General relativity at 100

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

Where is the cake?

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.

Shotest distance between London and New York

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…

Ripples in space time

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

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Interstellar—science and fiction

Interstellar black hole

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).

General relativity textbooks

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 and redshifting of an accretion disc around a black hole.

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.

Simulated Event Horizon Telescope image

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.