction in the vector of motion (that is, if the forward vector is slightly more contracted), motion through time will give rise to motion through space.

In terms of macroscopic physics, this means acceleration is a wave transformation of an object's gravitational field, and leads naturally to a speed of light limitation (as this is the point when the Lorentz Contraction, viewed as a wave, is a singularity).

Considering the effect on a given object's gravitational field as it falls into a gravitational field, an asymmetric contraction arises, from an outside perspective; because gravitational fields are a change in distance, from an outside perspective treated as occupying a flat space, the leading (downward) portion of the gravitational field of the falling object traverses more distance than the trailing (upward) portion relative to the observer than the observed distance, which itself is traversing more distance. As the object falls, even without considering Lorentz Contraction from a velocity perspective, an outside observer would observe a slightly asymmetric contraction increasing as the object falls. In order to arrive at Lorentz Contraction, it is only necessary that some portion of this transformation persists, and is the phenomenon we refer to as motion.

In this formulation, Lorentz Contraction is "real" in a particular sense that the observed contraction is a result of increased curvature in the vector of motion, which corresponds to greater surface area relative to a flat space. In much the same way that the interior space of a gravity well will exceed the exterior dimensions, the observed contraction arises because contracted gravitational field gives rise to greater interior dimensions than the exterior measure, from an outside observer's perspective. (The reverse holds as well, as relativity remains untouched.)

This is, I believe, technically incorrect with respect to conventional considerations of relativity, but I have been unable to find a case where they aren't equivalent. (It does require differentiating between time dilation caused by velocity from time dilation caused by acceleration, however.)
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What do you mean by a "wave transformation"?
 
The opening post is word salad. Start with the first sentence or two:

Assuming motion through time, Lorentz Contraction can be treated as the cause of motion, rather than the interaction of relativity and motion.

Specifically, given that gravity is contracted, a slight asymmetry in the contraction in the vector of motion (that is, if the forward vector is slightly more contracted), motion through time will give rise to motion through space.
​
What is "motion through time"? Who knows?
What is the method by what "Lorentz contraction" can be "treated" as a "cause of motion"? It is not specified.
It is not a "given" than "gravity is contracted". I don't even know what that's supposed to mean.
What is an "asymmetry in the contraction"? It is not specified.
What is a "vector of motion"? There is no definition or explanation.
Apparently, there is a "forward vector" and ... what? A "backwards vector"? A "sideways vector"? What is this stuff? There's no attempt at definition or explanation.

There's no point bothering with the rest.
 
I don't know what you are asking by "What is motion through time". Guessing, you aren't certain what I mean by it. I mean I am assuming the existence of a time dimension of some kind, and assuming that the constituent parts of the universe are moving through it at a (relatively) constant speed. I specify that this motion is assumed because I assume it; nothing here describes why things move in time. Effectively I am assuming geodesics work.

The mechanism by which Lorentz Contraction can cause motion in space is the subject of everything else. The shorter explanation is "An object's gravitational field is contracted asymmetrically, such that it falls down its own gravity well." For a given value of down, at least. Asymmetrically means, when we consider Lorentz Contraction, one side (there being, yes, two sides, forward and backward, relative to the motion observed) is more contracted than the other.

As for what it means for gravity to be contracted, again guessing at what you are asking, I mean that, from an outside perspective, gravity appears to behave as if it is traversing more space than it is observed to be traversing. From an inside perspective the space-time is flat (and it is everything else that is contracted), and gravity is observed to behave "normally". Does that answer your question?
 
Okay...?

So what are you doing? I mean, it isn't like I posted this somewhere where a reasonable person would expect it to be taken terribly seriously.
 
Michael 345 said:
:) I'm not biting

:)
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Adirian said:
Okay...?

So what are you doing? I mean, it isn't like I posted this somewhere where a reasonable person would expect it to be taken terribly seriously.
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If above directed to me (my not biting) may I suggest you check the thread Does TIME exist where my contention is that it does not exist

Crackpot in thread title does not appear serious along with the frequency of "assumings" below

Try "observations" and from observations make predictions

Adirian said:
I mean I am assuming the existence of a time dimension of some kind, and assuming that the constituent parts of the universe are moving through it at a (relatively) constant speed. I specify that this motion is assumed because I assume it; nothing here describes why things move in time. Effectively I am assuming geodesics work.
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Other than not making ANY sense in your posting your spelling appears OK

:)
 
I am entirely serious, I simply treat my ideas with the skepticism they deserve.

As for observation, I don't exactly have relativistic objects running around to examine. I'm currently working on trying to find a proportionality between the integral of the Schwarzschild dilation factor and the Lorentz Factor for an object in frictionless freefall, as, if I am correct, I should be able to model the Lorentz Factor as a persistent portion of the Schwarzschild factor. However I'm struggling a bit with the dimensional analysis, so the math will have to wait until I figure that out.

(To be clear, given that the Schwarzschild factor in this case is proportional to acceleration, and the Lorentz factor is proportional to velocity, that there is a relationship wouldn't in and of itself prove anything, I am well aware. I expect the relationship to be... hm. Suggestive? Of something more interesting.)
 
Adirian said:
I am entirely serious, I simply treat my ideas with the skepticism they deserve.

As for observation, I don't exactly have relativistic objects running around to examine. I'm currently working on trying to find a proportionality between the integral of the Schwarzschild dilation factor and the Lorentz Factor for an object in frictionless freefall, as, if I am correct, I should be able to model the Lorentz Factor as a persistent portion of the Schwarzschild factor. However I'm struggling a bit with the dimensional analysis, so the math will have to wait until I figure that out.

(To be clear, given that the Schwarzschild factor in this case is proportional to acceleration, and the Lorentz factor is proportional to velocity, that there is a relationship wouldn't in and of itself prove anything, I am well aware. I expect the relationship to be... hm. Suggestive? Of something more interesting.)
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Good luck

:)
 
Adirian:

Adirian said:
I mean I am assuming the existence of a time dimension of some kind, and assuming that the constituent parts of the universe are moving through it at a (relatively) constant speed.
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I don't know how the idea of motion applies to time. You speak about things having a speed through time. What is that? What are the units of measure of this speed-through-time thing? As I understand it, speed means distance over time.

The mechanism by which Lorentz Contraction can cause motion in space is the subject of everything else. The shorter explanation is "An object's gravitational field is contracted asymmetrically, such that it falls down its own gravity well."
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I don't understand what you mean in your short explanation.

Asymmetrically means, when we consider Lorentz Contraction, one side (there being, yes, two sides, forward and backward, relative to the motion observed) is more contracted than the other.
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Lorentz contraction of what?

Also, Lorentz contraction is an effect of relativity velocity, not gravity, isn't it?

As for what it means for gravity to be contracted, again guessing at what you are asking, I mean that, from an outside perspective, gravity appears to behave as if it is traversing more space than it is observed to be traversing.
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How does gravity traverse space? As I understand it, gravity is an effect of the curving of spacetime. I'm not understanding what your picture of gravity is, especially when whatever it is somehow moves around in space.
 
James R said:
Adirian:


I don't know how the idea of motion applies to time. You speak about things having a speed through time. What is that? What are the units of measure of this speed-through-time thing? As I understand it, speed means distance over time.


I don't understand what you mean in your short explanation.


Lorentz contraction of what?

Also, Lorentz contraction is an effect of relativity velocity, not gravity, isn't it?


How does gravity traverse space? As I understand it, gravity is an effect of the curving of spacetime. I'm not understanding what your picture of gravity is, especially when whatever it is somehow moves around in space.
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Moving in order:

Trying not to be misleading with this, because otherwise stuff I will do later in the crackpot physics will seem like I'm violating the rules I established here. By motion through time, I mean that time progresses. At object "at" t=1 will "move" to t=2. This motion can be expressed in meters per second - it is the speed of light - but that is merely a conversion factor between distances in time and distances in space. Everything "moves" through time at (relatively) the same rate. It is difficult to get much more specific without introducing a whole hell of a lot more questions, though.

The next three questions, I think, are basically the same question? Or at least have a similar answer.

Shifting gears for a moment, imagine the prototypical view of gravity in general relativity as a bowl-like depression. Now, put another, smaller gravity well somewhere about half of the distance from the center to the edge of your imaginary bowl. (There's not really an edge, just pick an arbitrary point.) Now, because as far as the inner bowl is concerned, it occupies flat space, it will be shortened on the axis connecting it to the center of the gravity well compared to the lateral axis - the curvature of the big gravity well "amplifies" the curvature of the inner gravity well, such that the inner gravity well has steeper sides toward the gravity well, and away from it.

Traversal is a metaphor for this amplification of curvature; gravity remains (within locality constraints) consistent with the actual distance, instead of the Minkowski distance.

This is identical, in this conceptual framework, to Lorentz Contraction. So Lorentz Contraction is the persistence of the effect of gravity on another gravitational field; a moving object has had its own space-time geometry altered by gravity such that "forward" has slightly more curvature than "behind".

Given that an object is progressing through time, this means the forward side is being tilted by the curvature into motion through space. I can explain this with various metaphors - a truck whose left side tires spin faster than it's right side tires, for instance - but this is probably too long already.
 
Moving slightly ahead, if Lorentz Contraction is a geometric change to gravity which gives rise to motion, and there is no inherent "velocity" or "inertia" property in matter, our other three forces must, like gravity, be geometric, in order to impart this geometric change.

Examining the nuclear force, of course, this immediately implies that atoms should be black holes. 25,000 Newtons of force, considered as acceleration on a mass the size of a proton, amounts to a degree of curvature light couldn't possibly escape from.

But curvature implies greater surface area - that is, the interior space is larger than the Minkowski metric. The interior time is correspondingly larger, as well. Since acceleration is distance over time squared, this means, if we want to consider the nuclear force as curvsture, we need to apply relativistic corrections.

Approximating the average of the nuclear force from .7 to 1 femtometers as one half the maximum (it isn't, but my best guess is that the inaccuracy is less than 5%, I get around .495 instead of .5 using sin(ln(x)/x) as an approximation of the nuclear force - I'll return to this function later), and recalling that the Schwarzschild time dilation factor is also a space dilation factor, we have enough information to solve the equation; the factor is around 10^14. The force experienced, if the nuclear force is relativistic, is around 10^-11 Newtons, and the distance from what we observe as .7 to 1 femtometer is around .22 meters. All in all, assuming my math is right, a hydrogen atom is around 2.5 meters in diameter, if measured from the inside.

Depending on whether you use the "corrected" distance, or the original measured distance, this either makes this part of the hierarchy problem significantly worse (for the corrected distance and corrected force), or significantly better (for the uncorrected distance and corrected force).
 
Sin(ln(x))/x is, I think, a useful approximation for the grand unified field equation in my crackpot physics.

A slightly more mathematically accurate version of this may be sin(si(x)-si(1)), for values of x greater than 1, where x is distance divided by the radius of the particle originating the field. For large values of x, I believe this is equivalent to sin(ln(x)).

It still isn't accurate enough - for one, it is wildly inaccurate, as the derivation of it involves vacillating between flat-space distance and curved-space distance. I honestly have no idea how to represent the idea correctly, mathematically. I know the approximate form I expect the final equation to take - a sinuisodal equation whose period is some multiple of distance, and whose amplitude also decays with x. Sin(ln(x))/x fulfills this quite nicely, and I use it to make guesses about what I expect. But it isn't correct.

The reason I have these expectations is relatively simple - I don't think we occupy a special place, in terms of scale. I don't think there is anything special about the size we happen to be, or the size of atoms.

And a sinuisodal wave with a period proportional to distance, and an amplitude inversely proportional to distance, exhibits a very interesting feature: The downward slope from a peak closely resembles a function whose value is inversely proportional to the square of the distance. That is, it looks like gravity. Until it doesn't.

The multiplier of the period is a problem. Right now my best guess is that the half period might be the distance times 10^6, approximately. If this is the case, gravity may only be accurate for distances greater than around 10^6l and less than 10^12, or greater than 10^18 and less than 10^24).

By extension (using Kaluza-Klein to extend this field to encompass electrical fields), electromagnetic fields may vanish, or behave oddly and weakly, at distances of around 10^6, and reverse apparent polarity at distances much larger than that. A prediction, albeit a weak one, as it is guesswork about the period.

(I am using the Kuiper Cliff as the basis of this guesswork, so it isn't evidence of anything.)

As for where sin(si(x) - si(1)) comes from, it is based on an abortive attempt to try to derive a proper equation, assuming mass exhibits curvature in flat space like sin(x)/x, assuming that this curvature is self-reinforcing; that is, the distance for the purposes of curvature is expected to take into account the curvature that mass produces, and integrating from 1 (the radius of the particle) to that distance. The equation I was playing with was adjusted distance n+1 = adjusted distance n + integral(sin(adjusted distance n)))[1,adjusted distance n]

It's recursive, and the equation that results includes an adjusted distance term I haven't actually figured out how to cancel out yet. So I'm not optimistic about that approach; currently working on developing a different approach that doesn't involve infinite recursion, and is less sensitive to the fact that the root of the equation is based on an assumption of an underlying flat space.
 
On the topic of Kaluza-Klein, I utilize it.

I both think it is important to why curvature may be shaped the way it is, and for making sense of time dilation in this model.

To begin, in this model, all fermions are either singularities, or made of other singularities. I don't have mass; what we call mass is just the amplitude of curvature here. My singularities are two-dimensional surfaces, with the third dimension warped by the singularity into a closed loop, which the two-dimensional surfaces move on at the speed of light.

Here words start to break down a bit; I have no idea how to convey the concepts here, and I am reduced to a very clumsy description like "The two dimensional surfaces occupy the entirety of three-dimensional space, mediated in terms of distance from the origin by the closed dimension." Word salad, basically. The singularity doesn't stop at the event horizon - it occupies the entire universe. The two dimensional surface of the event horizon isn't an actual two dimensional space, it is something like "amplitude", or at least that is what it ends up meaning.

But the rotation around the closed dimension is, in a sense, "real". Matter rotates in one direction, antimatter in the other. Electrons are antimatter in this description. The rotation is a special case of the Lorentz Contraction-based motion I previously described, and is caused by an asymmetry in the curvature of the closed dimension. The rotation is at light speed because the curvature is a singularity.

I think it may be possible to derive the correct field equation from these concepts, but I have not figured out how yet.

In terms of physics, I think this closed dimension is a large part of what is normally thought of as "time", and acceleration causes time dilation by changing the size of the closed dimension.

Velocity causes apparent time dilation - and contraction - by changing the relative size of the closed dimension.

Or, in terms of curvature, acceleration-curvature causes the closed dimension to bend asymmetrically, where velocity-curvature (Lorentz Contraction) results in a symmetric curvature, depending on the reference frame it is being observed from.
 
Now for something (probably) completely inaccurate.

Using 10^6 as the multiplier for our half period (each half period is approximately 10^6 times as large as the period before), let's look at what the universe might look like.

Moving up in scale from a meter, we expect protons to repel one another up to a distance of about 10^6. (Recalling that the nuclear force, if relativistic, has a half period of a little over a meter). We shouldn't expect objects significantly less in size than this to exist, unless chemically bound. Even at this size, gravity alone shouldn't be sufficient to hold anything together. At this scale we expect electrical fields to stop working, and at greater distances, to reverse in apparent polarity. So far so not-quite-right.

From 10^6 to 10^12 meters, gravity should behave as expected. At around 10^12, gravity drops (somewhat unexpectedly) out of existence, and orbits stop working. Kuiper belt. This is my normalization point, and where the guess of 10^6 comes from.

From 10^12 to 10^18, we have a repulsive force again. Because our force drops off with x, instead of x^2, the force at 10^18 is significantly stronger than we would expect, which means relativistic corrections are in order; we can salvage this in terms of the galactic core, if we assume gravity is strong enough that the 10^17 is actually a 10^18 after relativistic corrections.

And as the largest chunk of mass by far, it can hold the rest of the galaxy together with the next phase of attractive force from 10^18 to 10^24. We can even arrive at the galaxy rotation curve, maybe, if everything else in the galaxy is repelling neighbors, and the core's attractive force is holding everything together; an initially stronger gravity, counterbalanced by repulsive forces, might net a flatter rotation curve. The virgo supercluster is also in this range, so all the galaxies in our local supercluster are very weakly attracted.

Now we get to 10^24 to 10^30; we are back in a repulsive phase, and approaching the Hubble Radius. *Edited: Removed a section about light shifting because I can't figure out what I expect to happen.

Okay. It almost sort of works, if you squint just right. Or rather, on a macroscopic scale, we could build a universe that looks very similar to our own using this force.

What about a microscopic scale?

Well, the nuclear force straddles an inflection point. The curvature at these scales is sufficient to make intuitive understanding of what is going on difficult; again, if the nuclear scale is curvature, a hydrogen atom is about 2.5 meters in diameter, measured from the inside.

Moving down from the nuclear force, we have a repulsive force from about 10^-6 to about 1. Relativistic corrections imply that we will observe the 10^-6 to be significantly greater in magnitude, however. Where the relativistic correction for the attractive phase was around 10^14, this will be... well, a bigger magnitude of correction. I couldn't guess. But we are in the realm of quarks, now. The one notable thing is the density may seem strangely high.

There isn't a lot of point continuing down; I can say I expect the gluey behavior of gluons, and I do, but it doesn't even qualify as a guess at this point, but rather an intuition.

I'll talk instead about the cosmological model implicit in all of this.

The interesting thing about this model is that the universe has a finite age, but an infinite history. The force I describe is fractal; the history of the universe is likewise fractal. The cosmological model here is a slow collapse upwards in scale; in the first few milliseconds of the universe, tiny galaxies were born, torn apart, collapsed, and went cold, stabilizing into the building blocks of the scale we observe as the universe now. Our observable universe may be a handful of particles in a trillion years, just starting to slow down enough to start forming stable configurations on a scale we cannot conceptualize.
 
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I have so far kind of skipped past the microscopic, but just as you can build a universe that looks like ours (if you squint) on a macroscopic scale, you can build a universe that looks like ours (if you squint) on a microscopic scale, as well.

Starting with the atom, we have the nuclear force. And it already looks like our force, at least with respect to protons. To get electrons to behave as expected, we need our modified Lorentz Contraction, and we need them to be antimatter.

Antimatter in this model isn't particularly interesting; it is just matter that goes around the closed Kaluza-Klein dimension in the opposite direction. Alternatively constructed, antimatter is just matter that is half-period "off" from matter; where matter curves space one direction, antimatter curves it the other.

This curvature is with respect to the closed dimension; protons moving in one direction bend in one direction, electrons moving in the other direction bend in the other. So antimatter behaves exactly like matter, with respect to other antimatter; at scales matter repels matter, antimatter repels antimatter. But they attract each other at such scales.

This can be conceptualized as "antimatter travels backwards in time", which, if they travel in opposite directions in the closed dimension, means we can conceptualize the closed dimension as "time".

Anyways, the short of it is, electrons are attracted to protons at the scales protons repel.

My particles are all singularities. Specifically, stable particles are white holes; they begin as black holes, but as they accumulate mass, the distances shrink until the repulsive phase of the force on step out dominates, and only light can enter. Eventually they "fill up", which is the point at which light can no longer enter; a white hole.

(But why, you ask, doesn't the infinite distance implies by the original black hole just completely eliminate the force involved? It does, at least as far as the stuff being added to the black hole goes; none of the mass dropped into the black hole contributes any force outside the black hole. The gravity of the black hole has to arise from the singularity itself, rather than from the mass contained within it. And we are back to "A two dimensional surface mediated onto three dimensions by a closed third dimension", the word salad from a previous section I still can't convey. I'm working on it. It is the surface of a hypersphere, mapped onto our more conventional trio of directions? No? Ah well.)

Anyways, where a black hole looks like a pit, conceptualized in the usual two dimensional representation of curvature, a white hole looks like an empty space; there is a section of Minkowski space that has no coordinates at all. There is no "there", there.

This is my picture of an electron. Spin corresponds to the position of this lack of position on the closed dimension; the various attractive and repulsive forces gradually shove all particles into one of two relative positions, as the particles move in the closed dimension; opposing sides of the circle, such that they are equidistant in either direction. Mind that protons and electrons are rotating in opposite directions, so they align four times per rotation.

This describes hydrogen pretty well. What about neutrons?

I don't know.

Big bloody hole, but there it is. I don't have an informed guess about how neutrons behave in this model. Maybe they work, maybe they don't.

What about electron orbits, returning to something I can talk about? These are fairly straightforward: The orbits correspond to the points where attractive and repulsive forces balance out. It works, surprisingly well, at predicting the number of electrons in each shell.

Of course, we still have uncertainty, so the shells are just where you are likely to find them. Granted, our uncertainty is probably something like the Pilot Wave model, or MWI, both of which fit pretty neatly, although we lack conventional particles, since the equivalent of a particle in this model is a region of space where there isn't any space. I am uncertain whether the probability distribution of the position of this non-position is the same as the magnitude of it's field; I suspect this is the case, but my suspicions are usually wrong. (What, you think this crazy nonsense is crazy nonsense 1.0? This is somewhere around version 104.22)

I won't go further down than that. We have two thirds of an atomic model that, if you squint, looks like a real atom.

Note that we lose quantization in all of this. Yeah, I know, it is weird for a crackpot to keep relativity and uncertainty and lose quantization, but quantization violates the Copernican Please, where the other two don't, by implying that there is something special about our place in the universe with respect to our scale. You've seen my replacement for electron spin; relative position instead of quantized rotation. Other quantization will have to be addressed similarly piecemeal, but the general trend among all the treatments is "A finite set of stable configurations results in a finite set of expected energy emissions from configuration changes". That is, quantization is an emergent phenomenon, and it emerges in particular from the fact that, at microscopic scales, the timescales are so absurdly fast compared to our ability to observe them that stable equilibria are all we can ever observe in nature. (Even what looks unstable from our time scale perspective is basically stable.)
 
The unified field theory I am approximating as sin(ln(x))/x may have a geometric explanation. Considering the surface area of an event horizon, and extending it over distance via the closed dimension, I think can be expressed differently:

A 3-sphere, with a point removed (corresponding to the singularity), with one negative closed dimension corresponding both to distance from the origin (in normal three dimensional space) and (for some reason) time.

A fractal. Sin(ln(x))/x is easier on my brain, however.
 
A small update:
The current version can probably be tested by examining electrical fields at 10^6 meters. It expects them to behave incorrectly at that range.

The only specific case I can think of where this might be observed is in the solar wind (maybe), where some unusual/unexpected magnetic discontinuities might be observed; solar wind behaving "oddly" at around 10^6 meters away from asteroids, for instance
 
On further consideration, there isn't a point removed from the 3-sphere.

To explain that a bit, take a singularity in general relativity. Actually, consider a forming singularity. Consider a 3D set of Cartesian axes set one meter away from the singularity as it is forming, with x and y perpendicular to z, which corresponds to distance.

As the singularity begins to form, x and y curve around the origin, and z shortens; upon formation, x and y form a closed sphere of radius 0. Z also closes, albeit it is harder to see; a 3-sphere of zero radius.

Now, add a little more mass. What happens to our closed dimensions? I suggest they remain closed, but invert. Three negative closed dimensions. If we consider x and y in relation to z, they return to a spherical shape (a negative dimension over a negative dimension becomes positive). However, z exhibits different behavior.

I submit it forms a logarithmic spiral with respect to a complex plane; a negative closed dimension. One complex axis represents curvature, the other complex axis represents (maybe?) time.

Curvature, represented as i/z^2, becomes, substituting using polar coordinates, sin(ln(z))/z. What might be time becomes cos(ln(z))/z. (I can't think of anything else that might be considered orthogonal to curvature besides time.)

I expect the b parameter of the logarithmic spiral to be 3*pi^2, but I am still working on the geometries.
 
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