George Sparling, a mathematician at the University of Pittsburgh, has came up with an alternative space-time idea that involves six dimensions, with the extra two being time-like:
Sparling’s paper, which was published in the Proceedings of the Royal Society A, lays the groundwork for his theory. He explains how spatial dimensions contain positive signs (e.g., 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.
“In three dimensions, the formula reads s2 = x2 + y2 + z2,” Sparling explained to PhysOrg.com. “Our standard spacetime has four dimensions, but the formula has a critical minus sign: s2 = x2 + y2 + z2 - t2. The Lithuanian Hermann Minkowski invented this idea, which was published just six weeks before he died. Indeed, [Sir Roger] Penrose, for one, says that special relativity was not a finished theory until Minkowski's famous Raum und Zeit [‘Space and Time’] paper.”
Up until now, Sparling explains, most theories concerning extra dimensions have dealt 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..