The question often comes up: How can you recognize a reasoning-based
approach to physics, in contrast to the rote-only equation-hunting
plug-and-chug approach?
The annual aberration of starlight provides a good example. Suppose
you want to handle the case where the observer's velocity (relative
to the source) is not necessarily small compared to the speed of
light. The problem is usually posed in terms of angles and velocities
as measured in ordinary 3-dimensional Euclidean space. However, the
easy way to solve the problem is
a) Restate the problem in modern (post-1908) terms, i.e. spacetime.
b) Solve the problem, which is now very simple.
c) If necessary, convert back to lab-frame Euclidean coordinates.
If the goal is to /derive/ the result, the three-step process
involves /fewer/ steps than the brute-force pre-modern approach,
and each step is /easier/.
More importantly, the spacetime derivation is so easy to perform
and so easy to remember that it is easier to remember the outline
of the derivation than it is to remember the final formula. You
can rederive the formula whenever necessary.
Again: The derivation is cloyingly obvious:
a) Find the 4-momentum of the photon.
b) Boost the 4-momentum, just like any other 4-vector.
c) Project it onto lab coordinates.
Even more importantly: There is a fundamental idea here that
applies to an enormous range of problems, not just the aberration
calculation: Physics is very much simpler in spacetime. So
almost no matter what the question is, reframe the question in
terms of spacetime, answer the question, and then (if necessary)
convert back to the lab frame.
Converting to the Minkowski representation and back again may
seem like unnecessary work, but it's not. Several easy steps
is very often easier than one hard step. Would you like to
jump from the ground up to the second floor, or would you like
to walk up the stairs, step by step?
The reasoning-based approach to physics is like a Lego robotics
kit: There are a whole bunch of parts, most of which are very
simple, but you can snap them together to build some impressive
things. Equation-hunting gives you a formula that solves one
specific problem, whereas spacetime and 4-vectors solve lots
and lots of problems.
I don't now whether to laugh or cry when I see the various
"concept" inventories and the various state-mandated high-stakes
tests, because most of the questions can be answered in a single
step. The idea of multi-step reasoning seems to have been left
by the wayside. Yeah, I know that students need some basic facts
and concepts first, because otherwise there would be nothing to
reason about ... just like an elephant needs ankles. However,
the more tightly you focus attention on the ankles, the more
you lose sight of the overall elephant.
Visiting simple concepts in isolate is mostly OK as a starting
point, but you don't want to stay there any longer than necessary.
Furthermore, there is a proverb that says:
Utility is the best mnemonic.
That is to say, students are incomparably more likely to /retain/
the fundamental concepts if they see that the concepts can be
applied in zillions of different ways, as part of easy-but-powerful
multi-step arguments.