On the dimensionality of spacetime (1997) [pdf]
space.mit.edu
space.mit.edu
* 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...)
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.
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.
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 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.
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 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.
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.
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.
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.
[0]: https://en.wikipedia.org/wiki/Spacetime#Privileged_character...
“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.”
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.
Stable intelligences might be another condition to explore in such an analysis.
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
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.
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).
This appears to be the Bartini paper (1965) https://www.researchgate.net/publication/26408302_Relations_...
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 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?
This sort of intuition. Applies to electromagnetic waves, sound and gravity all alike.
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.
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