If I Fits, I Sits

Today's special guest is biologist/bioengineer Joanne Manaster.  In addition to lecturing at the University of Illinois-Urbana, Joanne does great science promotion through her blog Joanne Loves Science.  She writes, "How about you do a more thorough answer to my 'Cats in Sinks' video?"

The video, which illustrates the differences between theoretical and experimental work, asks, "How many cats can fit in a sink?"

If you've ever tried to fit all your personal belongings in a small U-Haul, you're familiar with the packing problem.  In its most basic form, the problem asks "How many X can you fit in a space Y?"  The packing of irregularly-shaped objects has been studied since antiquity, and there's a lot of physics involved.  Current research on packing has applications in cancer treatment, secure wireless networks, microelectronics, demolitions, and apparently putting cats in sinks.

Cats are certainly no strangers to packing themselves in tight places, and there are many ways to determine just how tightly they can pack themselves.  For example, cats are about as dense as water, so you could weigh the cat and use this density to find its volume.  A less scrupulous way of accomplishing the same goal would be to use a blender, but this inevitably leads to some cat juice slipping through the drain and throwing off the final number.

Since I don't have an experimental budget or a team of lawyers to defend me from animal cruelty charges, I'm going to tackle this problem theoretically.  Very theoretically.

You may have heard of Schrödinger's "Cat in a Box."  It's a quantum mechanical thought experiment, but here we'll be using a relativistic cat in box.  According to Einstein's theory of relativity, the length of a moving object contracts as it goes faster and faster.  For velocities in our everyday experience, the effect is too small to notice unless we make a very precise measurement.  The effect is, however, very noticeable for objects moving close to the speed of light.  According to Einstein, a moving object's length contracts by a factor of

f = (1 − v²/c²)^1/2,

where v is the speed of the object and c = 3×10⁸ m/s is the speed of light.  If 10% of people in the world have cats, there would be roughly one-billion (~10⁹) cats.  These cats could fit quite comfortably in a rocket ship that was one-billion meters long.  For this rocket ship to fit in the sink, it would need to contract by a factor of about three billion (f = 3.0×10^-10).  Solving for the velocity, we find that every cat in the world could fit in the sink if they were loaded on a rocket ship traveling with a speed

v =  99.999999999999999995% the speed of light.

Since this rocket ship is essentially traveling at the speed of light, the cats would, admittedly, not remain in the sink very long.  While you might think that cats moving at the speed of light would suffer just as much as the cat in a blender, I assure you they remain quite comfortable.  According to relativity, the cats (in their own frame) remain the same size as normal and go about as usual, blissfully unaware that they're traveling at the speed of light.  For them, it is the sink and the rest of the world that is contracted.  I'll leave it to the reader to see if he/she can resolve the seeming paradox of how one-billion cats can fit in a contracted sink.

Rarely do I get the opportunity to answer a question that involves relativity, hypothetical animal cruelty, and cats.  Thanks, Joanne.  The internet should be pleased.  To find out more about Joanne, visit her blog or follow her on twitter @sciencegoddess.

Aaron Santos is a physicist and author of the books How Many Licks? Or How to Estimate Damn Near Anything and Ballparking: Practical Math for Impractical Sports Questions.