is it true ..?

[khaled]

In mathematics, the shortest path between any two points is the straight line ...

 

But in physics, a friend told me that the shortest path between two points in space is Zero,

 

Is it true ?.. and how is that possible ??

[Mr Skeptic]

Distances in real life depend on how fast you are going. Going at almost the speed of light it would seem to you that the distances are almost zero. Time also gets distorted. These distortions are necessary for the speed of light to always be c for all observers despite them moving at different rates, as required by maxwell's equations and the principle of relativity.

[ajb]

In mathematics, the shortest path between any two points is the straight line ...

 

On [math]R^{n}[/math] ok. It is a bit more involved to discuss what is meant by the shortest path on a more general curved space.

 

But in physics, a friend told me that the shortest path between two points in space is Zero,

Is it true ?.. and how is that possible ??

 

Assuming the points are distinct.

 

Sort of. See null geodesics.

 

In essence, a vector, which represents a "small displacement" can have positive, negative or zero magnitude. This is because the metric (inner product) is not positive definite.

 

A null path (path light follows) between two points will always have zero space-time interval, so "zero length".

[timo]

I agree that khaled's buddy was likely to refer to something like relativistic space-time intervals, but I disagree that he has any point at all. The existence of a path from A to B with a length of X does not imply that the distance (in the sense of "any properly defined distance") between A and B is X.

[khaled]

A friend told me that if we draw two points on the opposing edges of a paper,

 

mathematically the shortest distance is the length of the paper,

 

but physically, if we can envelope the paper where the paper edges meet,

the paper will looks like a cylinder, both points will be at the same space\time

and the length will be Zero ...

 

 

what do you think ?

[ajb]

This is compactification. The thing is you are now identifying the points A and B. They are no longer distinct points.

 

This example is interesting. Can you think about the distinct types of loops (paths that start and end on a specified point) you can have?

[between3and26characterslon]

In mathematics, the shortest path between any two points is the straight line ...

 

But in physics, a friend told me that the shortest path between two points in space is Zero,

 

Is it true ?.. and how is that possible ??

 

Because the rules you use in mathematics are different to the rules you use in physics or more accuratley they are not always necessarily the same.

 

Is it true that the shortest distance between two points is a straight line? This question is meaningless. In Euclidian geometry a straight line is uniquely defined as the shortest distance between two points situated upon it. This is one of the axioms, you can not ask if it is true you can only say this is the rule. (There are also many other rules in geometry which have to be obeyed, you have to have (x, y, z) coordinates which are straight lines at right angles to each other which are marked off uniformly at equidistances).

 

i.e. You can not ask whether winning a race is true.

 

You can say the rule is the first person to cross the line is the winner and then ask the question; this person is the winner, true or false?

 

So if in your Euclidian framework of (x, y, z) coordinates (and obeying all the other rules) you draw two points and then connect these two points by the shortest possible distance you have, by definition, a straight line.

 

In physics your framework does not have to be constructed out of straight lines nor do the lines have to be marked off uniformly and equidistantly (though they do still have to be at right angles to each other)

 

Now you have two frameworks a Euclidian one made of straight lines which are marked off uniformly and your physics one made of say randomly curved lines marked off at varying distances. You can now, through maths I can't explain, map your physics framework onto your Euclidian framework. You can now find that two points seperated by a distance on one framework, when mapped onto the other framework, are not seperated by any distance.

 

So is a straight line the shortest distance between two points? If that's the rule then yes. If it's not the rule then no.

Edited February 3, 2011 by between3and26characterslon

[lemur]

A friend told me that if we draw two points on the opposing edges of a paper,

 

mathematically the shortest distance is the length of the paper,

 

but physically, if we can envelope the paper where the paper edges meet,

the paper will looks like a cylinder, both points will be at the same space\time

and the length will be Zero ...

In what empirical physical situation is this supposed to apply?

Edited February 3, 2011 by lemur

[steevey]

In what empirical physical situation is this supposed to apply?

 

When the fabric of space get's warped like in a worm hole? Not that wormholes exist, but that's basically what the paper is doing. It's curving a plane to make two separate points the same point, making the distance between them 0.

 

In mathematics, the shortest path between any two points is the straight line ...

 

But in physics, a friend told me that the shortest path between two points in space is Zero,

 

Is it true ?.. and how is that possible ??

 

As predicted by quantum mechanics, the true distance between two objects is 0, otherwise entanglement would not be able to occur (which it does and has been proven to exist). Real experiments have been done where entangled particles have been separated by over 100 kilometers and the properties of one particle as well as the determination of the two particles responded instantaneously, proving distance does not effect entanglement at all. How could that happen though unless distance wasn't really there?

Edited February 3, 2011 by steevey