General relativity – Bean Thinking

Outside Treelogy on Eastbourne Terrace, Paddington. The topiary could suggest a series of strontium atoms lining up on top of each other.

Good coffee near a mainline train station? It’s often difficult to find a good spot to take time to enjoy a coffee if you only have about 30 minutes (or less) before your train. Fortunately for coffee lovers in London, both Kings Cross (/St Pancras) and Paddington have several very good speciality coffee places nearby. There’s the cafe in the Pilgrm hotel just across the road from Paddington on London St, but Treelogy is perhaps even closer, directly opposite the buildings that house the new Elizabeth Line on Eastbourne Terrace.

Treelogy appears to have opened in April 2023. There does not seem to be much information online about it apart from Trip Advisor reviews so, having approximately 50 minutes before we needed to catch a train, we decided to stop at this new cafe. The interior is very modern and open. The counter is in front of you on the left as you enter with plenty of seats in the window and along the wall, as befits a cafe that is also close to a station. The style of some of the seats in the cafe and the fact that it is going to attract people who are about to embark on journeys (or have just come off a journey) means that there are elements here that could remind you of the scene in Edward Hopper’s Nighthawks. The coffee appears to be roasted by Treelogy themselves. There was a wide selection of pastries and breakfast bagels arranged on the counter and so we ordered two coffees to stay, and a bagel for the train.

The “Real Time” clock by Maarten Baas in Paddington. How much do they pay that man to be there all day?

We intended to sit on the bench just outside the cafe with our coffees but nonetheless we were offered our coffee in ceramic cups which was a nice touch. Inside, there was plenty to notice: circular lights on the wall leading to the back of the cafe which resembled ship lighting. A coffee dictionary book (and a book by Martin Wolf) that could offer a good read or a thought train on the physics of finance and the (Brownian-motion) links to coffee. The travellers with their roller bags going in and out of the cafe, who are they and where are they going? Yet, moving outside and settling down, the oat milk flat white and long black were both a very enjoyable way to spend time with a coffee.

As we were ‘spending the time’ with the coffees, the hands of the “REAL TIME, Paddington” Maarten Baas clock were being re-drawn every minute. Installed back in 2021, this clock appears as if the time is being painted onto the clock face by a man who seems trapped inside the clock. Each minute he erases the minute hand before redrawing it into its new correct position. Literally marking the minutes before our train is due.

For a physicist waiting for a train, an immediate thought may occur: what does ‘Real time’ mean? Admittedly, this question fades into the background again as the man wanders around, points at something on the clock, adjusts his position and then gets ready to move the clock hand again. The art is distracting from the question. But the question keeps surfacing: what is a minute, what is a second, is time absolute? There is perhaps a diversion that could be made here to a more philosophical question about the nature of time and our perception of it but we only had one long black and one flat white, the physics may take longer than that anyway!

A closer view of the man in the clock as he is erasing the minute hand of the clock. The colour bands on the clock face are not really there but are the result of the projected video onto the clock face and the way that the camera images that.

The physics bit remains because you may remember hearing about Einstein’s twin paradox, a thought experiment arising out of an aspect of his theory of Special Relativity. Relativity in general in physics refers to moving ‘frames of reference’, a classic case is that of a person on a train relative to a person on a station platform. For the person on the train, they are stationary, with respect to the train carriage. If they bounce a ball on the floor of a carriage, the ball bounces straight back up at them. They do not experience themselves moving (apart from when the train is accelerating or braking) and instead to them it appears that the person standing on the station platform is moving, backwards at the speed of the train.

Ordinarily our brains will process this and recognise that it is we who are on the train that are moving and we identify the ‘rest frame’ (the frame that is not moving) with the station platform. However we may all have experienced the sensation when on a train in a station next to another train. As the guard whistle blows the train moves but we cannot immediately tell whether it is our train that moves or the train next to us. This is the essence of relativity: all reference frames move relative to each other. The frame that is genuinely at rest is the one we define so (even the station platform is moving relative to the Sun, we just don’t notice this movement of the Earth at all).

Einstein’s theory of special relativity arises out of the special case when one of the moving frames is travelling at close to the speed of light, c. As the speed of light in a vacuum is constant, what would happen if someone travelling in a car at a speed just less than c looked at themselves in the rear view mirror? Einstein’s answer was that they would see themselves as anyone would because, relative to the reference frame of the car, the speed of light is still constant, it is still c. However an observer outside the car looking at the car and the person looking at themselves in the mirror also measures the speed of light as c, not nearly 2c (the speed of light plus the speed of the car). The speed of light in a vacuum is constant!

The explanation for this apparent problem is that our perception of time (and of distance) is not the same at different speeds. A person moving at a fast speed (relative to a person defined at rest) would have a wrist watch that was slow, relative to the person at rest – moving clocks go slow. This is the origin of the twin paradox which is that if one of a pair of twins travels away from Earth at close to the speed of light and returns, they will return younger than their twin who remained ‘at rest’ on Earth (but not relative to the twin who travelled who considered themselves at rest too so their earth bound twin should, to them, be younger).

Topiary at the entrance to Treelogy. The atomic clocks used in the study described in the text used super cold strontium ions positioned just above each other.

The solution of the twin paradox comes with Einstein’s second theory of relativity: General relativity. Special relativity only concerns the case when different frames of reference move at constant relative velocity to each other. General relativity extends the case to accelerating frames and gravity. In order to meet again, the twin in the space ship had to turn around (decelerate and accelerate again). This changes the situation from the case expected purely from special relativity. There is a lot of experimental evidence for both special and general relativity, but recently one test of general relativity tested the idea on a very small scale.

The theory of general relativity postulates that it is not just moving clocks that go slower. Clocks in strong gravitational fields will also run more slowly. The extreme example of this would be the event horizon of a black hole, but even on Earth, a clock closer to the centre of the Earth will tick more slowly than one that is further away. Remarkably this prediction has recently been verified using extremely accurate clocks by measuring time using atoms spaced just 1 mm apart. The ‘clocks’ of the atoms 1 mm lower moved slower than the clocks of the atoms 1 mm above. Absolutely astonishing! And yet absolutely expected because one remarkable and weird feature about physics is that it seems to be universally applicable: what happens at the event horizon of a black hole shares the same physics as what happens in conditions far less extreme, conditions found in a coffee cup.

The Real Time clock is 7.8m above the pavement where I was enjoying my coffee. These experiments mean that I can be confident that the clock is going very slightly faster than the time I experience sitting on the bench. However, I shouldn’t use this thought to justify enjoying my coffee much longer and thereby miss the train! It seems that our trains aren’t quite so precise as the deviations implied by the theory of General Relativity. It is still necessary to get through the barriers with several minutes to spare. Treelogy, and the clock man, will have to wait for a return visit.

Treelogy is at 48 Eastbourne Terrace, W2 6LG

More about Einstein’s theories of relativity can be found here or in a good book in a library.

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.