On the dimensionality of spacetime (1997) [pdf](space.mit.edu)
space.mit.edu
On the dimensionality of spacetime (1997) [pdf]
https://space.mit.edu/home/tegmark/dimensions.pdf
55 comments
> * For very short distances the so-called spectral dimension is theorized to get reduced from four to two in certain approaches to quantum gravity, see this post by Sabine Hossenfelder([…])
FTFY.
Hossenfelder's blog post says:
> The notion of dimension that is relevant for the effect of dimensional reduction is not the Hausdorff dimension, but instead the “spectral dimension.” The spectral dimension can be found by first getting rid of the Lorentzian signature and going to Euclidean space. And then to watch a random walker who starts at one point, and measure the probability for him to return to that point. The smaller the average return probability, the higher the probability he’ll get lost, and the higher the number of dimensions. One can define the spectral dimension from the average return probability.
(The Hausdorff dimension is what we typically mean when we say "spatial dimension.")
Moreover, all the papers cited are "approaches to quantum gravity", as she says, i.e. attempts at finding a theory of quantum gravity that works. No one has found one yet, though.
FTFY.
Hossenfelder's blog post says:
> The notion of dimension that is relevant for the effect of dimensional reduction is not the Hausdorff dimension, but instead the “spectral dimension.” The spectral dimension can be found by first getting rid of the Lorentzian signature and going to Euclidean space. And then to watch a random walker who starts at one point, and measure the probability for him to return to that point. The smaller the average return probability, the higher the probability he’ll get lost, and the higher the number of dimensions. One can define the spectral dimension from the average return probability.
(The Hausdorff dimension is what we typically mean when we say "spatial dimension.")
Moreover, all the papers cited are "approaches to quantum gravity", as she says, i.e. attempts at finding a theory of quantum gravity that works. No one has found one yet, though.
Imagine what kind of computers could you make if you could radiate heat into more than one dimension. Or maybe you would get greedy and use up all available dimensions for interconnects and use one remaining dimension for heat dissipation anyway.
A hypercube's surface area to volume ratio grows without bound with dimension, so radiating heat ought to be easier with more dimensions even if you build a fairly blobby (I'm sure that's the technical term) core, though an n-sphere's peaks at n=7, so you don't want to get too blobby in higher dimensions.
Took me a minute to figure out what you are saying. Then I realized you are referring to the 16pi^3/15 surface area ratio of a 7-sphere. After n>7 it does seems to infinitely move towards zero. But the heat equasion is different in higher dimensional space and governed by the parabolic Harnack inequalities. I'm not sure how surface area makes sense here.
A CPU, a planar object, does so by extending its fins in the three-dimensional direction, but not without limit. Similarly, even if the fourth dimension can be used, it cannot be used without limit. Also, just as there are no two-dimensional objects without thickness in three-dimensional space, there are no three-dimensional objects without thickness in four-dimensional space, and all three-dimensional objects are considered to have size in the fourth-dimensional direction in four-dimensional space. However, if there is such a simple fourth spatial dimension, it should be easily recognizable, so if there is a fourth spatial dimension, it should be closely attached to other known dimensions so that it cannot be found. Finding or creating a usable space out of a closely attached fourth spatial dimension would be a large cost.
Can’t we already radiate heat into 3 dimensions?
I think what he means is that chips are 2D, so heat is radiated orthogonal to the plane of the circuit -- up or down.
Yes. See: radiators
Heat from the chip to the radiator still flows in one direction. So you're effectively limited by the bandwidth of this joint even if you have radiators or liquid-cooling (remote radiators) later in the pipeline.
Wouldn't a 4th dimension allow connection between any two points in a 3D space? I think that's how that works but I'm no mathemagician.
Essentially, but you still have to travel the distance in a parallel dimension. Think about layers on a circuit board or road overpasses. You can connect two things separated by something by going "up" in the third dimension, but you still have to actually go the distance after avoiding the collision (and then go down).
so in current design you travel the distance in the time dimension
No, time has nothing to do with this. The point is that two points separated by an object in 3D space (say, a point inside a box and one outside the box) can be connected by something in 4D space, just like two points on a 2D plane on opposite sides of a river can be connected by a bridge in 3D space.
The time dimension is entirely separate. You always travel a distance in the time dimensions, in space the distance just exists, there is no concept of traveling without time. This is also visible in the math of SR - the time dimension is different from the three spatial dimensions (opposite sign in the distance metric).
The time dimension is entirely separate. You always travel a distance in the time dimensions, in space the distance just exists, there is no concept of traveling without time. This is also visible in the math of SR - the time dimension is different from the three spatial dimensions (opposite sign in the distance metric).
Sort of. I don't believe that time is considered a spatial dimension. But from a looser conceptual framework, basically yeah you have to spend some time to travel space.
Also note that even using 3 dimensions to traverse a 1-dimensional space doesn't mean there's guaranteed to be no distance between two points. A string can be super densely coiled in 3D space where there's basically no 3-dimensional distance between two points on the string, but it can also be stretched out or loosely crumpled and still have lots of 3d space between two points on the string.
Also note that even using 3 dimensions to traverse a 1-dimensional space doesn't mean there's guaranteed to be no distance between two points. A string can be super densely coiled in 3D space where there's basically no 3-dimensional distance between two points on the string, but it can also be stretched out or loosely crumpled and still have lots of 3d space between two points on the string.
It is possible by deforming it in the fourth dimension, like folding paper. However, the shapes that can be created by folding are considered limited.
My "cognitive white noise generator" has been locked on emergent spacetime for a decade now. The idea here, and I'm very loosely summarizing, is that the dimensions themselves are emergent properties of large scale entanglement - the mathematical "dimensions" resolve quite well into flesh and blood, "this is a chair" dimensions.
The paper that sort of kicked it off
https://arxiv.org/abs/1005.3035
Quite evocative, but would require spectacularly sensitive testing apparatus to falsify . . and even then, other explanations aplenty.
It is the fabric of space, after all, no shortage of theories.
The paper that sort of kicked it off
https://arxiv.org/abs/1005.3035
Quite evocative, but would require spectacularly sensitive testing apparatus to falsify . . and even then, other explanations aplenty.
It is the fabric of space, after all, no shortage of theories.
I'm right with you on this. It just makes so much sense to me. It makes sense imo, given any large system in distributed programming needs to be coherent to work. AFAIK coherence is entanglement. So in order for the system to have an actually meaningful, consistent structure, there needs to be some sort of synchronization/coherence. In distributed systems, you need both time and space (propagation delay and # coherent/up to date nodes for a given "quanta" of information exchange) before you can complete a "strongly consistent" read or write. Trying to ask the system for coherent state about anything in an update on timescales finer than the propagation delay or using less than the required # of nodes for consistency is kind of nonsensical.
Idk. In general, the idea of emergence seems to be the best answer we have for a ton of physical phenomena around us. I'm betting/putting my faith that gravity could end up one as well, unless/until someone proves otherwise.
Idk. In general, the idea of emergence seems to be the best answer we have for a ton of physical phenomena around us. I'm betting/putting my faith that gravity could end up one as well, unless/until someone proves otherwise.
The chart on the second page of the article used to be located on a Wikipedia page called "Privileged character of 3+1 spacetime". I think it's curious that it seems to be mirrored over a diagonal axis.
That's expected when you realize that time and space are not different things but just two aspects of the same, pseudo-Riemannisn structure that describes our universe. The only thing separating them is a sign which in turn is based purely on convention.
> That's expected when you realize that time and space are not different things [...] The only thing separating them is a sign which in turn is based purely on convention.
The fact that spatial and temporal dimensions require different signs in the metric signature (regardless of whether you take the +--- or the -+++ convention) should suggest that time and space dimensions are indeed, in some sense, different (in the sense of "not interchangeable").
There's a reason space-time is usually modelled as a 4-dimensional Lorentizan manifold, rather than a 4-dimensional Riemannian manifold.
The fact that spatial and temporal dimensions require different signs in the metric signature (regardless of whether you take the +--- or the -+++ convention) should suggest that time and space dimensions are indeed, in some sense, different (in the sense of "not interchangeable").
There's a reason space-time is usually modelled as a 4-dimensional Lorentizan manifold, rather than a 4-dimensional Riemannian manifold.
What do you mean with usually? Spacetime (i.e. the object of discussion in relativity) is by definition not positive definite. You said it yourself that space and time require different signs. But they are still interchangeable - without changing physics. On top of that, all that a Lorentz transformation (at least for boosts) does is rotate (hyperbolically) space and time into each other. Seeing space and time as two different things rather than interchangeable components of a more fundamental object is not in line with relativity.
Agree. tx bivectors squares to 1 giving hyperbolic rotations, while xy bivectors squares to -1 giving euclidean rotations.
saying time and space aren't different because spacetime exists sounds like saying electric fields and magnetic fields aren't different because EM field exists.
seems to miss the point of unification.
saying time and space aren't different because spacetime exists sounds like saying electric fields and magnetic fields aren't different because EM field exists.
seems to miss the point of unification.
>like saying electric fields and magnetic fields aren't different because EM field exists.
Funny you should say that, because this idea is well known as S-duality [1] in Quantum Field Theory (and more generalized in String Theory). In fact, if magnetic monopoles (read: charges) exist, you could already see that simply by looking at Maxwell's equations and replacing E->B and B->-E. If you also know a bit about the Maxwell bivector, it's easy to see how closely this duality mirrors how space and time are related.
[1] https://en.m.wikipedia.org/wiki/S-duality
Funny you should say that, because this idea is well known as S-duality [1] in Quantum Field Theory (and more generalized in String Theory). In fact, if magnetic monopoles (read: charges) exist, you could already see that simply by looking at Maxwell's equations and replacing E->B and B->-E. If you also know a bit about the Maxwell bivector, it's easy to see how closely this duality mirrors how space and time are related.
[1] https://en.m.wikipedia.org/wiki/S-duality
Very interesting sub-thread.
It is true from SR+Minkowski that space and time can be considered together as a manifold, where (relative motion) boosts rotate smoothly between the dimensions. The signature just decides the interval expression, but doesn't change physics.
But is also true (SR, esp GR) that the important thing about spacetime is the light-cone structure at each spacetime point. Space-like and time-like directions (2 for time-like, past and future) really are different. The forward and backward light cones define a null-surface with zero interval (everywhere, all at once, a la photon).
Relative velocities tilt light cones to bring the local and global views into agreement. Somehow...
However... Proper time is asymmetric, past and future really are different here and now. Local proper time seems to be inexorable, and ~independent of the universe. It is the clock tick we all feel. Proper time seems directed, but non-transformable. We always experience the same rate of 1s/s, looking at our local inertial atomic clocks.
It is confusing to hold all those ideas in my mind at the same time. I have not seen a good explanation that resolves the confusion (I have seen the equations :)
My guess is that local proper time is a real progression, perhaps by axiom, not dependent on cause-effect, or 2nd Law, or QM entanglement, or .. anything else. However, relative time is accurately described by SR, GR and some future QG.
Also, QM is fundamentally wrong, because it is background-dependent. GR showed us that the more profound approach is to include gravity, space, time, matter and energy as dynamic participants in the same framework of laws. QM (Schrodinger/Dirac/QED/QFT...) seems provincial, and provisional, because it assumes a spacetime background - which cannot possibly be true, as is.
I suppose I align with the GR-istas, especially Penrose, who seem to understand this conformal/twistor structure deeply, and use it to lead their intuition for speculative future theories, of cosmology, and also QG.
It is true from SR+Minkowski that space and time can be considered together as a manifold, where (relative motion) boosts rotate smoothly between the dimensions. The signature just decides the interval expression, but doesn't change physics.
But is also true (SR, esp GR) that the important thing about spacetime is the light-cone structure at each spacetime point. Space-like and time-like directions (2 for time-like, past and future) really are different. The forward and backward light cones define a null-surface with zero interval (everywhere, all at once, a la photon).
Relative velocities tilt light cones to bring the local and global views into agreement. Somehow...
However... Proper time is asymmetric, past and future really are different here and now. Local proper time seems to be inexorable, and ~independent of the universe. It is the clock tick we all feel. Proper time seems directed, but non-transformable. We always experience the same rate of 1s/s, looking at our local inertial atomic clocks.
It is confusing to hold all those ideas in my mind at the same time. I have not seen a good explanation that resolves the confusion (I have seen the equations :)
My guess is that local proper time is a real progression, perhaps by axiom, not dependent on cause-effect, or 2nd Law, or QM entanglement, or .. anything else. However, relative time is accurately described by SR, GR and some future QG.
Also, QM is fundamentally wrong, because it is background-dependent. GR showed us that the more profound approach is to include gravity, space, time, matter and energy as dynamic participants in the same framework of laws. QM (Schrodinger/Dirac/QED/QFT...) seems provincial, and provisional, because it assumes a spacetime background - which cannot possibly be true, as is.
I suppose I align with the GR-istas, especially Penrose, who seem to understand this conformal/twistor structure deeply, and use it to lead their intuition for speculative future theories, of cosmology, and also QG.
The link to Wikipedia page for the curious [0].
[0]: https://en.wikipedia.org/wiki/Spacetime#Privileged_character...
[0]: https://en.wikipedia.org/wiki/Spacetime#Privileged_character...
Not too surprising given time and space are pretty symmetric (or anti-symmetric).
It seems that this paper assumes all the dimensions are flat. I believe that most current theories assume the higher dimensions are folded.
https://en.wikipedia.org/wiki/Extra_dimensions
https://en.wikipedia.org/wiki/Extra_dimensions
See the footnote on page 1:
“Hereafter, we let n and m refer to the number of non-compactified space and time dimensions, or more generally to the effective spacetime dimensionality that is relevant to the low-energy physics we will be discussing later.”
“Hereafter, we let n and m refer to the number of non-compactified space and time dimensions, or more generally to the effective spacetime dimensionality that is relevant to the low-energy physics we will be discussing later.”
For example, in 2 dimensions embedded in our 3 the force potential decreasing with square distance would have given a hint to the 2-dim occupants that they are embedded in 3. We don't see the decreasing with cube, so we don't seems to be embedded in full-flat 4+. (even more - the force falls off linear with distance in galactics, i.e. "dark force"/MOND, which seems to be another indication of some holograghy-like "folding" 3->2 (the other being is the black hole holography))
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I feel like we are over simplifying. This all rides on the existence of tachyon particles.
Math used in certain ways can validate or simulate anything, but it doesn't mean a scenario of a dimension greater or smaller than ours exists except by theoretical representation. I am fine with that it is fun to think about.
Dimensions beyond what we can perceive even if it is less than or greater than our own is pure nightmare fuel. I will simply leave it to math.
Math used in certain ways can validate or simulate anything, but it doesn't mean a scenario of a dimension greater or smaller than ours exists except by theoretical representation. I am fine with that it is fun to think about.
Dimensions beyond what we can perceive even if it is less than or greater than our own is pure nightmare fuel. I will simply leave it to math.
Nice analysis. But the point about stability - in other configurations of the universe, the subset of those that do evolve intelligence in some fashion is going to have different physics from our own.
Stable intelligences might be another condition to explore in such an analysis.
Stable intelligences might be another condition to explore in such an analysis.
Such a universe would have to have so radically different a conception of physics that it doesn't really apply to this analysis. Orbits are a pretty fundamental concept and you lose those in n>3 spacetime.
The authors also didn't consider the possibility of different geometries - e.g. hyperbolic space or time[0], or perhaps Nil[1] space. However, I suspect they would all have the same stable orbit problem that n>3 space runs into.
More damning, IMO, are the ultrahyperbolic (n>1 & m>1) and elliptical (n=0 | m=0) PDE spacetimes. These are spacetimes in which you can't compute physics, period. No alternative rules of physics will save you.
[0] Not to be confused with the Hyperbolic Time Chamber in Dragon Ball Z
[1] An alternative non-Euclidean geometry built specifically to make Penrose stairs in
The authors also didn't consider the possibility of different geometries - e.g. hyperbolic space or time[0], or perhaps Nil[1] space. However, I suspect they would all have the same stable orbit problem that n>3 space runs into.
More damning, IMO, are the ultrahyperbolic (n>1 & m>1) and elliptical (n=0 | m=0) PDE spacetimes. These are spacetimes in which you can't compute physics, period. No alternative rules of physics will save you.
[0] Not to be confused with the Hyperbolic Time Chamber in Dragon Ball Z
[1] An alternative non-Euclidean geometry built specifically to make Penrose stairs in
> Such a universe would have to have so radically different a conception of physics that it doesn't really apply to this analysis. Orbits are a pretty fundamental concept and you lose those in n>3 spacetime.
Everyone keeps referencing the same single paper that made this claim, but it is only true under certain assumptions.
Most modern theories of physics have more than 3 dimensions, starting with 5 for Kaluza Klein theory and up to 10+ with string theories.
There's lots of ways of having more than the usual 3+1 dimensions and still having stable orbits, 1/r^2 laws, etc..
Curled up dimensions is a common approach, but not the only one.
For example, even in 3+1 dimensional space time, fields are radiated not from zero-dimensional points, but from one-dimensional world lines! By extension, additional dimensions of time would work if fields were radiated from higher dimensional surfaces or hyper-planes.
In other words, as long as in the bulk three spatial dimensions it appeared that sources of charges were zero dimensional points, the other dimensions can do "whatever" and everything generally works out.
Everyone keeps referencing the same single paper that made this claim, but it is only true under certain assumptions.
Most modern theories of physics have more than 3 dimensions, starting with 5 for Kaluza Klein theory and up to 10+ with string theories.
There's lots of ways of having more than the usual 3+1 dimensions and still having stable orbits, 1/r^2 laws, etc..
Curled up dimensions is a common approach, but not the only one.
For example, even in 3+1 dimensional space time, fields are radiated not from zero-dimensional points, but from one-dimensional world lines! By extension, additional dimensions of time would work if fields were radiated from higher dimensional surfaces or hyper-planes.
In other words, as long as in the bulk three spatial dimensions it appeared that sources of charges were zero dimensional points, the other dimensions can do "whatever" and everything generally works out.
As increasing dimensions adds to the possible arrangements of particles in a universe, then entropy has to be renormalized lower with every added dimension- more dimensions = more ways to achieve disorder.
Isn't 2+1 dimensions gravity a thing ? https://arxiv.org/abs/hep-th/9204099v3
Yes, and even 1+1 gravity theories, like JT-gravity (https://en.wikipedia.org/wiki/Jackiw%E2%80%93Teitelboim_grav...).
But 2+1 Einstein's gravity is locally trivial (i.e. there are no local degrees of freedom) and is purely topological (i.e. holes and global structure of spacetime is what matters).
But 2+1 Einstein's gravity is locally trivial (i.e. there are no local degrees of freedom) and is purely topological (i.e. holes and global structure of spacetime is what matters).
I think you end up with no degrees of freedom in 2+1 and below
I didn't read the paper yet, but the title reminds me of Robert Bartini who had a theory of time being 3-dimensional as well as space is.
Tegmark's paper does seem related in that it is arguing that proposals like Bartini's might produce a universe that is 'dead'. It seems that Tegmark's paper is suggesting that only 4 dimensional (3+1) universe produces a world with 'observers'. The math is beyond my ken.
This appears to be the Bartini paper (1965) https://www.researchgate.net/publication/26408302_Relations_...
This appears to be the Bartini paper (1965) https://www.researchgate.net/publication/26408302_Relations_...
Bartini the aircraft designer? I'd like to read more about that, but most of the Bartini-related stuff is about airplanes.
AFAIK he was also a mystic of a kind and what we would call a "fringe scienctist" in today language. Which makes a curious case when such a character actually builds serious sophisticated working products like cutting-edge airplane designs. The latter leading to a clue that his theories may be worth exploring (they still can be wrong, but there is a chance of finding some interesting food for thought there). Another example of an inventor of similar kind coming into my mind obviously is Nikola Tesla.
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Time has at least three dimensions. I consider at least six but possibly twelve.
A future imperfect that has yet to be experienced would be three relative time of origin, thus six, relative to the observer, allowing for an additional six and total of twelve.
A future imperfect that has yet to be experienced would be three relative time of origin, thus six, relative to the observer, allowing for an additional six and total of twelve.
I am glad that I only encountered this after reading the book series that starts with the Three Body Problem.
Couldn't be other than Max Tegmark
My personal take is that 3+1 is not privileged; but that the physics that takes place in the other combinations is either so uninteresting it doesn't meaningfully interact, or that the number of Feynman paths through the non-3+1 cases all (mostly?) cancel out.
I have never quite understood why there can't be stable stallites in dimensions above 3.
I mean, I know the argument that gravity inverse square law becomes inverse cube law in 4d, but what I do not understand is that what/why enforces that. Why in a hypothetical 4d world there just can not be a gravity-like force that is inverse square? Would that cause some kind of contradiction?
I mean, I know the argument that gravity inverse square law becomes inverse cube law in 4d, but what I do not understand is that what/why enforces that. Why in a hypothetical 4d world there just can not be a gravity-like force that is inverse square? Would that cause some kind of contradiction?
If you make a uniform flash of light in 3d space it spreads around you in a shape of a sphere of increasing radius. Energy gets distributed evenly across the sphere's surface which is growing with time proportional to a square of the radius. So energy density (think intensity) decreases as inverse square. In 4d space the sphere's surface grows with the cube of the radius.
This sort of intuition. Applies to electromagnetic waves, sound and gravity all alike.
This sort of intuition. Applies to electromagnetic waves, sound and gravity all alike.
But there are forces even in our 3d world that do not follow inverse square law (strong/weak nuclear forces). That kind of proves that all forces do not need to follow this intuition?
In particle physics, fundamental forces are generated by the exchange of virtual particles (I have been reading a quantum field theory textbook for the last few months to try to understand precisely what this means, among other questions, but this is accepted fact in the field). The Coulomb force comes from the exchange of photons. So this 1/r^2 argument for intensity leads to the Coulomb force falling off like 1/r^2.
This argument doesn't obviously apply to gravity (though presumably it would for a quantum theory of gravity), but the equations for gravity (general relativity) give the same result.
At a higher level, it turns out that when you try to combine quantum mechanics with special relativity, the resulting theories are highly constrained. It's not like classical mechanics, where you can just say 'suppose there's a 1/r^12 force.' You get mathematical inconsistencies if you stay too far. Weird stuff
This argument doesn't obviously apply to gravity (though presumably it would for a quantum theory of gravity), but the equations for gravity (general relativity) give the same result.
At a higher level, it turns out that when you try to combine quantum mechanics with special relativity, the resulting theories are highly constrained. It's not like classical mechanics, where you can just say 'suppose there's a 1/r^12 force.' You get mathematical inconsistencies if you stay too far. Weird stuff
All long-distance forces seem to follow it though. I think it's related to the energy conservation law.
It proves nothing of course. When we speak of these N+T universes, we try to imagine a system that follow the same "fundamental laws" but with different N and T. What exactly is fundamental is up to debate. You can even imagine a system that has different math, but it will be very hard to reason about it.
It proves nothing of course. When we speak of these N+T universes, we try to imagine a system that follow the same "fundamental laws" but with different N and T. What exactly is fundamental is up to debate. You can even imagine a system that has different math, but it will be very hard to reason about it.
Those forces are also mediated by particles (called gluons, W, and Z bosons). But these particles are massive, charged and interacting, which caused them to behave quite differently and only act on short length scales.
* If you're like me and are clueless about the terms "hyperbolic", "too simple" etc. used in the plot, take a lot at this SE answer that explains it (relatively) simply (https://physics.stackexchange.com/a/110880)
* In his books the hard-SF writer Greg Egan has explored worlds with more than one timelike dimensions, see the discussion on HN ( https://news.ycombinator.com/item?id=36431620) and this comment on SE (https://physics.stackexchange.com/a/14106)
* For very short distances space-time dimensions get reduced from four to two, see this post by Sabine Hossenfelder(http://backreaction.blogspot.com/2013/05/dimensional-reducti...)