### Tow the line

The other day at Obsidian Wings someone wrote "tow the line" instead of the slightly-mysterious phrase "toe the line". I said that since lines are massless, they're easy to tow.

Now having actually putting in a second's thought, I realize this isn't right. First off, one can (if I remember any math) smoothly deform a cube into a line by approaching the limits correctly, so a line could have any mass. And that's based on the incorrect model that matter is like pudding - continuous instead of made of discrete elements of perhaps zero volume - points or curves.

And massless objects move at the speed of light. (Not that "mass" here means "rest mass", the mass the object has when measured in a frame in which it's not moving.) The reason for this is that, as Einstein realized 100 years ago, you move at the speed of light or somebody can see you sitting still (by going as fast as you are) - and if your rest mass is zero and someone sees you in that state, you vanish in a puff of vacuum. Massless object are the sharks of physics - they can't stop moving or they die. Or the birds of physics - you can fly or you can have a big brain, but not both. Or whatever.

(The fact that light moves at the speed of light falls out of electromagnetism - c can be calculated from two other constants which appear in some I guess old formulations of Maxwell's equations; one of the constants is called "the permeability of free space" unless I'm making stuff up. Maxwell's equations don't care about which non-accelerating frame of reference you live in, so the speed of light is the same no matter how you look at it.)

As far as I know it would be a real pain to tow something moving at the speed of light.

Now having actually putting in a second's thought, I realize this isn't right. First off, one can (if I remember any math) smoothly deform a cube into a line by approaching the limits correctly, so a line could have any mass. And that's based on the incorrect model that matter is like pudding - continuous instead of made of discrete elements of perhaps zero volume - points or curves.

And massless objects move at the speed of light. (Not that "mass" here means "rest mass", the mass the object has when measured in a frame in which it's not moving.) The reason for this is that, as Einstein realized 100 years ago, you move at the speed of light or somebody can see you sitting still (by going as fast as you are) - and if your rest mass is zero and someone sees you in that state, you vanish in a puff of vacuum. Massless object are the sharks of physics - they can't stop moving or they die. Or the birds of physics - you can fly or you can have a big brain, but not both. Or whatever.

(The fact that light moves at the speed of light falls out of electromagnetism - c can be calculated from two other constants which appear in some I guess old formulations of Maxwell's equations; one of the constants is called "the permeability of free space" unless I'm making stuff up. Maxwell's equations don't care about which non-accelerating frame of reference you live in, so the speed of light is the same no matter how you look at it.)

As far as I know it would be a real pain to tow something moving at the speed of light.

## 3 Comments:

While a cube is, of course, smoothly deformable to a line, all bounded shapes are smoothly deformable to a point: specifically by scalar multiplication by, say, c, as c runs from 1 to 0. But usually mass is proportional to volume, and therefore any such "deformation" would likewise deform mass to zero, leaving lines still massless.

The remarkable result you might be alluding to is "Peano's space-filling curve," with which a line segment may be continuously deformed into a square. (see Topology by J. Munkres, Chapter 44) But, like the scalar multiplication alluded to earlier, this is just a continuous mapping rather than a "homeomorphism," meaning a continuous mapping with an inverse, which is used to indicate that two forms are topologically "the same." And even if two figures are topologically the same, it means very little as to whether their masses should be identical.

On the other hand, I've recently seen "giant lines" alluded to by string theorists in, for instance, the New Scientist:

http://www.newscientist.com/channel/fundamentals/mg18424781.400

This isn't my area of expertise, but it seems to be directly on point: gigantic galaxy-spanning lines that would be impossible to tow.

Your link, which I'll look at when I don't have to tuck in my fiancee - I assume this is about cosmic strings, which I visualize more as fractures in space than actual objects, but will take a look.

I remember wondering about something like the space-filling curve - I seem to recall there's a way of cutting a cube into two cubes each of the original volume. But I never liked measure theory or much understood it - the class was at 8:30 at the top of an icy hill.

But regardless, I don't understand how the deformation in question would reduce the mass of the object. I'm thinking, for a cylinder of radius r and length l, to keep l*r*r constant while letting r->0. Is that not a line in some sense?

For some reason, I had a blind spot to that particular deformation. Yes, you're right, one could extend a cylinder in such a way to get a line with finite mass.

That process is very similar the Dirac delta function and distributions in general. [Wikipedia] In brief, "nascent" delta functions are functions whose limit is a point mass; likewise, your cylinder example has a limit with line mass. Like Fourier analysis and calculus, it's a field created for physics that mathematicians later formalized.

(Cutting one cube into two cubes of equal volume is the Banach-Tarski Paradox [Wikipedia], which I've never seen proven. I personally think of it as being more in the realm of mathematical logic, instead of measure theory.)

Cheers!

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