![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
mmcirvinSomething I was musing about for no reason on the airplane:
When I took high school physics, a thing that bothered me for reasons I could not entirely articulate was the way kinetic energy varied under a change of frame of reference. In Newtonian physics, either things don't change at all (time, which is absolute) or they change in some simple linear way: under a change of frame that increases the velocity by v',
t' = t
x' = x + v't
p' = p + mv'
where the expressions hold even if you write the positions, velocities and momenta as vectors. This is nice because you can easily see, for instance, that momentum is conserved in every frame. Momenta just add like vectors, and if you add mv' to the initial and the final momentum, of course everything's still going to come out the same.
Kinetic energy, though, was this weird, nonlinear odd man out because it depended on the square of the velocity:
Ek = (1/2) m v2
Ek' = (1/2) m (v+v')2 = Ek + m v.v' + (1/2) mv'2
In an elastic collision, that's conserved in every frame too, but it wasn't intuitively obvious that this was so. It always seemed slightly miraculous to me, and it always seemed as if, when I truly understood what was going on here, I'd understand something very important.
Many years later I found out that my intuition had been correct, since energy becomes no longer an odd man out in the theory of relativity. In special relativity, using units where the speed of light is 1, there is a four-dimensional vector consisting of time and the three components of position, and a four-dimensional vector consisting of energy (no longer just kinetic energy) and the three components of momentum. And these two four-dimensional vectors transform under frames of reference in exactly the same way. In particular, if you define γ = (1/sqrt(1-v2)),
t' = γ (t + v'.x)
E' = γ (E + v'.p)
Energy transforms just like time! And total energy-momentum adds like a four-dimensional vector, and this "Lorentz transformation" is linear over addition of vectors, so conservation all works out.
And the reason energy seemed like such an odd man out in Newtonian physics was just that we were looking at different terms in an expansion. To second order in v', γ is approximately 1 + (1/2) v'2. So you get
t' = t + v'.x + (1/2) t v'2 + smaller terms
E' = E + v'.p + (1/2) E v'2 + smaller terms
We're using units where the speed of light is 1, remember, so for everyday speeds v' is tiny. So for Newtonian purposes, all those terms in the time expression drop out except for the first one and time seems universal and invariant.
But for the energy expression, we were always ignoring the enormous energy inherent in the mass, the famous mc2 (which in our units is just m). So we have E = m + Ek, where m is really gigantic, and doesn't change significantly in the Newtonian world. Ek is already tiny compared to m, so while those two additional terms I wrote are small compared to m, they're not small compared to Ek; instead of dropping out in the Newtonian limit, they just turn into those extra terms that worried me so much in high school. In the Newtonian world it all looks asymmetrical and weird, but relativity reveals a hidden elegance.
The thing that gets me is that I already knew some of the rudiments of special relativity back then, having read them in popular books. But I didn't realize at the time that I was knocking on the door of understanding something important about special relativity, and I didn't even entirely make the connection after I'd studied the subject in college "modern physics" class. In fact, I don't think I really understood this entirely until I got into graduate school and became a TA in a freshman physics class, and the professor (who was Sheldon Glashow) went through this on the blackboard. I'm not sure it really sank in for the students, though.
Part of the problem is that college freshmen are still pretty new to thinking about things in terms of power series expansions, and high-school students aren't accustomed to it at all. It seems odd now that I took at least six years to get the real insight out of something that had given me significant cognitive dissonance in the eleventh grade, but some mental machinery only comes the hard way.
When I took high school physics, a thing that bothered me for reasons I could not entirely articulate was the way kinetic energy varied under a change of frame of reference. In Newtonian physics, either things don't change at all (time, which is absolute) or they change in some simple linear way: under a change of frame that increases the velocity by v',
t' = t
x' = x + v't
p' = p + mv'
where the expressions hold even if you write the positions, velocities and momenta as vectors. This is nice because you can easily see, for instance, that momentum is conserved in every frame. Momenta just add like vectors, and if you add mv' to the initial and the final momentum, of course everything's still going to come out the same.
Kinetic energy, though, was this weird, nonlinear odd man out because it depended on the square of the velocity:
Ek = (1/2) m v2
Ek' = (1/2) m (v+v')2 = Ek + m v.v' + (1/2) mv'2
In an elastic collision, that's conserved in every frame too, but it wasn't intuitively obvious that this was so. It always seemed slightly miraculous to me, and it always seemed as if, when I truly understood what was going on here, I'd understand something very important.
Many years later I found out that my intuition had been correct, since energy becomes no longer an odd man out in the theory of relativity. In special relativity, using units where the speed of light is 1, there is a four-dimensional vector consisting of time and the three components of position, and a four-dimensional vector consisting of energy (no longer just kinetic energy) and the three components of momentum. And these two four-dimensional vectors transform under frames of reference in exactly the same way. In particular, if you define γ = (1/sqrt(1-v2)),
t' = γ (t + v'.x)
E' = γ (E + v'.p)
Energy transforms just like time! And total energy-momentum adds like a four-dimensional vector, and this "Lorentz transformation" is linear over addition of vectors, so conservation all works out.
And the reason energy seemed like such an odd man out in Newtonian physics was just that we were looking at different terms in an expansion. To second order in v', γ is approximately 1 + (1/2) v'2. So you get
t' = t + v'.x + (1/2) t v'2 + smaller terms
E' = E + v'.p + (1/2) E v'2 + smaller terms
We're using units where the speed of light is 1, remember, so for everyday speeds v' is tiny. So for Newtonian purposes, all those terms in the time expression drop out except for the first one and time seems universal and invariant.
But for the energy expression, we were always ignoring the enormous energy inherent in the mass, the famous mc2 (which in our units is just m). So we have E = m + Ek, where m is really gigantic, and doesn't change significantly in the Newtonian world. Ek is already tiny compared to m, so while those two additional terms I wrote are small compared to m, they're not small compared to Ek; instead of dropping out in the Newtonian limit, they just turn into those extra terms that worried me so much in high school. In the Newtonian world it all looks asymmetrical and weird, but relativity reveals a hidden elegance.
The thing that gets me is that I already knew some of the rudiments of special relativity back then, having read them in popular books. But I didn't realize at the time that I was knocking on the door of understanding something important about special relativity, and I didn't even entirely make the connection after I'd studied the subject in college "modern physics" class. In fact, I don't think I really understood this entirely until I got into graduate school and became a TA in a freshman physics class, and the professor (who was Sheldon Glashow) went through this on the blackboard. I'm not sure it really sank in for the students, though.
Part of the problem is that college freshmen are still pretty new to thinking about things in terms of power series expansions, and high-school students aren't accustomed to it at all. It seems odd now that I took at least six years to get the real insight out of something that had given me significant cognitive dissonance in the eleventh grade, but some mental machinery only comes the hard way.