t*t

Recently i was asked about the meaning of c² in the famous equation: E = mc². How could it be, that the maximum speed could be squared?

First of all – squares don’t change the origins, so it it actually doesn’t matter if the square is about a maximum or less. The interesting question therefore was the meaning of the square of speed.

And actually, the meaning of “meaning”. But regarding information, meaning is only something like the “amount of information”, we have about something of interest, because we understand something, we know its “meaning”, when we know its states and behaviors. We know, what a “tree” means, because we know how it looks and how it expands biosphere by offering some “extra space” for life, how it reacts on wind and storm: All that is told us by the simple word “tree”, is its “meaning”. Meaning gives us the ability to “reproduce” situations and processes, to trace the steps of the object of our interest and to foresee its actions and changes.

So the real question was, what the square of speed means. Looking at the “components” of speed, way and time, the question seems to be, what square of way and square of time means. Square of way is an area, but what is square of time?

Regarding the universality of time, i used the ansatz that the “time vector” cannot change direction, simply because the “direction” of change is the before-after-direction. Each and every change in each and every thinkable and unthinkable dimension will have only that simple direction: initial state -> end state.

So i looked at multiplications of vectors and decided to use the scalar product with cos(0) = 1. Then the question seems to be reduced to the question, what areas over time could mean and concluded that it means something like the speed of areas.

But honestly, i am not very happy about that reduction of the square of time, because that would mean something like t² = t, which is only correct in case, t would be something like a “normal” dimension, which sounds a little weird. Then there is the fact, that change is only something like “differential time”: change creates time, but it creates it differently, regarding the “velocity” of the system. Think of GPS systems which have to consider relativity’s time dilatation.

When i argue with change, i would have to prove, that the differential can define an integral, which follows the laws of vector multiplication – and then i should have to show, why i use the scalar product in case of time and the vector product in case of the way. Oh, it is not that i wouldn’t have an idea where to start. Remember the definition of information? It describes the requirements for change to be representable by mathematical formulas: elements of change/transformations for a system/quality/eigenschaft, which do not only change the state of the observed "thing", but do it repeatably and coherently – therefore creating a mathematical group and functions on the set of values/states of the observed thing. In that case, each progress, each behavior of that observed thing is a series of changes with a well defined “time’s arrow”. Alas, i haven’t much of an idea how to continue. Would it make sense to link the change directly to physical action, which is always the reason for each and every change in universe, which then may lead to some quantification of time according to Planck’s constant? Then add some pinch of Heisenberg’s uncertainty principle and cook it on low flame until it is a continuous set of transformations with some symmetries, so that it satisfies the requirements of Noether’s Theorem? Then, maybe, “time’s arrow” would take the form of a vector...

Or not...

But the conclusion – regarding E = mc², which is a proven correlation – is something like the maximum transportability of change of extendend objects (objects with surfaces), related to c as maximum transportability of change (speed of action) for massless (zero-dimensional?) objects, which could mean, that both the transportability of change of the “idealized” object, represented by its “center” and the protection of the coherence of the system, therefore the transportability of change for each subsystem under condition of conservation of the inner cohesion have to be considered (even with v = 0!)

Despite the fact, that this conclusion is based on the scalar product of a “time vector”, there seems to be some sense in that conclusion, because it would even give a hint, why our universe has just three continual and reversible dimensions – because it allows objects, described by only two parameters: center and surface, which is in case of balls center and radius. But that is also just an analogy from the two-body-problem.

Alas, those interesting conclusions depend on a very doubtful step – the reduction of the square of time. So question remains: what about t²?

Weird.