### 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.