He contradicts older theories

Apr 18, 2007 09:47 GMT  ·  By
Visual representation of the analytical structure underlying the spinorial theory
   Visual representation of the analytical structure underlying the spinorial theory

What is the nature of time ?

This is a fundamental question pondered since the time of Pythagoras, and still vexing scientists today. Mathematician George Sparling of the University of Pittsburgh, after analyzing different perspectives, offers an alternative idea: space-time may have six dimensions, with the extra two being time-like.

A splitting of five-dimensional spacetime into the Einstein equations and Maxwell equations in four dimensions was first discovered by Gunnar Nordstr?m in 1914, in the context of his theory of gravity, but subsequently forgotten.

In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set, and the phenomenon of having a space-time with compact dimensions is referred to as compactification.

In modern geometry, the extra fifth dimension can be understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U(1). Once this geometrical interpretation is understood, it is relatively straightforward to replace U(1) by a general Lie group.

Such generalizations are often called Yang-Mills theories. If a distinction is drawn, then it is that Yang-Mills theories occur on a flat space-time, whereas Kaluza-Klein treats the more general case of curved spacetime. The base space of Kaluza-Klein theory need not be four-dimensional space-time; it can be any (pseudo-)Riemannian manifold, or even a supersymmetric manifold or orbifold or even a noncommutative space.

Sparling explains how spatial dimensions contain positive signs ( Pythagoras' 3D space is expressed as the sum of the squares of the intervals in three directions, x, y, and z). Minkowski's time-like dimension, on the other hand, combines these three dimensions with the square of time displacement, which contains an overall negative sign.

Most theories concerning extra dimensions have dealt until now with space-like rather than time-like dimensions, which results in a "hyperbolic" rather than an "ultra-hyperbolic" geometry. However, Sparling notes that there are no a priori arguments for a hyperbolic geometry, and he looks into the possibility of a "spinorial" theory of physics, where six dimensions of space-time arise naturally.

Sparling's spinorial theory is based on Einstein's general relativity and Elie Cartan's triality concept, which can link space-time with two twistor spaces. Twistor spaces are mathematical spaces used to understand geometrical objects in space-time landscapes. Sparling explains spinors in the following way:

"In physics, the idea of a spinor stems from the finding that spectral lines of atoms seem to behave as if the angular momentum of the particles radiating photons was in half-integral units of the quantized spin (whose size is determined by Planck's constant). This was fully explained by Dirac's famous theory of the electron, which led him to successfully predict the existence of the positron."

"Consider this analogy: if you take a plate and hold it in one hand horizontally whilst twisting it under your arm backwards through 360 degrees, your arm ends up in the air after one rotation, and it needs another 360 degree rotation to get it back to the beginning," he said.

Twistors, then, are a special kind of spinor first introduced by Penrose (Sparling was a PhD student of Penrose). In Sparling's theory, the two twistor spaces are each six-dimensional, forcing space-time to also have six dimensions, in accordance with Cartan's unifying triality.

Because the twistor spaces' geometry is ultra-hyperbolic, the extra dimensions are time-like.

His theory opposes the "String Theory": "My work can be seen as a strong antidote to the present air of pessimism surrounding modern fundamental physics," Sparling said. "As is well-known, string theory has been roundly criticized for its lack of predictive power. String theorists have been reduced to an absurd reliance on the anthropic principle, for example. Here I have a clear-cut prediction, which goes against the common wisdom, which gives experimenters a target to go for: first find the extra dimensions, then decide their signature. Of course I could be proved wrong, but the effort to decide is surely worthwhile."