# A black hole Pokémon

The world is currently going mad for Pokémon Go, so it seems like the perfect time to answer the most burning of scientific questions: what would a black hole Pokémon be like?

Type: Dark/Ghost

Black holes are, well, black. Their gravity is so strong that if you get close enough, nothing, not even light, can escape. I think that’s about as dark as you can get!

After picking Dark as a primary type, I thought Ghost was a good secondary type, since black holes could be thought of as the remains of dead stars. This also fit well with black holes not really being made of anything—they are just warped spacetime—and so are ethereal in nature. Of course, black holes’ properties are grounded in general relativity and not the supernatural.

In the games, having a secondary type has another advantage: Dark types are weak against Fighting types. In reality, punching or kicking a black hole is a Bad Idea™: it will not damage the black hole, but will certainly cause you some difficulties. However, Ghost types are unaffected by Fighting-type moves, so our black hole Pokémon doesn’t have to worry about them.

Height: 0’04″/0.1 m

Real astrophysical black holes are probably a bit too big for Pokémon games.  The smallest Pokémon are currently the electric bug Joltik and fairy Flabébé, so I’ve made our black hole Pokémon the same size as these. It should comfortably fit inside a Pokéball.

Measuring the size of a black hole is actually rather tricky, since they curve spacetime. When talking about the size of a black hole, we normally think in terms of the Schwarzschild radius. Named after Karl Schwarzschild, who first calculated the spacetime of a black hole (although he didn’t realise that at the time), the Schwarzschild radius correspond to the event horizon (the point of no return) of a non-spinning black hole. It’s rather tricky to measure the distance to the centre of a black hole, so really the Schwarzschild radius gives an idea of the circumference (the distance around the edge) of the event horizon: this is 2π times the Schwarschild radius. We’ll take the height to really mean twice the Schwarzschild radius (which would be the Schwarzschild diameter, if that were actually a thing).

Weight: 7.5 × 1025 lbs/3.4 × 1025 kg

Although we made our black hole pocket-sized, it is monstrously heavy. The mass is for a black hole of the size we picked, and it is about 6 times that of the Earth. That’s still quite small for a black hole (it’s 3.6 million times less massive than the black hole that formed from GW150914’s coalescence). With this mass, our Pokémon would have a significant effect on the tides as it would quickly suck in the Earth’s oceans. Still, Pokémon doesn’t need to be too realistic.

Our black hole Pokémon would be by far the heaviest Pokémon, despite being one of the smallest. The heaviest Pokémon currently is the continent Pokémon Primal Groudon. This is 2,204.4 lbs/999.7 kg, so about 34,000,000,000,000,000,000,000 times lighter.

Within the games, having such a large weight would make our black hole Pokémon vulnerable to Grass Knot, a move which trips a Pokémon. The heavier the Pokémon, the more it is hurt by the falling over, so the more damage Grass Knot does. In the case of our Pokémon, when it trips it’s not so much that it hits the ground, but that the Earth hits it, so I think it’s fair that this hurts.

Gender: Unknown

Black holes are beautifully simple, they are described just by their mass, spin and electric charge. There’s no other information you can learn about them, so I don’t think there’s any way to give them a gender. I think this is rather fitting as the sun-like Solrock is also genderless, and it seems right that stars and black holes share this.

Ability: Sticky Hold
Hidden ability:
Soundproof

Sticky Hold prevents a Pokémon’s item from being taken. (I’d expect wild black hole Pokémon to be sometimes found holding Stardust, from stars they have consumed). Due to their strong gravity, it is difficult to remove an object that is orbiting a black hole—a common misconception is that it is impossible to escape the pull of a black hole, this is only true if you cross the event horizon (if you replaced the Sun with a black hole of the same mass, the Earth would happily continue on its orbit as if nothing had happened).

Soundproof is an ability that protects Pokémon from sound-based moves. I picked it as a reference to sonic (or acoustic) black holes. These are black hole analogues—systems which mimic some of the properties of black holes. A sonic black hole can be made in a fluid which flows faster than its speed of sound. When this happens, sound can no longer escape this rapidly flowing region (it just gets swept away), just like light can’t escape from the event horizon or a regular black hole.

Sonic black holes are fun, because you can make them in the lab. You can them use them to study the properties of black holes—there is much excitement about possibly observing the equivalent of Hawking radiation. Predicted by Stephen Hawking (as you might guess), Hawking radiation is emitted by black holes, and could cause them to evaporate away (if they didn’t absorb more than they emit). Hawking radiation has never been observed from proper black holes, as it is very weak. However, finding the equivalent for sonic black holes might be enough to get Hawking his Nobel Prize…

Moves:

Start — Gravity
Start — Crunch

The starting two moves are straightforward. Gravity is the force which governs black holes; it is gravity which pulls material in and causes the collapse  of stars. I think Crunch neatly captures the idea of material being squeezed down by intense gravity.

Level 16 — Vacuum Wave

Vacuum Wave sounds like a good description of a gravitational wave: it is a ripple in spacetime. Black holes (at least when in a binary) are great sources of gravitational waves (as GW150914 and GW151226 have shown), so this seems like a sensible move for our Pokémon to learn—although I may be biased. Why at level 16? Because Einstein first predicted gravitational waves from his theory of general relativity in 1916.

Level 18 — Discharge

Black holes can have an electric charge, so our Pokémon should learn an Electric-type move. Charged black holes can have some weird properties. We don’t normally worry about charged black holes for two reasons. First, charged black holes are difficult to make: stuff is usually neutral overall, you don’t get a lot of similarly charged material in one place that can collapse down, and even if you did, it would quickly attract the opposite charge to neutralise itself. Second, if you did manage to make a charged black hole, it would quickly lose its charge: the strong electric and magnetic fields about the black hole would lead to the creation of charged particles that would neutralise the black hole. Discharge seems like a good move to describe this process.

Why level 18? The mathematical description of charged black holes was worked out by Hans Reissner and Gunnar Nordström, the second paper was published in 1918.

Level 19 —Light Screen

In general relativity, gravity bends spacetime. It is this warping that causes objects to move along curved paths (like the Earth orbiting the Sun). Light is affected in the same way and gets deflected by gravity, which is called gravitational lensing. This was the first experimental test of general relativity. In 1919, Arthur Eddington led an expedition to measure the deflection of light around the Sun during a solar eclipse.

Black holes, having strong gravity, can strongly lens light. The graphics from the movie Interstellar illustrate this beautifully. Below you can see how the image of the disc orbiting the black hole is distorted. The back of the disc is visible above and below the black hole! If you look closely, you can also see a bright circle inside the disc, close to the black hole’s event horizon. This is known as the light ring. It is where the path of light gets so bent, that it can orbit around and around the black hole many times. This sounds like a Light Screen to me.

Light-bending around the black hole Gargantua in Interstellar. The graphics use proper simulations of black holes, but they did fudge a couple of details to make it look extra pretty. Credit: Warner Bros./Double Negative.

Level 29 — Dark Void
Level 36 — Hyperspace Hole

These are three moves which with the most black hole-like names. Dark Void might be “black hole” after a couple of goes through Google Translate. Hyperspace Hole might be a good name for one of the higher dimensional black holes theoreticians like to play around with. (I mean, they like to play with the equations, not actually the black holes, as you’d need more than a pair of safety mittens for that). Shadow Ball captures the idea that a black hole is a three-dimensional volume of space, not just a plug-hole for the Universe. Non-rotating black holes are spherical (rotating ones bulge out at the middle, as I guess many of us do), so “ball” fits well, but they aren’t actually the shadow of anything, so it falls apart there.

I’ve picked the levels to be the masses of the two black holes which inspiralled together to produce GW150914, measured in units of the Sun’s mass, and the mass of the black hole that resulted from their merger. There’s some uncertainty on these measurements, so it would be OK if the moves were learnt a few levels either way.

Level 63 — Whirlpool
Level 63 — Rapid Spin

When gas falls into a black hole, it often spirals around and forms into an accretion disc. You can see an artistic representation of one in the image from Instellar above. The gas swirls around like water going down the drain, making Whirlpool and apt move. As it orbits, the gas closer to the black hole is moving quicker than that further away. Different layers rub against each other, and, just like when you rub your hands together on a cold morning, they heat up. One of the ways we look for black holes is by spotting the X-rays emitted by these hot discs.

As the material spirals into a black hole, it spins it up. If a black hole swallows enough things that were all orbiting the same way, it can end up rotating extremely quickly. Therefore, I thought our black hole Pokémon should learn Rapid Spin as the same time as Whirlpool.

I picked level 63, as the solution for a rotating black hole was worked out by Roy Kerr in 1963. While Schwarzschild found the solution for a non-spinning black hole soon after Einstein worked out the details of general relativity in 1915, and the solution for a charged black hole came just after these, there’s a long gap before Kerr’s breakthrough. It was some quite cunning maths! (The solution for a rotating charged black hole was quickly worked out after this, in 1965).

Level 77 — Hyper Beam

Another cool thing about discs is that they could power jets. As gas sloshes around towards a black hole, magnetic fields can get tangled up. This leads to some of the material to be blasted outwards along the axis of the field. We’ve some immensely powerful jets of material, like the one below, and it’s difficult to imagine anything other than a black hole that could create such high energies! Important work on this was done by Roger Blandford and Roman Znajek in 1977, which is why I picked the level. Hyper Beam is no exaggeration in describing these jets.

Jets from Centaurus A are bigger than the galaxy itself! This image is a composite of X-ray (blue), microwave (orange) and visible light. You can see the jets pushing out huge bubbles above and below the galaxy. We think the jets are powered by the galaxy’s central supermassive black hole. Credit: ESO/WFI/MPIfR/APEX/NASA/CXC/CfA/A.Weiss et al./R.Kraft et al.

After using Hyper Beam, a Pokémon must recharge for a turn. It’s an exhausting move. A similar thing may happen with black holes. If they accrete a lot of stuff, the radiation produced by the infalling material blasts away other gas and dust, cutting off the black hole’s supply of food. Black holes in the centres of galaxies may go through cycles of feeding, with discs forming, blowing away the surrounding material, and then a new disc forming once everything has settled down. This link between the black hole and its environment may explain why we see a trend between the size of supermassive black holes and the properties of their host galaxies.

Level 100 — Spacial Rend
Level 100 — Roar of Time

To finish off, since black holes are warped spacetime, a space move and a time move. Relativity say that space and time are two aspects of the same thing, so these need to be learnt together.

It’s rather tricky to imagine space and time being linked. Wibbly-wobbly, timey-wimey, spacey-wacey stuff gets quickly gets befuddling. If you imagine just two space dimension (forwards/backwards and left/right), then you can see how to change one to the other by just rotating. If you turn to face a different way, you can mix what was left to become forwards, or to become a bit of right and a bit of forwards. Black holes sort of do the same thing with space and time. Normally, we’re used to the fact that we a definitely travelling forwards in time, but if you stray beyond the event horizon of a black hole, you’re definitely travelling towards the centre of the black hole in the same inescapable way. Black holes are the masters when it comes to manipulating space and time.

There we have it, we can now sleep easy knowing what a black hole Pokémon would be like. Well almost, we still need to come up with a name. Something resembling a pun would be traditional. Suggestions are welcome. The next games in the series are Pokémon Sun and Pokémon Moon. Perhaps with this space theme Nintendo might consider a black hole Pokémon too?

# Neutrino oscillations and Nobel Prizes

This year’s Nobel Prize in Physics was awarded to Takaaki Kajita and Arthur McDonald for the discovery of neutrino oscillations. This is some really awesome physics which required some careful experimentation and some interesting new theory; it is also one of the things that got me interested in astrophysics.

### Neutrinos

Neutrinos are an elusive type of subatomic particle. They are sometimes represented by the Greek letter nu $\nu$, and their antiparticle equivalents (antineutrinos) are denoted by $\bar{\nu}$. We’ll not worry about the difference between the two. Neutrinos are rather shy. They are quite happy doing their own thing, and don’t interact much with other particles. They don’t have an electric charge (they are neutral), so they don’t play with the electromagnetic force (and photons), they also don’t do anything with the strong force (and gluons). They only get involved with the weak force (W and Z bosons). As you might expect from the name, the weak force doesn’t do much (it only operates over short distances), so spotting a neutrino is a rare occurrence.

The charming bestiary of subatomic particles made by Particle Zoo.

There is a large family of subatomic particles. The electron is one of the most familiar, being a component of atoms (and hence you, me, cake and even marshmallows). The electron has two cousins: the muon (not to be confused with the moo-on) and the tau particle. All three have similar characteristics, with the only real difference being their mass. Electrons are the lightest, muons are about 207 times heavier, and tau are about 17 times heavier still (3477 times the mass of the electron). Each member of the electron family has a neutrino counterpart: there’s the electron-neutrino $\nu_e$, the muon-neutrino $\nu_\mu$ ($\mu$ is the Greek letter mu) and the tau-neutrino $\nu_\tau$ ($\tau$ is the Greek letter tau).

Neutrinos are created and destroyed in in certain types of nuclear reactions. Each flavour of neutrino is only involved in reactions that involve their partner from the electron family. If an electron-neutrino is destroyed in a reaction, an electron is created; if a muon is destroyed, a muon-neutrino is created, and so on.

### Solar neutrinos

Every second, around sixty billion neutrinos pass through every square centimetre on the Earth. Since neutrinos so rarely interact, you would never notice them. The source of these neutrinos is the Sun. The Sun is powered by nuclear fusion. Hydrogen is squeezed into helium through a series of nuclear reactions. As well as producing the energy that keeps the Sun going, these create lots of neutrinos.

The nuclear reactions that power the Sun. Protons ($p$), which are the nuclei of hydrogen, are converted to Helium nuclei after a sequence of steps. Electron neutrinos $\nu_e$ are produced along the way. This diagram is adapted from Giunti & Kim. The traditional names of the produced neutrinos are given in bold and the branch names are given in parentheses and percentages indicate branching fractions.

The neutrinos produced in the Sun are all electron-neutrinos. Once made in the core of the Sun, they are free to travel the 700,000 km to the surface of the Sun and then out into space (including to us on Earth). Detecting these neutrinos therefore lets you see into the very heart of the Sun!

Solar neutrinos were first detected by the Homestake experiment. This looked for the end results of nuclear reactions caused when an electron-neutrino is absorbed. Basically, it was a giant tub of dry-cleaning fluid. This contains chlorine, which turns to argon when a neutrino is absorbed. The experiment had to count how many atoms of argon where produced. In 1968, the detection was announced. However, we could only say that there were neutrinos around, not that they were coming from the Sun…

To pin down where the neutrinos were coming from required a new experiment. Deep in the Kamioka Mine, Kamiokande looked for interactions between neutrinos and electrons. Very rarely a neutrino will bump into an electron. This can give the electron a big kick (since the neutrino has a lot of momentum). Kamiokande had a large tank of water (and so lots of electrons to hit). If one got a big enough kick, it could travel faster than the speed of light in water (about 2/3 of the speed of light in vacuum). It then emits a flash of light called Cherenkov radiation, which is the equivalent of the sonic boom created when a plane travels faster than the speed of sound. Looking where the light comes from tells you where the electron was coming from and so where the neutrino came from. Tracing things back, it was confirmed that the neutrinos were coming from the Sun!

This discovery confirmed that the Sun was powered by fusion. I find it remarkable that it was only in the late 1980s that we had hard evidence for what was powering the Sun (that’s within my own lifetime). This was a big achievement, and Raymond Davies Jr., the leader of the Homestake experiment, and Masatoshi Koshiba, the leader of the Kamiokande experiment, were awarded the 2002 Nobel Prize in Physics for pioneering neutrino astrophysics. This also led to one of my all-time favourite pictures: the Sun at night.

The Sun at night. Solar neutrinos as detected by Super-Kamioknade looking through the Earth. I think this is the astronomical equivalent of checking if the fridge light does go off when you close the door. Credit: Marcus Chown & Super-Kamiokande.

### The mystery of the missing neutrinos

Detecting solar neutrinos was a big success, but there was a problem. There were only a fraction of the predicted number. This became known as the solar neutrino problem. There were two possibilities, either solar physicists had got their models wrong, or particle physicists were missing a part of the Standard Model.

The solar models were recalculated and tweaked, with much work done by John Bahcall and collaborators. More sophisticated calculations were performed, even folding in new data from helioseismology, the study of waves in the Sun, but the difference could not be resolved.

However, there was an idea in particle physics by Bruno Pontecorvo and Vladimir Gribov: that neutrinos could change flavour, a phenomena known as neutrino oscillations. This was actually first suggested before the first Homestake results were announced, perhaps it deserved further attention?

The first evidence in favour of neutrino oscillations comes from Super-Kamiokande, the successor to the original Kamiokande. This evidence came from looking at neutrinos produced by cosmic rays. Cosmic rays are highly energetic particles that come from space. As they slam into the atmosphere, and collide with molecules in the air, they produce a shower of particles. These include muons and muon-neutrinos. Super-Kamiokande could detect muon-neutrinos from cosmic rays. Cosmic rays come from all directions, so Super-Kamiokande should see muon-neutrinos from all directions too. Just like we can see the solar neutrinos through the Earth, we should see muon-neutrinos both from above and below. However, more were detected from above than below.

Something must happen to muon-neutrinos during their journey through the Earth. Super-Kamiokande could detect them as electron-neutrinos or muon-neutrinos, but is not sensitive to tau-neutrinos. This is evidence that muon-neutrinos were changing flavour to tau-neutrinos.

The Sudbury Neutrino Observatory detector, a 12-metre sphere containing 1000 tonnes of heavy water which is two kilometres underground. Credit: SNOLAB.

The solar neutrino problem was finally solved in 2001 through measurements of the Sudbury Neutrino Observatory (SNO). SNO is another Cherenkov detector like (Super-)Kamiokande, but it used heavy water instead of regular water. (High-purity heavy water is extremely expensive, it would have cost hundreds of millions of dollars for SNO to buy the 1000 tonnes it used, so it managed to secure it on loan from Atomic Energy of Canada Limited). Using heavy water meant that SNO was sensitive to all flavours of neutrinos. Like previous experiments, SNO found that there were not as many electron-neutrinos from the Sun as expected. However, there were also muon-neutrinos and tau-neutrinos, and when these were added, the total worked out!

The solar astrophysicists had been right all along, what was missing was that neutrinos oscillate between flavours. Studying the Sun had led to a discovery about some of the smallest particles in Nature.

### Neutrino oscillations

Experiments have shown that neutrino oscillations occur, but how does this work? We need to delve into quantum mechanics.

The theory of neutrino oscillations say that each of the neutrino flavours corresponds to a different combination of neutrino mass states. This is weird, it means that if you were to somehow weight an electron-, muon- or tau-neutrino, you would get one of three values, but which one is random (although on average, each flavour would have a particular mass). By rearranging the mass states into a different combination you can get a different neutrino flavour. While neutrinos are created as a particular flavour, when they travel, the mass states rearrange relative to each other, so when they arrive at their destination, they could have changed flavour (or even changed flavour and then changed back again).

To get a more detailed idea of what’s going on, we’ll imagine the simpler case of there being only two neutrino flavours (and two neutrino mass states). We can picture a neutrino as a clock face with an hour hand and a minute hand. These represent the two mass states. Which neutrino flavour we have depends upon their relative positions. If they point in the same direction, we have one flavour (let’s say mint) and if they point in opposite directions, we have the other (orange). We’ll create a mint neutrino at 12 noon and watch it evolve. The hands more at different speeds, so at ~12:30 pm, they are pointing opposite ways, and our neutrino has oscillated into an orange neutrino. At ~1:05 pm, the hands are aligned again, and we’re back to mint. Which neutrino you have depends when you look. At 3:30 am, you’ll have a roughly even chance of finding either flavour and at 6:01 pm, you’ll be almost certain to have orange neutrino, but there’s still a tiny chance of finding an mint one. As time goes on, the neutrino oscillates back and forth.

With three neutrinos flavours, things are more complicated, but the idea is similar. You can imagine throwing in a second hand and making different flavours based upon the relative positions of all three hands.

We can now explain why Super-Kamiokande saw different numbers of muon-neutrinos from different sides of the Earth. Those coming from above only travel a short distance, there’s little time between when they were created and when they are detected, so there’s not much chance they’ll change flavour. Those coming through the Earth have had enough time to switch flavour.

A similar thing happens as neutrinos travel from the core of the Sun out to the surface. (There’s some interesting extra physics that happens here too. A side effect of there being so much matter at the centre of the Sun, the combination of mass states that makes up the different flavours is different than at the outside. This means that even without the hands on the clock going round, we can get a change in flavour).

Neutrino oscillations happen because neutrino mass states are not the same as the flavour states. This requires that neutrinos have mass. In the Standard Model, neutrinos are massless, so the Standard Model had to be extended.

2015 Physics Nobel laureates, Takaaki Kajita and Arthur B. McDonald. Credit: Nobel Foundation.

### Happy ending

For confirming that neutrinos have mass, Takaaki Kajita of Super-Kamiokande and Arthur McDonald of SNO won this year’s Nobel Prize. It is amazing how much physics has been discovered from trying to answer as simple a question as how does the Sun shine?

Even though neutrinos are shy, they are really interesting characters when you get to know them.

Now that the mystery of the missing neutrinos is solved, what is next? Takaaki Kajita is now involved in another project in the Kamioka Mine, the construction of KAGRA, a gravitational-wave detector.

The control room of KAGRA, the gravitational-wave detector in the Hida Mountains, Japan. I visited June 2015. Could a third Nobel come out of the Kamioka Mine?

# General relativity at 100

General relativity, our best theory of gravitation, turns 100 this week!

Happy birthday general relativity! Einstein presented his field equations to the Prussian Academy of Science on 25 November 1915.

Gravity is the force which pulls us down towards the ground and keeps the Earth in orbit around the Sun. It is the most important force in astrophysics, causing gas clouds to collapse down to become stars; binding gas, stars and dark matter to become galaxies, and governing the overall evolution of the Universe.

Our understanding of gravity dates back to Isaac Newton. Newton realised that the same force that makes apples fall from trees also controls the motion of the planets. Realising that we could use physics to explain the everyday and the entire cosmos was a big leap! Newton’s theory was hugely successful, but he was never quite satisfied with it. In his theory gravity acted between distant objects (the Earth and an apple or the Earth and the Sun) instantaneously, without any explanation of what was linking them. The solution to this would come over 200 years later from Albert Einstein.

Einstein’s first big idea didn’t come from thinking about gravity, but thinking about electromagnetism. Electromagnetism is the force that is responsible for fridge magnets sticking, atoms binding to form molecules and the inner workings of whatever device you are currently reading this on. According to the rules of electromagnetism, ripples in electromagnetic fields (better known as light) always travel at a particular speed. This tweaked Einstein’s curiosity, as the rules didn’t say what this speed was relative to: you should measure the same speed if standing still, travelling at 100 miles per hour in a train or at a million miles per hour in a spacecraft. Speed is the distance travelled divided by the time taken, so Einstein realised that if the speed is always the same, then distances and times must appear different depending upon how you are moving! Moving clocks tick slower; at everyday speeds this effect is tiny, but we have confirmed that this is indeed the case. These ideas about space and time became known as Einstein’s theory of special relativity. Special relativity has a couple of important consequences, one is the infamous equation, the other is that the speed of light becomes a universal speed limit.

Special relativity says that no information can travel faster than the speed of light; this is a problem for Newton’s theory of gravitation, where the effects of gravity are transmitted instantaneously. Einstein knew that he would have to extend his theory to include gravity and freely falling objects, and he spend almost 11 years pondering on the problem. The result was general relativity.

In special relativity, space and time become linked, merging into one another depending upon how you are moving relative to what you are measuring. General relativity takes this further and has space–time distorted by the energy and matter. This idea can be a little tricky to explain.

In Newtonian mechanics, things (apples, light, billiard balls, etc.) like to travel in straight lines. They keep going at a constant speed in the same direction unless there is a force acting on them. Gravity is a force which pulls things away from their straight line, pulling the Earth into its circular orbit around the Sun, and accelerating an apple towards the ground. In general relativity, we take a different view. Things still travel in a straight line, but the effect of gravity is to bend space–time! A straight line in a curved space is a curve. If we don’t know about the curvature, it looks like the object is pulled off its straight line and there must be a force doing this, which we call gravity. Alternatively, we can say that gravity curves the space–time, and that the object follows its straight line in this. In general relativity, space–time tells matter how to move; matter tells space–time how to curve.

The shortest way to travel from London Heathrow airport to JFK International airport. On a long-distance flight, you may have noticed that it appears that you are moving along a curved line, but that is because the shortest distance across the Earth’s curved surface is a curve. We call this a geodesic, and the same idea applies to curved space–time in general relativity. Credit: Mr Reid.

General relativity solves Newton’s original worries. Objects are connected by space–time. This is not the rigid background of Newtonian physics, but a dynamic object, that is shaped by its contents. Space–time is curved by mass, and when the mass moves or reshapes itself, it takes time for the curvature everywhere else to readjust. When you drop a pebble into a pond, you disturb the surface, but it takes a while for the water further away to know about the splash; there’s a ripple that travels outwards, carrying the information about the disturbance. A similar thing happens for changes in gravity, there are ripples in space–time. Ripples in electromagnetic fields are electromagnetic waves, and these ripples in the gravitational fields are gravitational waves: both travel at the speed of light, in agreement with special relativity.

General relativity is not only a beautiful theory, it has so far passed every experimental test. Right from the start Einstein looked for checks of his theory. One of the calculations he did while formulating his theory was how the orbit of Mercury would change. Mercury is the planet closest to the Sun and so experiences the strongest gravity. Its orbit isn’t a perfect circle, but an ellipse so that Mercury is sometimes a little closer to the Sun, and is sometimes a little further. In Newtonian gravity, each orbit should trace out exactly the same path, but in general relativity there is some extra rotation. Each orbit is slightly shifted with respect to the last, so if you traced out many orbits, you’d end up with a Spirograph-like pattern. This is known as precession of the orbit, and is a consequence of there being slightly greater curvature closer to the Sun. This evolution of Mercury’s orbit had already been measured. Some thought it indicated there was a new planet inside Mercury’s orbit (which was called Vulcan but isn’t Spock’s home) that was giving it a little pull. However, Einstein calculated the general relativity predicted exactly the right amount of extra rotation!

The next test came in 1919. General relativity predicts that the path of light is bent by massive objects. This is gravitational lensing. At the time, the only object that could cause measurable bending was the Sun. If we could measure a change in the position of background stars when the Sun was in front of them, we could check if the amount of bending was as expected. There’s an obvious problem here: the Sun’s so bright that you can’t see stars around it. Arthur Eddington had the idea of making the measurement during an eclipse. He mounted an expedition and confirmed the prediction. This was big news and made Einstein a superstar.

Now, 100 years after Einstein proposed his theory, we are poised to make the most precise tests. There is currently a global effort to directly detect gravitational waves. Measuring the gravitational waves will tell us if ripples in space–time behave as Einstein predicted. The waves will also tell us about the systems that created them, this will give us an up-close glimpse of black holes. Black holes are the regions of strongest gravity; they are where the curvature of space–time becomes so immense that all straight lines lead inwards. Checking that the black holes of Nature match what we expect from general relativity, will test the theory in the most extreme conditions possible.

The Advanced LIGO detectors are currently listening for gravitational-wave signals from merging neutron stars or black holes, and next year Advanced Virgo plans join the hunt too. We don’t (yet) know how often such signals occur, so we can’t say when the first detection will be made. Perhaps this will be soon and we will learn something more about gravitation…

Merging black holes create ripples in space time. These can be detected with a laser interferometer. Credit: Gravitational Wave Group.

# Interstellar—science and fiction

Planet and accretion disc orbiting Gargantua, the black hole in Interstellar. Visual effects produced by the cunning people of Double Negative.

Interstellar is the latest film from Christopher Nolan. After completing his work with the Dark Knight, it seems he has moved onto even darker material: black holes. I have been looking forward to the film for some time, but not because of Nolan’s involvement (even though I have enjoyed his previous work). The film is based upon the ideas of Kip Thorne, an eminent theoretical physicist. Kip literally wrote the book on general relativity. He was a pioneer of gravitational-wave science, and earlier versions of the script included the detection of gravitational waves (I’m sad that this has been removed). Here, I’ll briefly discuss the film, before going on to look at it’s science (there are some minor spoilers).

My copies of Gravitation by Misner, Thorne & Wheeler, and General Theory of Relativity by Dirac. The difference in length might tell you something about the authors. MTW (as Gravitation is often called) is a useful textbook. It is so heavy that you can actually use it for experiments testing gravity.

Last week, my research group organised a meeting for our LIGO collaborators. We all got together in Birmingham to work on how we analyse gravitational-wave data. It was actually rather productive. We decided to celebrate the end of our meeting with a trip to see Interstellar. The consensus was that it was good. We were rather pleased by the amount of science in the film, undoubtedly due to Kip’s involvement (even if he doesn’t approve of everything)—we also liked how there was a robot called KIPP.

My favourite characters were, by far, the robots. They had more personality than any of the other characters: I was more concerned for their survival than for anyone else. (No-one was wearing red, but I thought it was quite obvious who was expendable). Michael Caine’s character is apparently based upon Kip—they do have similar beards.

The film is beautiful. Its visualisations have been much hyped (we’ll discuss these later). It shows an obvious debt to Kubrick’s 2001: A Space Odyssey. This is both for better and worse: mimicking the wonderful cinematography and the slow pacing. However, the conclusion lacks the mystery of 2001 or even the intelligence of Nolan’s earlier work Memento or Inception (both of which I would highly recommend).

I don’t want to say too much about the plot. I (unsurprisingly) approve of its pro-science perspective. There were some plot points that irked me. In particular, why on Earth (or off Earth) would a mission with the aim of continuing the human race only take one woman? Had no-one heard about putting all your eggs in one basket? Also, using Morse code to transmit complicated scientific data seems like a bad idea™. What if there were a typo? However, I did enjoy the action sequences and the few tense moments.

## Why so scientific?

I expect that if you were after a proper film critique you’d be reading something else, so let’s discuss science. There is a lot of science in Interstellar, and I can’t go through it all, so I want to highlight a couple of pieces that I think are really cool.

### Time is relative

An interesting story device is the idea that time is relative, and its passing depends upon where you are in gravitational field. This is entirely correct (and although time might flow at different apparent speeds, it never goes backwards). Imagine that you are tied to a length of extremely long and strong string, and lowered towards a black hole. (I wonder if that would make a good movie?) Let’s start off by considering a non-rotating black hole. The passage of time for you, relative to your friend with the other end of the string infinitely far away from the black hole, depends how close to the black hole you are. Times are related by

$\displaystyle \Delta T_\mathrm{infinity} = \left(1 - \frac{2 G M}{c^2 r}\right)^{-1/2} \Delta T_\mathrm{string}$,

where $M$ is the black hole’s mass, $G$ is Newton’s gravitational constant, $c$ is the speed of light, and $r$ measures how far you are from (the centre of) the black hole (more on this in a moment). If you were to flash a light every $\Delta T_\mathrm{string}$, your friend at infinity would see them separated by time $\Delta T_\mathrm{infinity}$; it would be as if you were doing things in slow motion.

You might recognise $2GM/c^2$ as the location of the event horizon: the point of no return from a black hole. At the event horizon, we would be dividing by zero in the equation above, time would appear to run infinitely slowly for you. This is rather curious, time continues to run fine for you, but watching from infinity you would fade to a complete stand-still.

Actually, you would also fade from view too. The frequency of light gets shifted by gravity. Light is a wave, it’s frequency is set by how fast it completes one cycle. The period of the wave gets stretched using the formula above. As you get closer to a black hole, light from you becomes more red (then infra-red, radio, etc.), and also becomes dimmer (as less energy arrives at your friend at infinity in a given time). You appear to fade to to black as you approach the event horizon. This stretching of light to lower frequencies is known as red-shifting (as red light has the lowest frequencies of the visible spectrum). I didn’t see much sign of it in Interstellar (we’ll see the effect it should have had below), although it has appeared in an episode of Stargate: SG-1 as a plot device.

The event horizon is also the point where the force on the string would become infinite. Your friend at infinity would only be able to pull you back up if they ate an infinite amount of spinach, and sadly there is not enough balsamic dressing to go around.

A technicality that is often brushed over is what the distance $r$ actually measures. I said it tells you how how you are from the centre of the black hole, but it’s not as simple as dropping a tape measure in the see where the singularity is. In fact, we measure the distance differently. We instead measure the distance around the circumference of a circle, and divide this by $2\pi$ to calculate $r$. The further away we are, the bigger the circle, and so the larger $r$. If space were flat, this distance would be exactly the same as the distance to the middle, but when considering a black hole, we do not have flat space!

This time stretching due to gravity is a consequence of Einstein’s theory of general relativity. There is another similar effect in his theory of special relativity. If something travels past you with a speed $v$, then time is slowed according to

$\displaystyle \Delta T_\mathrm{you} = \left(1 - \frac{v^2}{c^2}\right)^{-1/2} \Delta T_\mathrm{whizzing\:thing}$.

If it were to travel closer and closer to the speed of light, the passage of time for it would slow to closer and closer to a standstill. This is just like crossing the event horizon.

Imagine that while you were sitting on the end of your string, a planet orbiting the black hole whizzed by. Someone of the planet flashes a torch every second (as they measure time), and when you see this, you flash your torch to your friend at infinity. The passage of time on the planet appears slowed to you because of the planet’s speed (using the special relativity formula above), and the passage of time for you appears slowed because of gravity to your friend at infinity. We can combine the two effects to work out the total difference in the apparent passage of time on the planet and at infinity. We need to know how fast the planet moves, but it’s not too difficult for a circular orbit, and after some algebra

$\displaystyle \Delta T_\mathrm{infinity} = \left(1 - \frac{3 G M}{c^2 r}\right)^{-1/2} \Delta T_\mathrm{planet}$.

In Interstellar, there is a planet where each hour corresponds the seven years at a distance. That is a difference of about 61000. We can get this with our formula if $r \approx 3GM/c^2$. Sadly, you can’t have a stable orbit inside $r = 6GM/c^2$, so there wouldn’t be a planet there. However, the film does say that the black hole is spinning. This does change things (you can orbit closer in), so it should work out. I’ve not done the calculations, but I might give it a go in the future.

### Black holes

Interstellar does an excellent job of representing a black hole. Black holes are difficult to visualise, but the film correctly depicts them as three-dimensional: they are not a two-dimensional hole.

As nothing escapes from a black hole (and they don’t have a surface), they are dark, a shadow on the sky. However, we can see their effects. The image at the top shows a disc about the black hole. Material falling into a black hole often has some angular momentum: it doesn’t fall straight in, but goes off to the side and swirls about, exactly as water whirls around the plug-hole before falling in. This material swirling around is known as an accretion disc. In the disc, things closer to the black hole are orbiting faster (just as planets closer to the Sun orbit faster than those further away). Hence different parts of the disc rub against each other. This slows the inner layers (making them lose angular momentum so that they move inwards), and also heats the disc. Try rubbing your hands together for a few seconds, they soon warm up. In an accretion disc about a black hole, things can become so hot (millions of degrees) that they emits X-rays. You wouldn’t want to get close because of this radiation! Looking for these X-rays is one way of spotting black holes.

The video below shows a simulation from NASA of an accretion disc about a black hole. It’s not quite as fancy as the Interstellar one, but it’s pretty cool. You can see the X-rays being red-shifted and blue-shifted (the opposite of red-shifted, when radiation gets squashed to higher frequencies) as a consequence of their orbital motion (the Doppler effect), but I’m not sure if it shows gravitational red-shifting.

Black holes bend spacetime, so light gets bent as it travels close to them. The video above shows this. You can see the light ring towards the centre, from light that has wrapped around the black hole. You can also see this in Interstellar. I especially like how the ring is offset to one side. This is exactly what you should expect for a rotating black hole: you can get closer in when you’re moving with the rotation of the black hole, getting swept around like a plastic duck around a whirlpool. You can also see how the disc appears bent as light from the back of the disc (which has to travel around the black hole) gets curved.

Light-bending around a black hole. This is figure 15 from James, von Tunzelmann, Franklin & Thorne (2015). The top image shows an accretion disc as seen in Interstellar, but without the lens flare. The middle image also includes (Doppler and gravitational) red-shifting that changes the colour of the light. To make the colour changes clear, the brightness has been artificially kept constant. The bottom image also includes the changes in brightness that would come with red-shifting. The left side of the disc is moving towards us, so it is brighter and blue-shifted, the right side is moving away so it is red-shifted. You can see (or rather can’t) how red-shifting causes things to fade from view. This is what the black hole and accretion disc would actually look like, but it was thought too confusing for the actual film.

It’s not only light from the disc that gets distorted, but light from stars (and galaxies) behind the black hole. This is known as gravitational lensing. This is one way of spotting black holes without accretion discs: you watch a field of stars and if a black hole passes in front of one, it’s gravitational lensing will magnify the star. Spotting that change tells you something has passed between you and the star, working our its mass and size can tell you if it’s a black hole.

Looking at the shadow of a black hole (the region from which there is no light, which is surrounded by the innermost light ring) can tell you about the structure of spacetime close to the black hole. This could give you an idea of its mass and spin, or maybe even test if it matches the predictions of general relativity. We are hoping to do this for the massive black hole at the centre of our Galaxy. This will be done using the Event Horizon Telescope, an exciting project to use several telescopes together to make extremely accurate images.

False-colour image of what the Event Horizon Telescope could see when look at Sagittarius A* (Dexter et al. 2010). Red-shifting makes some part of the the disc appear brighter and other parts dimmer.

Interstellar is science fiction, it contains many elements of fantasy. However, it does much better than most on getting the details of the physics correct. I hope that it will inspire many to investigate the fact behind the fiction (there’s now a paper out in Classical & Quantum Gravity about the visualisation of the black hole, it comes with some interesting videos). If you’ve not seen the film yet, it’s worth a watch. I wonder if they could put the gravitational waves back in for an extended DVD version?

Score out of 5 solar masses: enough for a neutron star, possibly not enough for a black hole.

# How big is a black hole?

Physicist love things that are simple. This may be one of the reasons that I think black holes are cool.

Black holes form when you have something so dense that nothing can resist its own gravity: it collapses down becoming smaller and smaller. Whatever formerly made up your object (usually, the remains of what made up a star), is crushed out of existence. It becomes infinitely compact, squeezed into an infinitely small space, such that you can say that the whatever was there no longer exists. Black holes aren’t made of anything: they are just empty spacetime!

Daisy, a spherical cow, or “moo-on”. Spherical cows are highly prized as pets amongst physicists because of their high degree of symmetry and ability to survive in a vacuum. They also produce delicious milkshakes.

Black holes are very simple because they are just vacuum. They are much simpler than tables, or mugs of coffee, or even spherical cows, which are all made up of things: molecules and atoms and other particles all wibbling about and interacting with each other. If you’re a fan of Game of Thrones, then you know the plot is rather complicated because there are a lot of characters. However, in a single glass of water there may be 1025 molecules: imagine how involved things can be with that many things bouncing around, occasionally evaporating, or plotting to take over the Iron Throne and rust it to pieces! Even George R. R. Martin would struggle to kill off 1025 characters. Black holes have no internal parts, they have no microstructure, they are just… nothing…

(In case you’re the type of person to worry about such things, this might not quite be true in a quantum theory, but I’m just treating them classically here.)

Since black holes aren’t made of anything, they don’t have a surface. There is no boundary, no crispy sugar shell, no transition from space to something else. This makes it difficult to really talk about the size of black holes: it is a question I often get asked when giving public talks. Black holes are really infinitely small if we just consider the point that everything collapsed to, but that’s not too useful. When we want to consider a size for a black hole, we normally use its event horizon.

The event horizon is not actually sign-posted. It’s not possible to fix a sign-post in empty space, and it would be sucked into the black hole. The sign would disappear faster than a Ramsay Street sign during a tour of the Neighbours set.

The event horizon is the point of no return. Once passed, the black hole’s gravity is inescapable; there’s no way out, even if you were able to travel at the speed of light (this is what makes them black holes). The event horizon separates the parts of the Universe where you can happily wander around from those where you’re trapped plunging towards the centre of the black hole. It is, therefore, a sensible measure of the extent of a black hole: it marks the region where the black hole’s gravity has absolute dominion (which is better than possessing the Iron Throne, and possibly even dragons).

The size of the event horizon depends upon the mass of the black hole. More massive black holes have stronger gravity, so there event horizon extends further. You need to stay further away from bigger black holes!

If we were to consider the simplest type of black hole, it’s relatively (pun intended) easy to work out where the event horizon is. The event horizon is a spherical surface, with radius

$\displaystyle r_\mathrm{S} = \frac{2GM}{c^2}$,

This is known as the Schwarzschild radius, as this type of black hole was first theorised by Karl Schwarszchild (who was a real hard-core physicist). In this formula, $M$ is the black hole’s mass (as it increases, so does the size of the event horizon); $G$ is Newton’s gravitational constant (it sets the strength of gravity), and $c$ is the speed of light (the same as in the infamous $E = mc^2$). You can plug in some numbers to this formula (if anything like me, two or three times before getting the correct answer), to find out how big a black hole is (or equivalently, how much you need to squeeze something before it will collapse to a black hole).

What I find shocking is that black holes are tiny! I meant it, they’re really small. The Earth has a Schwarzschild radius of 9 mm, which means you could easily lose it down the back of the sofa. Until it promptly swallowed your sofa, of course. Stellar-mass black holes are just a few kilometres across. For comparison, the Sun has a radius of about 700,000 km. For the massive black hole at the centre of our Galaxy, it is 1010 m, which does sound a lot until you release that it’s less than 10% of Earth’s orbital radius, and it’s about four million solar masses squeezed into that space.

The event horizon changes shape if the black hole has angular momentum (if it is spinning). In this case, you can get closer in, but the position of the horizon doesn’t change much. In the most extreme case, the event horizon is at a radius of

$\displaystyle r_\mathrm{g} = \frac{GM}{c^2}$.

Relativists like this formula, since it’s even simpler than for the Schwarzscild radius (we don’t have to remember the value of two), and it’s often called the gravitational radius. It sets the scale in relativity problems, so computer simulations often use it as a unit instead of metres or light-years or parsecs or any of the other units astronomy students despair over learning.

We’ve now figured out a sensible means of defining the size of a black hole: we can use the event horizon (which separates the part of the Universe where you can escape form the black hole, from that where there is no escape), and the size of this is around the gravitational radius $r_\mathrm{g}$. An interesting consequence of this (well, something I think is interesting), is to consider the effective density of a black hole. Density is how much mass you can fit into a given space. In our case, we’ll consider the mass of the black hole and the volume of its event horizon. This would be something like

$\displaystyle \rho = \frac{3 M}{4 \pi r_\mathrm{g}^3} = \frac{3 c^6}{4 \pi G^3 M^2}$,

where I’ve used $\rho$ for density and you shouldn’t worry about the factors of $\pi$ or $G$ or $c$, I’ve just put them in case you were curious. The interesting result is that the density decreases as the mass increases. More massive black holes are less dense! In fact, the most massive black holes, about a billion times the mass of our Sun, are less dense than water. They would float if you could find a big enough bath tub, and could somehow fill it without the water collapsing down to a black hole under its own weight…

In general, it probably makes a lot more sense (and doesn’t break the laws of physics), if you stick with a rubber duck, rather than a black hole, as a bath-time toy.

In conclusion, black holes might be smaller (and less dense) than you’d expect. However, this doesn’t mean that they’re not very dangerous. As Tyrion Lannister has shown, it doesn’t pay to judge someone by their size alone.

# On symmetry

Dave Green only combs half of his beard, the rest follows by symmetry. — Dave Green Facts

Physicists love symmetry! Using symmetry can dramatically simplify a problem. The concept of symmetry is at the heart of modern theoretical physics and some of the most beautiful of scientific results.

In this post, I’ll give a brief introduction to how physicists think about symmetry. Symmetry can be employed in a number of ways when tackling a problem; we’ll have a look at how they can help you ask the right question and then check that your answer makes sense. In a future post I hope to talk about Noether’s Theorem, my all-time favourite result in theoretical physics, which is deeply entwined with the concept of symmetry. First, we shall discuss what we mean when we talk about symmetry.

## What is symmetry?

We say something is symmetric with respect to a particular operation if it is unchanged after that operation. That might sound rather generic, but that’s because the operation can be practically anything. Let’s consider a few examples:

• Possibly the most familiar symmetry would be reflection symmetry, when something is identical to its mirror image. Something has reflection symmetry if it is invariant under switching left and right. Squares have reflection symmetry along lines in the middle of their sides and along their diagonals, rectangles only have reflection symmetry along the lines in the middle of their sides, and circles have reflection symmetry through any line that goes through their centre.
The Star Trek Mirror Universe actually does not have reflection symmetry with our own Universe. First, they switch good and evil, rather than left and right, and second, after this transformation, we can tell the two universes apart by checking Spock’s beard.
• Rotational symmetry is when an object is identical after being rotated. Squares are the same after a 90° rotation, rectangles are the same after a 180° rotation, and circles are the same after a rotation by any angle. There is a link between the rotational symmetry of these shapes and their mirror symmetry: you can combine two reflections to make a rotation. With rotations we have seen that symmetries can either be discrete, as for a square when we have to rotate by multiples of 90°, or continuous, as for the circle where we can pick any angle we like.
• Translational symmetry is similar to rotational symmetry, but is when an object is the same when shifted along a particular direction. This could be a spatial direction, so shifting everything to the left, or in time. This are a little more difficult to apply to the real world than the simplified models that physicists like to imagine.
For translational invariance, imagine an infinite, flat plane, the same in all directions. This would be translational invariant in any direction parallel to the ground. It would be a terrible place to lose your keys. If you can imagine an infinite blob of tangerine jelly, that is entirely the same in all directions, we can translate in any direction we like. We think the Universe is pretty much like this on the largest scales (where details like galaxies are no longer important), except, it’s not quite as delicious.
The backgrounds in some Scooby-Doo cartoons show periodic translational invariance: they repeat on a loop, so if you translate by the right amount they are the same. This is a discrete symmetry, just like rotating my a fixed angle. Similarly, if you have a rigid daily routine, such that you do the same thing at the same time every day, then your schedule is symmetric with respect to a time translation of 24 hours.
• Exchange symmetry is when you can swap two (or more) things. If you are building a LEGO model, you can switch two bricks of the same size and colour and end up with the same result, hence it is symmetric under the exchange of those bricks. The idea that we have the same physical system when we swap two particles, like two electrons, is important in quantum mechanics. In my description of translational symmetry, I could have equally well have used lime jelly instead of tangerine, or even strawberry, hence the argument is symmetric under exchange of flavours. The symmetry is destroyed should we eat the infinite jelly Universe (we might also get stomach ache).
Mario and Luigi are not symmetric under exchange, as anyone who has tried to play multiplayer Super Mario Bros. will know, as Luigi is the better jumper and has the better moustache.

There are lots more potential symmetries. Some used by physicists seem quite obscure, such as Lorentz symmetry, but the important thing to remember is that a symmetry of a system means we get the same thing back after a transformation.

Sometimes we consider approximate symmetries, when something is almost the same under a transformation. Coke and Pepsi are approximately exchange symmetric: try switching them for yourself. They are similar, but it is possible to tell them apart. The Earth has approximate rotational symmetry, but it is not exact as it is lumpy. The spaceship at the start of Spaceballs has approximate translational invariance: it just keeps going and going, but the symmetry is not exact as it does end eventually, so the symmetry only applies to the middle.

## How to use symmetry

When studying for an undergraduate degree in physics, one of the first things you come to appreciate is that some coordinate systems make problems much easier than others. Coordinates are the set of numbers that describe a position in some space. The most familiar are Cartesian coordinates, when you use $x$ and $y$ to describe horizontal and vertical position respectively. Cartesian coordinates give you a nice grid with everything at right-angles. Undergrad students often like to stick with Cartesian coordinates as they are straight-forward and familiar. However, they can be a pain when describing a circle. If we want to plot a line five units from the origin of of coordinate system $(0,\,0)$, we have to solve $\sqrt{x^2 + y^2} = 5$. However, if we used a polar coordinate system, it would simply be $r = 5$. By using coordinates that match the symmetry of our system we greatly simplify the problem!

Pirates are trying to figure out where they buried their treasure. They know it’s 5 yarrrds from the doughnut. Calculating positions using Cartesian coordinates is difficult, but they are good for specifying specific locations, like of the palm tree.

Using polar coordinates, it is easy to specify the location of points 5 yarrrds from the doughnut. Pirates prefer using the polar coordinates, they really like using r.

Picking a coordinate system for a problem should depend on the symmetries of the system. If we had a system that was translation invariant, Cartesian coordinates are the best to use. If the system was invariant with respect to translation in the horizontal direction, then we know that our answer should not depend on $x$. If we have a system that is rotation invariant, polar coordinates are the best, as we should get an answer that doesn’t depend on the rotation angle $\varphi$. By understanding symmetries, we can formulate our analysis of the problem such that we ask the best questions.

At the end of my undergrad degree, my friends and I went along to an awards ceremony. I think we were hoping they’d have the miniature éclairs they normally had for special occasions. There was a chap from an evil corporation™ giving away branded clocks, that apparently ran on water. We were fairly convinced there was more to it than that, so, as now fully qualified physicists, we though we should able to figure it out. We quickly came up with two ideas: that there was some powder inside the water tank that reacted with the water to produce energy, or that the electrodes reacted in a similar way to in a potato clock. We then started to argue about how to figure this out. At this point, Peter Littlewood, then head of the Cavendish Laboratory, wandered over. We explained the problem, but not our ideas. Immediately, he said that it must be to do with the electrodes due to symmetry. Current flows to power the clock. It’ll either flow left to right through the tank, or right to left. It doesn’t matter which, but the important thing is the clock can’t have reflection symmetry. If it did, there would be no preferred direction for the current to flow. To break the symmetry, the two (similar looking) electrodes must actually be different (and hence the potato clock theory is along the right lines). My friends and I all felt appropriately impressed and humbled, but it served as a good reminder that a simple concept like symmetry can be a powerful tool.

A concept I now try to impress upon my students, is to use symmetry to guide their answers. Most are happy enough to use symmetry for error checking: if the solution is meant to have rotational symmetry and their answer depends on $\varphi$ they know they’ve made a mistake. However, symmetry can sometimes directly tell you the answer.

Lets imagine that you’ve baked a perfectly doughnut, such that it has rotational symmetry. For some reason you sprinkled it with an even coating of electrons instead of hundreds and thousands. We now want to calculate the electric field surrounding the doughnut (for obvious reasons). The electric field tells us which way charges are pushed/pulled. We’d expect positive charges to be attracted towards our negatively charged doughnut. There should be a radial electric field to pull positive charges in, but since it has rotational symmetry, there shouldn’t be any field in the $\varphi$ direction, as there’s now reason for charges to be pulled clockwise or anticlockwise round our doughnut. Therefore, we should be able to write down immediately that the electric field in the $\varphi$ direction is zero, by symmetry.

Most undergrads, though, will feel that this is cheating, and will often attempt to do all the algebra (hopefully using polar coordinates). Some will get this wrong, although there might be a few who are smart enough to note that their answer must be incorrect because of the symmetry. If symmetry tells you the answer, use it! Although it is good to practise your algebra (you get better by training), you can’t learn anything more than you already knew by symmetry. Working efficiently isn’t cheating, it’s smart.

Symmetry is a useful tool for problem solving, and something that everyone should make use of.