## On Scaled Patches and Time-Warping

Suppose we have an idealized canal of width , on which a fluid flow has been established in some remote past. Let us focus solely on the dynamics of the surface. Pick a spot along the canal which we will call . Next pick a spot along the width of the canal, which we will monitor. Pick a second spot down the canal, some distance from the original spot we picked. Now let us assume that, up the canal at some remote point, a paper boat has been released. We will only care about the boat if the boat passes through which we have picked, and we will write down the resultant position at . Let us do this a number of times with any number of boats, and obtain a distribution of the position of the boat at , saving it. Next let us repeat the experiment, this time focusing on , save the resultant distribution at , and so on and so forth, until we are comfortable having mapped the totality of positions at . Let us next put together (stack, respecting the natural order of ) all the distributions we obtained at . We now have a discrete surface which we can smooth to obtain a *Pasquali patch*.

Let us now look at position which is the same distance as is from . Having defined the dynamics of the system (from a single *Pasquali patch*), the dynamics at can be theoretically described by . We can therefore ascertain the probability that we will find the boat at along the width of the canal. In fact, at , very large, we can ascertain the probability that the boat will be at any position along the width. It should be close to . More importantly, a great distance from the origin (any distance, not necessarily a distance ), the position probability is aptly described by . See **Figure 1** and **Figure 2**.

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This simple thought experiment brings about several questions. What if the dynamics of the surface system are described by the *Pasquali patch*, but at points which are not a distance apart? In other words, what if the description is apt but at points that are not linear in distance? This curious situation suggests a time anomaly, and therefore a manner in which we can *measure* time warps (by measuring the actual time differences between *Pasquali patches*). See **Figure 3**.

Next, we looked at the surface dynamics of the system. If we add a depth variable to the canal, we can in theory produce a *Pasquali cube*, which would measure the dynamics of any point on the cross-section a discrete distance down the canal (and any distance very far from our origin).

A third question arises when we consider the same canal, but whose width opens by a scalar (linear) amount a distance from our chosen origin. There is no reason we cannot “renormalize” the width (set it equal to 1 again) at a point some set distance from our chosen origin, and proceed with our analysis as before. See **Figure 4**.

In a subsequent post I'd like to reconsider the slit experiment under this new light, see where it takes us.