I also understood this using parametric equations, although I simplified to t, x(t) and y(t), to aid in visualization. So then I was looking at my mental image, and I thought “but what About memory?” At any particular point on my curve, the observer in that point knows what the curve looks like in one direction(past), but not the other. I get that both directions along the curve are determined, but why would my mind contain information about exactly one?
This is off the top of my head, so it may be total bullshit. I find the idea of memory in a timeless universe slippery myself, and can only occasionally believe I understand it. But anyway...
If you want to implement a sort of memory in your 2D space with one particle, then for each point (x0,y0) in space you can add a coordinate n(x0,y0), and a differential relation
dn(x0,y0) = δ(x-x0,y-y0) sqrt(dx^2 + dy^2)
where δ is the Dirac delta. Each n(x0,y0) can be thought of as an observer at the point (x0,y0), counting the number of times the particle passes through. There is no reference to a time parameter in this equation, and yet there is a definite direction-of-time, because by moving the particle along a path you can only increase all n(x0,y0) for points (x0,y0) along that path.
A point in this configuration space consists of a “current” point (x,y), along with a local history at each point. If you don’t make any other requirements, these local histories won’t give you a unique global history, because the points could have been visited in any order. But if you impose smoothness requirements on x and y, and your local histories are consistent with those smoothness requirements, then you will have only one possible global history, or at most a finite number.
one small problem, nothing is moving. how can you have an observer when every bit of that observer is part of the existing static universe… observer implies a moving entity and no motion exists in a static universe.