What is the physical interpretation of imaginary lengths?
My question is about the meaning of imaginary lengths, which occur often in the solution to various numerical problems in Physics. Generally imaginary quantities are discarded as nothing but mathematical tools, but this also means our narrowmindedness, for in such a case we are escaping curiosity. The following is only an example of such a question.
A boat moves relative to water with a velocity $n$ times less than the river flow velocity. At what angle to the stream direction must the boat move to minimize drifting? Take the river width to be $d$. (Adopted from Problems in General Physics, <PERSON>, Mir Publishers, Moscow, 1979)
Let $\vec{v_0}$ be the stream velocity and $\vec{v'}$ be the velocity of the boat w.r.t water. Let $\frac {v_0}{v'}=n>0$ s.t. the boat is drifted. Let $\vec{v'}$ make an angle $\theta$ with $\vec{v_0}$. Then time taken to cross the river, $$ t=\frac {d}{v'\sin \theta} $$ In this time interval, drifting of the boat, $$ x=(v'\cos \theta+v_0)t $$ $$ =(\cot \theta + n\csc \theta)d $$ For minimum, $$ \frac {dx}{d\theta}=-d \csc\theta(\csc\theta +n\cot \theta) = 0 $$ Or, $$ \csc\theta +n\cot \theta=0 $$ Or, $$ \theta = \pi - \sec^{-1} (n) $$ In this case, drifting, $$ x_0 = d\sqrt {n^2 -1} $$
My Problem
I dared to investigate the other case, i.e. 397
when $\csc \theta =0$. Mathematically (as well physically, for there is no drift possible) this is impossible for $\theta \in \mathbb {R}$. But for $\theta \in \mathbb {C}$, this implies that the boat does have a drift, not a real and observable drift, but an imaginary drift $x_0 = id$. The problem begins here. As far as I know, physics is the subject of the observed and observable things. But then this definition of physics looks very naive when we come to acknowledge that electrons do exist, although we can never directly observe them in strict sense (or in other words, <PERSON>'s principle wouldn't allow us to do so!). But there are always indirect evidences. In the same spirit, does the above solution point towards the existence of an alternate dimension, in which the boat does drift a distance $d$? This is not my personal theory, rather I have been motivated by the fact that time is taken as an imaginary dimension in <PERSON> spacetime. So, should I take it to be the traversal of the body (in this case the boat) from one point of time to another, or a dilation or contraction of time? Any help would be appreciated. 397