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.

[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]

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.

[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.

[FNHinVA]

First, it is my understanding that Planck's distance is the smallest measurable distance - not necessarily the smallest distance. Second, I have no idea what difference that makes.

As to your comment: "the question of how a particle gets from one point to another is still unknown." Well, hell. That ruins my day. That is the whole (underlying) issue.

I'm going to go with speed-of-light (or instantaneous) state changes. Quantum mechanics must allow for it even if it doesn't explain it.

[Mano]

The fact that you can traverse the distance in a finite amount of time is not reliant on the quantum nature of spacetime. As stated before the sum of the distances 1/2 + 1/4 + 1/8 +... = 1. The time to traverse each of these distances also converges to a finite number - i.e. If your velocity is v, and you travel a distance d, it takes you d/v time units to cover the distance. So it takes 1/2v + 1/4v + 1/8v + ... = (1/2 + 1/4 + 18 + ..)/v = 1/v time units to cover the distance.

[usmhot]

In our Universe it is reliant on the quantum nature of space-time.
Only in a continuous space does the sum over infinite series actually apply.

[K C]

The simplest way to explain this seeming paradox is this. This example is construed such that it is set up to measure an infinite series short of completion. It's certainly much more of a "riddle" than any scientific problem. The reason of course that the series does not lead to completion is that the possibility of such is not built in to the example.

Given A and B as points in space of course there are an infinite amount of points in between, just as there are infinite points in time. I can for instance postulate the same thing from the time it takes from this very moment to a minute later : 10:26 to 10:27 for instance. Yet oddly enough 10:27 did occur as I'm writing this. And it did not involve an infinite amount of time for this to occur.


KC