Daily Archives: January 28, 2008

The Relativistic Van

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Who cares about gas mileage? This sucker warps time!

When relativity is discussed in popular literature it’s often couched in terms of affecting objects moving at a significant fraction of the speed of light, and that’s a true statement: kinematic time dilation cannot generally be ignored in that situation. But the implication that the opposite is true — that you can ignore these effects under other circumstances — doesn’t hold. At least, it doesn’t hold if you have some expensive toys at your disposal.

Let’s say you were going to drive across the US and back, and you had the aforementioned expensive toys. Maybe you wanted to calibrate clocks and check on the reliability of a satellite time-transfer system, and you have a mobile system that would do time transfer at the source and at the target site, allowing you to check on that calibration. Or something like that.

The time dilation in question gives a fractional frequency shift that goes with the square of the speed, as compared to the square of speed of light. That’s normally very small, and has to be under this approximation (c is big, v/c is small, (v/c) squared is reeaaally small), so you can usually ignore it, right? Not everyone can. The famous Hafele-Keating experiment that used airplanes and around–the–world travel was able to measure kinematic dilations. A trip across the US is ~2700 miles, and at 600 mph you’d get a frequency shift of 4 parts in 10^13 and a dilation of about 13 nanoseconds on your round-trip due to traveling at that speed. (one thing to note is that I’m using a different coordinate system than is used in the H-K writeup, in case you want to play along at home. The answer will be the same, but the east vs west contributions are accounted for differently, and I’m not showing that detail)

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The Geometry of a Cow

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There’s an old physics joke about the dairy (Motto: “Smell Our Dairy Air!”) that hires cosultant after consultant to help optimize their milk production but to no avail. Each and every time, the end result is that nothing has improved. Finally they hire a physicist, who comes in and takes all sorts of data, and then retires to his office, madly doing calculations. After waiting a while, the dairy contacts him, inquiring about his recommendations. He come in to do a presentation, with all sorts of papers and slides for the overhead (or powerpoint on his laptop if you want to update the joke). He puts up the first slide, and starts in with, “First, we assume a spherical cow…”

It’s funny if you know physics, or more specifically, physicists, who tend to idealize all sorts of things in their models. (frictionless surfaces, elephants whose mass a may be ignored, etc.)

Anyway, hop in the wayback machine to a few years ago, when several of us were having a discussion about problems on our comprehensive exams in grad school. My boss tells one about a cow on a tether attached to a point on the rim of a cylindrical silo of a given height and radius. You were supposed to calculate the grazing area available to the cow. I, being in a smartass mood (sarcasm is my ground state), ask, “Did you assume a spherical cow?” The didn’t-miss-a-beat response was, “No, I was able to use the point-cow approximation for this problem.” And I thought that was pretty funny, and something that works in a single-panel cartoon.

point_cow.jpg