Does the Universe Have Higher Dimensions? Part 1(backreaction.blogspot.com)
backreaction.blogspot.com
Does the Universe Have Higher Dimensions? Part 1
https://backreaction.blogspot.com/2021/04/does-universe-have-higher-dimensions.html
64 comments
Look at a game of Asteroids: https://www.youtube.com/watch?v=_TKiRvGfw3Q
It takes place on what is topologically a torus, because the top of the screen and the bottom are the same, and the left and right are the same. From the point of view of the player, the space never ends; if there were no asteroids and you simply went left forever, you'd never find the end. Nevertheless, the X and Y dimensions are finite, because that is the shape of space in that game. But note you could grow or shrink those all you like, and all you'd change is the size of the playspace; nothing else about such shrinking or growing would be impossible or fundamentally change anything.
There is more than one way to hook such structures together: https://www.youtube.com/watch?v=jj5lDmaQTuo&t=0s In that video the author plays with discrete cases of topologies like the asteroids case where otherwise flat space is just glued together in various configurations but there are continuous analogues to at least most, if not all of them, along with options on how to handle the curvature.
The video also deal with macroscopically-sized things, using a model of Earth as its example, but if you can imagine it being shrunk down arbitrarily small in one dimension you can get an idea. Obviously, if you shrink one of them down arbitrarily small you would get a "3D space" that is effectively only 2D, because one of the numbers would seem to be irrelevant since it is always very small. In the real universe this would be some set of dimensions beyond the usual 3 spatial dimensions.
It takes place on what is topologically a torus, because the top of the screen and the bottom are the same, and the left and right are the same. From the point of view of the player, the space never ends; if there were no asteroids and you simply went left forever, you'd never find the end. Nevertheless, the X and Y dimensions are finite, because that is the shape of space in that game. But note you could grow or shrink those all you like, and all you'd change is the size of the playspace; nothing else about such shrinking or growing would be impossible or fundamentally change anything.
There is more than one way to hook such structures together: https://www.youtube.com/watch?v=jj5lDmaQTuo&t=0s In that video the author plays with discrete cases of topologies like the asteroids case where otherwise flat space is just glued together in various configurations but there are continuous analogues to at least most, if not all of them, along with options on how to handle the curvature.
The video also deal with macroscopically-sized things, using a model of Earth as its example, but if you can imagine it being shrunk down arbitrarily small in one dimension you can get an idea. Obviously, if you shrink one of them down arbitrarily small you would get a "3D space" that is effectively only 2D, because one of the numbers would seem to be irrelevant since it is always very small. In the real universe this would be some set of dimensions beyond the usual 3 spatial dimensions.
Think of a cylinder whose radius is of atomic size. More or less that is what they mean.
That space is “felt” as 1-dimensional because there is no way to tell one point on a circumference from another on the same one.
That space is “felt” as 1-dimensional because there is no way to tell one point on a circumference from another on the same one.
If it's atomic size, it can be relatively easily detected, I suppose, by setting things in a spiral motion.
The tricky variant is when the compact dimension is seriously sub-atomic, so you have trouble moving anything easily detectable along it.
The tricky variant is when the compact dimension is seriously sub-atomic, so you have trouble moving anything easily detectable along it.
If two (neutral) objects were only slightly separated along that dimension, no matter how compact it is, wouldn't they be able to move through each other as seen from our perspective (as in, occupy the same position in the usual three dimensions at the same time, because they are separated along that additional spatial dimension)?
Only if the objects themselves are really small along that extra dimension relative to that extra dimension - if they occupy most of the dimension, they would "bump into each other" anyway.
Makes me think of quantum tunneling
Quantum stuff can already move through each other. (Bosons easily, fermions have the Pauli exclusion principle.)
Not if the strong nuclear force extends significantly beyond the small dimension.
But the curvature would be enormous.
"Curvature" in physics generally refers to the "intrinsic" curvature (see from within the space) as opposed to the extrinsic curvature (seen from a larger, embedding space). A cylinder has zero intrinsic curvature, the same as a flat sheet of paper.
> So an infinitely long but finitely thick cylinder would have one large (along) and one small (around) dimension?
Exactly. The mathematical concept behind this is the idea of compactness[0, 1]. A compact manifold always has finite volume. In the same way, a compact dimension always has finite length and this is what is meant here.[2]
Note that the converse does not hold: Not every dimension of finite length is necessarily compact. (The mathematical reason being that the metric tensor could become smaller and smaller ("fade out") towards the infinite ends.) But for the purposes of talking about string theory, you can usually equate the terms "finite length/volume" and "compact".
> Is "small" then always finite, but large can be large-but-finite or inifinte?
Yes, "small" is always finite (compact). As for "large" dimensions, I would say that's a matter of terminology and physicists tend to not be very precise with their terminology. It can both be large-but-finite (compact) or infinite (non-compact), depending on the context.
[0]: https://en.wikipedia.org/wiki/Compact_space
[1]: https://mathworld.wolfram.com/CompactManifold.html
[2]: This is also what string theorists mean when they "compactify a dimension": They take this infinite dimension and "wrap it around" a circle (of a given circumference) and end up with a cylinder or some more complicated object (depending on what they started with).
Exactly. The mathematical concept behind this is the idea of compactness[0, 1]. A compact manifold always has finite volume. In the same way, a compact dimension always has finite length and this is what is meant here.[2]
Note that the converse does not hold: Not every dimension of finite length is necessarily compact. (The mathematical reason being that the metric tensor could become smaller and smaller ("fade out") towards the infinite ends.) But for the purposes of talking about string theory, you can usually equate the terms "finite length/volume" and "compact".
> Is "small" then always finite, but large can be large-but-finite or inifinte?
Yes, "small" is always finite (compact). As for "large" dimensions, I would say that's a matter of terminology and physicists tend to not be very precise with their terminology. It can both be large-but-finite (compact) or infinite (non-compact), depending on the context.
[0]: https://en.wikipedia.org/wiki/Compact_space
[1]: https://mathworld.wolfram.com/CompactManifold.html
[2]: This is also what string theorists mean when they "compactify a dimension": They take this infinite dimension and "wrap it around" a circle (of a given circumference) and end up with a cylinder or some more complicated object (depending on what they started with).
We live in a numpy array.
If spacetime is indeed quantized, it's even a finite-precision array.
> spacetime is quantized.
Doesn't that just adds more evidence for us living in a simulation, and not the nice kind either.
Would be interesting to see when the hardware is upgraded, physcists would suddenly discover that spacetime isn't as quantized as they initially thought.
Doesn't that just adds more evidence for us living in a simulation, and not the nice kind either.
Would be interesting to see when the hardware is upgraded, physcists would suddenly discover that spacetime isn't as quantized as they initially thought.
Quantizing space and time would not necessarily make simulations simpler - just like quantizing forces actually turns out to probably make the simulation exponentially harder on a classical computer - quantum computers can be simulated by classical computers in exponential time with the best algorithms that we know of, though it is still possible that Polynomial algorithms could exist.
Of course, nothing at all can actually prove/disprove that we live in a simulation in the scientific sense - it's just a transcendental model for people who don't like theistic transcendental models, but no more "scientific" than "creation science" or "the gods did it" theories.
Of course, nothing at all can actually prove/disprove that we live in a simulation in the scientific sense - it's just a transcendental model for people who don't like theistic transcendental models, but no more "scientific" than "creation science" or "the gods did it" theories.
> physcists would suddenly discover that spacetime isn't as quantized as they initially thought
I don't think the majority of physicists think that spacetime is quantized. Sure, something is going to happen at the Planck scale but whether that's quantization of time and space is very much an open question.
I don't think the majority of physicists think that spacetime is quantized. Sure, something is going to happen at the Planck scale but whether that's quantization of time and space is very much an open question.
It wasn't until Columbus came along that they finally bothered to move us from the "flat" default starter server to the spherical one.
No, that was already patched before Erastothenes. Some simple branch predictions based on behavior patterns of the Bronze Age civilizations were enough to get the change through management.
What does it mean for a dimension to sit inside our 3 dimensions? I would assume additional dimensions would be outside of our 3, but it sounds like what we’re talking about here is just a surface inside 3d space. Why isn’t my body, or anything for that matter, considered an extra dimension in 3d space?
As far as I understand, the idea is that the extra dimension is everywhere, but only has a tiny bit of wiggle room, and once you get to the end of it you come back from the other side, like a circle.
It's not an object in space, it's a direction that all objects can move in. If you're standing somewhere, you can move N/S, E/W, and jump up/down. The extra dimension is another direction particles can move in, except when they do they stay in the same place, and their momentum in that non-direction works out to be the same energy as electric charge.
More of a magic trick than anything else.
It's not an object in space, it's a direction that all objects can move in. If you're standing somewhere, you can move N/S, E/W, and jump up/down. The extra dimension is another direction particles can move in, except when they do they stay in the same place, and their momentum in that non-direction works out to be the same energy as electric charge.
More of a magic trick than anything else.
> What does it mean for a dimension to sit inside our 3 dimensions?
Could you clarify where in the video/text this is said?
Could you clarify where in the video/text this is said?
I’m probably just grossly misinterpreting this, but it sounds like the extra dimension she mentions here is just a microscopic tube. My naive question is how does that differ from, let’s say, a microscopic piece of penne.
“This problem was solved few years later by Oskar Klein, who assumed that the 4th dimension of space has to be rolled up to a small radius, so you can’t get lost in it. You just wouldn’t notice if you stepped into it, it’s too small. This idea that electromagnetism is caused by a curled-up 4th dimension of space is now called Kaluza-Klein theory.”
“This problem was solved few years later by Oskar Klein, who assumed that the 4th dimension of space has to be rolled up to a small radius, so you can’t get lost in it. You just wouldn’t notice if you stepped into it, it’s too small. This idea that electromagnetism is caused by a curled-up 4th dimension of space is now called Kaluza-Klein theory.”
sdwr already explained this very well in their sibling comment: The extra dimension is another direction that you can move in, apart from the three directions you're already familiar with. A piece of penne doesn't provide you with such an extra direction because movement along the penne can still be described by a tuple of 3-dimensional position + velocity vectors.
There are certain proofs of the dimensionality of space at different scales.
The equipartition theorem details that thermal energy is divided down by degrees of freedom. In higher dimensional space there are different ratios of translations to rotations, so observations of the triple state diagrams of diatomic vs monatomic molecules would diverge from observation.
The time it takes spheres to settle after falling into a square container is a function of the dimensionality of the container. Observations as we scale real experiments with steel ball bearings suggest that the balls can't move in any other dimensions.
In higher dimensional space the great geodesics are longer, we would expect a deviation in the relativistic corrections to the GPS system that we don't observe.
The interesting question however is at the largest scales. The universe might have much higher dimensions, we just happen to be in a relatively "compact" part of it...
The equipartition theorem details that thermal energy is divided down by degrees of freedom. In higher dimensional space there are different ratios of translations to rotations, so observations of the triple state diagrams of diatomic vs monatomic molecules would diverge from observation.
The time it takes spheres to settle after falling into a square container is a function of the dimensionality of the container. Observations as we scale real experiments with steel ball bearings suggest that the balls can't move in any other dimensions.
In higher dimensional space the great geodesics are longer, we would expect a deviation in the relativistic corrections to the GPS system that we don't observe.
The interesting question however is at the largest scales. The universe might have much higher dimensions, we just happen to be in a relatively "compact" part of it...
Additional tiny "curled up" dimensions wouldn't have such direct macroscopic effects. Atoms would be many trillions of times the size of the extra dimensions
Heat capacity of deuterium and ortho/para hydrogen too (if spin would be affected)
Can someone who knows more physics than I tell me if the question is even meaningful? My naive assumption is that we'd have as many dimensions as there are distinct things (atoms, I suppose).
Do I know more physics than you? Problynot. But he’s my attempt to explain how I understood the article:
We experience a universe with 3 spatial dimensions, if we imagine an axis through space, there are two other axes orthogonal to the first axis and to each other, that an object can move on any of the axes independently, without moving in any of the others. We can’t add a fourth axis, there is no fourth line we can choose along which an object can move without also moving in at least one other dimension - or at least that is our experience. We also know from seeing 2d shadows of 3D objects, that a higher dimensional object can be projected into lower dimensions. We can twist an object and see changes in a shadow that make perfect sense if we know the shadow is a 2d projection of a rotating 3D object. There are features of our universe that make sense if we think about the universe as a 3D shadow of a higher dimension system. So what does that mean? Are higher dimensional models a nice math trick, or do the models work because they describe the true higher dimensional nature of the universe, that we experience as a lower dimensional projection. Ok time to pass the baton who can take this deeper.
I think it helps build an intuition of how higher dimensions can help us explain physics, if instead of a shadow, you think of projection as a cross-section, like the ones you see in brain scans, where you only see one "layer" of the brain at a time. Objects that are moved into the plane of the scanner (so they intersect the plane) will essentially "pop into existence" from the perspective of someone looking only at that cross-section - and then that cross-section morphs and changes shape as the object is moved so different parts of it intersect the scanner. What if the same thing is happening when elementary particles pop into existence and later annihilate, that we're seeing the cross-section of a higher-dimensional object as it intersects our 3 dimensions? What if matter and antimatter are like the outer edges of such a cross-section, appearing in a single point as the object touches our plane, moving apart as the bulk of the object passes through our plane, and then finally coming together in a single point before disappearing (annihilation), as the object stops intersecting our plane?
Good answer, a 4th dimension and the shadow projection analogy is brilliantly exemplified by Carl Sagan in this popular video: https://www.youtube.com/watch?v=UnURElCzGc0
I think it's also the most important analogy, more than things like a 2D being meeting a 3D being. The goal of these physics models is to better explain the world. It's a hint that a model is better when it simplifies things (not that the model itself is necessarily simple).
Think of the crazy things astronomers drew to explain the paths of the planets in the sky with the Earth in the center and circular orbits. But what if the Sun is in the center and the orbits are ellipses? Things are way easier.
Same thing happens if you look at the projection of a cube. It looks like chaos. But if you are aware of a third dimension, it all makes a lot of sense.
Think of the crazy things astronomers drew to explain the paths of the planets in the sky with the Earth in the center and circular orbits. But what if the Sun is in the center and the orbits are ellipses? Things are way easier.
Same thing happens if you look at the projection of a cube. It looks like chaos. But if you are aware of a third dimension, it all makes a lot of sense.
> There are features of our universe that make sense if we think about the universe as a 3D shadow of a higher dimension system.
Which one would this be?
Which one would this be?
fair call,
I was trying to make it clear that my comments were 'how is understood the article and vid' and not my Wolfram/Weinstein 'physics is wrong I'm going to fix it' manifesto.
So let me try improve that line:
<strike>There are features of our universe that make sense if we think about the</strike>
<i>Kaluza-Klein theory proposed that electromagnetism could be modelled using an additional closed fourth dimension. Kerner generalised this approach to model gravity, time, electromagnetism and the strong and week forces using a total of 11 dimensions. I have understood these models as describing our experience of the </I>universe as a 3D <strike>shadow</strike> <i>projection</i> of a higher dimension system.
Kaluza-Klein was the exact point at which I lost interest in pursuing university physics, so the sentence is marks the point at which the ship of my knowledge hits the rocks. I loved first year physics, but in my second year at university a tutor convinced that physicists now worked exclusively on string theory, which he explained beginning with Kaluza-Klein theory. What he described reminded me of Copernican epicycles, and I felt so little enthusiasm for it, and was deeply disappointed after having loved first year physics and it was the end of my imagining I might be a physicist. Much later I discovered he was suicidally depressed the dry bloodless descriptions he gave were him trying to hang on and keep himself together while describing a subject he had come to hate.
I was trying to make it clear that my comments were 'how is understood the article and vid' and not my Wolfram/Weinstein 'physics is wrong I'm going to fix it' manifesto.
So let me try improve that line:
<strike>There are features of our universe that make sense if we think about the</strike>
<i>Kaluza-Klein theory proposed that electromagnetism could be modelled using an additional closed fourth dimension. Kerner generalised this approach to model gravity, time, electromagnetism and the strong and week forces using a total of 11 dimensions. I have understood these models as describing our experience of the </I>universe as a 3D <strike>shadow</strike> <i>projection</i> of a higher dimension system.
Kaluza-Klein was the exact point at which I lost interest in pursuing university physics, so the sentence is marks the point at which the ship of my knowledge hits the rocks. I loved first year physics, but in my second year at university a tutor convinced that physicists now worked exclusively on string theory, which he explained beginning with Kaluza-Klein theory. What he described reminded me of Copernican epicycles, and I felt so little enthusiasm for it, and was deeply disappointed after having loved first year physics and it was the end of my imagining I might be a physicist. Much later I discovered he was suicidally depressed the dry bloodless descriptions he gave were him trying to hang on and keep himself together while describing a subject he had come to hate.
Thank you for answering.
You don't need to know more physics. If you see something (a particle fe) disappear, and reappear somewhere else then a possible explanation is that it got there via another dimension. So if you eliminate all other possible explanations, the only remaining one is that there must be more dimensions. However, since things appearing and disappearing does not happen on macroscopic scale, the dimensions must be small.
So this is how UFOs move!
In this context, the dimensions are the degrees of freedom that a particle can move in. I think it becomes meaningful when you consider the metric (ie. definition of distances) and the laws of nature inside that space, and if additional dimensions help in formulating more phenomena with a less complex system of equations.
-> Classically, there are 3 spatial dimensions (directions of movement) with Euclidean metric, and all the classical laws (Newton's laws, gravity, etc.). Quite successful, but breaking down in extreme cases (high energy, ie. velocity/mass/etc). +there is a whole range of seemingly unrelated laws necessary.
-> with relativity theory (both special and general), time becomes a 4th dimension, but it has a special status in the metric (opposed sign), making spacetime non-Euclidean. One effect of this is suddenly you don't need a law of gravity anymore. Things just follow geodesics in this space. It also explains some effects that could not be derived from classical gravity. Essentially explaining more phenomena with less "overhead", in exchange for more dimensions.
-> Classically, there are 3 spatial dimensions (directions of movement) with Euclidean metric, and all the classical laws (Newton's laws, gravity, etc.). Quite successful, but breaking down in extreme cases (high energy, ie. velocity/mass/etc). +there is a whole range of seemingly unrelated laws necessary.
-> with relativity theory (both special and general), time becomes a 4th dimension, but it has a special status in the metric (opposed sign), making spacetime non-Euclidean. One effect of this is suddenly you don't need a law of gravity anymore. Things just follow geodesics in this space. It also explains some effects that could not be derived from classical gravity. Essentially explaining more phenomena with less "overhead", in exchange for more dimensions.
That is what is called by the name of degrees of freedom.
The dimensions of space-time are to the degrees of freedom as the basis vectors of a coordinate system are to the homogeneous matrix. Or something like that.
The dimensions of space-time are to the degrees of freedom as the basis vectors of a coordinate system are to the homogeneous matrix. Or something like that.
That implies there ARE distinct things, doesn't it?
Not at all serious but... maybe all electrons have the same mass, charge, etc. because there is only one electron, bouncing backwards and forwards through time. As it passes backwards we see it as the anti-electron (positron). When they meet they annihilate from our perspective, but that's just the electron being reflected and becoming a positron heading backwards.
To be clear this is all not at all consistent with observations - just a fun(?) thought experiment.
To be clear this is all not at all consistent with observations - just a fun(?) thought experiment.
Out of interest, what observations is this not consistent with?
The argument against is summarized in the Wikipedia article.
Basically it is this:
We measure electrons in different places all the time.
Due to the speed of light it can't instantly move from A to B, so for this to actually be one electron it would have to travel back in time to be at some place at the right time.
However, an electron traveling back in time would appear as a positron, so if that what was going on we should be seeing a fairly equal number of positrons as we do electrons, as the one electron rushes around to appear as an electron where it needs to.
Except we don't, electrons outnumber positrons by a huge margin.
Basically it is this:
We measure electrons in different places all the time.
Due to the speed of light it can't instantly move from A to B, so for this to actually be one electron it would have to travel back in time to be at some place at the right time.
However, an electron traveling back in time would appear as a positron, so if that what was going on we should be seeing a fairly equal number of positrons as we do electrons, as the one electron rushes around to appear as an electron where it needs to.
Except we don't, electrons outnumber positrons by a huge margin.
> However, an electron traveling back in time would appear as a positron, so if that what was going on we should be seeing a fairly equal number of positrons as we do electrons, as the one electron rushes around to appear as an electron where it needs to.
What about this part of the article though...?
What about this part of the article though...?
"According to Feynman he raised this issue with Wheeler, who speculated that the missing positrons might be hidden within protons."As noted, the discussion between Wheeler and Feynman was in 1940, long before the development of the Standard Model[1].
In the Standard Model there is a sea of virtual particles in the nucleus, but they're virtual and hence not real in the sense that the positron in the One Electron model would have to be. At least that's my understanding.
Also, electrons can travel over large distances, CRT monitors do that all the time for example. So I'm not entirely sure how Wheeler imagined hiding the positrons in the nucleus would solve the whole positron problem.
[1]: https://en.wikipedia.org/wiki/Quark#History
In the Standard Model there is a sea of virtual particles in the nucleus, but they're virtual and hence not real in the sense that the positron in the One Electron model would have to be. At least that's my understanding.
Also, electrons can travel over large distances, CRT monitors do that all the time for example. So I'm not entirely sure how Wheeler imagined hiding the positrons in the nucleus would solve the whole positron problem.
[1]: https://en.wikipedia.org/wiki/Quark#History
The imbalace in observed particles could come from the fact that we move in the same (time) direction as electrons but the opposite (time) direction of positrons, couldn't it?
'Distinct in some context', I guess. Eg: a person could take a photograph of clouds in the sky and come up with list of rules to categorize them into a number of discrete clouds.
Don't take my comment too seriously, although it is based on serious speculation and thought.
I only wanted to refer on the one hand to the philosophical or at least linguistic problem, what a 'thing' is and the possibility of the unification to one field of the, up to now still multiform, thing in the quantum field theory we call Universe.
Just a mind game.
Or not...
;-)
What is meant in the article is that one needs currently 11 coordinates in a mathematical space which is supposed to be a mapping of our physical reality to describe a point in it.
I only wanted to refer on the one hand to the philosophical or at least linguistic problem, what a 'thing' is and the possibility of the unification to one field of the, up to now still multiform, thing in the quantum field theory we call Universe.
Just a mind game.
Or not...
;-)
What is meant in the article is that one needs currently 11 coordinates in a mathematical space which is supposed to be a mapping of our physical reality to describe a point in it.
[deleted]
are you saying: what if dimensions are not discrete, but fields.?
Ah, no no no.
Oh, God, what have I done!?
What I wanted to indicate was firstly, as a physical speculation the possibility of a universal field in the sense of the quantum field theory, which shows additional dimensionality with breaking of its symmetry in hierarchical levels and thus there is in a certain sense only one thing, the universal field.
Secondly, the question when something is considered as an independent 'thing', which is a philosophical question.
Third, the question of the interpretation of the relation of mathematical apparatus and reality.
What does the dimensionality of the mathematical model mean? A scientific-theoretical question.
And the gluons of this all are just linguistics and semantics.
;-)))
Oh, God, what have I done!?
What I wanted to indicate was firstly, as a physical speculation the possibility of a universal field in the sense of the quantum field theory, which shows additional dimensionality with breaking of its symmetry in hierarchical levels and thus there is in a certain sense only one thing, the universal field.
Secondly, the question when something is considered as an independent 'thing', which is a philosophical question.
Third, the question of the interpretation of the relation of mathematical apparatus and reality.
What does the dimensionality of the mathematical model mean? A scientific-theoretical question.
And the gluons of this all are just linguistics and semantics.
;-)))
Perhaps what you mean is the dimensionality of hilbert space
> Can someone who knows more physics than I tell me if the question is even meaningful?
Any new physical property could potentially be exploited to improve some existing idea. The discovery of the electromagnetic force eventually gave way to WiFi and 5G. The discovery of the quantum realm for example has yielded the concept of quantum computing. If we can discover high dimensions, we can eventually interact with them and use them.
Any new physical property could potentially be exploited to improve some existing idea. The discovery of the electromagnetic force eventually gave way to WiFi and 5G. The discovery of the quantum realm for example has yielded the concept of quantum computing. If we can discover high dimensions, we can eventually interact with them and use them.
So... where is Part 2? I can't find it...
Edit: Ah the content is 2 days old, it will come. In any case: Great content!
Edit: Ah the content is 2 days old, it will come. In any case: Great content!
Doesn't the holographic principle state that the universe is actually 2D?
The holographic principle arises basically from two facts:
* The AdS/CFT correspondence, which states briefly that a universe having properties our universe doesn't have is mathematically equivalent do a different universe with one fewer dimension and different properties our universe also doesn't have. This is the most celebrated result in string theory, by the way.
* The entropy of a black hole is proportional to its surface area, not its volume.
From these two results, one can speculatively extrapolate that the information density of the universe uses one fewer than dimension than the actual interaction volume we experience, and this extrapolation is the holographic principle.
* The AdS/CFT correspondence, which states briefly that a universe having properties our universe doesn't have is mathematically equivalent do a different universe with one fewer dimension and different properties our universe also doesn't have. This is the most celebrated result in string theory, by the way.
* The entropy of a black hole is proportional to its surface area, not its volume.
From these two results, one can speculatively extrapolate that the information density of the universe uses one fewer than dimension than the actual interaction volume we experience, and this extrapolation is the holographic principle.
> The entropy of a black hole is proportional to its surface area, not its volume.
Does the following argument hold? The Schwarzschild radius of a black hole is proportional to the mass. So the surface area is proportional to M^2.
How can bigger holes store ever more entropy per unit mass?
Does the following argument hold? The Schwarzschild radius of a black hole is proportional to the mass. So the surface area is proportional to M^2.
How can bigger holes store ever more entropy per unit mass?
Maybe this by the same author may help answer your question. [1]
[1] https://backreaction.blogspot.com/2021/03/is-universe-really...
[1] https://backreaction.blogspot.com/2021/03/is-universe-really...
If there were an infinite number of dimensions, would that mean there was also a path, if we could find it, from any point in 3d space to any other point in 3d space that was of arbitrary length?
Additional dimensions don't give you additional ways to be closer to something, they give you additional ways to be further away. If we're displaced from each other by X, Y and Z units along axes in the usual 3 spatial dimensions, the closest we can be is if we're at a coincident position in each of the additional dimensions. Any displacement along those additional axes just moves us further away from each other.
In general no unless there's one that's warped just the right way. Which is unlikely. (Also it comes down to cardinality of the involved infinities. If we have countably many infinite dimensions but uncountably many points in each one of them, then you have a "lot more" point-point pairs than dimensions.)
Step 1: Every dimension has an observer.
Step 2: "An observer watches something" happens in Zero dimension,
Step 3: "An observer watching an observer" happens in One dimension,
Step 4: "An observer watching an observer watching an observer" happens in Two dimension, . . . . . Step n: it can go on till the n dimension.
Let's twist the rule to itself.
An observer (zero dimension)
An observer watching itself, (one dimension)
An observer watching itself observing, (two dimension)
An observer watching itself observing itself, (three dimension)
An observer watching itself observing itself,.................. (n dimension)
And there can be any level of observer,
so there can be any level of dimensions.
Step 2: "An observer watches something" happens in Zero dimension,
Step 3: "An observer watching an observer" happens in One dimension,
Step 4: "An observer watching an observer watching an observer" happens in Two dimension, . . . . . Step n: it can go on till the n dimension.
Let's twist the rule to itself.
An observer (zero dimension)
An observer watching itself, (one dimension)
An observer watching itself observing, (two dimension)
An observer watching itself observing itself, (three dimension)
An observer watching itself observing itself,.................. (n dimension)
And there can be any level of observer,
so there can be any level of dimensions.
Step 5. The longer the buffering, the greater the sneeze.
How are dimensions larger and smaller? To me, a dimension is usually something you measure in. My height is my extent in the vertical diredction (if standing up). What would it mean for that dimension to be small?
Our 3 spatial dimensions I assume are "large", meaning what? The axes extend to some apparent infinity without repeating? Does a "small" dimension behave like a longitude, that after a certain while it repeats itself? So the two dimensions of a sphere surface are "smaller" than the 2 dimensions of an infinite plane, is that how to interpret the small/large? So [the surface of] an infinitely long but finitely thick cylinder would have one large (along) and one small (around) dimension? Is "small" then always finite, but large can be large-but-finite or inifinte?