# Is 3+1 spacetime as privileged as is claimed?

I've often heard the argument that having 3 spatial dimensions is very special. Such arguments are invariably based on certain assumptions that do not appear to be justifiable at all, at least to me. There is a summary of arguments on Wikipedia.

For example, a common argument for why >3 dimensions is too many is that the gravitational law cannot result in stable orbital motion. A common argument for <3 dimensions being too few is that one cannot have a gastrointestinal tract, or more generally, a hole that doesn't split an organism into two.

Am I being overly skeptical in thinking that while the force of gravity may not be able to hold objects in stable orbits, there most certainly exist sets of physical laws in higher dimensions which result in formation of stable structures at all scales? It may be utterly different to our universe, but who said a 4D universe must be the same as ours with one extra dimension?

Similarly, isn't it very easy to conceive of a 2D universe in which organisms can feed despite not having any holes, or not falling apart despite having them? For example, being held together by attractive forces, or allowing certain fundamental objects of a universe to interpenetrate, and thus enter a region of the body in which they become utilized. Or, conceive of a universe so incomprehensibly different to ours that feeding is unnecessary, and self-aware structures form through completely different processes.

While I realise that this is sort of a metaphysical question, is 3+1 dimensions really widely acknowledged to be particularly privileged by respected physicists?

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Well, it seems like the main point of your question is whether it's possible for stable bound structures to form in higher-dimensional spaces, which is a perfectly fine (and not at all metaphysical) question. –  David Z Jun 1 '11 at 1:15
If one wants to ask whether 3+1 is privileged, one should also consider the possibility of more than 1 temporal dimensions. :-) –  Willie Wong Sep 23 '11 at 17:58
Nature likes to minimize things, its a universal feature. Imagine living in a 3343432411111111110122-dimensional universe, that would require an explanation, not 3+1. –  Hobo Dec 2 '12 at 14:54
You may replace 'likes to' by 'tends to'. And 3+1 is very special in beeing very small, so she is very privileged. –  Hobo Dec 2 '12 at 15:44
Related: My answer to: physics.stackexchange.com/q/41109 –  Dimensio1n0 Jun 17 '13 at 10:23

No. While there are some arguments for why 3 spatial dimensions are a good place to live in, the answer to the question why our universe has 3 large spatial dimensions is presently not known.

Karch & Randall wrote a paper on the issue some years back: http://xxx.lanl.gov/abs/hep-th/0506053 They consider some higher dimensional space filled with objects of different dimensions that have some interactions among each other and argue that 3 dimensional ones are among those most likely to dominate. It's an argument though that is not widely accepted due to the assumptions they have to make for this to work.

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Science fiction writer (but also published physicist) Greg Egan has put quite a bit of work into investigated a universe with 4+0 dimensions: http://gregegan.customer.netspace.net.au/ORTHOGONAL/ORTHOGONAL.html Some of it is quite ingenious, eg. assuming a compact universe guarantees that the (modified) wave equation doesn't have exponentially growing solutions and time appears, without the -1 in the spacetime metric, as the local gradient of entropy.

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but how does he prevent a moving object from rotating so that it is going backwards in time? There is no way to prevent this without disconnecting the rotation group. –  Ron Maimon Sep 9 '11 at 17:29
He's a bit quiet on this topic: google.com/… I'm expecting this topic to be touched in the sequel :-) –  Dan Piponi Sep 9 '11 at 18:49

I would like to share my view on this issue.

I think some answers with the word "anthropic" need not to be dismissed, but could be interpreted them in a deeper sense.

Anthropic should not be something derogatory, "just humans", as if we were not part of universe, instead perhaps it could be treated as concepts like "inertial frame of reference" are treated. A measure, a way to measure, a point of view, a frame of reference.

An imagination exercise:

Suppose one day a networking software is self aware.

Then is make some self replicas, and they ask themselves :

"Why we are on layer 7 of the OSI model?"

"Does it have something special?"

One of them would say "Because we can't live in lower layers then if the universe would be lower layered we wouldn't be asking things like this"

Another might say : "To live in layer 7, previous layer must exist to allow us, but, think on layer 0, our conversation are ultimately travelling through a cable for example, then we are at the same time, layer 0, layer 1, ... layer 7, the universe is not layer 7!!, its one or all layer at same time, depending "who" is measuring it, we can see it till layer 7, but the top we see doesn't mean it's the whole that exist, perhaps there are higher layers than 7, and lower than 0, that are forbidden to us, and can't be known at all"

I think 3D+1 is the top that our natural senses are aware of, with technology we could know or suspect other dimensions, as far we know, "conscious beings" can't rise in lower dimensions, but that perhaps is a prejudice, because whatever we call 3D+1 perhaps can be parsed in just 1D! (similar as in the above story), so we should review our statements, of course beings could exist in higher dimensions too (if they do not exist already, they would).

A single matrix in a paper although is within a 3D+1 it could contain higher dimensions, of course a matrix in a paper is not conscious, but nobody knows if a computer program will be aware of itself someday, that day, it will "live" and even "measure" a higher dimension, and again as the matrix in the paper, we would know that it coexist in a lower dimension too.

what are dimensions?

Regards

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Not sure if an adequate answer, but at least a very innovative one! –  arivero Sep 15 '11 at 14:01
I find the analogy between spatial dimensions and layers of abstraction in protocols to be... questionable, if not wholly inapplicable. –  Timwi Jul 31 '12 at 12:56

If you look at http://en.wikipedia.org/wiki/Static_forces_and_virtual-particle_exchange#Inverse_square_law you'll see a line of reasoning that doesn't seem to depend on the number of spacelike dimensions, yet still arrives at an inverse square law. I realize this isn't exactly a rigorous QED calculation (for which I feel far too stupid) but it makes me reconsider my former belief in non-privilege. If d = 3 is the only case that allows both radiation and conservation of energy, then that's just... wow.

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I think that the uncertainty principle is a pretty strong assumption to start with if you're trying to justify three spatial dimensions. –  Eric Aug 29 '11 at 18:58
The argument on Wikipedia is totally wrong. It is assuming that the particle absorption is independent of the geometry, and it is assuming that to get from point A to point B a virtual particle needs to travel at the speed of light. Unfortunately, these two idiotic mistakes cancel out each other in three dimensions. –  Ron Maimon Sep 10 '11 at 20:14
It is the minimum dimension required for the Weyl Tensor $C_{abcd}$to exist in the decomposition of the (completely covariant) Riemann Curvature Tensor $R_{abcd}$. That is kind of privileged. Or else, there would be no gravity in a vacuum (and thus, no long distance gravity, and no orbits, no free-fall)! And if it were any more, the gravity would weaken too quick (the inverse square law would become the inverse cube law, etc.)