November | 2014 | Christopher Berry

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 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 and the massive black hole of the galaxy Messier 87 (M87). 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.

Update: The Event Horizon Telescope Team did it! They have an image of M87’s black hole. It compares nicely to predictions. I’m impressed (definitely cake-worthy). Science has taken another bite out of science fiction.

The Event Horizon Telescope's image of M87*

The shadow of a black hole reconstructed from the radio observations of the Event Horizon Telescope. The black hole lies at the center of M87, and is about 6.5 billion solar masses. Credit: Event Horizon Team