I am now. This is exactly the topic pointer I needed to formalize the relationships between things. And "oh god" at the degree to which this is generalized, but also "oh god I actually needed those generalizations".
I'm uncertain about the connection to quantum/observer effects; I'm currently digesting some new information I got from somebody, who I drew some pictures for, who recognized the topology of what I was calling a "negative dimension" - it's, uh. A Riemann Sphere.
Which is fascinating, in no small part because of the entirely ass-backwards way I reasoned into it. But it's a complex dimension, not a negative dimension - although, given how close I got to a correct description, given my near-complete lack of knowledge of the topic, reasoning entirely based on how I would expect a "negative dimension" to behave and not employing any reasoning about complex dimensions, I think maybe there's something there.
(The features of a Riemann sphere don't -quite- match my intuitions - in particular rotation doesn't quite behave the way I expect. Maybe; I'm digesting, and now digging into different areas of mathematics. One notable thing that came up, as I dug into the Riemann Sphere, is that there is a three-dimensional metric that can be defined on it. And it's. Uh. A metric that is employed in both relativity and quantum mechanics. Which makes me slightly annoyed, because, if my crackpottery is correct - somebody else really should have noticed this already, and saved me the twenty-odd years I've spent messing about with it. Well, maybe the last ten years of messing about when I noticed the idea might not be entirely nonsensical; the first ten years were just for fun)
For entropy, kind of broadly, yes, I think that's in the right ballpark. I have ... complicated opinions of entropy, but in a general way, I see it as an expression of the idea that "Physical laws apply the way physical laws apply" - if we reversed all the laws of physics, threw everything into reverse, and what we call entropy decreased instead of increased - we'd still face exactly the same issues we have today: Only so much work could be extracted from this a given system before you -ran out- of entropy.
So, in a nontrivial sense, entropy is kind of a measurement of time itself. The relationship to information isn't accidental, but that gets into "Complicated" territory, where I struggle to explain things, but ... I'll try anyways.
Entropy is maximally compressed information, encoded into whatever it can embed itself into, generally configurations of matter or energy. Time flows outward from particles, in my crackpot model - the history of a particle is moving away from it at the speed of light. This is deeply related to curvature, again in my crackpot model (actually just assume everything I saw is crackpot for the purposes of this comment) - because curvature is "created" by this flow of a particle's history. (This statement is somewhat nonsensical, the causal relationship is bidirectional, but for the purposes of explanation let's go with that). Information is the compactification of history, created by - well, negative curvature. Gravity. Information gets caught along with everything else. In a sense.
(What exactly happens in positive-curvature areas - where the force is repulsive instead of attractive - is somewhat interesting here. I don't think it's quite correct to say entropy is reduced in these areas - entropy still approaches a maximum ... but that maximum is much lower for the vacuum state. And the information still compacts somewhere, so a closed system under positive curvature doesn't behave much differently with respect to entropy than one under negative curvature. But an open system may exhibit entropy-reducing effects locally.)
Alas, I have not seen Johnny Mnemonic, so I have no idea what those things mean!
It's something called the Fubini-Study metric, which never quite rose to my attention before. It's now on my short list of "things I need to understand"; right now I don't, really.
To describe my understanding of the Fubini-Study metric at current - which to be clear is still developing, so everything here may be wrong! - it maps onto the Riemann sphere in a somewhat complex (no pun intended) way. The easiest mapping, in polar coordinate terms (one distance dimension and two rotation dimensions), is the distance dimension, which is made up of a vertical line connecting the poles. The two rotation dimensions are somewhat harder; considering the bundle of vertical lines, their position with respect to the surface of the sphere is given by two numbers: A complex coordinate and a real coordinate. The complex coordinate kind of maps onto one rotation, and the real coordinate kind of maps onto the other.
That's, I think, an unnecessary specification, because the mapping is in a sense more arbitrary than that. Trying to reason more abstractly: Since there are two complex coordinates on the sphere, and each has a real and a complex component (so four total coordinates), normally we'd expect eight degrees of freedom in principle. However, two of these are redundant, because the topology of the sphere means that certain linear transformations are equivalent, so we end up with six degrees of freedom instead. Three dimensions. How we map the dimensions to the degrees of freedom is somewhat arbitrary, but treating the poles as a real dimension, and separating out the rotation into a real and complex part, is more intuitive.
I'm pondering the Riemann Sphere, and while it's *really really* close to the thing in my head, the thing in my head might, maybe, have an important divergence - the poles, in the thing in my head, are the same point. Topologically, the thing in my head is a horn torus, not a sphere. I'm unsure if this matters - from a certain perspective, the critical topological feature this creates is already kind of conveyed in the complex components of a rotation, which, if I understand the behavior correctly, create a requirement that any rotation about the Riemann Sphere requires a 720 degree rotation to arrive at the same point; it's, in a certain sense, a spinor. So the topology in my head treats the vertical stripes as real, and the rotation as complex, and recovers the spinor behavior with respect to the real component by making the topology more complex. (This really only matters when it comes to understanding particle spin, however, and maybe there's a more elegant way of handling this I just don't grok yet.)
There's another important way that my topology potentially differs, but it's possibly kind of arbitrary: I don't think a straight line between the poles is a simple vertical stripe, but a kind of spherical spiral, whose characteristic behavior is that each rotation about the sphere covers an equal measure. So, supposing that measure is 1 for the sake of argument, the first rotation of the spiral moves, measuring the distance from the pole, from 0 to 1, the second from 1 to 2. Because the density of the points is something like hyperbolic, this means the spiral winds loosely at the 0 pole, and tightens to infinity as you approach the infinity pole. (But because "0" and "infinity" are poorly defined/arbitrary, there are two potential mappings of any given spiral to the sphere, depending on how you assign the poles. I think this is important, and gives rise to the characteristic relation between time and distance-in-polar-coordinates - as I think this gives rise to two dimensions, but only two degrees of freedom, so the dimensions are not distinct.)
This is more or less how I backwards-reasoned my way into something like the Riemann Sphere in the first place, by reasoning about the topology this spiral necessarily crosses. I hadn't actually made the jump from "Single curve" to "Bundle of curves" necessary to notice that this maps the surface of a sphere, however, and conceptualized it as a helix. (You don't need to glue the ends of the cylinder-space this crosses unless you have a bundle; all the other paths through the cylinder are basically kind of theoretical, so the fact that all the points on either end of the cylinder are 0 or infinity, or some weird superposition of both, isn't meaningful information.)
There would definitely be some advantages in something which measures "external" length, as opposed to "internal" length - but it might actually be more useful to have both, and to examine the difference between them. Depends on what "external" and "internal" mean, though, and I'm kind of guessing.
If they mean what I think they mean - something akin to the distinction between the difference in distance between flat and curved space - I typically think about curved space in terms of an imaginary flat space; how the flat space maps to the curved space. And the "negative" dimension idea came from examining what happens to a flat space superimposed on, and interpreted through, curved space, as that flat space passes through a singularity. So I kind of think in both.
But I'm uncertain if that is a correct interpretation / analogy of "internal" versus "external".
I am now. This is exactly the topic pointer I needed to formalize the relationships between things. And "oh god" at the degree to which this is generalized, but also "oh god I actually needed those generalizations".
Thank you.
I'm uncertain about the connection to quantum/observer effects; I'm currently digesting some new information I got from somebody, who I drew some pictures for, who recognized the topology of what I was calling a "negative dimension" - it's, uh. A Riemann Sphere.
Which is fascinating, in no small part because of the entirely ass-backwards way I reasoned into it. But it's a complex dimension, not a negative dimension - although, given how close I got to a correct description, given my near-complete lack of knowledge of the topic, reasoning entirely based on how I would expect a "negative dimension" to behave and not employing any reasoning about complex dimensions, I think maybe there's something there.
(The features of a Riemann sphere don't -quite- match my intuitions - in particular rotation doesn't quite behave the way I expect. Maybe; I'm digesting, and now digging into different areas of mathematics. One notable thing that came up, as I dug into the Riemann Sphere, is that there is a three-dimensional metric that can be defined on it. And it's. Uh. A metric that is employed in both relativity and quantum mechanics. Which makes me slightly annoyed, because, if my crackpottery is correct - somebody else really should have noticed this already, and saved me the twenty-odd years I've spent messing about with it. Well, maybe the last ten years of messing about when I noticed the idea might not be entirely nonsensical; the first ten years were just for fun)
For entropy, kind of broadly, yes, I think that's in the right ballpark. I have ... complicated opinions of entropy, but in a general way, I see it as an expression of the idea that "Physical laws apply the way physical laws apply" - if we reversed all the laws of physics, threw everything into reverse, and what we call entropy decreased instead of increased - we'd still face exactly the same issues we have today: Only so much work could be extracted from this a given system before you -ran out- of entropy.
So, in a nontrivial sense, entropy is kind of a measurement of time itself. The relationship to information isn't accidental, but that gets into "Complicated" territory, where I struggle to explain things, but ... I'll try anyways.
Entropy is maximally compressed information, encoded into whatever it can embed itself into, generally configurations of matter or energy. Time flows outward from particles, in my crackpot model - the history of a particle is moving away from it at the speed of light. This is deeply related to curvature, again in my crackpot model (actually just assume everything I saw is crackpot for the purposes of this comment) - because curvature is "created" by this flow of a particle's history. (This statement is somewhat nonsensical, the causal relationship is bidirectional, but for the purposes of explanation let's go with that). Information is the compactification of history, created by - well, negative curvature. Gravity. Information gets caught along with everything else. In a sense.
(What exactly happens in positive-curvature areas - where the force is repulsive instead of attractive - is somewhat interesting here. I don't think it's quite correct to say entropy is reduced in these areas - entropy still approaches a maximum ... but that maximum is much lower for the vacuum state. And the information still compacts somewhere, so a closed system under positive curvature doesn't behave much differently with respect to entropy than one under negative curvature. But an open system may exhibit entropy-reducing effects locally.)
Alas, I have not seen Johnny Mnemonic, so I have no idea what those things mean!
It's something called the Fubini-Study metric, which never quite rose to my attention before. It's now on my short list of "things I need to understand"; right now I don't, really.
To describe my understanding of the Fubini-Study metric at current - which to be clear is still developing, so everything here may be wrong! - it maps onto the Riemann sphere in a somewhat complex (no pun intended) way. The easiest mapping, in polar coordinate terms (one distance dimension and two rotation dimensions), is the distance dimension, which is made up of a vertical line connecting the poles. The two rotation dimensions are somewhat harder; considering the bundle of vertical lines, their position with respect to the surface of the sphere is given by two numbers: A complex coordinate and a real coordinate. The complex coordinate kind of maps onto one rotation, and the real coordinate kind of maps onto the other.
That's, I think, an unnecessary specification, because the mapping is in a sense more arbitrary than that. Trying to reason more abstractly: Since there are two complex coordinates on the sphere, and each has a real and a complex component (so four total coordinates), normally we'd expect eight degrees of freedom in principle. However, two of these are redundant, because the topology of the sphere means that certain linear transformations are equivalent, so we end up with six degrees of freedom instead. Three dimensions. How we map the dimensions to the degrees of freedom is somewhat arbitrary, but treating the poles as a real dimension, and separating out the rotation into a real and complex part, is more intuitive.
I'm pondering the Riemann Sphere, and while it's *really really* close to the thing in my head, the thing in my head might, maybe, have an important divergence - the poles, in the thing in my head, are the same point. Topologically, the thing in my head is a horn torus, not a sphere. I'm unsure if this matters - from a certain perspective, the critical topological feature this creates is already kind of conveyed in the complex components of a rotation, which, if I understand the behavior correctly, create a requirement that any rotation about the Riemann Sphere requires a 720 degree rotation to arrive at the same point; it's, in a certain sense, a spinor. So the topology in my head treats the vertical stripes as real, and the rotation as complex, and recovers the spinor behavior with respect to the real component by making the topology more complex. (This really only matters when it comes to understanding particle spin, however, and maybe there's a more elegant way of handling this I just don't grok yet.)
There's another important way that my topology potentially differs, but it's possibly kind of arbitrary: I don't think a straight line between the poles is a simple vertical stripe, but a kind of spherical spiral, whose characteristic behavior is that each rotation about the sphere covers an equal measure. So, supposing that measure is 1 for the sake of argument, the first rotation of the spiral moves, measuring the distance from the pole, from 0 to 1, the second from 1 to 2. Because the density of the points is something like hyperbolic, this means the spiral winds loosely at the 0 pole, and tightens to infinity as you approach the infinity pole. (But because "0" and "infinity" are poorly defined/arbitrary, there are two potential mappings of any given spiral to the sphere, depending on how you assign the poles. I think this is important, and gives rise to the characteristic relation between time and distance-in-polar-coordinates - as I think this gives rise to two dimensions, but only two degrees of freedom, so the dimensions are not distinct.)
This is more or less how I backwards-reasoned my way into something like the Riemann Sphere in the first place, by reasoning about the topology this spiral necessarily crosses. I hadn't actually made the jump from "Single curve" to "Bundle of curves" necessary to notice that this maps the surface of a sphere, however, and conceptualized it as a helix. (You don't need to glue the ends of the cylinder-space this crosses unless you have a bundle; all the other paths through the cylinder are basically kind of theoretical, so the fact that all the points on either end of the cylinder are 0 or infinity, or some weird superposition of both, isn't meaningful information.)
There would definitely be some advantages in something which measures "external" length, as opposed to "internal" length - but it might actually be more useful to have both, and to examine the difference between them. Depends on what "external" and "internal" mean, though, and I'm kind of guessing.
If they mean what I think they mean - something akin to the distinction between the difference in distance between flat and curved space - I typically think about curved space in terms of an imaginary flat space; how the flat space maps to the curved space. And the "negative" dimension idea came from examining what happens to a flat space superimposed on, and interpreted through, curved space, as that flat space passes through a singularity. So I kind of think in both.
But I'm uncertain if that is a correct interpretation / analogy of "internal" versus "external".