To the Subject, Time is relative, and Space is absolute; to the Object, Space is relative, and Time is absolute.
The definition of Velocity and Entropy; a comparison
I have written about my misgivings about the original, now somewhat discarded, 150 year old definition of entropy - and its consequences.
Those consequences being, in part:
The inability to come to grips with reality without throwing exception after exception, having to define such a thing as "negative" entropy to describe something positive, and getting into contradiction after contradiction and argument after argument when trying to define physical processes and conditions.
What, for instance, correctly and unequivocally defines thermodynamic closure? What are the universal conditions for the Second Law of Thermodynamics? How is entropy defined at low temperatures? At the beginning of the universe?
The original definition
My take is that things would have been and would be different, in the general concept of the world, if Rudolf Clausius had defined entropy the other way around;
not
S=Q/T (Quantity of heat, or energy, by / Temperature),
but
S=T/Q (Temperature by / Quantity of heat, or energy),
As division by zero is forbidden, the factor that may theoretically and independently reach that value (in this case, T) generally should not be put in the denominator of a fraction.
However, this was done; and more than just once.
Why?
The general situation
Starting from a blank state, defining a relationship between two factors as one divided by the other can be done in two, inverse ways; and this without misinterpreting reality, while laden with dire consequences in both cases.
So the question is primarily a question of practicability - with a profound effect on the perception of nature.
You could, for instance, describe the number of chairs (4) to a table (1) as 1 table to 4 chairs or 4 chairs to 1 table.
Or, a bit more intricate, the volume of air in a room (m³) per person (p) as
a) m³/p or
b) p/m³
(using m³ to visualize)
The first equation (a) is not defined for 0 persons (and rightly so), while the second (b) is ("there is no-one in the room").
In case of the volume of air being 0, the first expression spells suffocation or no room, while the second makes no sense.
Of course, in a room of fixed dimensions, people entering or leaving are the more independent variable, and can easily reach zero.
So it seems clever not to take them as the denominator, and define the relationship as persons per volume or p/m³ instead.
However - and that is the point - this is not done.
The necessary or resulting amount of air per person is usually expressed in as volume per person or m³/p - and never mind the zero, as any statement concerning an empty room is useless, and sensible use of the equation begins with the first person ("1") entering it.
Of course, one could enhance the expression by calculation 0.5 persons (children) or pets (0.25 persons) all the way down to one amoeba, but that only proves the point:
Although mathematically silly, we use the "wrong" one of the two possible ways to express a relationship between two variables.
Why?
Because it is easier, more intuitive and practicable in everyday life, and never mind the fringe silliness.
But more importantly, the question is:
Can you define the zero condition?
A more technical example
Wikipedia states that the current definition of speed was first arrived at by Galileo Galilei, who laid down that "speed" (v) should furthermore be defined as "distance by time", or:
v = d / t
Now, it is of minor importance if it was indeed Galileo, or someone else, or when speed was defined exactly. The importance lies in the fact that:
This was a decision to make; the relationship between space and time could have been set in two ways:
v = d / t ( = distance by time )
or
v = t / d ( = time by distance )
Before one definition was chosen over the other for all future, physical speed, other than "entropy" some centuries later, was not a new concept; it had been an issue over millennia, in the realms of sports, military, commerce and many more.
That said, since it obviously up to a certain point in history had not been defined in a mathematical equation, how had it been defined before?
And had this historical definition been enhanced or discarded by Galileo, if we credit him with the new mathematical definition?
And why did he favor "distance by time" over "time by distance"?
We will have to speculate
There are two possibilities:
Galileo formulated the definition as it was being used, by giving contemporary common sense a mathematical expression.
He revolutionized science (as he was wont to) by defining it contrary to contemporary common sense.
This is just for clarification; the historical common sense definition of speed is of less importance than the consequences each definition would have.
And there are, again, two options for that definition:
v = d / t or v = t / d
Let's have a look at these two options:
Velocity (v) or rather speed is now defined as distance by time or d / t, and measured in km/h, m/s, mph or something equivalent.
Wikipedia: "Italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time. In equation form, that is v=d/t"
However, I suppose Roman legions, comparing troop movements, and even ice age hunters used similar equations - it's a natural definition for a sentient being, at once a predator and prey, to cover movement in time and space, and defining it by the relation of distance to time.
Speed, defined as distance by time or d/t, increases with distance and decreases with time. This worked for eons; we instinctively appreciate speed that does so, and that speed does so.
But if you wish to validate a definition or formula, you test for the extremes - infinity and zero; zero is usually enough.
And what do we find?
for d = 0 ( immobility )
Speed = v = d/t; 0/x = 0
That's OK, but
for t = 0
Speed = v = d/t; x/0 is *not defined*
- making the formula v = d/t invalid at or near that point - and with that, universally.
For millennia, that had bothered absolutely no-one; t = 0 was an improbable condition - it meant that an object could be in two places at the same time.
And when it finally did, Einstein came along and defined the speed of light as the ultimate speed with ~300.000.000 m/s; in other words, t can never be 0, not even for the shortest of distances.
Now, whenever v = d / t nears the speed of light, relativity becomes relevant and strange things happen to time and space.
Both ends of the spectrum - zero speed ( immobility, d = 0 ) and ultimate speed ( the speed of light ) are now validly defined.
Problem solved
Now, imagine an alternative, perhaps more leisure society, where an alternative Galileo Galilei again decides to measure speed by considering the distance covered and the time it takes, but this time as Duration or "votever" (vo), the time it takes to cover a distance.
If this is hard to imagine, think of duration as a "two day's march" for a legion of soldiers or a certain stretch of ocean as a "two week's cruise" - a perfectly valid and comparable expression of - well, speed. Rapidity. Duration.
And it is in no way out of the question that this was, at least, one pre-mathematical definition of speed, again, for eons.
In equation form, that would be vo=t/d to cover movement in time and space, defined by the relation of time to space ( = time by distance ). This definition is *just as valid* as the non - alternative one.
Duration, thus defined, decreases with distance and increases with time.
Again we test for the extremes -
And what do we find?
for t = 0 (it's over already)
Duration = vo = t/d; 0/x = 0
i. e., if the time needed to cover a certain distance is zero, then the duration to cover this distance is zero; this may be duh, and mathematically valid - but really impossible, as the alternative Einstein will invariably find out, and define the duration for light as the minimal duration of ~1/300.000.000 s/m; again, t can never be 0, not even for the shortest of distances.
So far, so good - or rather, bad; for as it seems, t=0 would in this case be mathematically valid - but not physically.
But, before that there is again the problem of being in two places at the same time, if the distance is not 0 as well; this is because an alternative definition has no effect on reality - there is no danger of that - just on it's perception.
And, almost immediately, one can see that this variant definition is somehow much more convoluted than the primary one. Already it seems wiser not to pick it.
But even worse,
for d = 0 ( immobility )
Duration = vo = t/d; x/0 is *not defined*
In other words, in this definition of speed, immobility is not defined.
And that is bad, as immobility exists.
Well, not really;
In the realms of infinity and eternity pertaining to the universe, there is indeed no such thing as *true* immobility; given enough space and time, everything moves against everything else.
If only the time frame is chosen big enough, no two bricks, no two rocks, no two molecules have ever not moved relative to one another; and off we go into the arguments and definitions, and those ex negativo, and qualifications, and exceptions - hey, we may even have to construe something as esoteric as "negative duration" to get out of this mess we created, by defining speed the wrong way around.
Get my drift?
Turn it around, define v as d / t, and without having misrepresented reality one way or another, all you need to throw now is one exception only, which is real, and let Einstein take care of that.
Was Galileo just lucky to pick the right one out of two possibilities, the one where the zero condition is excepted by nature (with the end speed of light), or genius enough to think it through?
We may never know. But then, again, other than entropy, speed was already known.
And so, that is what I *think* will happen if we turn Clausius' mangled definition of entropy on its head:
The mess clears up instantly
And then all we have to do is calculate *everything* once more.
Just once more. ; - )