The Relativistic Van

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)

But what about a van? At 60 mph this will take you ~90 hours of road time round-trip. (It doesn’t matter if you do this all at once, so feel free to stop off and take pictures of the Grand Canyon or drop a few dollars in Vegas if that’s your thing.) This gives a change in the frequency of your clock of 4 parts in 10^15. So far, so good — the speed dropped by an order of magnitude and our frequency shift is the square of that, i.e. 100 times smaller. But here’s what I glossed over before: for the total time shift you have to integrate over the whole trip. If you are going 1/10 of the speed, the trip will take you 10 times longer. It turns out that because of this, the total time delay goes linearly with speed, not quadratically. Your time delay is 1.3 ns, which is still something that you can measure with an atomic clock, and is something for which you have to account if you are doing clock calibrations.

I haven’t mentioned gravitational time dilation at all, which is a much larger effect and more easily measured. And lest you think that only government labs would do this exercise, there are amateurs out there who do this sort of thing, like measure the 22 nanosecond gravitational time dilation due to a day-trip to Mt. Ranier.

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