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.
Monday, June 3, 2019
Potential Proper Equation
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.
--------------
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
-------------
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.
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.
--------------
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
-------------
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.
Wednesday, May 29, 2019
A Crackpot's Vision of Future Physics: Quarks
Quarks are hard. Really, really hard. I don't understand them at more than a superficial level; I don't know anybody who does. The math is notoriously difficult, and the philosophy appears nearly nonexistent, limited largely to naming the phenomena the math predicts. So this section is going to be wrong in a lot more ways than normal.
First, my suspicions: Quarks are extremely complex configurations of an absurdly large number of smaller particles, typically called gluons, across six phases of our mass wave. Which is to say, three pairs of attractive and repulsive phases of the wave, corresponding roughly to what in chromodynamics is referred to as "color". There are both matter and antimatter configurations, although that is an incomplete description, because I think all the configurations are composed of both matter and antimatter in part. Because these configurations are larger than a single phase of the gravity wave, you get an interlocking behavior we don't see much of at larger scales until you get to molecules, in which the different phases of the mass wave are, at a comprehensive scale, often in opposition; pull on opposing ends of a quark and there's a lot of give to it, stretching it out, as the conflicting waves give a stretchiness to the underlying behavior until they are no longer in conflict; at which point an abrupt and massive increase in the energy required to continue stretching occurs, no longer assisted by the conflicting wave patterns, beyond which the quark will "snap" in half. So much energy that new particles will be created to fill in the gaps.
We see this behavior; it is only the description of the phenomena that differs. Behavioral particulars aren't really addressed here except to say that they're complex, because the particles are complex.
A Crackpot's Vision of Future Physics: Quantum Physics
Yes, this model suggests quantum physics isn't quite right, or perhaps more accurately is a simplification. How do we go about using quantum physics?
First, the perturbation model is key. That model works quite perfectly in this framework.
Second, Hamiltonian operators, the mathematical mechanism of incorporating the "quantum" from quantum physics... also continue to work perfectly. They're still the only non-neglible energy under consideration. All energy below the quantum level has no macroscopic ramifications, until and unless it achieves the quantum minimum, at which point a configuration change occurs, and a quantum event - specifically some energy emission at a quantum scale - takes place.
Virtual particles get interesting, or rather, stop being interesting, which I think is more interesting. Since all interactions between mass in this model is the result of the interactions between mass waves, most interactions have a mutuality component. The virtual particle, traveling backwards in time, is a mathematical fix for a model which represents interactions as arising from particle interactions - A affects B simultaneously to B affecting A, and a mathematical representation of the interaction that describes information passing from A to B necessarily needs B to transmit information to A. Because the information has locality limits, however, the math only works if the virtual particle travels backwards in time to arrive back at A at the same time it "emitted the force carrier particle" that would eventually arrive at B. Virtual particles are just a fix to make the mathematical model arrive at the correct conclusion.
By and large, quantum mechanics remains surprisingly unchanged. The subtle shift from "Energy only comes in multiples of X" to "Energy below X has no effect" would be far more significant if not for uncertainty; with uncertainty, the model already accounts for occasional unexpected energy emissions.
What about uncertainty? I am honestly not certain. It seems to persist in electron double-slit experiments, if nowhere else. We'll devote a section to uncertainty however.
First, the perturbation model is key. That model works quite perfectly in this framework.
Second, Hamiltonian operators, the mathematical mechanism of incorporating the "quantum" from quantum physics... also continue to work perfectly. They're still the only non-neglible energy under consideration. All energy below the quantum level has no macroscopic ramifications, until and unless it achieves the quantum minimum, at which point a configuration change occurs, and a quantum event - specifically some energy emission at a quantum scale - takes place.
Virtual particles get interesting, or rather, stop being interesting, which I think is more interesting. Since all interactions between mass in this model is the result of the interactions between mass waves, most interactions have a mutuality component. The virtual particle, traveling backwards in time, is a mathematical fix for a model which represents interactions as arising from particle interactions - A affects B simultaneously to B affecting A, and a mathematical representation of the interaction that describes information passing from A to B necessarily needs B to transmit information to A. Because the information has locality limits, however, the math only works if the virtual particle travels backwards in time to arrive back at A at the same time it "emitted the force carrier particle" that would eventually arrive at B. Virtual particles are just a fix to make the mathematical model arrive at the correct conclusion.
By and large, quantum mechanics remains surprisingly unchanged. The subtle shift from "Energy only comes in multiples of X" to "Energy below X has no effect" would be far more significant if not for uncertainty; with uncertainty, the model already accounts for occasional unexpected energy emissions.
What about uncertainty? I am honestly not certain. It seems to persist in electron double-slit experiments, if nowhere else. We'll devote a section to uncertainty however.
A Crackpot's Vision of Future Physics: Uncertainty
We've lost quantum mechanics. We have not lost uncertainty, although we see a lot less of it.
There are a few reasons to expect to see something like uncertainty in the universe; the most important is locality. Information can't exceed the speed of light, yet phenomena exist in the universe which must simultaneously be observed, yet simultaneously prohibit observation. I am, of course, talking about black holes.
Within the event horizon, where does the singularity lay? Impossible to say. If the singularity were to move, the update wave to move the event horizon couldn't reach it to move it in turn.
Black holes are messy, mathematically. White holes more so, being unapproachable from either side; matter can neither enter nor escape. I don't know if white holes can even be said to be connected to our universe anymore; the spacial density interpretation suggests an effectively infinite distance involved, which implies to me that maybe the contents of white holes aren't meaningfully in our universe anymore, since spacetime should be conserved, and an infinite distance would otherwise fail to conserve it.
Black holes, by comparison, are still connected, albeit in a curious manner; there is still an infinite distance involved, however. I think the interior edge of the event horizon of a singularity of either type can be described as a cylindrical dimension, and more, I think this cylindrical dimension may be the Kaluza-Klein dimension. The surface area of an event horizon is two-dimensional, after all; I think the third dimension gets closed on itself.
If, as I posit in the section on time, the Kaluza-Klein dimension is what physics refers to as time, this means time may be a phenomenon that arises from singularities themselves. Thus, particles are three dimensional; two dimensions form a surface area that connects their spacetime to the rest of the universe, and the third dimension gets bound up and closed. In fact, all three dimensions are effectively closed, but this is a little hard to conceptualize.
When we remember that mass is a wave, and in particular a standing wave that occupies the entire universe, however, the nature of the connection between our closed three dimensional construct and the open three dimensions we are more familiar with gets a little... odd. We could say it is connected where it is connected, and just posit an origin, a particular space it occupies. But we come into this already knowing that there's going to be some oddities involving position, because we have observed them.
There's a simple solution, of sorts. The closed three dimensional area of a singularity particle is only partially connected to the universe, through its own mass wave; there is an origin, but the particle actually occupies the entire universe, proportional to the amplitude of its mass wave at that position. (Well, there are zeroes in its wave, places it doesn't occupy.) This is equivalent, in some significant respects, to uncertainty; it is less that mass has an uncertain position, however, so much that position ceases to be a particularly meaningful concept.
This ends up working surprisingly well, in a specific way: Acceleration isn't a discrete operation. You don't accelerate "the particle", you accelerate some portion of it's mass wave. It will self-correct over time, at lightspeed, but if there are multiple possible end destinations (as with the double slit experiment), you can temporarily split the mass wave, and it will self-interfere until it stabilizes. The section on Collisions describes part of this process.
This hints at an explanation for the fact that position and velocity have a similar factor in uncertainty: Velocity is only ever a transformation of a part of the mass wave.
There are a few reasons to expect to see something like uncertainty in the universe; the most important is locality. Information can't exceed the speed of light, yet phenomena exist in the universe which must simultaneously be observed, yet simultaneously prohibit observation. I am, of course, talking about black holes.
Within the event horizon, where does the singularity lay? Impossible to say. If the singularity were to move, the update wave to move the event horizon couldn't reach it to move it in turn.
Black holes are messy, mathematically. White holes more so, being unapproachable from either side; matter can neither enter nor escape. I don't know if white holes can even be said to be connected to our universe anymore; the spacial density interpretation suggests an effectively infinite distance involved, which implies to me that maybe the contents of white holes aren't meaningfully in our universe anymore, since spacetime should be conserved, and an infinite distance would otherwise fail to conserve it.
Black holes, by comparison, are still connected, albeit in a curious manner; there is still an infinite distance involved, however. I think the interior edge of the event horizon of a singularity of either type can be described as a cylindrical dimension, and more, I think this cylindrical dimension may be the Kaluza-Klein dimension. The surface area of an event horizon is two-dimensional, after all; I think the third dimension gets closed on itself.
If, as I posit in the section on time, the Kaluza-Klein dimension is what physics refers to as time, this means time may be a phenomenon that arises from singularities themselves. Thus, particles are three dimensional; two dimensions form a surface area that connects their spacetime to the rest of the universe, and the third dimension gets bound up and closed. In fact, all three dimensions are effectively closed, but this is a little hard to conceptualize.
When we remember that mass is a wave, and in particular a standing wave that occupies the entire universe, however, the nature of the connection between our closed three dimensional construct and the open three dimensions we are more familiar with gets a little... odd. We could say it is connected where it is connected, and just posit an origin, a particular space it occupies. But we come into this already knowing that there's going to be some oddities involving position, because we have observed them.
There's a simple solution, of sorts. The closed three dimensional area of a singularity particle is only partially connected to the universe, through its own mass wave; there is an origin, but the particle actually occupies the entire universe, proportional to the amplitude of its mass wave at that position. (Well, there are zeroes in its wave, places it doesn't occupy.) This is equivalent, in some significant respects, to uncertainty; it is less that mass has an uncertain position, however, so much that position ceases to be a particularly meaningful concept.
This ends up working surprisingly well, in a specific way: Acceleration isn't a discrete operation. You don't accelerate "the particle", you accelerate some portion of it's mass wave. It will self-correct over time, at lightspeed, but if there are multiple possible end destinations (as with the double slit experiment), you can temporarily split the mass wave, and it will self-interfere until it stabilizes. The section on Collisions describes part of this process.
This hints at an explanation for the fact that position and velocity have a similar factor in uncertainty: Velocity is only ever a transformation of a part of the mass wave.
A Crackpot's Vision of Future Physics: The Big Bang
Okay. It's turtles all the way down; or particles, at least. Where did it begin?
Strictly speaking, in this model, it didn't. The universe had simultaneously existed for a finite amount of time, and also never actually started. This is because of the curious, scale-symmetric properties of the speed of light.
The speed of light - the speed of propagation through spacetime - is both the maximum on speed, and also a throttle on the passage of time. The speed of events at a scale of observation is limited by the speed of light. To make sense of this, imagine a brain the size of our galaxy. Imagine the time it takes a piece of information to travel from one point to another; because of how long it takes information to move, from this superbrain's perspective, events at our scales happen absurdly quickly. It would observe the motion of galaxies as something happening quickly.
The same principle applies when you get smaller in scale, although it is perhaps harder to visualize. The distances are closer; light traverses small distances at the same speed it traverses large distances, so events at a smaller scale happen more quickly, from our perspective. From a sufficiently small-scale perspective, the first few milliseconds of the big bang were, scale-relative, billions of years. Galaxies formed, grew cold, and died. In that time, any beings at that scale would have wondered about when it all started, and where it is all going. They perhaps may have discussed the beings who might have existed in the first few milliseconds of the big bang.
Because the big bang wasn't an event, and it isn't over. Zoom out, in both time and space, and you will eventually reach a point where the universe looks like a dense hot plasma. There was no beginning, even if it has only been going on for a finite amount of time, because the meaningful concept of the relevant scale of time is constantly increasing. There is no end. The big bang, and the universe - because they are the same - are an ongoing process.
Strictly speaking, in this model, it didn't. The universe had simultaneously existed for a finite amount of time, and also never actually started. This is because of the curious, scale-symmetric properties of the speed of light.
The speed of light - the speed of propagation through spacetime - is both the maximum on speed, and also a throttle on the passage of time. The speed of events at a scale of observation is limited by the speed of light. To make sense of this, imagine a brain the size of our galaxy. Imagine the time it takes a piece of information to travel from one point to another; because of how long it takes information to move, from this superbrain's perspective, events at our scales happen absurdly quickly. It would observe the motion of galaxies as something happening quickly.
The same principle applies when you get smaller in scale, although it is perhaps harder to visualize. The distances are closer; light traverses small distances at the same speed it traverses large distances, so events at a smaller scale happen more quickly, from our perspective. From a sufficiently small-scale perspective, the first few milliseconds of the big bang were, scale-relative, billions of years. Galaxies formed, grew cold, and died. In that time, any beings at that scale would have wondered about when it all started, and where it is all going. They perhaps may have discussed the beings who might have existed in the first few milliseconds of the big bang.
Because the big bang wasn't an event, and it isn't over. Zoom out, in both time and space, and you will eventually reach a point where the universe looks like a dense hot plasma. There was no beginning, even if it has only been going on for a finite amount of time, because the meaningful concept of the relevant scale of time is constantly increasing. There is no end. The big bang, and the universe - because they are the same - are an ongoing process.
A Crackpot's Vision of Future Physics: Energy
You may, at this point, have started to gather that this model doesn't have quanta (that is, an energy "atom", an amount of energy which is the minimum amount possible). This isn't quite accurate; quanta still exist, but they are more a macroscopic (from a certain perspective) phenomena than a property of the universe. A quantum of energy is simply the smallest amount of energy necessary to move from one stable configuration to another, and vice versa, the amount released when moving from a higher-energy stable state to a lower-energy stable state.
More, energy levels below the level of the quantum stop looking like "energy". Something that looks like negative energy becomes possible at that level; since everything is the interaction between waves, some of those interactions will move configurations towards stable configurations, and some will move those configurations away. Energy below the level of the quantum is effectively negligible, from our macroscopic perspective, because the net effect is generally nil.
Generally, but not always. Under certain circumstances, sub-quantum energy suddenly becomes important, because fluctuations in the density of this energy can make it easier for a given, say, electron, to move to a different stable configuration. Say, when we are shining a very weak amount of light on a phosphorous surface. Under these circumstances, random variations in sub-quantum energy will result in some electrons being more energetic, and thus more likely to change configurations, and emit light in return. This looks, from a macroscopic perspective, like randomness.
From a certain perspective, however, sub-quantum energy isn't energy at all. We can't utilize it, as far as I can see, and even if we could there's not enough of it to be worth doing. We can't even measure it, and I don't see that changing in the next century or two; in order to measure it with current technology, we'd have to somehow get it to condense to the point where it could cause a configuration shift, because that is the smallest event we can currently measure.
This is to be expected. Given the scales involved, sub-quantum energy is effectively and permanently in a state of maximal entropy; any work it could do was expended in the first few seconds after the big bang. Although sub-quantum energy may exist, this doesn't imply that subatomic events are dynamic; the subatomic galaxy-equivalents were cold and entropically dead long before atoms could form at our scales. Our observed universe will be cold and entropically dead long before what we observe forms part of some still-larger configuration. Entropy marches upward, in terms of scale.
More, energy levels below the level of the quantum stop looking like "energy". Something that looks like negative energy becomes possible at that level; since everything is the interaction between waves, some of those interactions will move configurations towards stable configurations, and some will move those configurations away. Energy below the level of the quantum is effectively negligible, from our macroscopic perspective, because the net effect is generally nil.
Generally, but not always. Under certain circumstances, sub-quantum energy suddenly becomes important, because fluctuations in the density of this energy can make it easier for a given, say, electron, to move to a different stable configuration. Say, when we are shining a very weak amount of light on a phosphorous surface. Under these circumstances, random variations in sub-quantum energy will result in some electrons being more energetic, and thus more likely to change configurations, and emit light in return. This looks, from a macroscopic perspective, like randomness.
From a certain perspective, however, sub-quantum energy isn't energy at all. We can't utilize it, as far as I can see, and even if we could there's not enough of it to be worth doing. We can't even measure it, and I don't see that changing in the next century or two; in order to measure it with current technology, we'd have to somehow get it to condense to the point where it could cause a configuration shift, because that is the smallest event we can currently measure.
This is to be expected. Given the scales involved, sub-quantum energy is effectively and permanently in a state of maximal entropy; any work it could do was expended in the first few seconds after the big bang. Although sub-quantum energy may exist, this doesn't imply that subatomic events are dynamic; the subatomic galaxy-equivalents were cold and entropically dead long before atoms could form at our scales. Our observed universe will be cold and entropically dead long before what we observe forms part of some still-larger configuration. Entropy marches upward, in terms of scale.
A Crackpot's Vision of Future Physics 4: Light and Other Waves
The model doesn't have a collection of fields; it has a single wave. This leaves us with something that is, with respect to the Standard Model, problematic. I've briefly described light, but let's discuss it in more depth.
Light in this model, like everything, is a wave. It is specifically a wave in a wave; it is the change in the mass wave, propagating at lightspeed. A change in the change of density is itself a change in density. Which means we can simplify the idea a bit: Light can be treated as a wave in spacetime itself. It has some particular properties arising from it's origins in specific subatomic particles, and more specifically a particular range of sizes of particles, namely that it has a range of frequencies limited to Rhydberg's model of light, of resonant frequencies. Mostly.
Bosons in general are waves in spacetime of varying frequency, depending on the size and resonant frequency of the originating particles.
Note to self for future: Light as continuous Fourier transform of zero-width, infinite amplitude Lorentz Contraction, hence "speed of light". Tie into other sections explaining speed of light as irrelevant constant (speed of light doesn't matter, mediates all interactions including clocks timing it's speed)
Note to self for future: Light as continuous Fourier transform of zero-width, infinite amplitude Lorentz Contraction, hence "speed of light". Tie into other sections explaining speed of light as irrelevant constant (speed of light doesn't matter, mediates all interactions including clocks timing it's speed)
A Crackpot's Vision of Future Physics 3: Motion
If you've grokked what I've written so far, you may have noticed something. Without a particle, without mass, without a "thing" the property of "velocity" can be attached to, velocity becomes kind of difficult to reconcile with the universe as we observe it. What is motion? How can what is, basically, a wrinkle in spacetime be said to meaningfully "move"? As it turns out, we already have the answer in relativity.
Lorentz Contraction is the phenomenon in relativity in which objects contract along their axis of motion relative to the direction of motion. It isn't just the objects themselves, either; everything contracts, even the objects' own gravitational fields. By "contract", I mean that they get flatter, from an outside observer's perspective, along the direction of motion, as a proportion of lightspeed; an object somehow moving at the speed of light would appear as a perfectly flattened object.
At first, this might appear to be a product of subjectivity, rather than something that is "really" true. The trick in understanding Lorentz Contraction is to notice two things: First, that the gravitational field is also contracted, and second, that this means the density of spacetime is being distorted. Which is to say, the object isn't "really" flattening, but rather it is occupying a space that is bigger lengthwise (where length is parallel to the direction of motion) than the surrounding space.
And it isn't symmetric lengthwise; the space forward of the origin is slightly denser than the backwards section. Which is to say, space is denser forward of the ship than behind it. If you'll recall how gravity moves things, this is the same principle. We typically think of Lorentz Contraction as being a property of motion, but it is equally valid to think of motion as a property of Lorentz Contraction. And once you notice this, the notion of motion causing Lorentz Contraction becomes unnecessary; they're the same phenomenon, with Lorentz Contraction being the more basic.
Once we start thinking in these terms, we can start to redefine "motion" as a wave distortion in spacetime, or more specifically a wave distortion in the wave that is mass. This is a useful concept, but we aren't quite done yet; we need a mechanism of imparting this wave distortion on our mass-wave.
It's already been invented. I mean, I figured it out, but somebody figured it out long before I did, so I can't claim any credit. When objects are accelerating, they get an additional kind of distortion applied to them, which looks - not without accident - exactly like gravity. This phenomenon is called "Rindler Coordinates".
Now, I just made a massive leap, so let's take a quick step backwards. Let's try to figure out what gravity looks like to an object in it's influence, and what acceleration looks like.
Imagine two point masses in space. We'll simplify our mass wave to a single gauge, gravity. Point A, Point B, and us an observer at Point C. From Point C, the region between the two masses is denser. No surprise there. The interesting thing happens when we only examine the gravitational field of one of the two points, completely ignoring the other. Let's say we are looking at Point B.
Because the density between the two points is higher, this means that, from our outside perspective, the nice neat even distribution of spacial density, cleanly increasing as you approach Point B, no longer applies. It is flattened on the side facing Point A, because on that side, it has further to go, so weakens more quickly. More importantly, on its own, without any consideration of Point B's own gravity now, spacial density on the side facing Point B rises more quickly as we approach Point B. That is, even if we removed Point A entirely at this point, Point B's own gravitational field is already distorted such that it will pull -itself- towards Point A, through the simple expedient of moving through time and the side facing Point A being denser and thus more distance. (We are, for now, going to take motion through time as a given. It is the only motion we take for a given. Everything else arises from this single motion.)
Acceleration, and Rindler Coordinates, look exactly the same. Flattened more on the side oriented toward motion. This makes sense, if you think in terms of the speed of light; if you had reaction-less acceleration, that is, you weren't throwing particles out the back end of the ship, you'd get the same shape, just because the lightspeed "update" to the gravitational field takes time to propagate, such that, from an outside perspective, the forward region of fiethe gravitational field is always slightly behind, time-wise, the backward gravitational field. It takes time for light to reach things.
Of course, that is reaction-less acceleration; it presumes acceleration through some other mechanism causes this effect. I think this is slightly backwards; acceleration doesn't cause this shape, this shape is the phenomenon we call açceleration. This distortion is temporary, caused by the interaction of fields, such as gravity. But to explain this, we need to look at collisions, and more generally acceleration that isn't reaction-less. We haven't discovered reaction-less acceleration anyways, and this may help to illustrate why.
Collisions will require us to expand our gauge to include the repulsive phase of the wave just below gravity; gravity pulls things in, this phase pushes them away. Just as the attractive phase distorts the mass wave to be shorter on the near side, the repulsive phase distorts the mass wave to be longer, the opposite distortion, imparting acceleration.
If the two objects are approaching each other at speed, they'll get that much closer before the velocity is reversed, and have that much more time to accelerate away; greater velocity coming in means greater velocity going out, all things being equal.
Now, this acceleration has an interesting property that distinguishes it in relativity from Newtonian acceleration - it is relative.
Lorentz Contraction is the phenomenon in relativity in which objects contract along their axis of motion relative to the direction of motion. It isn't just the objects themselves, either; everything contracts, even the objects' own gravitational fields. By "contract", I mean that they get flatter, from an outside observer's perspective, along the direction of motion, as a proportion of lightspeed; an object somehow moving at the speed of light would appear as a perfectly flattened object.
At first, this might appear to be a product of subjectivity, rather than something that is "really" true. The trick in understanding Lorentz Contraction is to notice two things: First, that the gravitational field is also contracted, and second, that this means the density of spacetime is being distorted. Which is to say, the object isn't "really" flattening, but rather it is occupying a space that is bigger lengthwise (where length is parallel to the direction of motion) than the surrounding space.
And it isn't symmetric lengthwise; the space forward of the origin is slightly denser than the backwards section. Which is to say, space is denser forward of the ship than behind it. If you'll recall how gravity moves things, this is the same principle. We typically think of Lorentz Contraction as being a property of motion, but it is equally valid to think of motion as a property of Lorentz Contraction. And once you notice this, the notion of motion causing Lorentz Contraction becomes unnecessary; they're the same phenomenon, with Lorentz Contraction being the more basic.
Once we start thinking in these terms, we can start to redefine "motion" as a wave distortion in spacetime, or more specifically a wave distortion in the wave that is mass. This is a useful concept, but we aren't quite done yet; we need a mechanism of imparting this wave distortion on our mass-wave.
It's already been invented. I mean, I figured it out, but somebody figured it out long before I did, so I can't claim any credit. When objects are accelerating, they get an additional kind of distortion applied to them, which looks - not without accident - exactly like gravity. This phenomenon is called "Rindler Coordinates".
Now, I just made a massive leap, so let's take a quick step backwards. Let's try to figure out what gravity looks like to an object in it's influence, and what acceleration looks like.
Imagine two point masses in space. We'll simplify our mass wave to a single gauge, gravity. Point A, Point B, and us an observer at Point C. From Point C, the region between the two masses is denser. No surprise there. The interesting thing happens when we only examine the gravitational field of one of the two points, completely ignoring the other. Let's say we are looking at Point B.
Because the density between the two points is higher, this means that, from our outside perspective, the nice neat even distribution of spacial density, cleanly increasing as you approach Point B, no longer applies. It is flattened on the side facing Point A, because on that side, it has further to go, so weakens more quickly. More importantly, on its own, without any consideration of Point B's own gravity now, spacial density on the side facing Point B rises more quickly as we approach Point B. That is, even if we removed Point A entirely at this point, Point B's own gravitational field is already distorted such that it will pull -itself- towards Point A, through the simple expedient of moving through time and the side facing Point A being denser and thus more distance. (We are, for now, going to take motion through time as a given. It is the only motion we take for a given. Everything else arises from this single motion.)
Acceleration, and Rindler Coordinates, look exactly the same. Flattened more on the side oriented toward motion. This makes sense, if you think in terms of the speed of light; if you had reaction-less acceleration, that is, you weren't throwing particles out the back end of the ship, you'd get the same shape, just because the lightspeed "update" to the gravitational field takes time to propagate, such that, from an outside perspective, the forward region of fiethe gravitational field is always slightly behind, time-wise, the backward gravitational field. It takes time for light to reach things.
Of course, that is reaction-less acceleration; it presumes acceleration through some other mechanism causes this effect. I think this is slightly backwards; acceleration doesn't cause this shape, this shape is the phenomenon we call açceleration. This distortion is temporary, caused by the interaction of fields, such as gravity. But to explain this, we need to look at collisions, and more generally acceleration that isn't reaction-less. We haven't discovered reaction-less acceleration anyways, and this may help to illustrate why.
Collisions will require us to expand our gauge to include the repulsive phase of the wave just below gravity; gravity pulls things in, this phase pushes them away. Just as the attractive phase distorts the mass wave to be shorter on the near side, the repulsive phase distorts the mass wave to be longer, the opposite distortion, imparting acceleration.
If the two objects are approaching each other at speed, they'll get that much closer before the velocity is reversed, and have that much more time to accelerate away; greater velocity coming in means greater velocity going out, all things being equal.
Now, this acceleration has an interesting property that distinguishes it in relativity from Newtonian acceleration - it is relative.
A Crackpot's Vision of Future Physics 2: Time
It's time to talk about time.
First, some basic conventional physics background: First, time passes slower in a gravity well than in flat space. Second, time passes slower when in accelerating than in flat space. As we'll see, these two circumstances are very similar.
The major deviation from conventional physics here is more a simplification. In the Standard Model of physics, there is included a theory which attempts to explain electrical fields without electrical charge, called Kaluza-Klein. Kaluza-Klein, in effect, posits an additional dimension (that is, a direction) which has the unique property of being closed, which is to say, it is a loop. Travel some distance in that direction and you end up back where you started. You can think of it as a circle if that helps.
It is often called a "cylindrical dimension"; we can make sense of this by thinking in terms of a single dimension out of the three we move around in, and adding the closed dimension of Kaluza-Klein; the space created looks like a cylinder. You can move clockwise or counterclockwise, or backwards and forwards down the length; if you go far enough clockwise or counterclockwise, you arrive back at the same point.
The modification in this model is that "time", as we think of it in physics, is, in fact, this cylindrical dimension. Remember how I mentioned electrons would get more discussion in this chapter? Well, Kaluza-Klein explains electrical force as arising from rotation in this closed dimension, plus lots of math. People who understand the math say it works, so that's good enough for me. Electrons rotate around the circle in one direction - let's say counterclockwise - and protons rotate another - let's say clockwise. This both mathematically causes interesting things to happen - namely, electrical forces - and also helps explain both neutron behavior and electrons. Electrons, for their part, have two possible positions per orbit - this isn't merely opposing sides of the orbital sphere, as I described previously, but also opposing sides of the Kaluza-Klein closed dimension. Because all matter on the planet interacts with all other matter, we end up with two diametrically opposed positions in the circle, everything pushed or pulled by everything else into synchronization. And electrons have two possible positions; they can move from one position to the other, but it is energetically expensive, because they have to "push past" all the forces holding them in one of the two stable points. The energy is then freed up for other things to use, but it has to be there in the first place, so electrons resist being moved. Neutrons, meanwhile, are stuck; an electron on one end, a proton on the other, each trying to move in opposing directions, holding them mostly static in the Kaluza-Klein dimension, and thus electrically inert.
This isn't a stable configuration, however; the proton will eventually push the electron back out, unless there are enough other protons in the vicinity holding everything stable. (Sometimes, the electron might even be pushed in the opposite direction of it's desired direction, and become a positron.)
This is important because the Standard Model also has math that says antimatter goes backwards in time, and if you recall, I claimed electrons are antimatter. Once we have a closed dimension as time, suddenly this stops seeming so absurd, since forwards and backwards in time is the same thing.
Although maybe it still seems absurd, if you're having trouble getting past what the hell it means for time to be a closed dimension. It's a small closed dimension, too; a second would quite a few rotations worth of "distance".
Okay, that doesn't help. It may be more helpful if I say that "time" in this case may not be "time" as we humans think of it. See, we think of "time" as the thing that is full of history, and maybe even the future; this "time" isn't full of history or future, it is full of now. Think of it instead as a gear in a clock; the gear must rotate in order to cause the clock to move forward (or backward).
And if you are familiar with gears, you may be aware of an important quality of gears; if you have two gears, one large, and one small, attached to one another, the small gear rotates faster than the large gear. The circumferences - the outer edge of the gears - move at the same speed, but the smaller gear must make more rotations than the large gear, because it's circumference (the outer edge) is shorter.
This, I advance, is the cause of time in a gravity well, such as our planet, moving slower than time in flat space. Gravity, if you will recall, is just a higher density of space - or rather, a higher density of spacetime. Higher density in this case means more distance in less flat-space. In the case of our closed dimension, this translates to a greater distance, or a larger circle. The wave that is mass moves space around, and some of that moved space ends up in the Kaluza-Klein dimension.
Why does the number of rotations matter, in terms of perceived time by beings far larger in scale than we are considering? I'm not sure. This is one of the areas I am still thinking about. I have some ideas, but I don't understand them well enough to translate them into language yet.
None of this is, strictly speaking, all that interesting. Closed timelike curves are expected to arise around singularities; I think the Kaluza-Klein dimension is this closed curve, and the existence of it, if particles are all singularities or composites of singularities, isn't surprising.
First, some basic conventional physics background: First, time passes slower in a gravity well than in flat space. Second, time passes slower when in accelerating than in flat space. As we'll see, these two circumstances are very similar.
The major deviation from conventional physics here is more a simplification. In the Standard Model of physics, there is included a theory which attempts to explain electrical fields without electrical charge, called Kaluza-Klein. Kaluza-Klein, in effect, posits an additional dimension (that is, a direction) which has the unique property of being closed, which is to say, it is a loop. Travel some distance in that direction and you end up back where you started. You can think of it as a circle if that helps.
It is often called a "cylindrical dimension"; we can make sense of this by thinking in terms of a single dimension out of the three we move around in, and adding the closed dimension of Kaluza-Klein; the space created looks like a cylinder. You can move clockwise or counterclockwise, or backwards and forwards down the length; if you go far enough clockwise or counterclockwise, you arrive back at the same point.
The modification in this model is that "time", as we think of it in physics, is, in fact, this cylindrical dimension. Remember how I mentioned electrons would get more discussion in this chapter? Well, Kaluza-Klein explains electrical force as arising from rotation in this closed dimension, plus lots of math. People who understand the math say it works, so that's good enough for me. Electrons rotate around the circle in one direction - let's say counterclockwise - and protons rotate another - let's say clockwise. This both mathematically causes interesting things to happen - namely, electrical forces - and also helps explain both neutron behavior and electrons. Electrons, for their part, have two possible positions per orbit - this isn't merely opposing sides of the orbital sphere, as I described previously, but also opposing sides of the Kaluza-Klein closed dimension. Because all matter on the planet interacts with all other matter, we end up with two diametrically opposed positions in the circle, everything pushed or pulled by everything else into synchronization. And electrons have two possible positions; they can move from one position to the other, but it is energetically expensive, because they have to "push past" all the forces holding them in one of the two stable points. The energy is then freed up for other things to use, but it has to be there in the first place, so electrons resist being moved. Neutrons, meanwhile, are stuck; an electron on one end, a proton on the other, each trying to move in opposing directions, holding them mostly static in the Kaluza-Klein dimension, and thus electrically inert.
This isn't a stable configuration, however; the proton will eventually push the electron back out, unless there are enough other protons in the vicinity holding everything stable. (Sometimes, the electron might even be pushed in the opposite direction of it's desired direction, and become a positron.)
This is important because the Standard Model also has math that says antimatter goes backwards in time, and if you recall, I claimed electrons are antimatter. Once we have a closed dimension as time, suddenly this stops seeming so absurd, since forwards and backwards in time is the same thing.
Although maybe it still seems absurd, if you're having trouble getting past what the hell it means for time to be a closed dimension. It's a small closed dimension, too; a second would quite a few rotations worth of "distance".
Okay, that doesn't help. It may be more helpful if I say that "time" in this case may not be "time" as we humans think of it. See, we think of "time" as the thing that is full of history, and maybe even the future; this "time" isn't full of history or future, it is full of now. Think of it instead as a gear in a clock; the gear must rotate in order to cause the clock to move forward (or backward).
And if you are familiar with gears, you may be aware of an important quality of gears; if you have two gears, one large, and one small, attached to one another, the small gear rotates faster than the large gear. The circumferences - the outer edge of the gears - move at the same speed, but the smaller gear must make more rotations than the large gear, because it's circumference (the outer edge) is shorter.
This, I advance, is the cause of time in a gravity well, such as our planet, moving slower than time in flat space. Gravity, if you will recall, is just a higher density of space - or rather, a higher density of spacetime. Higher density in this case means more distance in less flat-space. In the case of our closed dimension, this translates to a greater distance, or a larger circle. The wave that is mass moves space around, and some of that moved space ends up in the Kaluza-Klein dimension.
Why does the number of rotations matter, in terms of perceived time by beings far larger in scale than we are considering? I'm not sure. This is one of the areas I am still thinking about. I have some ideas, but I don't understand them well enough to translate them into language yet.
None of this is, strictly speaking, all that interesting. Closed timelike curves are expected to arise around singularities; I think the Kaluza-Klein dimension is this closed curve, and the existence of it, if particles are all singularities or composites of singularities, isn't surprising.
A Crackpot's Vision of Future Physics 1: Mass
Mass is the simplest concept you will find here, yet terribly unintuitive. This framework has nothing we could call "mass", mass is an emergent property of the structure of the universe. Mass is very simple: It is a wave. It is a fairly simple wave; I suspect it might be simply sin(x)/x^2, where x is distance from the origin. In practical terms we will never see this simplified wave, and what we observe is closer to sin(ln(x))/x^2, which, if you don't have a graphing calculator handy, is a sin wave in which the wave gets wider and wider as you move away from the origin, and shorter and shorter as you get closer. Likewise, each wave (or rather, the period of the wave) gets shorter and shorter as you move away.
If mass is a wave, what is it a wave in? Space itself. We can call it curvature, but I think it is more helpful to think of it as the density of space itself, which is to say, it is a change in the distance between two points. It is very difficult to think in terms of distance itself being variable, because in order to make comparisons, you must make comparisons against an imaginary "flat" space, either in terms of curvature or density. This change in distance is why it looks more like sin(ln(x)) than sin(x), because x, that is, the distance, isn't flat; all the mass around it is simultaneously changing the distances involved.
All mass in this framework has this property, but not all mass is identical. There are, broadly, two kinds of relevant mass; composite mass, which is to say, something like a proton, made up of even more smaller pieces; and unified mass, like an electron, which is only a single wave. Composite mass has more interesting properties than unified mass, most specifically that the sum of the waves can have multiple points in a given stretch of space where the magnitude of the wave is at zero, or close to it. Why does this matter? Because zeroes are where interesting things happen.
When space is becoming denser as you get closer - that is, when the sine wave has a positive value - it becomes an apparently attractive force. When it is becoming less dense - when the sine wave has a negative value - it becomes an apparently repulsive force. This is the same principle as relativity - indeed, this model is a relativistic model, although I may often, for purposes of conveying ideas, talk as if it weren't. Let's talk about relativity for a moment.
In relativity, the variable density of spacetime is often called "curvature". I'm sure you have seen the bowl-shaped model of gravity; this often confuses people because it feels like the orientation of gravity changes from pulling towards mass, to pulling "down" in the bowl. Curvature, while mathematically elegant, is also misleading. The important thing about the bowl shape isn't that things fall down into the bowl, it is that the bowl has greater surface area than a flat circle. If you were to draw, say, triangles of a given size, you could draw more triangles on a bowl than a circle. So how does gravity in relativity really work? Well, to understand, think of a spaceship orbiting the planet.
Because space is getting denser as you get closer to the origin of the wave - the planet, in this case - we arrive at an interesting situation in which the near side of the spaceship is traversing more space than the far side, if it were to go in a straight line. Because the entire ship is traveling the same speed, this can't be the case - they must traverse the same amount of space. But the same amount of space, in this case, is "shorter", relative to an imaginary flat space, on one side than the other. This pulls the ship towards the planet.
But wait, you say, that makes sense when the spaceship is orbiting, but why do things fall when they aren't moving? The answer is that they are moving - through time. It is exactly the same behavior, it's just really hard for most people to think in four dimensional terms, so it is easier to think in terms of the spaceship orbiting, and then intuit that movement through time provokes a similar effect. We'll discuss this in more detail in the chapter on time.
Attractive forces and repulsive forces both use the same principle - the near side and far side exhibit different distances. This has some really interesting characteristics that I won't go into detail about here, but a thought experiment for you to consider is to think about what this means for, for example, the moon. What if the moon had its own rotation? What would happen, as the rotation is going "faster" on one side than the other?
Back to zeroes. There are, broadly, two types of zeroes to consider. When the sine wave is rising, and when it is falling. When it is rising, the nearer side of the zero is repulsive, and the far side is attractive. Everything will tend to fall into the zero boundary and probably oscillate slightly, but generally stay there. The other kind of zero, when the sine wave is falling, has an attractive force on the near side, and a repulsive force on the far. Things will tend to fall away from this zero boundary. One boundary is stable, one is unstable. I think of these as spheres of stable orbit, and spheres of unstable orbit. Really it is the stable orbits that are most interesting, because matter will tend to congregate there. Like electrons.
Now, without a discussion of time, I can't really give a full explanation of behavior here, and the explanation I have in mind for time is still later, but let's try anyways.
First, I am going to assume that electrons are antimatter. This gets into the time thing, for reasons physicists should already know, but we're going to tiptoe past that for now. All this means for our current purposes is that electrons are repulsive where protons are attractive, and vice versa - their sine waves are upside down. So at the distance from a proton at which electrons occupy a stable orbit, the electron's own wave is repulsive; the electron is in a stable orbit, but the proton is in an unstable one. Since the proton is so much bigger than the electron this doesn't end up matter much for the proton for our purposes here, but it matters a lot for other electrons - the orbit gets a lot less stable when occupied by an electron. Future electrons that happen to fall into our proton's orbit will be in a much less tenable position, at least as long as there is only one proton there. If there are two, we get an interesting new effect: There are now two stable orbits, because each proton's stable zone will be originating from a different point. Imagine two spheres which largely overlap; more, they reinforce, because the attractive force further out from the nucleus of the atom is that much stronger. Thus, we can now fit two electrons there. Technically, four; for now, imagine the extra two occupy opposing positions on the far sides of the spheres, which while not quite accurate, is good enough for now.
As we add more protons, the overlapping spheres get more and more complex, and eventually we reach a point where some of the spheres aren't quite stable without more electrons in the nearer spheres pushing outward. This corresponds to metals, in the periodic table. As we continue to add more protons this happens again, then again, each time corresponding to another group of metals.
What about neutrons? Well, this gets into time again. We'll discuss them in that chapter. For now, imagine an electron is suspended in a proton, which is a composite particle made up of smaller particles and thus has room. Again, not quite accurate, but close enough for now. This electron, with its wave canceling out part of the proton's wave, makes for an orbit that isn't strong enough to support an electron - and again, this isn't accurate, but is close enough to move forward.
I think you can see how a simple sin(ln(x))/x^2 could get us to a fairly simple atom at this point. Left unaddressed are the smaller particle which make up the proton; we'll discuss quarks, and leptons other than electrons, in a later chapter. For now, let's see what happens when we run the universe in fast forward in this model, because it is illustrative of how the model works on a larger scale.
I'm sure you have heard of black holes. They still exist in this model, and have largely the same behavior - with one critical difference. The attractive force that forms them, gravity, isn't alone. Farther out from the origin than gravity is another repulsive force, which we call a number of names; I'll call it the cosmological constant, because that's what Einstein called it.
In conventional physics, I believe the current consensus is that the cosmological constant is just that - a constant. In this model, however, it is a force like gravity, or rather, just another part of the same wave of which gravity is one part. Our repulsive force is really far away right now, but the stronger gravity gets, remembering that gravity is density of space, the closer, from an outside perspective, that repulsive force gets, because gravity increasing means there is a greater and greater distance between the origin and a given flat-space point. And the stronger gravity gets, because that repulsive force is part of the same wave, the stronger that force gets.
Eventually, if you fast forward through a black hole swallowing matter, the repulsive force gets so strong, nothing else can get in, not even light. We get a white hole. We'll discuss the mechanics of white holes later, but for now, it is enough to make one observation: The white hole looks like a really, really big electron (or a positron, it's regular matter counterpart).
Which is the pattern of the whole; made up of smaller pieces, all following a simple rule, sin(ln(x)). We'll come back to this idea a few times, fleshing out the universe a little more each time. But this should begin to give an idea of what the model looks like.
For now, I'm going to turn to a part of the model that I think works exceptionally well; the translation of energy into mass. This model only really has spacetime; there's no mass, there's no energy, just variations in the density of spacetime. Light is a variation in the density of spacetime; mass is a variation in the density of spacetime; even kinetic energy is a variation in the density of spacetime, which I'll get to in more detail in a bit. But let's use light as an example. Light is a moving variation in the density of spacetime, and specifically, it is an "update" to the mass wave itself, as its origin moves, and that change propagates away. Concentrate enough light in an area, and the increased (or decreased) density of spacetime creates a microscopic region of space which is, in effect, a tiny black (or white) hole, an area of spacetime in which light either can't enter - a white hole - or can't escape - a black hole. Either is self-sustaining, once created; a white hole, I think, will end up being antimatter at the scales we are considering, and a black hole will end up being matter. There's some stuff involving time here, relating to why zero-density white holes are probably self-sustaining as opposed to evaporating, but this is an area I won't return to here, because I don't understand my conceptualization well enough to translate it into language. I might understand it well enough to explain in a few years, assuming I don't drop it out of the model entirely.
In relativity, the variable density of spacetime is often called "curvature". I'm sure you have seen the bowl-shaped model of gravity; this often confuses people because it feels like the orientation of gravity changes from pulling towards mass, to pulling "down" in the bowl. Curvature, while mathematically elegant, is also misleading. The important thing about the bowl shape isn't that things fall down into the bowl, it is that the bowl has greater surface area than a flat circle. If you were to draw, say, triangles of a given size, you could draw more triangles on a bowl than a circle. So how does gravity in relativity really work? Well, to understand, think of a spaceship orbiting the planet.
Because space is getting denser as you get closer to the origin of the wave - the planet, in this case - we arrive at an interesting situation in which the near side of the spaceship is traversing more space than the far side, if it were to go in a straight line. Because the entire ship is traveling the same speed, this can't be the case - they must traverse the same amount of space. But the same amount of space, in this case, is "shorter", relative to an imaginary flat space, on one side than the other. This pulls the ship towards the planet.
But wait, you say, that makes sense when the spaceship is orbiting, but why do things fall when they aren't moving? The answer is that they are moving - through time. It is exactly the same behavior, it's just really hard for most people to think in four dimensional terms, so it is easier to think in terms of the spaceship orbiting, and then intuit that movement through time provokes a similar effect. We'll discuss this in more detail in the chapter on time.
Attractive forces and repulsive forces both use the same principle - the near side and far side exhibit different distances. This has some really interesting characteristics that I won't go into detail about here, but a thought experiment for you to consider is to think about what this means for, for example, the moon. What if the moon had its own rotation? What would happen, as the rotation is going "faster" on one side than the other?
Back to zeroes. There are, broadly, two types of zeroes to consider. When the sine wave is rising, and when it is falling. When it is rising, the nearer side of the zero is repulsive, and the far side is attractive. Everything will tend to fall into the zero boundary and probably oscillate slightly, but generally stay there. The other kind of zero, when the sine wave is falling, has an attractive force on the near side, and a repulsive force on the far. Things will tend to fall away from this zero boundary. One boundary is stable, one is unstable. I think of these as spheres of stable orbit, and spheres of unstable orbit. Really it is the stable orbits that are most interesting, because matter will tend to congregate there. Like electrons.
Now, without a discussion of time, I can't really give a full explanation of behavior here, and the explanation I have in mind for time is still later, but let's try anyways.
First, I am going to assume that electrons are antimatter. This gets into the time thing, for reasons physicists should already know, but we're going to tiptoe past that for now. All this means for our current purposes is that electrons are repulsive where protons are attractive, and vice versa - their sine waves are upside down. So at the distance from a proton at which electrons occupy a stable orbit, the electron's own wave is repulsive; the electron is in a stable orbit, but the proton is in an unstable one. Since the proton is so much bigger than the electron this doesn't end up matter much for the proton for our purposes here, but it matters a lot for other electrons - the orbit gets a lot less stable when occupied by an electron. Future electrons that happen to fall into our proton's orbit will be in a much less tenable position, at least as long as there is only one proton there. If there are two, we get an interesting new effect: There are now two stable orbits, because each proton's stable zone will be originating from a different point. Imagine two spheres which largely overlap; more, they reinforce, because the attractive force further out from the nucleus of the atom is that much stronger. Thus, we can now fit two electrons there. Technically, four; for now, imagine the extra two occupy opposing positions on the far sides of the spheres, which while not quite accurate, is good enough for now.
As we add more protons, the overlapping spheres get more and more complex, and eventually we reach a point where some of the spheres aren't quite stable without more electrons in the nearer spheres pushing outward. This corresponds to metals, in the periodic table. As we continue to add more protons this happens again, then again, each time corresponding to another group of metals.
What about neutrons? Well, this gets into time again. We'll discuss them in that chapter. For now, imagine an electron is suspended in a proton, which is a composite particle made up of smaller particles and thus has room. Again, not quite accurate, but close enough for now. This electron, with its wave canceling out part of the proton's wave, makes for an orbit that isn't strong enough to support an electron - and again, this isn't accurate, but is close enough to move forward.
I think you can see how a simple sin(ln(x))/x^2 could get us to a fairly simple atom at this point. Left unaddressed are the smaller particle which make up the proton; we'll discuss quarks, and leptons other than electrons, in a later chapter. For now, let's see what happens when we run the universe in fast forward in this model, because it is illustrative of how the model works on a larger scale.
I'm sure you have heard of black holes. They still exist in this model, and have largely the same behavior - with one critical difference. The attractive force that forms them, gravity, isn't alone. Farther out from the origin than gravity is another repulsive force, which we call a number of names; I'll call it the cosmological constant, because that's what Einstein called it.
In conventional physics, I believe the current consensus is that the cosmological constant is just that - a constant. In this model, however, it is a force like gravity, or rather, just another part of the same wave of which gravity is one part. Our repulsive force is really far away right now, but the stronger gravity gets, remembering that gravity is density of space, the closer, from an outside perspective, that repulsive force gets, because gravity increasing means there is a greater and greater distance between the origin and a given flat-space point. And the stronger gravity gets, because that repulsive force is part of the same wave, the stronger that force gets.
Eventually, if you fast forward through a black hole swallowing matter, the repulsive force gets so strong, nothing else can get in, not even light. We get a white hole. We'll discuss the mechanics of white holes later, but for now, it is enough to make one observation: The white hole looks like a really, really big electron (or a positron, it's regular matter counterpart).
Which is the pattern of the whole; made up of smaller pieces, all following a simple rule, sin(ln(x)). We'll come back to this idea a few times, fleshing out the universe a little more each time. But this should begin to give an idea of what the model looks like.
For now, I'm going to turn to a part of the model that I think works exceptionally well; the translation of energy into mass. This model only really has spacetime; there's no mass, there's no energy, just variations in the density of spacetime. Light is a variation in the density of spacetime; mass is a variation in the density of spacetime; even kinetic energy is a variation in the density of spacetime, which I'll get to in more detail in a bit. But let's use light as an example. Light is a moving variation in the density of spacetime, and specifically, it is an "update" to the mass wave itself, as its origin moves, and that change propagates away. Concentrate enough light in an area, and the increased (or decreased) density of spacetime creates a microscopic region of space which is, in effect, a tiny black (or white) hole, an area of spacetime in which light either can't enter - a white hole - or can't escape - a black hole. Either is self-sustaining, once created; a white hole, I think, will end up being antimatter at the scales we are considering, and a black hole will end up being matter. There's some stuff involving time here, relating to why zero-density white holes are probably self-sustaining as opposed to evaporating, but this is an area I won't return to here, because I don't understand my conceptualization well enough to translate it into language. I might understand it well enough to explain in a few years, assuming I don't drop it out of the model entirely.
Foreword
This is the beginning of a comprehensive update to the model reflecting new things I've learned, and trying to explain everything. This may get edited heavily, or I may just post new versions as I develop them, I am undecided.
What we call physics is really two distinct practices; physicists themselves only think of one as "physics", I think in large part because the other part has fallen so far behind. There is mathematics - what physicists think of as physics - and philosophy.
What we call physics is really two distinct practices; physicists themselves only think of one as "physics", I think in large part because the other part has fallen so far behind. There is mathematics - what physicists think of as physics - and philosophy.
It is pretty common today for physicists to have no idea what the math means. Partly this is because the math had gotten insanely complex, and partly because the math is more useful and so gets more attention - and also, the philosophy is hard. If you've ever taken basic physics coursework, you may be familiar with the "click", the moment the math suddenly makes sense. As the mathematics get increasingly complex, the click gets harder and harder to achieve. Hell, it wasn't until I started studying quantum physics that I understood what a partial derivative meant, and I worked with them in my coursework in college without issue - I just didn't worry about what it meant, because I didn't need to know.
Physics thus has developed a near-spiritual nature; understanding is generally limited to a superficial high-level view, or to a narrow-focused low-level view (mathematics), and the high level and the low level view don't have much bridging them. I've seen PhD physicists who get basic relativity wrong (for example, by thinking velocity, rather than acceleration, is responsible for twin paradox situations); this isn't because they don't know the math, and if you point out that they need to integrate the equation they're using they'll figure out what you meant pretty quickly, but rather because the explanations attached to the mathematics frequently get key details wrong in subtle ways.
Physics thus has developed a near-spiritual nature; understanding is generally limited to a superficial high-level view, or to a narrow-focused low-level view (mathematics), and the high level and the low level view don't have much bridging them. I've seen PhD physicists who get basic relativity wrong (for example, by thinking velocity, rather than acceleration, is responsible for twin paradox situations); this isn't because they don't know the math, and if you point out that they need to integrate the equation they're using they'll figure out what you meant pretty quickly, but rather because the explanations attached to the mathematics frequently get key details wrong in subtle ways.
I think this is pretty much the routine attitude in physics today, fostered by an attitude that seems increasingly prevalent that the universe doesn't work by human logic, and it may be unreasonable to expect humans to be able to make sense of it.
I think this view is wrong.
But the philosophy remains a hard problem. This is yet another attempt to create a philosophy of physics. It is a crackpot's attempt - it is heavily influenced by my own, to put it mildly, unconventional ideas about physics. I have a grand unified field theory here - sin(ln(x))/x^2. It's probably wrong, but the point isn't actually to get it right, the point is to get it right enough to convey a view of physics, a philosophy of physics, that I think may be helpful.
The Copernican Principle says, basically, that we aren't special, and the place we occupy in the universe isn't special either. Modern physics says we are, or rather that we do - specifically, it holds that the scale, the size, we happen to occupy holds a special place in the universe, exactly some multiple of the fundamental, special size.
This is a natural outcome of quantum physics, the most successful theory of physics since the last one. At its core is the belief in the quantum, the fundamental unit of energy. So, being arrogant enough to think I could figure out something better, I started on it. It's been a hobby since I was seven or eight, although until a couple of years ago, the ideas had too many problems. Over the last couple of years, I have had surprising-to-me success in resolving these problems, and I suddenly have a philosophical framework for physics that appears to maybe work.
Why is this important? I mean, strictly speaking, the mathematics works. It works really, really well, even if we have no idea what any of it means, and even without knowing what any of it means we've made surprising progress over the last century. It staggers me that we continue to make progress. So why does it matter?
I think it matters because philosophy gives direction to mathematics. A century ago, people made progress in physics through thought experiments; you could advance the field just by thinking carefully about it; Einstein invented the concepts of relativity, and only then set about inventing the math to describe it. That doesn't seem to be the case anymore.
So here is a radically different perspective on physics, diverging, like most crackpots, around the time of Einstein. Unlike many, I think he was right. I just don't think he took his ideas quite far enough. Specifically, I think relativity extends to scale, which is to say, large objects have the same relationships to each other as small objects have to each other. I think physics can be described purely in terms of size.
I'd say the Copernican Principle is the zeroeth law of physics. We aren't special. We don't occupy a special place in the universe. Quantum physics gives us a special place. There was an alternative - many alternatives, actually - at the time, being developed by Johannes Rhydberg. Modern physicists are probably most familiar with Rhydberg through the concept of Rhydberg Atoms, which are atoms which obey the rules Rhydberg was attempting to create; as far as I can tell, he gave up on the idea when quantum physics solved the problem. In an alternate universe in which he finished his work, I think physics would look very, very different.
The problem he was trying to solve was that of light emission; specifically, the previous models didn't describe why electrons in a given atom in a given configuration only emitted certain wavelengths of light, and suggested, in fact, that they should emit all the wavelengths of light, which also happened to be an infinite amount of energy. It was, at the time, a serious problem.
Quantum Physics solved the problem by asserting that energy had a minimum value, and that it only came in multiples of this value. Mathematically, this worked out really well, and the rest is history.
Rhydberg's solution was to suggest that light was emitted in the frequency of the resonant frequency of the electron in that configuration. As far as his work went, it worked. Unfortunately, the mathematics was really, really hard, and the problem was solved in quantum physics shortly after he had figured it out for the simplest case, a hydrogen atom.
I think he was right. And also, I think quantum physics, or at least the mathematics of it, were right. There's a concept in mathematics called isomorphy; it means, approximately, that two things are at a fundamental level the same, even if they are structured differently. I think quantum physics is isomorphic to Rhydberg's unfinished work. I'll talk more about this later, like much of this.
Fundamentally, the problem in this enterprise is that a working philosophy that addresses both relativistic and quantum scales can't just tinker around the edges; either the macroscopic philosophy has to give in it's entirety, or the microscopic, or both. I think microscopic philosophy is the one that has to give the most, and I have to not only conceptualize what, for example, a quark is in this new model, but also to be able to explain that to you, the reader. This is, given the totality of the extent to which everything is getting subtly redefined, a problem, not just in terms of explaining it, but organizing the explanation to minimize things that are obviously wrong with the explanation at each step. An explanation of neutrons requires a new understanding of what "time" in physics refers to; do I start with time? Time in turn requires an understanding of what motion now means, and motion, in turn, requires understanding what mass now means.
At each step, there are lots of dependencies left hanging, each of which is going to be a big "But what about..." for you, the reader. I am acutely aware I don't get infinite lassitude on this point; if at any point I leave too many things hanging, unexplained, I'm going to lose your interest. So the chapter on mass can't be complete; I can't really explain neutrons purely in terms of the mechanics described there. Every chapter is going to be incomplete, even the last one; I don't know everything. My ultimate goal isn't to explain the universe, to answer every question, because I don't know the questions to answer. My goal is to provide a framework; ultimately, if you're not comfortable thinking in terms of the framework, if you aren't willing to try to figure out an answer for yourself, none of this will be of any use to you.
Discussing my crackpot grand unified field equation - which basically amounts toreplacing G in General Relativity with sin(ln(x)), although I think since it is a tensor it is more complicated than I am making it sound - the exact equation doesn't matter too much. This framework is designed around the requirements I used to develop that equation, the equation is entirely secondary to those requirements.
The requirements aren't complicated. First, the equation must be relativistic - in particular, it must pertain to the shape of spacetime, rather than to additional phenomena. Second, it must describe a recursive or fractal geometry. And third, the geometry should reflect an alternating series of attractive and repulsive phases.
There are reasons for all three requirements, of course. The first is necessary for compatibility with relativity itself. The second is necessary to comply with Copernican non-specialness. And the third requirement is necessary to ensure that the spacetime metric is conserved - the conservation law which I believe exists beneath all conservation laws, whose symmetry is, I believe, on the axis of scale itself. As above, so below.
My earliest version, as a teenager, was literally just an infinite nested series of forces with alternating polarity and an ever-increasing rate of decay. Later I realized this simplified to a sine wave with a decaying period and amplitude, and had something like sin(1/x^2), which works great, but only for x<1.
Sin(ln(x)) has stuck with me since I discovered it, in large part owing to it's possibly coincidental characteristic shape looking an awful lot like the characteristic shape of the strong (and weak) nuclear force, the force I think we, from our perspective, have the most insight into, seeing the fullest range of. Gravity is a good second contender, but I don't have the skill to tease out what the shape of gravity would look like without dark matter being used to fill in the apparent holes.
The important thing here isn't the equation, it is the requirements, because they, not the equation, are what this approach to understanding physics is based upon. Everything here works regardless of the actual equation, so long as the requirements themselves are a valid description of the universe. I think they are.
I'd say the Copernican Principle is the zeroeth law of physics. We aren't special. We don't occupy a special place in the universe. Quantum physics gives us a special place. There was an alternative - many alternatives, actually - at the time, being developed by Johannes Rhydberg. Modern physicists are probably most familiar with Rhydberg through the concept of Rhydberg Atoms, which are atoms which obey the rules Rhydberg was attempting to create; as far as I can tell, he gave up on the idea when quantum physics solved the problem. In an alternate universe in which he finished his work, I think physics would look very, very different.
The problem he was trying to solve was that of light emission; specifically, the previous models didn't describe why electrons in a given atom in a given configuration only emitted certain wavelengths of light, and suggested, in fact, that they should emit all the wavelengths of light, which also happened to be an infinite amount of energy. It was, at the time, a serious problem.
Quantum Physics solved the problem by asserting that energy had a minimum value, and that it only came in multiples of this value. Mathematically, this worked out really well, and the rest is history.
Rhydberg's solution was to suggest that light was emitted in the frequency of the resonant frequency of the electron in that configuration. As far as his work went, it worked. Unfortunately, the mathematics was really, really hard, and the problem was solved in quantum physics shortly after he had figured it out for the simplest case, a hydrogen atom.
I think he was right. And also, I think quantum physics, or at least the mathematics of it, were right. There's a concept in mathematics called isomorphy; it means, approximately, that two things are at a fundamental level the same, even if they are structured differently. I think quantum physics is isomorphic to Rhydberg's unfinished work. I'll talk more about this later, like much of this.
Fundamentally, the problem in this enterprise is that a working philosophy that addresses both relativistic and quantum scales can't just tinker around the edges; either the macroscopic philosophy has to give in it's entirety, or the microscopic, or both. I think microscopic philosophy is the one that has to give the most, and I have to not only conceptualize what, for example, a quark is in this new model, but also to be able to explain that to you, the reader. This is, given the totality of the extent to which everything is getting subtly redefined, a problem, not just in terms of explaining it, but organizing the explanation to minimize things that are obviously wrong with the explanation at each step. An explanation of neutrons requires a new understanding of what "time" in physics refers to; do I start with time? Time in turn requires an understanding of what motion now means, and motion, in turn, requires understanding what mass now means.
At each step, there are lots of dependencies left hanging, each of which is going to be a big "But what about..." for you, the reader. I am acutely aware I don't get infinite lassitude on this point; if at any point I leave too many things hanging, unexplained, I'm going to lose your interest. So the chapter on mass can't be complete; I can't really explain neutrons purely in terms of the mechanics described there. Every chapter is going to be incomplete, even the last one; I don't know everything. My ultimate goal isn't to explain the universe, to answer every question, because I don't know the questions to answer. My goal is to provide a framework; ultimately, if you're not comfortable thinking in terms of the framework, if you aren't willing to try to figure out an answer for yourself, none of this will be of any use to you.
Discussing my crackpot grand unified field equation - which basically amounts toreplacing G in General Relativity with sin(ln(x)), although I think since it is a tensor it is more complicated than I am making it sound - the exact equation doesn't matter too much. This framework is designed around the requirements I used to develop that equation, the equation is entirely secondary to those requirements.
The requirements aren't complicated. First, the equation must be relativistic - in particular, it must pertain to the shape of spacetime, rather than to additional phenomena. Second, it must describe a recursive or fractal geometry. And third, the geometry should reflect an alternating series of attractive and repulsive phases.
There are reasons for all three requirements, of course. The first is necessary for compatibility with relativity itself. The second is necessary to comply with Copernican non-specialness. And the third requirement is necessary to ensure that the spacetime metric is conserved - the conservation law which I believe exists beneath all conservation laws, whose symmetry is, I believe, on the axis of scale itself. As above, so below.
My earliest version, as a teenager, was literally just an infinite nested series of forces with alternating polarity and an ever-increasing rate of decay. Later I realized this simplified to a sine wave with a decaying period and amplitude, and had something like sin(1/x^2), which works great, but only for x<1.
Sin(ln(x)) has stuck with me since I discovered it, in large part owing to it's possibly coincidental characteristic shape looking an awful lot like the characteristic shape of the strong (and weak) nuclear force, the force I think we, from our perspective, have the most insight into, seeing the fullest range of. Gravity is a good second contender, but I don't have the skill to tease out what the shape of gravity would look like without dark matter being used to fill in the apparent holes.
The important thing here isn't the equation, it is the requirements, because they, not the equation, are what this approach to understanding physics is based upon. Everything here works regardless of the actual equation, so long as the requirements themselves are a valid description of the universe. I think they are.
Cosmic Expansion
Noting my own thoughts here more than anything else.
I'm uncertain if cosmic expansion is necessary in this model, as that is slightly beyond the areas of the model I have tried to comprehend, but it is definitely possible.
We have a repulsive phase of the force on the scale of galaxies, the next phase out from gravity. Now, I can reconcile this with relativity to enable cosmic expansion - from any given galaxy, on average every other galaxy within sight is experiencing a repulsive force, which means we should expect them to move away. Conservation of spacetime, however, needs some additional thought; on the one hand, we are converting potential energy into spacetime, which should be allowed. On the other hand, we need an actual explanation of what is happening, as opposed to a simple "It's allowed".
One way of viewing the problem is that, at a basic level, a repulsive force is a reduction in the density of spacetime, so as galaxies move apart as a result of these density variances, the intermediate spaces get subjected to less of this lessening of density, resulting in more proper distance over a given Minkowski distance. That presumes, of course, that the sine wave has passed its apex. But again, conservation of spacetime - where is the actual spacetime coming from? Somewhere has to lose spacetime in order for somewhere else to gain it.
One possibility is that it arises from the nature of motion itself - that velocity, representing a bend in spacetime, itself represents some portion of spacetime, and that velocity, or energy, is being lost somewhere. Given that galaxies are accelerating away from each other, this doesn't seem correct. But let's try handling velocity as subtraction, rather than addition - that is, suppose we define velocity as a -loss- of spacetime along a vector. We can get the same results - the important quality of velocity in this model is its assymetry. We can treat velocity as either a scrap of added spacetime, or a scrap of lost spacetime in an opposing orientation, and we arrive at the same geometric shape, the same geometry capable of giving rise to motion.
So maybe the extra space between galaxies is coming from the galaxies themselves. One way to think about this is that Lorentz Contraction from the galaxies moving away from each other is causing a contraction of the repulsive phase of the mass wave, resulting in increased spacetime density (or increased curvature, in traditional nomenclature) at intergalactic scales; I think this is slightly misleading, because it arises from and encourages a tendency to think purely in terms of mass wave densities as describing distance. But I don't think it is necessarily wrong, either; the difference only really matters, as far as a can tell, in a finite universe with a finite amount of matter. In an infinite universe, I don't expect there to be any difference between these two perspectives, and the one I view as misleading and incorrect is sort of a this-model version of string theory.
I'm uncertain if cosmic expansion is necessary in this model, as that is slightly beyond the areas of the model I have tried to comprehend, but it is definitely possible.
We have a repulsive phase of the force on the scale of galaxies, the next phase out from gravity. Now, I can reconcile this with relativity to enable cosmic expansion - from any given galaxy, on average every other galaxy within sight is experiencing a repulsive force, which means we should expect them to move away. Conservation of spacetime, however, needs some additional thought; on the one hand, we are converting potential energy into spacetime, which should be allowed. On the other hand, we need an actual explanation of what is happening, as opposed to a simple "It's allowed".
One way of viewing the problem is that, at a basic level, a repulsive force is a reduction in the density of spacetime, so as galaxies move apart as a result of these density variances, the intermediate spaces get subjected to less of this lessening of density, resulting in more proper distance over a given Minkowski distance. That presumes, of course, that the sine wave has passed its apex. But again, conservation of spacetime - where is the actual spacetime coming from? Somewhere has to lose spacetime in order for somewhere else to gain it.
One possibility is that it arises from the nature of motion itself - that velocity, representing a bend in spacetime, itself represents some portion of spacetime, and that velocity, or energy, is being lost somewhere. Given that galaxies are accelerating away from each other, this doesn't seem correct. But let's try handling velocity as subtraction, rather than addition - that is, suppose we define velocity as a -loss- of spacetime along a vector. We can get the same results - the important quality of velocity in this model is its assymetry. We can treat velocity as either a scrap of added spacetime, or a scrap of lost spacetime in an opposing orientation, and we arrive at the same geometric shape, the same geometry capable of giving rise to motion.
So maybe the extra space between galaxies is coming from the galaxies themselves. One way to think about this is that Lorentz Contraction from the galaxies moving away from each other is causing a contraction of the repulsive phase of the mass wave, resulting in increased spacetime density (or increased curvature, in traditional nomenclature) at intergalactic scales; I think this is slightly misleading, because it arises from and encourages a tendency to think purely in terms of mass wave densities as describing distance. But I don't think it is necessarily wrong, either; the difference only really matters, as far as a can tell, in a finite universe with a finite amount of matter. In an infinite universe, I don't expect there to be any difference between these two perspectives, and the one I view as misleading and incorrect is sort of a this-model version of string theory.
Monday, March 4, 2019
Minor Note
As it transpires, "Acceleration and gravity look alike" isn't an interesting result, and is part of relativity already (where it is variably called Rindler Motion or Rindler Coordinates).
This is both frustrating and reassuring - frustrating, because it was one of the things I thought could be relatively easily tested coming out of the model, and reassuring, because it mean my mental models aren't entirely off-base. The equivalence of motion and Lorentz Contraction is on slightly firmer ground. (Lorentz Contraction isn't symmetric about the axis of motion, and spacetime is slightly denser forward of the object in motion, meaning movement along the Kaluza-Klein cylindrical dimension is sufficient to produce linear motion. If we posit that Kaluza-Klein is time, we arrive at a description of reality without "motion", only geometry.)
I've also concluded that the next step for me to figure out validity, given the absence of relatively inexpensive tests, will need to be mathematical. If I am correct, the hierarchy problem should go away if you assume that all fields exhibit general relativistic behavior with regard to spacial distortions, which is to say, given that spacetime curvature means that a gravity well has greater interior than exterior volume, the same very may well be true of smaller particles, inwhichcase the hierarchy problem may entirely arise from the fact that we're measuring a space that is substantially bigger on the inside than the outside from which we are measuring it: There may be no hierarchy problem, at all.
I'll be focusing on tensors as my next area of study, for lack of a clear idea what I need to actually study in order to make this determination.
This is both frustrating and reassuring - frustrating, because it was one of the things I thought could be relatively easily tested coming out of the model, and reassuring, because it mean my mental models aren't entirely off-base. The equivalence of motion and Lorentz Contraction is on slightly firmer ground. (Lorentz Contraction isn't symmetric about the axis of motion, and spacetime is slightly denser forward of the object in motion, meaning movement along the Kaluza-Klein cylindrical dimension is sufficient to produce linear motion. If we posit that Kaluza-Klein is time, we arrive at a description of reality without "motion", only geometry.)
I've also concluded that the next step for me to figure out validity, given the absence of relatively inexpensive tests, will need to be mathematical. If I am correct, the hierarchy problem should go away if you assume that all fields exhibit general relativistic behavior with regard to spacial distortions, which is to say, given that spacetime curvature means that a gravity well has greater interior than exterior volume, the same very may well be true of smaller particles, inwhichcase the hierarchy problem may entirely arise from the fact that we're measuring a space that is substantially bigger on the inside than the outside from which we are measuring it: There may be no hierarchy problem, at all.
I'll be focusing on tensors as my next area of study, for lack of a clear idea what I need to actually study in order to make this determination.
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