by Thorne, Kip
There are two very special stars in Gargantua’s sky with gravitational lensing turned off. One lies directly above Gargantua’s north pole; the other directly below its south pole. These are analogs of the star Polaris, which resides directly above the Earth’s north pole. I placed five-pointed stars at the primary (red) and secondary (yellow) images of Gargantua’s pole stars. All the stars on the Earth’s sky appear to circulate around Polaris as we humans are carried around by the Earth’s rotation. Similarly, all of Gargantua’s primary stellar images circulate around the red pole-star images as the camera orbits the hole, but their circulation paths (for example, the two red eddy curves) are highly distorted by the whirl of space and gravitational lensing. Similarly, all the secondary stellar images circulate around the yellow pole-star images (for example, along the two distorted yellow curves).
Why, for a nonspinning hole (Figure 8.4), did the secondary images appear to emerge from the black hole’s shadow, swing around the hole, and descend back into the shadow, instead of circulating around a closed curve as for Gargantua (Figure 8.5)? They actually do circulate around closed curves for a nonspinning hole. However, the inner edge of the closed curve is so close to the shadow’s edge that it can’t be seen. Gargantua’s spin makes space whirl, and that whirl moves the inner Einstein ring outward, revealing the secondary images’ full circulatory pattern (yellow curves in Figure 8.5), and revealing the inner Einstein ring.
Inside the inner Einstein ring, the streaming pattern is more complicated. The stars in this region are tertiary and higher-order images of all the stars in the universe—the same stars as appear as primary images outside the outer Einstein ring and secondary images between the Einstein rings.
In Figure 8.6, I show five small pictures of Gargantua’s equatorial plane, with Gargantua itself in black, the camera’s orbit in dashed purple, and a light ray in red. The light ray brings to the camera the stellar image that is at the tip of the blue arrow. The camera is moving counterclockwise around Gargantua.
You can get a lot of insight into the gravitational lensing by walking yourself through these pictures, one by one. Take note: The actual direction to the star is upward and rightward (see outer ends of the red rays). The camera and beginning of each ray point toward the stellar image. The tenth image is very near the left edge of the shadow and the right secondary image is near the right edge; comparing the directions of the camera for these images, we see that the shadow subtends about 150 degrees in the upward direction. This despite the fact that the actual direction from camera to center of Gargantua is leftward and upward. The lensing has moved the shadow relative to Gargantua’s actual direction.
Fig. 8.6. Light rays that bring images of the stars at the tips of the blue arrows. [From the same Double Negative simulation as Figures 8.1 and 8.5.]
Creating Interstellar’s Black-Hole and Wormhole Visual Effects
Chris wanted Gargantua to look like what a spinning black hole really looks like when viewed up close, so he asked Paul to consult with me. Paul put me in touch with the Interstellar team he had assembled at his London-based visual-effects studio, Double Negative.
I wound up working closely with Oliver James, the chief scientist. Oliver and I talked by phone and Skype, exchanged e-mails and electronic files, and met in person in Los Angeles and at his London office. Oliver has a college degree in optics and atomic physics and understands Einstein’s relativity laws, so we speak the same technical language.
Several of my physicist colleagues had already done computer simulations of what one would see when orbiting a black hole and even falling into one. The best experts were Alain Riazuelo, at the Institut d’Astrophysique in Paris, and Andrew Hamilton, at the University of Colorado in Boulder. Andrew had generated black-hole movies shown in planetariums around the world, and Alain had simulated black holes that spin very, very fast, like Gargantua.
So initially I planned to put Oliver in touch with Alain and Andrew and ask them to provide him the input he needed. I lived uncomfortably with that decision for several days, and then changed my mind.
During my half century physics career I put great effort into making new discoveries myself and mentoring students as they made new discoveries. Why not, for a change, do something just because it’s fun, I asked myself, even though others have done it before me? And so I went for it. And it was fun. And to my surprise, as a byproduct, it produced (modest) new discoveries.
Using Einstein’s relativistic laws of physics and leaning heavily on prior work by others (especially Brandon Carter at the Laboratoire Univers et Théories in France and Janna Levin at Columbia University), I worked out the equations Oliver needed. These equations compute the trajectories of light rays that begin at some light source, for example, a distant star, and travel inward through Gargantua’s warped space and time to the camera. From those light rays, my equations then compute the images the camera sees, taking account not only of the light’s sources and Gargantua’s warping of space and time, but also the camera’s motion around Gargantua.
Having derived the equations, I implemented them myself, using user-friendly computer software called Mathematica. I compared images produced by my Mathematica computer code with Alain Riazuelo’s images, and when they agreed, I cheered. I then wrote up detailed descriptions of my equations and sent them to Oliver in London, along with my Mathematica code.
My code was very slow and had low resolution. Oliver’s challenge was to convert my equations into computer code that could generate the ultra-high-quality IMAX images needed for the movie.
Oliver and I did this in steps. We began with a nonspinning black hole and a nonmoving camera. Then we added the black hole’s spin. Then we added the camera’s motion: first motion in a circular orbit, and then plunging into a black hole. And then we switched to a camera around a wormhole.
At this point, Oliver hit me with a minibombshell: To model some of the more subtle effects, he would need not only equations describing the trajectory of a ray of light, but also equations describing how the cross section of a beam of light changes its size and shape during its journey past the black hole.
I knew more or less how to do this, but the equations were horrendously complicated and I feared making mistakes. So I searched the technical literature and found that in 1977 Serge Pineault and Rob Roeder at the University of Toronto had derived the necessary equations in almost the form I needed. After a three-week struggle with my own stupidities, I brought their equations into precisely the needed form, implemented them in Mathematica, and wrote them up for Oliver, who incorporated them into his own computer code. At last his code could produce the quality images needed for the movie.
At Double Negative, Oliver’s computer code was just the beginning. He handed it over to an artistic team led by Eugénie von Tunzelmann, who added an accretion disk (Chapter 9) and created the background galaxy with its stars and nebulae that Gargantua would lens. Her team then added the Endurance and Rangers and landers and the camera animation (its changing motion, direction, field of view, etc.), and molded the images into intensely compelling forms: the fabulous scenes that actually appear in the movie. For further discussion, see Chapter 9.
In the meantime, I puzzled over the high-resolution film clips that Oliver and Eugénie sent me, struggling to extract insights into why the images look like they look, and why the star fields stream as they stream. For me, those film clips are like experimental data: they reveal things I never could have figured out on my own, without those simulations—for example, the things I described in the previous section (Figures 8.5 and 8.6). We plan to publish one or more technical papers, describing the new things we learned.
Imaging a Gravitational Slingshot
Although Chris chose not to show any gravitational slingshots in Interstellar, I wondered what they would look like to Cooper as he piloted the Ranger toward Miller’s planet. So I used my equations and Mathematica
to simulate them and produce images. (My images have far lower resolution than Oliver’s and Eugénie’s due to my code’s slowness.)
Figure 8.7 shows a sequence of images, as Cooper’s Ranger swings around an intermediate-mass black hole (IMBH) to initiate its descent toward Miller’s planet—in my scientist’s interpretation of Interstellar. This is the slingshot described in Figure 7.2.
Fig. 8.7. Gravitational slingshot around an IMBH, with Gargantua in the background. [My own simulation and visualization.]
In the top image, Gargantua is in the background with the IMBH passing in front of it. The IMBH grabs light rays from distant stars that are headed toward gargantua, swings the rays around itself, and ejects them toward the camera. This explains the donut of starlight that surrounds the IMBH’s shadow. Although the IMBH is a thousand times smaller than Gargantua, it is far closer to the Ranger than is Gargantua, so it looks only modestly smaller.
As the IMBH appears to move rightward, as seen by the slingshot-moving camera, it leaves Gargantua’s primary shadow behind itself (middle picture in Figure 8.7), and it pushes a secondary image of Gargantua’s shadow ahead of itself. These two images are completely analogous to the primary and secondary images of a star gravitationally lensed by a black hole; but now it is Gargantua’s shadow that is being lensed, by the IMBH. In the bottom picture, the secondary shadow is shrinking in size, as the IMBH moves onward. By this time the slingshot is nearly complete, and the camera, on board the Ranger, is headed downward, toward Miller’s planet.
As impressive as these images may be, they can be seen only up close to the IMBH and Gargantua, not from the great distance of Earth. To astronomers on Earth, the most visually impressive things about gigantic black holes are jets that stick out of them and the light from brilliant disks of hot gas that orbit them. To these we’ll now turn.
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20 See Figures 6.4 and 6.5.
9
Disks and Jets
Quasars
Most of the objects seen by radio telescopes are huge clouds of gas, clouds far larger than any star. But in the early 1960s a few tiny objects were found. Astronomers named these objects quasars for “quasi-stellar radio sources.”
In 1962 the Caltech astronomer Maarten Schmidt, looking through the world’s largest optical telescope on Palomar Mountain, discovered light coming from a quasar called 3C273. It looked like a bright star with a faint jet shooting out of it (Figure 9.1). This was weird!
When Schmidt split 3C273’s light into its various colors (as is sometimes done by sending light through a prism), he saw the set of spectral lines in the bottom of Figure 9.1. At first sight, these were unlike any spectral lines he had ever seen. But in February 1963, after a few months’ struggle, he realized the lines were unfamiliar simply because their wavelengths were 16 percent larger than normal. This is called the Doppler shift; it was caused by the quasar’s moving away from Earth at 16 percent the speed of light, about c/6. What could cause that ultrafast motion? The least crazy explanation Schmidt could find was the expansion of the universe.
Fig. 9.1. Top: Photograph of 3C273 taken by NASA’s Hubble Space Telescope. The star (upper left) looks big only because the photo is overexposed in order to see the faint jet (lower right). It is actually so small that its size cannot be measured. Bottom: Maarten Schmidt’s spectral lines from 3C273 (upper panel) compared with spectral lines of hydrogen measured in an Earth laboratory. The quasar’s three lines are the same as hydrogen’s lines called Hβ, Ηγ, and Ηδ, but with wavelengths increased by 16 percent. (The images of the spectral lines are photographic negatives: black lines are really bright.)
As the universe expands, objects far from Earth move apart from us at very high speed, and objects nearer move away more slowly. 3C273’s enormous speed, one-sixth that of light, meant that 3C273 was 2 billion light-years from Earth, nearly the farthest object that had ever been seen at that time. From its brightness and its distance, Schmidt concluded that 3C273 puts out 4 trillion times more power in light than the Sun, and a hundred times more power than the brightest galaxies!
This prodigious power fluctuated on times as short as a month, so most of the light must be coming from an object so small that the light can travel across it in one month’s time—far smaller than the distance from Earth to the nearest star, Proxima Centauri. And other quasars with almost as much power fluctuated on times of a few hours, so they had to be not much larger than our solar system. One hundred times the power of a bright galaxy, coming from a region the size of our solar system; that was phenomenal!
Black Holes and Accretion Disks
How could so much power come out of a region so small? When we think about the fundamental forces in Nature, there are three possibilities: chemical energy, nuclear energy, or gravitational energy.
Chemical energy is the energy released when molecules combine together to make new kinds of molecules. An example is burning gasoline, which combines oxygen from the air with gasoline molecules to make water and carbon dioxide, and a lot of heat. The power from that would be far, far, far too little though.
Nuclear energy results when atomic nuclei combine together to make new atomic nuclei. Examples are an atomic bomb, a hydrogen bomb, and the burning of nuclear fuel inside a star. Though this can be far more powerful than chemical energy (think of the difference between a gasoline fire and a nuclear bomb), astrophysicists couldn’t see any plausible way for nuclear energy to power quasars. It was still too puny.
So the only possibility left was gravitational energy, the same kind of energy we were driven to, when navigating the Endurance around Gargantua. For the Endurance, gravitational energy was harnessed by a slingshot around an intermediate-mass black hole (Chapter 7). The black hole’s intense gravity was key. For quasars, similarly, the power must come from a black hole.
For several years, astrophysicists struggled to figure out how a black hole could do the job. The answer was found in 1969, by Donald Lynden-Bell at the Royal Greenwich Observatory in England. A quasar, Lynden-Bell hypothesized, is a gigantic black hole surrounded by a disk of hot gas (an accretion disk) that is threaded by a magnetic field (Figure 9.2).
Hot gas in our universe is almost always threaded by magnetic fields (Chapter 2). These fields are locked into the gas; the gas and fields move together, in lockstep.
When threading an accretion disk, a magnetic field becomes a catalyst for converting gravitational energy into heat and then light. The field provides ultrastrong friction21 that slows the gas’s circumferential motion, reducing the centrifugal force that holds it out against the pull of gravity, so the gas moves inward, toward the black hole. As the gas moves inward, the hole’s gravity speeds up its orbital motion by even more than the friction slowed it. In other words, gravitational energy is converted into kinetic energy (energy of motion). Magnetic friction then converts half that new kinetic energy into heat and light, and the process repeats.
The energy comes from the black hole’s gravity. The agents for extracting it are magnetic friction and the disk’s gas.
The quasar’s bright light, seen by astronomers, comes from the disk’s heated gas, Lynden-Bell concluded. Moreover, the magnetic field accelerates some of the gas’s electrons to high energies; and the electrons then spiral around the magnetic force lines, emitting the quasar’s observed radio waves.
Lynden-Bell worked out the details of all this using a combination of the Newtonian, relativistic, and quantum laws of physics. He easily explained everything about quasars that astronomers had seen, except their jets. His technical article describing his reasoning and his calculations (Lynden-Bell 1979) is one of the great astrophysics articles of all time.
The Jets: Extracting Power from Whirling Space
Over the next few years, astronomers discovered many more jets sticking out of quasars and studied them in great detail. It soon became clear that they
are streams of hot, magnetized gas ejected from the quasar itself: from the black hole and its accretion disk (Figure 9.2). And the ejection is extremely powerful: the gas travels out the jets at nearly the speed of light. As it travels, and when it plows into material far from the quasar, the gas emits power in light, in radio waves, in X-rays, and even in gamma rays. The jets are sometimes as bright as the quasar itself, a hundred times brighter than the brightest galaxies.
Fig. 9.2. Artist’s conception of an accretion disk around a black hole, and jets emerging from near the hole’s poles. [Drawing by Matt Zimet based on a sketch by me; from my book Black Holes & Time Warps: Einstein’s Outrageous Legacy.]
Astrophysicists struggled for nearly a decade to explain how the jets are powered and what makes them so fast, so narrow, and so straight. The answers came in several variants, with the most interesting in 1977 from Roger Blandford at the University of Cambridge, England, and his student Roman Znajek, building on foundations laid by the Oxford physicist Roger Penrose; see Figure 9.3.
The accretion disk’s gas gradually spirals into the black hole. When crossing the hole’s event horizon, each bit of gas deposits its bit of magnetic field onto the horizon, and then the surrounding disk holds it there, Blandford and Znajek concluded. As the black hole spins, it drags space into whirling motion (Figures 5.4 and 5.5), and the whirling space makes the magnetic field whirl (Figure 9.3). The whirling magnetic field generates an intense electric field like in a dynamo at a hydroelectric power station. The electric field and the whirling magnetic field together fling plasma (hot, ionized gas) upward and downward at near light speed, creating and powering two jets. The jets’ directions are held steady (when averaged over years) by the black hole’s spin, which is steady due to gyroscopic action.