Properties of Negative Lengths

[studiot]

HalfWit has given you (post#13) the mathematical definition and reason why a straightforward distance (length) cannot be negative (by definition).

 

What I am asking you to consider is

 

The difference between two distances (lengths) is still technically a distance (length), so can this be negative?

 

That is what happens to (L₂- L₁) when L₂ < L₁ ?

Edited November 19, 2013 by studiot

[Endercreeper01]

The change in x would be negative, but that doesn't mean that the length would be negative.

Basically with the spring, you have this:

||||||||||||||||

When it expands, it is now this:

| | | | | | | | | | | | |

When you compress it again, it goes back to it's previous state.

Now, let's explain this with vectors:

| | | | | | | | | | | | | Compression <-- --> Expansion

If you assume the direction of the compression is the same as the expansion, then it would be negative. But, since it is not, then it would be positive in that direction, and negative of the expansion vector.

In fact, it is more like a 1 dimensional vector field.

[studiot]

 

The change in x would be negative

 

You didn't understand.

 

What do you measure x in? metres? miles?

 

So what do you measure the change in x in? metres? miles?

 

Why is that not a form of negative length?

[ajb]

We have versions of metric that are not positive definite, in that sense we have a negative length. These things are used in physics all the time in the form of a pseudo-Riemannian metric on a manifold we call space-time!

 

The lack of positive definiteness geometrically is no real problem, but when it comes to analysis this can be a real pain.

[michel123456]

You'd be going backwards in time

 

(...)

Yes.

negative length would produce in physics

1_negative velocity (v=d/s)

2_negative acceleration (a=d/s^2)

 

Whatever it means.

And I guess a whole bunch of negative phenomena.

 

What I find very interesting, and intriguing, is that negative distance does not exist and negative time does not exist too. The concept of distance is linked to the concept of time and together form the concept of spacetime.

Now, if one reverts theoretically both the concepts of distance AND time, it is like making a pirouette and nothing happens. Velocity keeps positive. Acceleration though

Edited November 19, 2013 by michel123456

[Unity+]

We have versions of metric that are not positive definite, in that sense we have a negative length. These things are used in physics all the time in the form of a pseudo-Riemannian metric on a manifold we call space-time!

 

The lack of positive definiteness geometrically is no real problem, but when it comes to analysis this can be a real pain.

Could you link an article to this? This seems quite interesting to me.

[ajb]

Could you link an article to this? This seems quite interesting to me.

About the difficulties with non-strictly positive metrics from an analysis point of view? Or Pseudo-Riemannian geometry?

 

On the second there are plenty of books on general relativity that will cover the basics of that. I like the lecture notes of Carroll as a starting place, but you will soon want a more rigorous approach. For the first one this is outside my area of expertise and so I am not sure what the good introductions are.

[decraig]

Here’s something rough, in the context of differential geometry.

 

We have a smooth parametric curve AB, on some manifold of arbitrary dimension, n.

 

We want to define a length along the curve from A to B, such that it is natural to define the ‘length’ from B to A to be the negative of the length AB.

 

Use coordinate basis (to keep it simple). At each point on the curve attach a coordinate system. The coordinate system at point A is arranged so that a dual basis vector ds⁽⁰⁾ points along the curve.

 

We need another coordinate system displaced along the curve by ds⁽⁰⁾. It also needs a dual basis vector. Call it ds⁽¹⁾. And so forth.

 

We want keep propagating in one direction along the curve, instead of bouncing back and forth or changing to some other dual basis vector.

 

To do this we need to demand that the coordinate transformation matrix is a continuous function of our parameter.

 

Most of the machinery seems to be set-up. Now integrate ds.

 

[math] L_{AB} = \int_A^{B} ds[/math]

 

[math] L_{BA} = \int_B^{A} ds[/math]

 

There’s something still missing (I’m doing this off the top of my head). We have a parametric curve, and the parameter is missing from the expressions. I suppose that’s where the transformation matrices come in.

 

Finally, we would probable like something to relate this back to the tangent space along the curve.

Edited November 22, 2013 by decraig

[ajb]

There’s something still missing (I’m doing this off the top of my head). We have a parametric curve, and the parameter is missing from the expressions.

You can think about what you have suggested as "running time backwards". This would introduce minus signs in the right places and the length would be the same in either direction.

[michel123456]

Sort of.

 

An object that is at a distance from you lies in your past. The longest the distance, the longest in the past.

An object that is at distance zero is in your present.

An hypothetical object at negative distance would lie in your future.

[studiot]

Mathematically distance and length are different, although the distinction is more blurred in Physics, this thread was posted in the Maths section.

The original question is about length and many posters (myself included) have been disregarding the distinction.

 

I think it is time to set that straight.

 

Distance refers to how close (or far) apart two objects are. It is defined only by the endpoints and makes no reference to or provides any information about the points in between, if indeed there are any.

 

For instance in statistics we often measure how close one set of numbers is to another and compute the 'distance' between the sets. There are no intermediate points in this case.

 

Length, on the other hand, includes both the endpoints and all the points in between.

This is best exemplified by considering intervals on sets of real numbers.

 

An interval consists of all the numbers (x) satisfying one of the following conditions

 

For any real numbers a, b with [math]a < b[/math]
[math]a \le x \le b[/math]
[math]a < x < b[/math]
[math]a < x \le b[/math]
[math]a \le x < b[/math]

[math]x \le a[/math]
[math]x < a[/math]
[math]x \ge a[/math]
[math]x > a[/math]

For the first four the length of the interval is defined as the real number b-a.

This can be seen to be positive definite when a is not equal to b and zero when a = b