# How 'far' is far apart?

1. Feb 4, 2008

### rohanprabhu

I'm totally new to quantum mech... i've just started reading a few articles on that. I came across the 'Locality Principle' on wikipedia..

"The following idea characterises the relative independence of objects far apart in space (A and B): external influence on A has no direct influence on B; this is known as the Principle of Local Action, which is used consistently only in field theory. If this axiom were to be completely abolished, the idea of the existence of quasienclosed systems, and thereby the postulation of laws which can be checked empirically in the accepted sense, would become impossible."

- Albert Einstein

So my questions are:

i] What exactly does Einstein mean when he says 'far apart'?
ii] What does this imply for fields? Does it imply that all fields are inversely proportional to some power of seperation i.e. $F \propto \frac{1}{r^n}$ such that the field vanishes at infinite seperation?

forgive me if the 2nd statement seems too vague to you.. as i'm still hooked to the classical model of physics...

2. Feb 8, 2008

### Reid

I can contrubute with an answer for the first one...

i) I would say that far apart means "sufficiently far apart such that the objects doesn't interact".

3. Feb 8, 2008

### Brainv2.1beta

But isn't that impossible? Gravity has an infinite reach, if my teachings have been correct...

4. Feb 8, 2008

### Reid

The influence on A from B is sufficiently small if they are far apart since the gravitational potential tends to zero as one over distance. If I remember correctly...

'doesn't interact' means an interaction that is sufficiently small.

5. Feb 8, 2008

### peter0302

I believe all Einstein means here is that objects cannot interact within a relevant time period "t" if their spatial separation is greater than "ct" - in other words, if they are so far apart that an influence travelling from one to the other could not reach the other in time to be relevant to the experiment.

So in sum he means you can't have influence travel faster than light.

Gravity has an unlimited distance but still travels at the speed of light. The gravitational pull of the Sun depends on where the sun was 7 minutes or so ago. If someone suddenly teleported the Sun away, we wouldn't feel that influence for 7 minutes.

6. Feb 8, 2008

### rohanprabhu

phew.. some reply finally. So.. what einstein means is that.. if 't' is large enough such that the observations of an experiment are concluded within 't', then objects 'ct' distance away would have no influence on it?

7. Feb 8, 2008

### peter0302

Well, I'm just interpreting his remarks, but I have to believe that's what he means. There is no single measure of distance beyond which is considered "far apart". I don't think he meant the Planck Length (if that even was a concept at the time).

Einstein's entire career was essentially built around the inviolability of the speed of light. He understood and fervently believed that fields such as the EM field (and, we now know, gravity) could not propagate faser than 'c'. So when it came to "nonlocal action" the problem he had was that the influence appeared to require no time (dt = 0). Well, obviously any time anything happens with dt equaling zero means faster than light - so this was something he could not accept. But he certainly understood that spatially separated objects could influence one another _given sufficient time_, the minimum time being distance / c. Hence, my interpretation of his statement, i.e., what he means by "far apart" depends on the timeframe of the experiment (t), and in no case can x exceed ct.

To further answer your question, what this implies for fields is that the effect a field has upon a particle is solely determined by the value of the field at that particle's position. If something "far away" (distance = x) alters the field (i.e., my earlier example, someone teleports away the sun) that alteration cannot be felt until x/c seconds have passed.

Last edited: Feb 8, 2008
8. Feb 8, 2008

### vanesch

Staff Emeritus
Yes. It means that the action of things are determined as much as can be by the state of other things "in the neighbourhood" of it, understood in a topological sense. That means, for instance, that the action of a field on another field at P0 can depend on that field, but also on its derivative, its second derivative etc.... So if we say that it depends on the field at P0, that is not strictly correct, because in order to have the derivative (gradient), we need a small neighbourhood (topologically speaking) of P0 over which we know the field.