Ask a Physicist: Rudolph the Redshifted Reindeer

Visiting every house in the world in one night is a tough job, even when you don’t count the difficulty of squeezing down a chimney after eating a few million Christmas cookies. Just from a logistics perspective, it’s a nightmare: finding the most efficient route between a bunch of points on a map (the so-called “Traveling Salesman” problem) is such a notoriously difficult nut to crack that it seems we’ve got no shot at doing it efficiently without quantum computing.

For a sense of scale, one of the largest Traveling Salesman problems ever tackled looked at 86,000 points, and involved so many calculations that it would have taken a single 500MHz computer processor 136 years to complete. Even with the best programs for solving the problem, each point you add can double the calculation’s complexity and the amount of time it takes to run, so the scale of the problem Santa’s solving—about 92,000,000 homes according to a Gizmodo estimate—becomes jaw-dropping, even with a hefty dose of magic.

Some say Santa Claus was once a mortal man, but stumbled upon a generalized solution for the traveling salesman problem with N arbitrary nodes and was bound into servitude by the elder gods of Mathematics for his hubris.

But any attempt to talk about the physics of Christmas is inevitably an exercise in picking very specific aspects of the legends to take issue with, so hopefully you’re ready to play along and imagine that Santa knows his route by heart. After all, he’s made this trip plenty of times before, and the homes of all the good boys and girls on his list might not change too much from year to year. That would mean the NP-hard Traveling Salesman problem might reduce to a slightly more palatable perturbation theory problem based on the previous year’s list. Regress that all the way back to when the tradition started, and voila!

But that still leaves the problem of getting to all these houses in one night. The same Gizmodo article from earlier estimates that he’d have to move around 650 miles per second, meaning his arrival would be heralded not by the soft jingling of silver bells, but rather by a window-shattering sonic boom. That might be followed by some very strange noises—the same Doppler effect that’s responsible for the characteristic zoom of a race car going by would drop the pitch of Santa’s characteristic laugh by at least a few octaves as he flew off into the night.

But thinking about the sonic Doppler effect brings us to another interesting question: would we see the same thing happening for light? If you’re familiar with Einstein’s theory of special relativity, you know that strange things can start to happen at extreme speeds, to keep things consistent with the universal law that light always seems to move the same speed, no matter what speed you’re going. Even if the sleigh is moving incredibly fast, the light coming off of Rudolph’s nose can’t move any faster than c—so, just like the sound waves, it’d be frequency-shifted: higher (bluer) as the sleigh approaches, and lower (redder) as it recedes into the distance. At 650 miles per second, would Rudolph’s nose even still appear red?

Perhaps surprisingly, the answer is yes! Although 650 miles per second is fast enough that we have to completely ignore air resistance in order to avoid incinerating Santa like a meteor, it’s still only about 0.3% of the speed of light. And while a “Ho Ho Ho!” would sound positively demonic at that speed (if it were even still in the range of human hearing), the color of light coming off Rudolph’s nose wouldn’t change much at all. This also means that the passage of time wouldn’t be altered appreciably; Santa’s got plenty of helpers to make his journey, but Einstein’s relativity isn’t one of them. It’s also worth noting that, if he were moving at relativistic speed, it’d mean time passed slower for him compared to us—so he’d have less time in between houses to rummage around in his bag, rather than more.

But 650 miles per second is fast. Just consider: to travel at that kind of speed, Santa’s sleigh would have to have stirrups built in, or else the sled couldn’t follow the curvature of the Earth without its driver being ejected by the centrifugal force. If his foot slipped out at full speed, his trajectory wouldn’t just carry him all the way out of Earth’s gravitational field, but all the way out of the entire galaxy, never to return.

To all the good boys and girls who’ll be getting presents this year, and especially to the elves who engineered the handling on the ride that makes it all possible, Merry Christmas from PhysicsCentral!

P.S. If you’re having instant cocoa with your family this year, check out our piece on the “Hot Cocoa effect” for some tasty acoustic physics!

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