Space, Time & Stephen Hawking Jive

[FNHinVA]

Actually, one of Stephen Hawking's graduate assistants.

I have a physics question that has baffled me for many years. I posed it to a few physicists and got unsatisfactory answers. So I decided to email Stephen Hawking. (Why fool around with amateurs?)

As expected, he did not answer. But one of his graduate assistants did. But first, the question...

I am going to do a "time trial" over the distance A to B (A|B). I will maintain a constant rate of speed. Obviously, in order to traverse A|B, I must first traverse half of A|B which I will do in half the time. Just as obviously, I must also traverse half of the half of A|B. (You see where this is going...)

Since I have in front of me an infinite number of "halves" I must traverse (and take time doing it), how will I ever pass the B finish line? Obviously, it will take forever. But, because I know I can, in fact, traverse A|B in a finite amount of time, I know it doesn't take forever.

The answer from Mr. Hawking's graduate assistant involved calculus, Planck lengths and the uncertainty principle.

Essentially, what all of this (and he) said was "when things get that small, we can no longer measure them so we don't know what the hell is going on."

Anyone have a better answer?

[kpux]

This is Zeno's paradox. It's an ancient Greek riddle. It's easily solvable with infinite series.

Basically, you can add up an infinite number of positive real numbers and have it come out to a finite sum. For simplicity, let's say the distance A|B = 1. In this example, 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 1. Even though you are traversing an infinite number of halves, the total distance is still finite. Just because you can express the number 1 as an infinite sum of smaller real numbers, doesn't mean that A|B somehow becomes an infinite distance.

[PairTheBoard]

Just because you measure time in smaller increments doesn't mean you're slowing it down.

PairTheBoard

[drudman]

[ QUOTE ]
This is Zeno's paradox. It's an ancient Greek riddle. It's easily solvable with infinite series.

Basically, you can add up an infinite number of positive real numbers and have it come out to a finite sum. For simplicity, let's say the distance A|B = 1. In this example, 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 1. Even though you are traversing an infinite number of halves, the total distance is still finite. Just because you can express the number 1 as an infinite sum of smaller real numbers, doesn't mean that A|B somehow becomes an infinite distance.

[/ QUOTE ]

[usmhot]

Absolutely right. Mathematically this is an old problem which is solved exactly as indicated, by a sum over infinite series. It applies to a continuous space - i.e. one in which two points can be an arbitrarily small distance apart.

However, the rest of what Hawking's assistant was talking about is ...
according to Quantum Mechanics space is _not_ continuous - there is a finite distance which is the smallest distance that can be between two points. So in the real Universe there are only a finite number of sub-points (distances) between two points. Thus, in the real Universe you can get from A to B in finite time because there are a finite number of time intervals from sub-point to sub-point.

[FNHinVA]

You are correct. But that is not the issue (or the question). The question is how do I traverse an infinite series of shortening distances in any amount of time. The rate of time has nothing to do with it.

[PairTheBoard]

[ QUOTE ]
You are correct. But that is not the issue (or the question). The question is how do I traverse an infinite series of shortening distances in any amount of time. The rate of time has nothing to do with it.

[/ QUOTE ]

You haven't made the distance longer by dividing it up into infinitely many small increments. Neither have you made the time longer by doing the same with it.

PairTheBoard

[FNHinVA]

OK - now we're getting somewhere (so to speak). This is exactly the answer I was looking for, and didn't really get from Hawking's assistant or the others I asked.

I have a very lame math background so when the answers are in calculus and other languages I don't speak, I get kind of lost.

The concept of a finite measure of time (and distance) is what I have theorized. Obviously, time and space cannot - in reality - be divided into infinity. I don't care about math proofs; reality says it has to stop somewhere or we can never get anywhere.

My theory is that at the ultimate and finite level, motion becomes state changes, rather like the frame changes of a motion picture. These would - I believe - occur at the speed of light.

Now I need to work on the implications of this - if any.

Then, I'm on to World Peace and poverty.

[FNHinVA]

I'm not trying to make anything longer. I'm trying to understand what happens. The math doesn't really explain anything. See my response to usmhot above.

[usmhot]

That argument for space being discrete does not hold - if space were continuous then you would still get from A to B in finite time, as indicated by the post on Zeno's paradox - provable using infinite series. In fact, up until Quantum Mechanics it was assumed that space was continuous and the infinite series proof was totally acceptable as an answer to that original question.

But, there is evidence to suggest that space is not continuous, so the sum over infinite series does not apply. (Not least the energies involved in continuous space becomne infinite.)

As to what 'causes' motion, which is what you're trying to get at ... I think that's a very interesting question. Given that (it seems very likely that) space is discrete and between two points that are a Plank's distance apart (the smallest distance there can be) there is effectively nothing at all, the question of how a particle gets from one point to another is still unknown.