Richard Feynman’s “Time and Distance” – Hunter Ash

Erwin Schrödinger (1887 – 1961) made several contributions to our understanding of physics including the creation of wave mechanics, the famous Schrödinger’s cat thought experiment, and won the Nobel Prize for Physics for the Schrödinger Equation.

Erwin Schrödinger (1887 – 1961) made several contributions to our understanding of physics including the creation of wave mechanics, the famous Schrödinger’s cat thought experiment, and won the Nobel Prize for Physics for the Schrödinger Equation.

I’m a sophomore physics major, so I’ve taken a fair if not extravagant number of physics classes. For the most part, they have been taught well enough. They’ve all had some bad days, lectures during which my mind inexorably drifted away from whatever tedious and unenlightening derivation was being scrawled on the board. However, they’ve also had moments that made my eyes widen at the bizarre, cold, alien beauty of the world, moments reminding me why I study physics and restocking my enthusiasm to carry me through the intervening tedium. The concepts that trigger this feeling of awe necessarily connect to my intuition in some way. They show how things that make immediate sense, that any child would know who gave it a moment’s thought, things you can see and touch, lead inevitably to conclusions so elegant and strange that they make me look around at my immediate surroundings as if I had never seen them before.

Pure abstraction, at least for me, cannot trigger this feeling. It has its own beauty, but it is of a very different sort. Mathematically, the Schrödinger equation isn’t that special. It’s member of a whole class of similar equations, and solving it isn’t much different than solving any others like it. It is only when I remember that it describes how the real world really works, when I look at my paper and realize that every particle of it (and me) is an utterly bizarre object, which doesn’t necessarily have any location whatsoever, that I remember how amazing that equation really is.

Feynman’s brilliance as a teacher lies precisely in building these kinds of connections. Reading his lecture on time and distance, I was struck again and again by the realness of everything he says. Every concept is built, carefully but enthusiastically, from the ground up, from elementary notions we all share and can all understand.

The Schrödinger Equation. From the Wikipedia page: A partial-differential equation in which “Ψ is the wave function of the quantum system, i is the imaginary unit, ħ is the reduced Planck constant, and H is the Hamiltonian operator, which characterizes the total energy of any given wavefunction and takes different forms depending on the situation.”

The Schrödinger Equation. From the Wikipedia page: A partial-differential equation in which “Ψ is the wave function of the quantum system, i is the imaginary unit, ħ is the reduced Planck constant, and H is the Hamiltonian operator, which characterizes the total energy of any given wavefunction and takes different forms depending on the situation.”

The chapter begins with a discussion of time and how to measure it. While Feynman gives an obligatory nod to the infinite regress inherent in trying to define time, he doesn’t waste space or the reader’s attention span with a lengthy philosophy-of-science interlude. Rather, he makes a commonsense, seeing-and-touching-based case for using events that seem to happen over and over again regularly as a way to measure time. Beginning with the day as a unit, he works his way down to smaller and smaller scales, each step believably derived from the last.

All of this, while particularly well written and clear, is standard enough. What made this section sparkle to me was his treatment of very short amounts of time. Most lecturers would be content to say 10^-12 seconds is 1 part in 10^12 of a second and leave it at that. But Feynman points out that, since we can’t eyeball anything close to a time so short, since the techniques used to measure times at that scale are fundamentally different that those used to measure more everyday times, we have implicitly invoked a new definition of time, specifically distance divided by speed. This is the kind of subtlety that reveals the strength of Feynman’s mental architecture, as well as his commitment to the democratic nature of science: he never asks you to just accept anything, even things most people would have agreed to without a second thought. He wants his audience to be skeptical of him, to question every step and pick apart every claim.

Raphael’s The School of Athens (c. 1510) depicts Euclid giving a geometry lesson. Euclid lived around 300 BCE and his book, Elements, provided a coherent account of geometry, number theory, and mathematical proofs that remains important today.

Raphael’s The School of Athens (c. 1510) depicts Euclid giving a geometry lesson. Euclid lived around 300 BCE and his book, Elements, provided a coherent account of geometry, number theory, and mathematical proofs that remains important today.

The section on distance measurement is very similar to the section on time. After briefly dealing with measurements on a human scale, done in the usual way using standard lengths to which other lengths are compared, Feynman dives into the more interesting problem of measuring very large distances. One of the a known distance apart, compare the angles observed from the locations, and use the geometric properties of triangles to find the desired length. He makes the initially bizarre claim that using geometry like this to measure distance constitutes a definition of distance different from the one based on standardized sticks. He playfully hints at the reason when he says, “space is more or less what Euclid thought it was” (my bolding). What he is referring to is a line of reasoning so subtle and counterintuitive that it took mathematicians after Euclid 2000 years to think of it: space can bend, and when it does the familiar rules of geometry don’t apply. The interior angles of triangles don’t add up to 180°, parallel lines can converge, etc. By referring to this strange result so offhandedly, he spares the casual reader the long and abstract explanation required to understand non-Euclidean geometry, while simultaneously planting a seed of curiosity in the more serious student.

In non-Euclidean spaces, for instance the surface of a sphere, shapes don’t have the same geometrical properties taught in high-school classes.

In non-Euclidean spaces, for instance the surface of a sphere, shapes don’t have the same geometrical properties taught in high-school classes.

Overall, I was amazed at how interesting I found this chapter. I have come away from lectures on relativity less stimulated than I was at the end of this short discussion on what it means to say how long something lasts or how big it is. Feynman’s enthusiasm for even these most basic of notions is palpable and contagious. The satisfaction I felt reading it was partially an appreciation of his mind, but the lion’s share of it was the sheer pleasure of real, deep understanding. Even a seemingly simple thing, well and truly understood, is a joy, and I think Feynman’s greatest asset was the depth and thoroughness of his understanding, the clarity of his thought. His contributions to theoretical physics were great, but I believe that the grasp of fundamental concepts he conveyed through his teaching will ultimately prove his greatest contribution to science.

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