LOL, Nigel. Reminds me of a time when I was in Scotland and the people (Aussies) I was travelling with *insisted* that we go to the flashy Loch Ness visitor centre, whereas I preferred to sit by the lake.

I’ve just finished reading Prof. Smolin’s *Trouble with Physics*, which I’ve read in a random, non-linear way in bits and pieces over a long time. He has a very deep understanding of some vital concepts underlying spacetime which I find helpful to clarifying the issue of the role of time dimension(s).

Pages 42-43 are really useful. Prof. Smolin explains curvature of spacetime very simply there, especially figure 3 which plots the deceleration of a car as space (i.e. distance in direction of motion) versus time.

The curvature of the line (e.g. for space = time^2), is “curved spacetime”.

I think this is a very good way to explain the curvature of spacetime! Quite often, you hear criticisms that nobody has ever seen the curvature of spacetime, but this makes it clear that the general relativity is addressing physical facts expressed mathematically.

It also makes it clear that “flat spacetime” is simply a non-curved line on a graphical plot of space versus time. Because special relativity applies to non-accelerating motion, it is restricted to flat spacetime. Profl Smolin writes (p42):

“Consider a straight line in space. Two particles can travel along it, but one travels at a uniform speed, while the other is constantly accelerating. As far as space is concerned, the two particles travel on the same path. *But they travel on different paths in spacetime*. The particle with a constant speed travels on a straight line, not only in space but also in spacetime. The accelerating particle travels on a curved path in spacetime (see Fig. 3).

“Hence, just as the geometry of space can distinguish a straight line from a curved path, the geometry of spacetime can distinguish a particle moving at a constant speed from one that is accelerating.

“But Einstein’s equivalence principle tells us that the effects of gravity cannot be distinguished, over small distances, from the effects of acceleration. Hence, by telling which trajectories are accelerated and which are not, the geometry of spacetime describes the effects of gravity. The geometry of spacetime is therefore the gravitational field.”

What I like most about it is that Prof. Smolin is explaining spacetime by matching up one spatial dimension with one time dimension.

Extend this to three spatial dimensions, and you would naively expect to require three time dimensions, instead of just one.

The simplification that there appears to be just one time dimension surely arises because the time dimensions are all expanding uniformly, so there is no mathematical difference between them.

In three spatial dimensions, if all the spatial dimensions are indistinguishable it is a case of spherical symmetry. In this case, x = y = z = r, where r is radial distance from the middle.

Hence, three dimensions can be treated as one, *provided that they are similar:* t_1 = t_2 = t_3 = t.

So the reason why three time dimensions can normally be treated as one time dimension is that time dimensions are symmetric to one another (unlike spatial dimensions). So the symmetry orthagonal group SO(3,3) is equivalent to SO(3,1), provided that the three time dimensions are identical.

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