Beach Thoughts

At the Santa Barbara beach I sit on a picnic bench (afraid to get my shorts more sandy than they already are), looking into the distance. The pier extends out to my left and affords spectacular views, but the views are irrelevant to what I find compelling about this place—

Definition: A Riemannian manifold $(\mathcal M, g)$ is geodesically complete iff every geodesic (i.e. curve $\gamma: (I \subset \mathbb{R}) \to \mathcal M$ satisfying $\nabla_{\dot\gamma}\dot\gamma = 0$) is defined for all $\mathbb R$.

Fascinating that beachy places seem to inspire quite visceral feelings of awe in otherwise unimpressionable people. Feynman himself — the Feynman; physicist, lockpicker, skeptic extraordinaire; he of the legendary dysteleological dismissiveness towards the fluffy artsy types — even he was so moved as to attempt poetry when confronted with the beach.

The seduction of the beach is particularly dangerous for physicists, who are cursed with having exactly enough ontological rope to hang themselves with. The smooth horizon out beyond is suggestive of hydrostatic equilibrium, but even the smallest amounts of vertical momentum out there translate into fully developed nonlinear turbulence when transported here, at the boundary of land and sea. One so inclined could spend ages marvelling at the exact moments when the Rayleigh-Taylor instability takes over and the fluid equations go nonlinear, again and again but never exactly the same; at eddies cascading angular momentum smallwards down Kolmogorov's ladder, vanishing into viscosity; at the foam and the sound and the sand adding surface tension and acoustic and thermal coupling to the atmosphere and substrate.

But I am no fluid dynamicist.

Claim: All compact Riemannian manifolds are geodesically complete.

This particular beach terminates, quite abruptly, at a cliff, and my picnic bench is on top of it. A sign — "warning: eroded bluff" — discourages me from approaching the water directly, although I will venture below the following day and walk along the sand at low tide. I will see exposed strata, proclaiming the geological history of the Goleta bedrock to all who will and can listen.

But I am no geologist¹.

Instead, I find myself surprised, and a little amused, at how I'm reacting to this.

Claim: $S^2$ is compact.

I have been various persons at various points in my academic career. Undergraduate-me, the differential geometer, chimes in with useless factoids about Riemannian manifolds (existence theorem: pick any two points on a geodesically complete manifold; there exists a geodesic passing through both). The stellar astrophysicist, tired and cranky from a long day of numerical stellar evolution², and unknowingly pre-empting Friday's computational exercise³, reminds me that we are all going to die a slow and scorching death when the Sun finally runs out of hydrogen and eats us all (assuming climate change doesn't get us first). The asteroseismologist idly wonders what kind of oscillation signature a planetary accretion event would have (after the initial fireworks). The sometime exoplaneteer idly wonders what kind of spectroscopic signature climate change would have.⁴

Right now I am none of these people.

Observation: The Earth's shape seems to be (coarsely speaking) isometric to $S^2$.

I am myself no stranger to beachy places in the USA, but this is my first time on the West Coast. That might explain the staring — right now, I am:

1. homesick,
2. staring in the general direction of home,
3. via the short end of the geodesic,
4. over the correct ocean.

It is quite something. I am not on the pier, but the view is sufficiently spectacular for my purposes.

I stay this way for perhaps half an hour or so before tearing myself away and rejoining the dinner party.⁵