Trying to derive a "natural" b coefficient for a logarithmic spiral, assuming that it is a negative closed dimension.
Given that I am attempting to define a negative dimension x- as being -1/x+, I tried solving for b where r=1/n (where n is the number of turns); this translates in polar coordinates to r=1/theta. Which...
Didn't work. That's just a hyperbolic spiral. Which I don't think will work.
(As obvious as this should have been, I went through the process of canceling out terms to arrive back at the same equation.)
Back to the drawing board.
I am pretty sure a spiral in the complex plane is the right way to represent a closed dimension in which the distance back to where you started, in either direction, is infinity. But without an origin this is challenging my relatively limited mathematical knowledge.
Monday, October 28, 2019
Friday, October 25, 2019
The Preconditions for All This To Work
I should probably elaborate a bit on what would mean these ideas work, and would would mean they don't.
If forces alternate between attractive and repulsive for a given particle type (over distance), the ideas work.
If there is an attractive force which fades out over distance, then another attractive force, without an intermediate repulsive force, they don't. Likewise, a repulsive force followed by another repulsive force rules all this out.
That's the "big" one.
If electrical forces between particles persist at gravitational scales - if protons continue to be electrically repulsive at distances we can observe gravity at - these ideas don't work, at least without serious modification. A major assumption here is that the apparently electrical repulsion between protons is, in fact, just a repulsive phase of gravity, which I expect to end somewhere between 10^4 and 10^6 meters. Magnetism is fine, electrical forces still dominating gravitational forces isn't.
Non-ferrous, non-molecular objects shouldn't exist below a certain margin of size. It they do, that either requires explanation, or rules all this out. Specifically, we definitely should not observe objects which require gravity to hold them together, at scales gravity is supposed to be repulsive at in this model. Likewise, we shouldn't see non-paramagnetic materials adhering to small bodies in a vacuum, generally speaking, although specific cases may be expected.
Attractive forces should generally be observed to be both stronger, and occupy smaller scales, than repulsive forces, particularly but not exclusively in composite particles.
Stars above a certain size should start to exhibit paradoxical gravitational behavior; in particular, a star close to collapsing into a singularity should have nearly no gravitational effect on neighboring objects.
Black holes, therefore, should have apparently more mass, from orbital velocities, than the stars that formed them.
If forces alternate between attractive and repulsive for a given particle type (over distance), the ideas work.
If there is an attractive force which fades out over distance, then another attractive force, without an intermediate repulsive force, they don't. Likewise, a repulsive force followed by another repulsive force rules all this out.
That's the "big" one.
If electrical forces between particles persist at gravitational scales - if protons continue to be electrically repulsive at distances we can observe gravity at - these ideas don't work, at least without serious modification. A major assumption here is that the apparently electrical repulsion between protons is, in fact, just a repulsive phase of gravity, which I expect to end somewhere between 10^4 and 10^6 meters. Magnetism is fine, electrical forces still dominating gravitational forces isn't.
Non-ferrous, non-molecular objects shouldn't exist below a certain margin of size. It they do, that either requires explanation, or rules all this out. Specifically, we definitely should not observe objects which require gravity to hold them together, at scales gravity is supposed to be repulsive at in this model. Likewise, we shouldn't see non-paramagnetic materials adhering to small bodies in a vacuum, generally speaking, although specific cases may be expected.
Attractive forces should generally be observed to be both stronger, and occupy smaller scales, than repulsive forces, particularly but not exclusively in composite particles.
Stars above a certain size should start to exhibit paradoxical gravitational behavior; in particular, a star close to collapsing into a singularity should have nearly no gravitational effect on neighboring objects.
Black holes, therefore, should have apparently more mass, from orbital velocities, than the stars that formed them.
Trying to De-Gibberish
Rereading some of these, I get the impression they will make no sense to anyone who isn't me.
So, to iterate my two major assumptions:
All fields are relativistic, by which I mean that they are curvature in space-time.
Measurements of these fields are complicated by this, because the frames of reference differ. In particular distance, from the perspective of the particle giving rise to a given field, is a lot different than it looks to us.
Curvature can be negative or positive, where negative curvature correlates to attraction, and positive curvature correlates to repulsion.
Negative curvature is an increase in distances, positive curvature is a decrease in distances, from a given reference frame.
Curvature is also relative, but I tend to convey the ideas without conveying relativity.
Negative curvature corresponds to a metric that is "bigger on the inside"; that is, an observer looking at a ruler that is sitting in relatively flat space will see it as longer than a ruler that is sitting in relatively negatively curved space.
So the negative curvature portion of the nuclear force can be seen as covering a distance of 2.5m "on the inside", and 10^-15m "on the outside", by which I mean a 1.5m ruler could "fit" in the span of the attractive phase of the nuclear force. (Except a ruler made out of atoms wouldn't. It would strictly be an imaginary ruler.)
Repulsive phases are quite the opposite; they have less "internal" space, and the distances they cover are less than an outside observer would perceive them as covering.
Which distance is "real", for the purposes of asking how distance affects a field?
I think the answer depends. I think the sum of these fields for a particle is it's own reference frame. So gravity doesn't "consider" how it curves space; it curves space as if it wasn't curving space. The outside distance is the correct one. When considering a single particle.
However, this is only true for a given instance, for lack of a better term, of the set of fields. Put two particles - two sets of fields - together, and the behavior changes, so gravity from particle A affects the distance for gravity from particle B, and likewise the reverse.
Composite particles, therefore, have different field behavior than singular particles. They behave as if part or all of their field curvature is self-interfering.
Considering something the size of a planet, we can basically treat it as entirely self-interfering. Which means the effective distance for gravity should include how gravity changes the distance involved.
Which is what I mean when I say we can potentially recover orders of magnitude of the period of our force; if the inside-frame-of-reference distance to the surface of the sun is, on average, 10^18 meters, the paltry 10^12 meter range of planets is easier to fit in the portion of sin(ln(x))/x that approximates an inverse square law; the distances are all basically 10^18, plus a rounding error.
This also potentially accounts for the hierarchy problem; the problem exists entirely in how we are measuring distance (in both space and time).
So, to iterate my two major assumptions:
All fields are relativistic, by which I mean that they are curvature in space-time.
Measurements of these fields are complicated by this, because the frames of reference differ. In particular distance, from the perspective of the particle giving rise to a given field, is a lot different than it looks to us.
Curvature can be negative or positive, where negative curvature correlates to attraction, and positive curvature correlates to repulsion.
Negative curvature is an increase in distances, positive curvature is a decrease in distances, from a given reference frame.
Curvature is also relative, but I tend to convey the ideas without conveying relativity.
Negative curvature corresponds to a metric that is "bigger on the inside"; that is, an observer looking at a ruler that is sitting in relatively flat space will see it as longer than a ruler that is sitting in relatively negatively curved space.
So the negative curvature portion of the nuclear force can be seen as covering a distance of 2.5m "on the inside", and 10^-15m "on the outside", by which I mean a 1.5m ruler could "fit" in the span of the attractive phase of the nuclear force. (Except a ruler made out of atoms wouldn't. It would strictly be an imaginary ruler.)
Repulsive phases are quite the opposite; they have less "internal" space, and the distances they cover are less than an outside observer would perceive them as covering.
Which distance is "real", for the purposes of asking how distance affects a field?
I think the answer depends. I think the sum of these fields for a particle is it's own reference frame. So gravity doesn't "consider" how it curves space; it curves space as if it wasn't curving space. The outside distance is the correct one. When considering a single particle.
However, this is only true for a given instance, for lack of a better term, of the set of fields. Put two particles - two sets of fields - together, and the behavior changes, so gravity from particle A affects the distance for gravity from particle B, and likewise the reverse.
Composite particles, therefore, have different field behavior than singular particles. They behave as if part or all of their field curvature is self-interfering.
Considering something the size of a planet, we can basically treat it as entirely self-interfering. Which means the effective distance for gravity should include how gravity changes the distance involved.
Which is what I mean when I say we can potentially recover orders of magnitude of the period of our force; if the inside-frame-of-reference distance to the surface of the sun is, on average, 10^18 meters, the paltry 10^12 meter range of planets is easier to fit in the portion of sin(ln(x))/x that approximates an inverse square law; the distances are all basically 10^18, plus a rounding error.
This also potentially accounts for the hierarchy problem; the problem exists entirely in how we are measuring distance (in both space and time).
Thursday, October 24, 2019
More Geometries
Generally I have been considering one particle for geometries here, in which infinity is at the origin, and magnitudes decrease with distance; -1/x, basically.
Considering the space between two particles, we have two infinities. But it has finite extent (the proper distance between the two particles), and curvature remains.
We arrive at an interesting parallel to hyperbolic spaces, and our metric regains something like positive-dimensional properties, considered from an outside perspective. However, the curvature isn't constant in this model, so it isn't actually hyperbolic.
---
Pi^3 probably doesn't work, although it provides nice values on the scale of the solar system. What it utterly fails to do is provide a half-period between the nuclear force and gravity, which is a significant failure. Even if relativistic corrections might recover the cosmological constant at appropriate scales, the maximum period that is feasible is probably around 10^15. I can potentially recover some orders of magnitude in relativistic corrections when the resulting force is attractive, but I have to lose some when it is repulsive, and I need a repulsive phase between the nuclear force and gravity for any of this to work.
---
I think I might have missed an important element in the logarithmic relationship - I think r may need to be the complex dimension. This makes things somewhat easier for me to conceptualize. Still trying to wrap my head around a brick there, though.
---
I think I might have missed an important element in the logarithmic relationship - I think r may need to be the complex dimension. This makes things somewhat easier for me to conceptualize. Still trying to wrap my head around a brick there, though.
Monday, October 21, 2019
Another Update to Geometries
On further consideration, two of the three complex dimensions are basically identical, and correspond to the relationships between Xi and Yi with Zi. That is, two correspond to curvature, and one corresponds to something else, potentially time.
Additionally, there are four potential positions of the origin. Calling X and Y the orthogonal dimensions, and collectively referring to them as O, we have:
O, Z; O-,Z; O-,Z-, and O, Z-. Each minus represents a symmetry flip; thus, O,Z and O-,Z- represent the same profile (let's call this matter), and O-,Z and O,Z- represent a flipped profile (let's call this antimatter).
I'm guessing the two identical profiles may represent spin; O,Z represents S+, and O-,Z- represents S-.
Note that this is a departure from earlier guesses that there may be movement in a closed dimension corresponding to Kaluza-Klein; this description implies instead that these are relatively fixed, mostly identical configurations.
Thus, antimatter is matter rotated by 90° in the complex dimensions, and equivalent spin is separated by 180°.
Additionally, I think I can rule out some lower guesses for b; in particular I don't think it can be lower than π^2/2. π^3 or 3π^2 remain viable. This is because there is no constant c such that sin(ln(c*r)/b) that is consistent with gravity at distance r for the solar system; in particular no constants are consistent with the orbital velocities of the planets while also being consistent with the existence of Earth. (Specifically, gravity on Earth would be negative.)
ETA:
Pi^3 seems oddly feasible; I can get constants that are off, on average, by around 2.5% from observed velocities. These deviations are small enough to potentially be explainable in relativistic terms (distance not being exactly as observed).
This corresponds to a far longer period than I find desirable - r*7*10^12 has some potentially great explanatory power for a variety of phenomena, ranging from the galaxy rotation curve to the Kuiper Cliff to a particular relationship between proton and electron mass-charge ratios. However, explaining those things while failing to explain why Earth doesn't just fly apart isn't a good trade-off.
Pi^3 results in a period of around r*10^73. Far, far higher than is desirable.
Note that by period I mean the distance between peaks, rather than the instantaneous period, which varies within a given period quite considerably.
Additionally, there are four potential positions of the origin. Calling X and Y the orthogonal dimensions, and collectively referring to them as O, we have:
O, Z; O-,Z; O-,Z-, and O, Z-. Each minus represents a symmetry flip; thus, O,Z and O-,Z- represent the same profile (let's call this matter), and O-,Z and O,Z- represent a flipped profile (let's call this antimatter).
I'm guessing the two identical profiles may represent spin; O,Z represents S+, and O-,Z- represents S-.
Note that this is a departure from earlier guesses that there may be movement in a closed dimension corresponding to Kaluza-Klein; this description implies instead that these are relatively fixed, mostly identical configurations.
Thus, antimatter is matter rotated by 90° in the complex dimensions, and equivalent spin is separated by 180°.
Additionally, I think I can rule out some lower guesses for b; in particular I don't think it can be lower than π^2/2. π^3 or 3π^2 remain viable. This is because there is no constant c such that sin(ln(c*r)/b) that is consistent with gravity at distance r for the solar system; in particular no constants are consistent with the orbital velocities of the planets while also being consistent with the existence of Earth. (Specifically, gravity on Earth would be negative.)
ETA:
Pi^3 seems oddly feasible; I can get constants that are off, on average, by around 2.5% from observed velocities. These deviations are small enough to potentially be explainable in relativistic terms (distance not being exactly as observed).
This corresponds to a far longer period than I find desirable - r*7*10^12 has some potentially great explanatory power for a variety of phenomena, ranging from the galaxy rotation curve to the Kuiper Cliff to a particular relationship between proton and electron mass-charge ratios. However, explaining those things while failing to explain why Earth doesn't just fly apart isn't a good trade-off.
Pi^3 results in a period of around r*10^73. Far, far higher than is desirable.
Note that by period I mean the distance between peaks, rather than the instantaneous period, which varies within a given period quite considerably.
Wednesday, October 16, 2019
An Update to Geometries
Fiddling around with alternative expressions of the basic ideas, I've come up with an entirely distinct way of conceptualizing the basic idea.
Take general relativity as it is. And, specifically, examine a forming singularity, in terms of a 3D Cartesian set of axes set some short distance away from the singularity, with each axis being (for the sake of argument) a meter long. Align Z with the forming singularity, such that it represents distance, and X and Y orthogonally.
As the singularity forms, X and Y curve around it. Z shortens.
When it has formed, X and Y close - that is, they form a sphere. Z, meanwhile, has been reduced in length to 0, meaning it has also closed. We have a 3-sphere of radius 0, wrapping around the singularity.
Add more mass. I submit that the axes turn inside out - that is, they become negative closed dimensions. I submit further this is equivalent to the transformation of each axis into a logarithmic spiral, where arc length is equivalent to distance in that dimension. Treating X and Y as measured along Z, they continue to form a sphere (a negative dimension measured over a negative dimension appears positive), and Z forms a logarithmic spiral in two, possibly three complex dimensions.
One complex dimension is equivalent to curvature - we can derive the equation sin(ln(distance))/distance from simple trigonometry.
An additional complex dimension may correspond to time, and might be given by cos(ln(distance))/distance.
The third complex dimension, if it exists, may correspond to the Kaluza-Klein dimension, which is no longer truly closed.
If this is geometrically correct - I am uncertain - I think the value of b in our Z logarithmic spiral may be pi^3. I don't know why yet, but this gives feasible values for curvature. I suspect Euler's Identity may be at work, if any of this works at all.
This is a promising approach, providing roughly the expected values for a unified field theory, for what may be entirely geometric reasons. I'm still working on it.
Take general relativity as it is. And, specifically, examine a forming singularity, in terms of a 3D Cartesian set of axes set some short distance away from the singularity, with each axis being (for the sake of argument) a meter long. Align Z with the forming singularity, such that it represents distance, and X and Y orthogonally.
As the singularity forms, X and Y curve around it. Z shortens.
When it has formed, X and Y close - that is, they form a sphere. Z, meanwhile, has been reduced in length to 0, meaning it has also closed. We have a 3-sphere of radius 0, wrapping around the singularity.
Add more mass. I submit that the axes turn inside out - that is, they become negative closed dimensions. I submit further this is equivalent to the transformation of each axis into a logarithmic spiral, where arc length is equivalent to distance in that dimension. Treating X and Y as measured along Z, they continue to form a sphere (a negative dimension measured over a negative dimension appears positive), and Z forms a logarithmic spiral in two, possibly three complex dimensions.
One complex dimension is equivalent to curvature - we can derive the equation sin(ln(distance))/distance from simple trigonometry.
An additional complex dimension may correspond to time, and might be given by cos(ln(distance))/distance.
The third complex dimension, if it exists, may correspond to the Kaluza-Klein dimension, which is no longer truly closed.
If this is geometrically correct - I am uncertain - I think the value of b in our Z logarithmic spiral may be pi^3. I don't know why yet, but this gives feasible values for curvature. I suspect Euler's Identity may be at work, if any of this works at all.
This is a promising approach, providing roughly the expected values for a unified field theory, for what may be entirely geometric reasons. I'm still working on it.
Subscribe to:
Posts (Atom)