Just for fun and perhaps some edification, let's look at a very simple FRLW style metric expansion:

ds^2 = dt^2 - a(t)*dx^2

This would be a 1T, 1D space-time using a positive time-like convention. a(t) here is a function of time, generally called the scale factor. What does this say in the most general possible case? It says, the "proper distance", which I would tend to call the metric distance because "proper" there just isn't quite proper, between space-like events (those that occur at constant coordinate time, t = somethng), varies with time according to our a(t) function.

Ie, if we have events at (x1, t) and (x2, t), the metric distance between them is going to depend on t. If a(t) is an increasing function of time, then we can say "space is expanding", as the metric distance between those events increases with time.

Now, what would be meant by "time is expanding" in the same sense? Well, it would suggest we put a b(t) on the time part of our metric:

ds^2 = b(t) *dt^2 - a(t) * dx^2

That would mean the proper time between events at constant x would depend on the time coordinate t. Now, that's just a "silly clock", indeed. That's a clock whose hands are speeding up or slowing down with time-- proper time is proper time, and that's what a proper clock measures. So all we do there is make a substitution, dT^2 = b(t)*dt^2, and replace t by T and get rid of that silly clock. Doing that would change a(t) into some a(T), of course.

So making the time part depend on time is sort of a silly operation that can be transformed away -- and there would be an equivalent thing to do on the space side, have some a(x) where the metric distance depended on the x coordinate, and that's done all the time, Schwarzschild for instance. That doesn't seem so silly, of course, and that's sort of the difference in how we think of time coordinates and spatial coordinates. (yes, space and time certainly get mixed, but there is a difference between time-like and space-like coordinates. One observer's time is some mixture of the time and space of another observer, but his time is still time-like, IOW)

So making "time depend on time" just doesn't make much sense, although it might have some mathematical utility in some situations I can imagine. But what is the difference between this and expanding space? Well, that means distance depends on time. The reverse of that would be "time depends on distance".

The former is distance goes as a(t)*dx^2 and the latter would be simply time goes as b(x)*dt^2. And that happens all that time with various metrics. In Schwarzschild, our b(x) is simply (1 - R/r). That is, the proper time between events occuring at the same coordinate spatial location depends on the spatial location. That is the analog of the "proper/metric" distance between events occuring at the same coordinate time depends on the time.

We call the latter "expanding space", but we don't call the former expanding time. But you might say "time expands with distance" if you like there, but the notion of expandING, that "ING" implies a process occuring with time, not with space.

-Richard