Galilean principle of relativity

[roineust]

DH,
I don't know in which sense, since i still don't understand the use of 'invertibility'.

 

[Dale]

What is the problem with multiple time at the same state?

If you have multiple times at the same state then you cannot say:

time depends on "the state of the matter"

If you have multiple times for one state then there is something in addition to the state which is needed to determine the time. In other words, the time depends on something else in addition to just the state.

I think that it is always correct to say "the state of matter depends on time", assuming a system which has more than one state. In addition, it is sometimes correct to say "the time depends on the state of matter". It is never correct to say "the time depends on the state of matter and not vice versa".

 

[roineust]

DaleSpam,

What is the reason that it is OK to say the above parts but NOT OK to say the other above part? Does it have to do with math, experimental results, both of them or philosophical view? What ever kind of reason it is, please elaborate, since myself, i do not exactly understand, to what kind of dark street corner of 'physics city' i got myself in.

 

[Dale]

I am not sure what is unclear. The English phrase "y depends on x" translates to the mathematical expression $y=f(x)$. You can always have $State=f(time)$ but you cannot always have $time=f(State)$.

 

[roineust]

You are saying that something i am suggesting, misuses the very basic definition of the term function? Because, one reply back and further up, if i am not wrong, you address the term 'invertibility', which is something not exactly the same as having something that is for sure, not a function, if i understand correctly, is that right or wrong and why? Is the property of invertibility, in this case we are discussing, a matter of an axiomatic definition or is it only based on having the math correct or on experimental results? If i decide, that because of my world view ("No more than 3 dimension in the universe - but yes, more than one time in every state of matter"), do i arbitrarily decide about something that is not-invertible, which other views (such as "Yes there are more than 3 dimensions in the universe") do not do - or is it, that by this view, i am not deciding arbitrarily, axiomatically, that something is NOT invertible, but actually, i am erroneously misusing the very basic definition of the term function? Am i trying to invert a function that is not invertible and in this sense i am misusing the definition of a function, if so, why and how?

 

[Dale]

Yes, you are misusing the very basic definition of a function. The very first sencence of the Wikipedia entry covers this:
http://en.wikipedia.org/wiki/Function_(mathematics )

"In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output" (emphasis added).

Again, consider a periodic state. Each value of time maps to one and only one state, but each state maps to an infinite number of times. So the map from time to states has the property that each input is related to exactly one output, but the map from states to time does not have that property. So the map from states to time is not a function meaning there is no function f such that time=f(State), and therefore it is not correct to say that "time depends on the state" in this case.

 

[roineust]

OK,
Lets take a person riding his bicycle sometimes pedaling, sometimes not.

When you are saying "...but each state maps to an infinite number of times..." what you mean is, that for all the snapshot pictures i will take of him riding his bike, there might be more than one picture that his legs pedaling, are positioned the same - since there is a cycle of pedaling the bike forward, and hence, there will be more than one snapshot ("state"?) that look exactly the same (if it doesn't have a time tag printed on it) but on the other hand, you say, for every time tag
i will choose, there can be only one snapshot picture that can be attached to that time tag ("...Each value of time maps to one and only one state...") ??

Did i understand correctly?

 

[Dale]

Yes, assuming that the position of the pedal captures all of the information about the state.

 

[roineust]

OK,
So now, i am asking - why can't we switch this whole view:

Up until now, we had a certain time resolution, that we decided in advance, every 'snapshot' was at a different 'jump' from the one before and the one after, in the magnitude of your decided time resolution, be it 0.1 seconds or 10^-6 seconds or 10^-50 seconds.

Now, for that time axis, let's call it the 'first' axis, we have something else, new - every time there is an acceleration change, there is taken a new snapshot - that is our new rule - 'snapshots', means acceleration change. These 'snapshots' of acceleration changes, can be taken, between different accelerations or between acceleration and constant speed. They don't necessarily have to be at the same constant 'pace', as with the time tags, in the previous view of things, it can even be a million years between one snapshot and the one that comes right after it (poor biker), and then again it can also be 10^-38, between one snapshot and the next one.

Now, in this 'switched' axis view of things, on the second axis, you can order the snapshots, according to the assumption that within each snapshot, all the information about previous snapshots, is already included - Every old acceleration changes 'snapshots', are already included in each and every new snapshot taken - A snapshot, includes all the previous accelerations before it, so from the amount of information included in each 'snapshot', you can tell if a snapshot comes before another snapshot or after it.

So, if you replace order of identical 'snapshots', you have extra or a lack of information, just as with the 'regular' function, that we discussed above - In that situation we discussed above, if you have the exact same position of the legs on the pedals, still, actually, each and every one of them should have a time tag printed on them, so if, because we have two identical 'snapshots', where the paddling legs are at the exact same position, and i switch an older 'snapshot', with a newer one, i get one 'snapshot' with extra time tag information and another with a lack of time tag information.

Well, just the same is with the axis switched view, that i describe here, only you have the 'pool ripples', which are the 'state' of the bicycle matter, that includes within it, the latest acceleration that was paddled in, and also all the previous information of that sort, that is, instead of the time tags information.

Have i made a too big mess of the whole scenario, or pushed the analogy to a place where the ability to mathematically describe it rigorously, is damaged, or neither and just again, included a mathematical error somewhere in this description?

 

[Dale]

Up until now, we had a certain time resolution, that we decided in advance, every 'snapshot' was at a different 'jump' from the one before and the one after, in the magnitude of your decided time resolution, be it 0.1 seconds or 10^-6 seconds or 10^-50 seconds.

There is no mainstream physical theory in which time is discrete like this. There are no "snapshots".

Now, for that time axis, let's call it the 'first' axis, we have something else, new - every time there is an acceleration change, there is taken a new snapshot - that is our new rule - 'snapshots', means acceleration change. These 'snapshots' of acceleration changes, can be taken, between different accelerations or between acceleration and constant speed.

Similarly, there is no mainstream physical theory in which states are discrete. Even in quantum mechanics the states themselves are vectors, and it is operators on the states which may be quantized.

Now, in this 'switched' axis view of things, on the second axis, you can order the snapshots, according to the assumption that within each snapshot, all the information about previous snapshots, is already included - Every old acceleration changes 'snapshots', are already included in each and every new snapshot taken - A snapshot, includes all the previous accelerations before it, so from the amount of information included in each 'snapshot', you can tell if a snapshot comes before another snapshot or after it.

It would really be better if you would use standard terminology instead of trying to invent new terminology. When I think of a snapshot I think of a photograph, which has information about the position of objects, but no information about their velocity. Is that what you are trying to convey?

The state of a system is determined by both the (generalized) positions and the velocities. So there is not enough information in a snapshot to reconstruct the state even at the time of the snapshot, let alone at previous times.

If you mean to include the velocities in what you are calling a "snapshot" then you should use the standard term "state" which describes all of the position and velocity information about the system at a given point in time.

EDIT: also, I have completely lost the connection between this present discussion and the Galilean transform.