General relativity

The universe in a cup of coffee

black coffee, Vagabond, Highbury
The universe in a cup of coffee, but how much can we take this literally?

When people ask, what is Bean Thinking about, they often get the reply, it’s about “the universe in a cup of coffee”*. And it is perfectly true, much of the physics of the coffee cup is mirrored by the physics of the universe: you could think about the Black body radiation and the Cosmic Microwave Background, or the steam from the cup and cloud formation, but what about General Relativity? Could it really be that physics such as that of General Relativity mirrored in a coffee cup?

It could, perhaps, initially appear a ludicrous idea. Einstein’s theory of General Relativity explains the gravitational attractions of massive objects such as stars and planets through the curvature of space-time. And although what occurs on the planetary scale must also be valid on the scale of the coffee cup, we would surely expect classical, Newtonian physics to dominate here. But that would be to neglect the equally ludicrously named “Cheerios effect” and a paper that was published in Nature Communications earlier this month.

The cheerios effect is the phenomenon that you may have noticed on your tea or coffee whereby two floating objects on the surface are attracted to each other (and named after observations of the effect in a breakfast bowl). Two bits of a dropped biscuit come together or two bubbles bounce to form a pair. The effect occurs because both objects dent the surface of the drink by bending the surface of the liquid through surface tension effects. Consequently, the two objects don’t float on a flat coffee surface but a curved one and when they get close enough together, the surface tension effects bring the objects together into one big indentation rather than two smaller ones.

You can see surface tension effects from the curvature of the coffee around the edge of a cup. It is also visible around objects that float on top of the coffee.

On the face of it, this has similarities with the ‘cartoon version’ (or schematic) of the idea of gravity in general relativity. Each massive object (ie. any object with mass) bends the space-time around it, the more massive an object, the more the space-time is bent. This has the effect of seeming to bend light and leads to gravitational attraction. And yet there are very many differences. A liquid surface is 2D, planets clearly move in at least 4D, the way the surface bends owing to surface tension is surely not the same as the way that space time bends owing to its distortion through massive objects. It could go on only it turns out that some of the maths is quite similar: the surface is distorted proportional to the mass of the object in a cup of coffee, the attraction between the objects is a product of both masses (as it is with gravity). Indeed, it has even been proposed that studying the cheerios effect could be a way of gaining insight into some of the problems of general relativity. But there was always a catch: Friction.

On the surface of a coffee, although the floating object is bending the surface proportional to its mass, it is in some sense in contact with the fluid. When the object moves, there is a frictional resistance to the movement caused by the object’s interaction with the coffee. This makes it quite different from the situation in space. And so you would have been correct in your suspicion that general relativity would not be easily found in a coffee cup, but only for reasons of friction.

Which is where the recent Nature Communications paper comes in. Rather than float objects on coffee, the researchers floated silicone oil droplets on liquid nitrogen. Being a liquid, the nitrogen is subject to surface tension effects just like coffee, but being a very cold liquid (196 C below freezing point), it shows a second effect when the (room temperature, ie. warm) oil droplets are floated onto it: the inverse Leidenfrost effect.

Coffee, Van Gogh
What do you see in your coffee cup?

Again, you may have seen the Leidenfrost effect while frying eggs (or tofu if you’re vegan). When the frying pan is very hot, drops of water sprinkled into the pan will immediately vaporise in the layer between the pan and the droplet causing the drop to dance around the pan as if it is flying. The inverse Leidenfrost effect is, perhaps unsurprisingly, the inverse of this. When the liquid is very cold and a hot object is introduced to its surface it will instantaneously vaporise meaning that the hot object on the surface will skip over the cold liquid, without friction.

The reason that this is relevant to the idea of general relativity in a coffee cup is that this bending of the surface of the liquid nitrogen, coupled with the inverse Leidenfrost effect effectively levitating the drops means that you have a warped liquid surface, like the bending of space-time, but the floating object moves with absolutely no friction, because there is no contact between it and the liquid beneath. Clever.

And so what happens when you introduce two droplets to the nitrogen surface? How do they interact? Well, they attract each other and can even orbit each other like planets until, as the friction effects start to grow even in this system, the drops cease behaving as planets and can collide. It is a fascinating observation but one with relevance to biological self-organisation rather than an immediate extension to general relativity. That will be for another study, perhaps one with super-cold brew coffee.

So, the universe in a cup of coffee? Perhaps. But sometimes not strictly literally.

You can read the paper in Nature Communications here (it’s open access), or the summary in Physics.Org here.

*With suitable acknowledgement of the Feynman anecdote that you can see here.