Is time slowing down?

Jose Senovilla suggests that the reason why the expansion of the universe appears to be accelerating is that time is slowing down, prior to a geometrical 'signature-change' in which the existing time dimension becomes another spatial dimension.

Senovilla's idea is expressed in terms of braneworld cosmology, but the basic idea can be explained more simply. Suppose that the geometry of a 4-dimensional universe is specified in terms of the following metric tensor field:

f(t)dt² + g

where g is the Riemannian (spatial) geometry on a 3-dimensional manifold. Suppose that the coordinate t ranges from - ∞ to + ∞.

If f(t) is negative everywhere, say f(t) = -1, then t is a timelike coordinate everywhere, and the universe has one temporal dimension everywhere. However, suppose that f(t) is only negative for t < 0, and suppose that it approaches 0 at t = 0, and becomes positive for t > 0. In this case, t is a timelike coordinate for t < 0, but a spacelike coordinate for t > 0. In that region of the universe in which t < 0, there are 3 spatial dimensions, and 1 temporal dimension. In that region of the universe in which t > 0, there are 4 spatial dimensions, and no temporal dimension. The signature change hypersurface which divides the two regions, is the set of points for which t = 0.

Now, an observer is represented in general relativity by a timelike curve γ, and the 'proper time' which lapses for an observer is represented by the integral along the timelike curve, ∫_t √(|< γ',γ' >|) dt. One can detect the approach of a signature-change hypersurface because f(t) becomes smaller and smaller, with the consequence that the proper time which lapses along timelike curves becomes smaller and smaller. This is most clearly seen in the case of those timelike curves in which the spatial coordinates are fixed, and the lapse of proper time is therefore ∫_t √(|<∂_t,∂_t>|) dt = ∫_t √(|f(t)|) dt.