Spacetime - Privileged Character of 3+1 Spacetime

Privileged Character of 3+1 Spacetime

There are two kinds of dimensions, spatial (bidirectional) and temporal (unidirectional). Let the number of spatial dimensions be N and the number of temporal dimensions be T. That N = 3 and T = 1, setting aside the compactified dimensions invoked by string theory and undetectable to date, can be explained by appealing to the physical consequences of letting N differ from 3 and T differ from 1. The argument is often of an anthropic character.

The implicit notion that the dimensionality of the universe is special is first attributed to Gottfried Wilhelm Leibniz, who in the Discourse on Metaphysics suggested that the world is "the one which is at the same time the simplest in hypothesis and the richest in phenomena." Immanuel Kant argued that 3-dimensional space was a consequence of the inverse square law of universal gravitation. While Kant's argument is historically important, John D. Barrow says that it "...gets the punch-line back to front: it is the three-dimensionality of space that explains why we see inverse-square force laws in Nature, not vice-versa." (Barrow 2002: 204). This is because the law of gravitation (or any other inverse-square law) follows from the concept of flux and the proportional relationship of flux density and the strength of field. If N = 3, then 3-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of radius r has area of 4πr ². More generally, in a space of N dimensions, the strength of the gravitational attraction between two bodies separated by a distance of r would be inversely proportional to rN−1.

In 1920, Paul Ehrenfest showed that if we fix T = 1 and let N > 3, the orbit of a planet about its sun cannot remain stable. The same is true of a star's orbit around the center of its galaxy. Ehrenfest also showed that if N is even, then the different parts of a wave impulse will travel at different speeds. If N > 3 and odd, then wave impulses become distorted. Only when N = 3 or 1 are both problems avoided. In 1922, Hermann Weyl showed that Maxwell's theory of electromagnetism works only when N = 3 and T = 1, writing that this fact "...not only leads to a deeper understanding of Maxwell's theory, but also of the fact that the world is four dimensional, which has hitherto always been accepted as merely 'accidental,' become intelligible through it." Finally, Tangherlini showed in 1963 that when N > 3, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.

Max Tegmark expands on the preceding argument in the following anthropic manner. If T differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if T > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.) If N > 3, Ehrenfest's argument above holds; atoms as we know them (and probably more complex structures as well) could not exist. If N < 3, gravitation of any kind becomes problematic, and the universe is probably too simple to contain observers. For example, when N < 3, nerves cannot cross without intersecting.

In general, it is not clear how physical law could function if T differed from 1. If T > 1, subatomic particles which decay after a fixed period would not behave predictably, because time-like geodesics would not be necessarily maximal. N = 1 and T = 3 has the peculiar property that the speed of light in a vacuum is a lower bound on the velocity of matter; all matter consists of tachyons.. However, signature (1,3) and (3,1) are physically equivalent. To call vectors with positive Minkowski "length" timelike is just a convention that depends on the convention for the sign of the metric tensor. Indeed, particle phyicists tend to use a metric with signature (+−−−) that results in positive Minkowski "length" for timelike intervals and energies while spatial separations have negative Minkowski "length". Relativists, however, tend to use the opposite convention (−+++) so that spatial separations have positive Minkowski length.

Hence anthropic and other arguments rule out all cases except N = 3 and T = 1 (or N = 1 and T = 3 in different conventions) — which happens to describe the world about us. Curiously, the cases N = 3 or 4 have the richest and most difficult geometry and topology. There are, for example, geometric statements whose truth or falsity is known for all N except one or both of 3 and 4. N = 3 was the last case of the Poincaré conjecture to be proved.

For an elementary treatment of the privileged status of N = 3 and T = 1, see chpt. 10 (esp. Fig. 10.12) of Barrow; for deeper treatments, see §4.8 of Barrow and Tipler (1986) and Tegmark. Barrow has repeatedly cited the work of Whitrow.

String theory hypothesizes that matter and energy are composed of tiny vibrating strings of various types, most of which are embedded in dimensions that exist only on a scale no larger than the Planck length. Hence N = 3 and T = 1 do not characterize string theory, which embeds vibrating strings in coordinate grids having 10, or even 26, dimensions.

The Causal dynamical triangulation (CDT) theory is a background independent theory which derives the observed 3+1 spacetime from a minimal set of assumptions, and needs no adjusting factors. It does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. It shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time, but spacetime becomes 3+1-d in scales significantly larger than Planck. So, CDT may become the first theory which doesn't postulate but really explains observed number of spacetime dimensions.