### Why 3-D?

I recently listened to an interview on Public Radio with physicist Lisa Randall. She is a top theorist on foundational theory of why the universe is the way it is. That means string, branes, and such. One of the venerable questions is: why is space three-dimensional? It may seem natural to have three dimensions of space and one of time, but mathematically there can be any number of dimensions (think of specifying points using 4, 10, etc. variables.) Physicists, including Lisa, say they can't see why space *had* to have three dimensions. (Check out http://arxiv.org/abs/hep-th/0506053 ) However, they have come up with reasons why three large-scale dimensions would be more likely to expand out of a larger set (usually thought of as 10 or 11) of original, perhaps tiny dimensions. Below follows a statement adapted from my post to http://www.radioopensource.org/the-holy-grail-of-physics/ in response to the interview, and outlining my own efforts to answer this question. (I’ll also find out if the automatic editing puts in links for me.) Also, happy Cinco de Mayo, late as it is!

I have been working myself on the question, why are there three *large* dimensions of space? (There are probably more, like a total of 10 or 11 space dimensions, but the rest are curled up very small or otherwise inaccessible.) After extrapolating electromagnetic interactions to spaces of other dimensions, I found at least two arguments:

1. In spaces with other than one or three dimensions, an oscillating charge does not project the same *average* field along the axis of oscillation as the rest value. That is due to two things: the combination of "projection" of its retarded distance - where it would be had it continued at the velocity it had when light left it - and the distortion of the field due to Lorentz contraction, which weakens it to

gamma^(1-N) the value it has at rest. N is the number of large space dimensions. (We also must take into account the Doppler shift of projection intervals. Heh, it’s not quite as complicated as it sounds.) Remember that the Coulombic electric field intensity is given as E = qr^(1-N) due to field spreading. This amplifies the effect of the oscillating charge’s apparent position being close (projected from approaching cycle) to a second “target” charge at rest. It increasingly swamps the weakening effect of the gamma factor as N goes above three and is incorrect when N = 2. That would impose a net force on a second "target" charge unequal to that on the oscillating charge, and violate conservation of momentum and energy. The one-dimensional case is ruled out due to infinite potential energy as is the 2-D case (why didn’t A. K. Dewdney realize that about the 2-D Planiverse?)

2. Let two charges be connected by a reasonably rigid rod. Then, accelerate the rod along its length. The combined force between the charges will be derived from the sort of considerations given in (1.), as the projected field of each charge catches up to the other charge. Then we must take into account the extra force created by the action of acceleration on the relativistic stress-correction to the momentum and energy of the rod. Only in three dimensions of space does that equal in net the effective inertia the charges should have given their potential energy. (In higher dimensions, taking the integral of f = q1q2/r^(N-1), that potential w.r.t. infinity is: -q1q2r^(2-N)/(2-N). )

I hope I can publish the full development of this before long. I don’t think anyone else has an explicit proof that N *must* equal three, only reasons it was more likely to form, or oddities like being unfriendly to life, distorted wave propagation (see Barrow and Tipler’s _The Anthropic Cosmological Principle_ for great discussion of this.)

Neil Bates

## 7 Comments:

"Why 3-D?"Energy conservation.

Hi again, Neil,

I'm looking for somebody to collaborate on a series of high-impact papers on the "entropic" anthropic principle, as this interpretation is intricately linked to what you've discovered here.

You know where my blog is, but my email is island5@earthlink.net.

Let me know if you are interested in rocking the boat... a LOT... ;)

hai neil..

stopping by and just wanna say hi... :D

best regards,

-na

Maybe the fact that 3D is the only space that has stable atoms and planetary orbits is a clue.

And 3D is also the only space which can have knots is perhaps a clue too.

Recently I say an article that used a method to actually calculate the number of dimensions in a computer simulation using a quantum mechanical model.

(can't find the link anymore, sorry).

Hi Neil,

Why is space 3-dimensional? Well, first of all the space-time we all live in is actually 4-dimensional, right?

And, the reason space is not only 2-dimensional, or less, is quite simple; a two-dimensional organism's mouth is also its anus. Even an amoeba is not a two-dimensional organism.

Why not 10 large dimensions? I think the answer is the same. Imagine a 10-dimensional organism…

It seems that the 4-dimensional space-time is all a little amoeba can handle without eating its own shit for breakfast.

http://upload.wikimedia.org/wikipedia/commons/5/55/8-cell-simple.gif

Anonymous, thanks for commenting. Your answer to the question of why space has three dimensions is an example of "anthropic reasoning," that is, the anthropic principle. That means viewing the way the universe is in terms of viability for life and especially sentient intelligent life (not just humans, the "anthro" is a metaphor for observers in the broad sense.) So you are saying, if the universe had other than three space dimensions, then conditions would be difficult or impossible for living (and especially for intelligent beings.)

Most experts agree with you on that point. One of the first to make that same point was G. J. Whitrow, in 1955. A quote from his paper (Br. J. Phil. Sci.

6, 13 sounds much like your comment:"I suggest that a possible clue to the elucidation of this [the number of space dimensions] problem is provided by the fact that physical conditions on the Earth have been such that the evolution of Man has been possible ... this fundamental typological property of the world ... could be inferred as the unique natural concomitant of certain other contingent characteristics associated with the evolution of the higher forms of terrestrial life, in particular Man, the formulator of the problem."

The trouble is, what's to stop a universe from existing anyway even if it can't support life? I have no trouble with a philosophical notion that some ultimate reality or even "God" would purpose the universe such as to support life, hence creating it as 4-D space-time.

However, I wanted to find a purely physical rationale for why space had to have three dimensions only. I did so, in terms of whether conservations laws could be consistent when electromagnetic interactions were extrapolated to spaces where N <> 3 (i.e., number of observable space dimensions was not three.) I found internal contradictions as explained in the main post. Hence, such spaces cannot exist if we believe nature expresses coherent and consistent laws. Note that proving why space must have three "large" dimensions does not limit the number of tiny "compact" dimensions as per modern string theory. That's because the scale of such dimensions is so tiny that classical interactions (such as I used in my proof) are not relevant.

PS: Keep on speculating!

Also, Rob: sorry I have been away for so long, and didn't mention you as well. Your point is similar to that of Anonymous, and broader (not just about life but relevant to that as well) yet that still doesn't say why such a non 3-D universe could not even exist.

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