The science fiction film, Interstellar, tells the story of a team of astronauts searching a distant galaxy for habitable planets to colonize. Dust storms, failing crops, and famines have led them to a grim realization: humanity must find a new home amongst the stars.
Interstellar’s story draws heavily from contemporary science. The film makes reference to a range of topics, from established concepts such as fastspinning black holes, accretion disks, tidal effects, and time dilation, to far more speculative ideas such as wormholes, time travel, additional space dimensions, and the theory of everything.
The promotion for Interstellar emphasized the film’s scientific credentials and realism. Particular mention was made of the involvement of Kip Thorne as Interstellar’s scientific consultant and executive producer. Thorne has written a popular book explaining how he tried to ensure that the depictions of science were as accurate as possible, despite the sometimes exorbitant demands of the filmmakers.^{1} He did his best.
An Artificial Wormhole
During a scene in the first part of Interstellar, the chief scientist in the film, Professor Brand, describes the discovery some fifty years before of an artificial wormhole near Saturn. Wormholes, long a staple of science fiction, arise as solutions to the equations of general relativity. A technique known as embedding can be used to create a pictorial representation of a wormhole. One space is visualized by embedding it in another space with an additional dimension. This mapping is useless for the 4dimensional spacetime of general relativity because it would require a 5dimensional space, and that is impossible to visualize. But, in the case of a nonrotating black hole, spacetime geometry remains fixed; and no information is lost by looking at equatorial slices passing through the center of the sphere. The result is a 2dimensional surface embedded in a 3dimensional Euclidean space—a knife passing through an orange.
The embedding surface consists of a paraboloidshaped throat linking two distinct sheets of spacetime.^{2} The Schwarzschild throat, or Einstein–Rosen bridge, has a minimum radius equal to the Schwarzschild radius r = 2GM/c^{2}, where M is the mass of the black hole, G the gravitational constant, and c the speed of light. This radius corresponds to the event horizon, the immaterial boundary of the black hole, reduced now to a circle. For a typical stellar black hole of M = 10 solar masses, r = 30km. This is the size assumed in Interstellar. The Schwarzschild throat joins the perfectly symmetrical upper and lower sheets that we are at liberty to interpret as parallel universes. In the upper universe, the throat has the appearance of a black hole consuming matter, light and energy; below, it appears as a white fountain expelling matter, light and energy.
I passed a little further on and heard a peacock say:
Who made the grass and made the worms and made my feathers gay,
He is a monstrous peacock, and he waveth all the night
His languid tail above us, lit with myriad spots of light.^{3}
General relativity determines only the local, and not the global curvature of spacetime. In particular, general relativity allows the two distinct sheets of spacetime to be different regions of the same universe: a black hole and a white fountain in the same spacetime at an arbitrary distance from one another, linked by a stretchedout throat.^{4} Thus, a wormhole can be thought of as a tunnel with two ends at separate points in spacetime. The tunnel would, in effect, function as a shortcut.^{5}
As Interstellar unfolds, the viewer comes to understand that a series of exploratory manned missions, dispatched a decade beforehand, had already traversed the wormhole in search of planets suitable for colonization. Twelve potential worlds were discovered, three of which are located in a single planetary system. A new mission must now depart for this system, once again worming its way through the wormhole.
There are two major problems with this scenario. There is in the first place no wormhole without a black hole. The most common black holes are formed by the gravitational collapse of massive stars and are only a few solar masses in size. The radius of these black holes and their wouldbe wormholes is on the order of only a few kilometers. The tidal forces they generate are so huge that any spacecraft would be torn apart well before reaching, let alone penetrating, their event horizon.
Black holes are, in fact, quite rare. It is estimated that the average density of stellarmass black holes in our region of the Milky Way is on the order of 0.00001 per cubic lightyear. It is improbable that a stellar black hole could be found close to earth, if only because its gravitational field would have long destroyed the stability of the solar system. As a remedy for a local environmental crisis, a black hole in the vicinity of Saturn seems extreme. For the most obvious of reasons the filmmakers have turned to science fiction, presenting the artificial wormhole near Saturn as the creation of a highly advanced civilization.^{6}
One problem now gives rise to another. Calculations for a nonrotating black hole demonstrate that a wormhole is strangled at its middle by the infinite gravitational field of a singularity at r = 0, and thus bears a disconcerting resemblance to a roach motel. All is not lost. Natural black holes, like stars, must rotate. Their geometry then becomes much more subtle and complex, matters first investigated during the 1960s by Brandon Carter and Roger Penrose.^{7} The central singularity is no longer reduced to a central point at r = 0, but instead has the shape of a ring lying in the equatorial plane of the rotating black hole. The ring no longer defines an edge of the spacetime geometry, because travelers, aside from the dangers of tidal forces, could come safely within a hair’s breadth of the annular singularity and even pass through it.
Penrose–Carter diagrams^{8} suggest fascinating possibilities for exploration.^{9} Allowable trajectories show that it is theoretically possible to penetrate the interior of a rotating black hole so that a spacecraft is not destroyed by tidal forces. Flying above the plane of the singular ring, it might escape the black hole and emerge into an exterior universe. In such cases, Kerr black holes would have open wormholes, offering fantastic possibilities for spacetime travel.
Alas, the Penrose–Carter diagrams are idealized representations of space and time. In the real universe, physical processes such as gravitational collapse form astrophysical black holes, both rotating and static. Relativistic calculations indicate that their wormholes are unstable. As soon as they are formed, they collapse.
This is not quite the end of the story. Quantum processes now come into play. In 1988, Kip Thorne proposed that wormholes laced with matter or energy exerting an enormous negative pressure could be stable and traversable.^{10} Such forms of matter or energy are termed exotic. In the framework of quantum mechanics, some energy states of the quantum vacuum obtained by the Casimir effect may well produce exotic energy. This is all highly speculative, but not theoretically impossible. Nonetheless, the amount of negative energy required to keep the wormhole open would be greater than the total energy emitted by the sun each year.
Other types of traversable wormholes have been discovered as solutions to the equations of general relativity.^{11} Interstellar’s screenwriters imagined that a highly advanced civilization might have the ability to construct a negative wormhole, perhaps by growing a microscopic negative wormhole, as permitted by quantum mechanics, and then using it effectively as a spacetime shortcut. Even allowing for this fantastic idea, there is nothing to suggest that a spacecraft could safely cross the region of negative energy.
The astronomer Carl Sagan took advantage of this theoretical possibility in Contact, his characters using wormholes to communicate with extraterrestrial civilizations.^{12} Sagan’s advisor was Kip Thorne. Qui se ressemble s’assemble. For Interstellar, Thorne went much further. To help the visual effects team, he tried to calculate the basis for a journey through an artificially created wormhole. Except for the spherical shape of the wormhole, the filmmakers discarded most of the scientific calculations.
With our presentday knowledge, we can be certain that a traversable wormhole is not only very improbable, but even if it did exist, crossing it would not in any way resemble the depiction in Interstellar.^{13}
A FastSpinning Supermassive Black Hole
Having safely traversed Interstellar’s artificial wormhole, the spacecraft Endurance emerges into a threeplanet system orbiting Gargantua, a supermassive black hole. At first glance, the close proximity of the planets to the black hole might seem unlikely.
Supermassive black holes, with masses ranging from one million to several billion solar masses, are suspected to lie at the center of most galaxies.^{14} Our own Milky Way galaxy probably harbors such an object, Sagittarius A*, indirectly measured as four million solar masses.^{15} According to Thorne, Gargantua would be similar to an even larger black hole of 100 million solar masses at the center of the Andromeda galaxy.^{16}
Gargantua is depicted as a fastspinning supermassive black hole. Its rate of rotation is dependent on two parameters: its mass M, and angular momentum J. In contrast to stars, which are in differential rotation, these black holes rotate with perfect rigidity. All the points on their surface, the event horizon, move with the same angular velocity. There is, however, a critical angular momentum J_{max}, above which the event horizon breaks up. This limit corresponds to the event horizon possessing a spin velocity equal to the speed of light. For such extremal black holes, the gravitational field at the event horizon would be cancelled, the inward pull of gravity balanced by huge repulsive centrifugal forces. It is possible that most of the black holes formed in the real universe have an angular momentum rather close to this critical limit.^{17}
In Interstellar, Gargantua is assumed to have an angular momentum as close as 10^{10} to the critical value J_{max}. Although theoretically possible, this configuration is unrealistic. The faster a black hole rotates, the harder it is to capture material orbiting in the same direction because of centrifugal forces; material orbiting in the opposite direction would be easily sucked into the hole, slowing the speed of rotation. As a result, a toofastspinning black hole would tend to slow to an equilibrium velocity smaller than that assumed for Gargantua. General relativistic calculations indicate that black holes spin no faster than about 0.998J_{max}.
For the purposes of Interstellar’s plot, a fastspinning black hole has two important advantages. The first: planets may orbit close to the event horizon without being swallowed. The second: the planet closest to the event horizon may experience a huge dilation in time. For a black hole spinning very close to the critical limit J_{max}, the innermost stable circular orbit can be as close as the event horizon itself, 100 million kilometers. This is the reason why the closest planet to Gargantua, Miller, can safely orbit very close to the event horizon without being swallowed.
It is important to note that a Kerr black hole is not comparable to a spinning top revolving in a fixed exterior space. As it rotates, the black hole drags the entire fabric of spacetime along with it. As a consequence, Miller must orbit Gargantua at a velocity close to the speed of light.
Illuminated Planets
There are three planets orbiting Gargantua. From where do they get their heat and light? In principle, a star would be needed for this purpose, but none is present. The requisite heat cannot come from the black hole itself, either in the form of Hawking radiation or the recently proposed firewall phenomenon.^{18} These effects are purely quantum in nature, and while they could be noticeable for microscopic black holes, they are completely negligible for astrophysical black holes.
Could the necessary light and heat come from the gaseous ring known as an accretion disk that orbits Gargantua?^{19} The theory of accretion disks was first developed several decades ago,^{20} and is in agreement with recent astrophysical measurements using gravitational lensing.^{21} Due to the incredible forces involved, accretion disks are extremely hot, on the order of millions of degrees, so brilliant that they can be seen millions of lightyears away, and they emit enough radiation to completely destroy any normal material. Interstellar’s astronauts should have been destroyed as soon as the Endurance emerged from the artificial wormhole. Toward the end of the film, Interstellar’s main character, Cooper, ventures inside the black hole again. It is an encounter that he, but perhaps not the viewer, survives.
When questioned on this point, Thorne claimed that the requisite light and heat could come from an anemic accretion disk that had cooled to the temperature of the sun, around 5,500°C.^{22} By anemic, Thorne meant that the disk had not been fed by new gases, say from a tidally disrupted star, in the last million years. The accretion rate onto the black hole, a critical parameter for the luminosity of the disk, would then be extremely low. In such a scenario, a quiescent accretion disk could be relatively safe for human beings.
I doubt that an anemic accretion disk could provide enough heat and light for a planet to be habitable. An anemic accretion disk would be optically thin, whereas our sun’s photosphere is optically thick.
Visualizing an Accretion Disk
Interstellar is the first major Hollywood film to attempt to depict a black hole as it would appear to a nearby observer. Perhaps the single most arresting image in the film is the visualization of Gargantua, with a glowing accretion disk that spreads above, below, and in front of the black hole.
A black hole causes extreme deformations of spacetime. Thus it also creates the strongest possible deflections of light rays passing in its vicinity. This gives rise to spectacular optical illusions known as gravitational lensing. In order to depict this effect, the visual effects company, Double Negative, in collaboration with Kip Thorne, developed rendering software that could model equations describing lightray propagation in the curved spacetime of a black hole.^{23} The equations generated for the film can describe the gravitational lensing of distant stars as viewed by a camera near the event horizon.^{24} Due to the distances involved, and attendant lack of spatial resolution, no detailed image has yet been captured of an accretion disk.^{25} In 1979, I was the first to simulate (in black and white) the appearance of a thin accretion disk gravitationally lensed by a nonspinning black hole as viewed from the side, either by a distant observer or a photographic plate.^{26}
In Euclidean space, curvature is weak. This is the case for the solar system when one observes the planet Saturn from a viewpoint slightly above the plane. Parts of Saturn’s rings are hidden behind the planet, but one can mentally reconstruct their elliptical outlines easily enough. Around a black hole, optical deformations due to spacetime curvature lead to a very different viewing experience. Strikingly, one can see the top of the accretion disk in its totality from any viewing angle. The rear part of the disk is not hidden by the black hole, as might be expected, since its images are enhanced by curvature, and are visible to a distant observer. One can also see part of the bottom of the gaseous disk. The light rays that would normally propagate downwards climb back to the top, supplying a highly deformed picture of the bottom of the disk.^{27} Thus, when I viewed Interstellar for the first time, I wasn’t surprised to see the accretion disk spreading above, below, and in front of Gargantua’s silhouette. While the team involved in creating these images should be proud of their work, I was puzzled to read in press releases that this image was the first and most realistic image of a black hole accretion disk ever made. A number of basic visual effects were obviously missing. In my 1979 simulation, I had also accounted for the physical properties of the gaseous disk: rotation, temperature, and emissivity.
For a thin accretion disk, the intensity of radiation emitted from a given point on the disk is dependent on its distance from the black hole. Therefore, the brightness of the disk cannot be uniform, as depicted in Interstellar. The maximum brilliance comes from the inner regions closest to the event horizon where the gas is hottest.
The apparent luminosity of the disk is very different from its intrinsic luminosity. The radiation picked up at a great distance is frequency and intensity shifted. There are two types of shift effects: first, the Einstein effect, in which the gravitational field lowers the frequency and decreases the intensity; and second, the Doppler effect, whereby the displacement of the source with respect to the observer causes amplification as the source approaches and attenuation as the source retreats. In this case, the disk rotating around the black hole causes the Doppler effect. The regions of the disk closest to the black hole rotate at a velocity approaching that of light. Thus the Doppler shift is considerable and drastically modifies the image as seen by a faraway observer. The impression of the disk’s rotation is such that matter recedes from the observer on, say, the righthand side of the photograph, and approaches on the lefthand side. As the matter recedes, the Doppler deceleration is added to the gravitational deceleration, implying a very strong attenuation on the righthand side. In contrast, on the lefthand side, the two effects tend to cancel each other out, so the image more or less retains its intrinsic intensity. In any case, a realistic image must show a strong asymmetry in the disk’s brightness, with one side far brighter than the other.^{28}
All of this affects color. This could not be seen in my 1979 black and white (bolometric) image, but visualizations of ever increasing sophistication have been created in the intervening years. Jun Fukue and Takushi Yokoyama added colors to the disk.^{29} S. U. Viergutz made the black hole spin, and produced colored images, including the disk’s secondary image, wrapping under the black hole.^{30} The work of JeanAlain Marck^{31} laid the foundations for an animation during which the camera viewpoint moves around the disk, and includes higher order images.^{32} Sophisticated raytracing software and accretion flow models have recently been developed, including 3D simulations of accretion flows and accompanying images.^{33}
Kip Thorne did not ignore these effects. On the contrary, the filmmakers decided to omit the Einstein and Doppler shifts, as well as the physical properties of the disk, and depict an accretion disk with the right shape, but not the right lopsidedness. In an email to me, Thorne explained that these changes were made at the behest of Interstellar’s director to avoid confusing a general audience. To simplify matters further, the filmmakers also chose to apply their calculations to a black hole smaller than Gargantua and with a much more moderate rotation speed. The visual effects would otherwise have become extremely weird and likely incomprehensible, even for physicists.
In order to fully exploit the raytracing software developed for Interstellar, Thorne and his collaborators at Double Negative have published a technical paper, including all the corrections.^{34}
Tidal Force and Tidal Waves
After landing on Miller, a water world covered by a shallow ocean, Interstellar’s astronauts are confronted by immense periodic tidal waves that sweep across the planet. Two aspects of this scenario have proven controversial. Could a planet orbit so closely to a black hole without being torn apart by tidal forces? Are the kilometerhigh tidal waves depicted in the film physically realistic?^{35}
When an object, such as a star or a planet, orbits around a black hole, the gravitational force of the black hole is exerted more strongly on the side of the object nearest to the black hole than on the farthest side. The tidal force is the difference in strength between gravitational forces. If the object is moving along a circular orbit at a reasonable distance from a black hole, tidal forces remain small, the object is able to adjust its internal configuration to external forces, and adopts an elongated shape oriented towards the black hole. However, if the object is moving along an eccentric orbit, as the distance r from the black hole decreases, the tidal forces increase rapidly, on the order of r^{3}. Eventually a point is reached when these forces become as large as the forces binding the object together. The object no longer has time to adjust its internal configuration, and begins to break apart.
During the 1980s, I spent a great deal of time working on the process by which full stars are disrupted by massive black holes.^{36} In extreme cases, when a star moving along a parabolic orbit grazes the event horizon of a black hole without being swallowed, I predicted the occurrence of “flambéed stellar pancakes,” releasing large amounts of radiative energy.^{37} Telescopes have since captured such scenes. But these events occur only when the object moves within a critical radius from the black hole, a minimum distance known as the Roche limit.^{38}
As it turns out, Interstellar is marginally correct on this point. In effect, the Roche limit depends on the mass of the black hole and the average density of the external object: R_{R} ~ (M/ρ*)^{1/3}, where M is the mass of the black hole, and ρ* the density of the object. Applying this formula to Gargantua (M = 10^{8} solar masses) and a water planet (ρ* ~1 g/cm^{3}), we obtain R_{R} ~ 10^{13}cm. The gravitational radius of Gargantua, GM/c^{2}, is also of order 10^{13}cm. Miller must therefore experience large tidal forces, but of insufficient strength for the planet to be destroyed.^{39}
The origins of the vast tidal waves seen on the planet are not explained, but we can assume they are caused by Miller’s proximity to Gargantua. As depicted, the wavelength of the tidal waves is much greater than the depth of the water itself. This scenario is suitable for applying the Navier–Stokes shallow water equations. These equations are coupled, nonlinear partial differential equations whose solutions depend on the planet’s surface gravity, rate of rotation, viscous drag forces, and so on. We are told that the acceleration due to gravity on Miller is 130% that of Earth’s, meaning that g = 9.81 × 1.30 = 12.75m/s^{2}. The other parameters will be influenced by internal forces in the planet’s structure, combined with complex external effects due to the gravitational field of the rotating black hole. The information given in the film is not sufficient to allow anyone to solve the underlying equations numerically.
I suspect that the film is inconsistent on this point. A tidal wave is a bulge of water fixed in space, always oriented in the same configuration. The astronauts on Miller rotate in and out of that bulge, observing a wave approaching, passing, and then moving away from them: a high tide and a low tide. The waves appear every hour or so. Because there are two high tides for each rotation, the planet rotates once every two waves. With such huge tides, the planet should, in theory, quickly become tidally locked on Gargantua, always presenting the same face to the black hole. The timescale for tidal locking between a black hole of 10^{8} solar masses and a planet with a surface gravity of about 13m/s^{2} is 1 millisecond. Once the planet is tidally locked to the black hole, it spins only once per revolution, and on the planet’s surface the water stays in place, pulled always toward the black hole.
Time Dilation
Einstein’s theory of special relativity predicts that observers in differently accelerated frames of reference will perceive time differently. The wellknown phenomenon of time dilation has been experimentally verified to a high level of accuracy. The consequences of time dilation are felt throughout Interstellar’s story.
Close to the event horizon of a black hole, where the gravitational field is huge, time dilation is also huge. Clocks will be strongly slowed in comparison to clocks farther away. One hour on Miller (Miller Time) is equivalent to seven years on earth. This corresponds to a time dilation factor of 60,000. Although time dilation tends toward infinity when a clock tends toward the event horizon, a time dilation factor of 60,000 is impossible for a planet with a stable orbit around a black hole.
In his book, The Science of Interstellar, Kip Thorne explains that a time dilation factor of this magnitude was a nonnegotiable requirement from the film’s director.^{40} After a few hours of calculation, Thorne came to the conclusion that the scenario, although very unlikely, was marginally possible. The key is the black hole’s rate of rotation. A rotating Kerr black hole behaves rather differently from a static Schwarzschild black hole. The time dilation equation derived from the Kerr metric takes the form:
$$1\left(\mathrm{d}\mathrm{\tau}/\mathrm{d}t\right)=2\mathrm{G}\mathrm{M}r/{c}^{2}{\mathrm{\rho}}^{2}\text{,where}{\mathrm{\rho}}^{2}={r}^{2}+{\left(\mathrm{J}/\mathrm{M}c\right)}^{2}{\mathrm{c}\mathrm{o}\mathrm{s}}^{2}\mathrm{\theta}.$$
Substituting for dτ = 1 hour and dt = 7 years, one obtains:
$$\frac{1.334\times {10}^{10}{\mathrm{M}}^{3}r}{8.98755\times {10}^{16}{\mathrm{M}}^{2}{r}^{2}+{\mathrm{J}}^{2}{\mathrm{c}\mathrm{o}\mathrm{s}}^{2}\mathrm{\theta}}=\frac{3369802499}{3369802500}$$
This equation describes a black hole of mass M, rotating with angular momentum J, as observed at a radial coordinate r and an angular coordinate θ. The fraction on the right depicts the 1 hour = 7 years dilation effect. From the Schwarzschild metric, the orbital radius should be no smaller than three times the gravitational radius. A time dilation of this magnitude would thus be unachievable for the planet in question. However, the Kerr metric allows for stable orbits much closer to the event horizon. Calculations indicate that for M = 10^{8} solar masses, r = 1.48 × 10^{13}cm, θ = π, and J = 8.80275 × 10^{57}J.s. This implies that the requisite black hole must have an angular momentum close to the maximal possible value; the planet must have a circular orbit lying in the equatorial plane, and an orbit radius practically equal to the black hole’s gravitational radius.
This is theoretically possible, but not realistic.
The Penrose Process
Running low on fuel, and with one planet left to explore, the Endurance harnesses a particularly efficient form of gravitational assistance in order to continue its journey. Described in the film as a slingshot, the Endurance plunges perilously close to Gargantua’s event horizon before escaping with increased energy. This mechanism, depicted correctly in Interstellar, is known as the Penrose process.
According to the laws of Kerr black hole physics, although a black hole prevents any radiation or matter from escaping, it can release some of its rotational energy. A key role in the process is played by the ergosphere, a region between an outer boundary known as the static or stationary limit, and the event horizon; within the ergosphere, spacetime is itself irresistibly dragged along, as if in a maelstrom, in the direction of the black hole’s rotation.^{41}
As part of a 1969 thought experiment, Roger Penrose theorized that if a projectile entered the ergosphere and disintegrated into two fragments, the first, rotating in the opposite direction to the black hole, would fall into the event horizon, while the second might escape, carrying greater energy than the initial projectile.^{42} Replace the projectiles with a spacecraft that sheds some part of its structure into the black hole along a carefully chosen retrograde orbit and le tour est joué.
Calculations indicate that energy could be extracted equivalent to the restmass energy of the fragment lost in the black hole, which is already huge, given that E = mc^{2}. Additional energy could be extracted from the spinning black hole. How much energy? About 29% of its mass.
Time Travel inside Gargantua
During a scene in the final part of the film, Cooper plunges into Gargantua, in order to ensure that the Endurance reaches the third and final planet. Despite the threat posed by tidal forces, Cooper survives the encounter. He is lucky. Tidal forces become infinite as r approaches 0. So, even for a supermassive black hole like Gargantua, once safely past the event horizon and approaching the central singularity, any object would ultimately be destroyed. Fortunately, Gargantua is a fastspinning black hole and its lethal singularity has an avoidable ring shape. Cooper uses the Kerr black hole as a wormhole and is transported to another region in spacetime, a fivedimensional universe the film refers to as the Tesseract.
Much research has been undertaken to determine whether the laws of physics permit travel backwards in time. Black hole physics provides interesting results but no firm answers. As shown in Penrose–Carter diagrams, a rotating black hole could connect a number of wormholes to different parts of spacetime. Since two events can differ in time, as well as in space, it would be possible to travel along a carefully chosen trajectory and arrive at the same position, but at a different time. A black hole could thus become a time travel machine of sorts.
The notion of a journey backwards in time is an affront to common sense, and violates the law of causality. That causes must always precede their effects is implicit in special relativity, where there is no gravitation. Travel into the past would require motion faster than light, and is absolutely forbidden. In general relativity, the universe is curved by gravitation; spacetime can be distorted by a wormhole associated to a rotating black hole. Time travel no longer requires speeds faster than light.
Do such strange time warps exist in the universe? Perhaps at the quantum scale. If microscopic black holes, governed by the laws of quantum physics, were created soon after the Big Bang, microscopic wormholes would also have been created.^{43} It is not impossible that very special conditions in the early universe led to the formation of miniwormholes that have grown to become macroscopic following a phase of primordial inflation.
Gargantua could be a giant wormhole, an open gate leading to other regions of the cosmos, or to parallel universes.
A Fifth Dimension
Following Cooper’s arrival in the fivedimensional Tesseract, the notions embraced by the film become ever more speculative. General relativity describes our universe in terms of three dimensions of space and one dimension of time. This model, referred to as 3+1 space, provides a very accurate description of the universe that we observe.
Nonetheless, theorists examine alternative models to see how they differ from regular general relativity. They might consider, for example, 2+1 space, a kind of flat geometry with a time dimension. These alternative models can be useful because they help us gain a deeper understanding of gravity. There is brane cosmology,^{44} for example, which embodies the speculation that
the visible, fourdimensional universe is restricted to a brane inside a higherdimensional space, called the “bulk” (also known as “hyperspace”). … In the bulk model, at least some of the extra dimensions are extensive (possibly infinite), and other branes may be moving through this bulk. Interactions with the bulk, and possibly with other branes, can influence our brane and thus introduce effects not seen in more standard cosmological models.^{45}
An influential paper by Lisa Randall and Raman Sundrum proposed a model based on brane cosmology.^{46} There are two different versions of the model, RS1 and RS2, but both assume that our 4dimensional universe is a brane inside the bulk, a 5dimensional spacetime. In a Randall–Sundrum universe, matter and light cannot propagate in the fifth dimension. Gravitational waves are the only physical entities able to propagate in the bulk. This is the model depicted in Interstellar: the screenwriters have imagined a highly advanced civilization born into the bulk, who, having mastered of the laws of the gravity, are able to create wormholes and influence our brane by means of gravitational waves.
The Final Equation
In a scene near the end of Interstellar, the scientist Murph can be seen writing an equation aimed at resolving the incompatibility between general relativity and quantum mechanics. In the background are a series of blackboards covered with diagrams and iterative equations supposedly building towards an ultimate equation, a theory of everything. The fate of humanity hinges on its solution. Notwithstanding the naïveté of such a plot device, an interesting question to consider is whether the equations are meaningful.
At first glance, the lengthy series of equations appear dubious. The unification of general relativity and quantum mechanics remains as yet unsolved. A number of approaches to the problem, such as loop quantum gravity, string theory, and noncommutative geometry, are the subject of intensive and ongoing research.^{47}
In imagining a future solution to this problem, the filmmakers have depicted a successful outcome from the most fashionable theoretical framework for unifying all fundamental interactions: string theory.
String theory contends that the fundamental constituents of matter are not pointlike particles, but open or closed strings on the scale of the Planck length (10^{33}cm), whose vibrational modes define the properties of particles. In this framework, spacetime becomes a derived concept that only makes sense at a scale larger than that of the strings. String theory, of which there are five different varieties, aroused such keen interest in the 1990s, that it was believed capable of yielding a theory of everything. The mathematical difficulties remain formidable.^{48}
The five different string theories have led to a larger unifying vision: Mtheory. The ultimate equation seen briefly on Murph’s blackboard has a form similar to the following, if memory serves:
$$\mathrm{S}=\int \sqrt{{g}_{5}}{d}^{5}x\left\{{\mathcal{L}}_{bulk}+\dots \right\}$$
Physicists familiar with string theory will recognize in this equation the effective action of Mtheory in the lowest approximation of its perturbative development. This is a hint, a nod from Kip Thorne to string theorists, that a future theory of everything will be similar to Mtheory. I do not share this viewpoint, but it is surprising to find such a sophisticated message in a Hollywood film.
The g_{5} and d^{5}x terms indicate that, as in Randall–Sundrum models, this is a 5dimensional theory: one time dimension and four space dimensions. The analogue of the cube in a 4dimensional space without curvature is the hypercube, or tesseract. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is stunningly depicted in Interstellar as a multidimensional space inside which time appears as a spatial dimension.^{49} From inside the Tesseract, Cooper is able to travel back in time and communicate with his daughter Murph by means of gravitational signals, providing the data she needs to solve the final equation.^{50}
Denouement
Released some months before Interstellar, the science fiction thriller Gravity was lauded for its impressive depictions of weightlessness and the hostile nature of space.^{51} Most of the science shown in Gravity could be understood in the framework of Newtonian theory, published more than 400 years ago. In sharp contrast, understanding most of the phenomena depicted in Interstellar requires some understanding of general relativity, along with quantum mechanics, and even a little bit of string theory. The result is an intriguing and noteworthy film, one that attempts to combine a great story and accurate science.
But Interstellar is a work of science fiction. Artistic license and scientific extrapolation are part of the game. Assessing the point at which science gives way to science fiction is a worthwhile educational endeavour all by itself.
 Kip Thorne, The Science of Interstellar (New York: W. W. Norton & Company, 2014). ↩
 Ludwig Flamm discovered the paraboloid shape in 1916. Albert Einstein and Nathan Rosen first studied the Schwarzschild throat in 1935. See Albert Einstein and Nathan Rosen, “The Particle Problem in the General Theory of Relativity,” Physical Review 48 (1935): 73–77. ↩
 William Yeats, “The Indian Upon God,” in The Collected Poems of W.B. Yeats, ed. Richard Finneran (New York: Scribner, 1996), 13. ↩
 John Wheeler coined the term “wormhole” to describe this configuration in 1957. Wheeler is also credited with popularizing the term “black hole.” ↩
 Charles Misner found a solution to the wormhole equations allowing for travel from the solar system to the nearest star, Proxima Centauri, located 4.2 lightyears away, in less than 4.2 years, importantly, without exceeding the speed of light. See Charles Misner, “Wormhole Initial Conditions,” Physical Review 118 (1960): 1,110–11. ↩
 At the end of Interstellar, it is revealed that these aliens were, in fact, advanced humans from the future. The future humans created the black hole and associated wormhole, manipulating time and events so that events had to unfold the way they are portrayed in the film. ↩
 Bryce and Cécile DeWitt, eds., Black Holes (Les Houches Summer School 1972) (New York: Gordon Breach 1973). ↩
 Penrose–Carter diagrams, also referred to as conformal diagrams by their inventor Roger Penrose, are 2dimensional spacetime diagrams that are often used to illustrate the spacetime environment of black holes and wormholes. Penrose–Carter diagrams capture the causal relations between different points in spacetime. ↩
 JeanPierre Luminet, Black Holes (Cambridge: Cambridge University Press, 1992) ch. 12. For French readers, the updated version is JeanPierre Luminet, Le destin de l’univers, trous noirs et énergie sombre (Paris: Fayard 2006). ↩
 Michael Morris, Kip Thorne, and Ulvi Yurtsever, “Wormholes, Time Machines, and the Weak Energy Condition,” Physical Review Letters 61, no. 13 (1988): 1,446–49. ↩
 Matt Visser, Lorentzian Wormholes: From Einstein to Hawking (New York: American Institute of Physics, 1996). ↩
 Carl Sagan, Contact (New York: Simon & Schuster, 1985). ↩
 Scientific visualizations of traversable wormholes were calculated in 2006 by Alain Riazuelo at the Institut d’Astrophysique de Paris. The result is much more spectacular than the artistic rendering shown in Interstellar. See Voyage au cœur d’un trou noir (Journey at the heart of a black hole), directed by Patrice Desenne (Paris, France: Sciences et Avenir, 2008), DVD. ↩
 The biggest black hole detected to date is located in the galaxy NGC 1277, some 250 million lightyears away. Its mass could be as large as 17 billion solar masses and its size would encompass the orbit of Neptune. See Remco van den Bosch et al., “An OverMassive Black Hole in the Compact Lenticular Galaxy NGC 1277,” Nature 491 (2012): 729–31. ↩
 Fulvio Melia, The Galactic Supermassive Black Hole (Princeton: Princeton University Press, 2007). ↩
 Ralf Bender et al., “HST STIS Spectroscopy of the Triple Nucleus of M31: Two Nested Disks in Keplerian Rotation around a Supermassive Black Hole,” Astrophysical Journal 631, no. 1 (2005): 280–300. ↩
 For example, a typical stellar black hole of three solar masses, believed to be the engine of many binary Xray sources, must rotate at almost 5,000 revolutions per second. ↩
 Ahmed Almheiri et al., “Black Holes: Complementarity or Firewalls?” Journal of High Energy Physics 2 (2013): 1–20. ↩
 Accretion disks have been detected in some doublestar systems that emit Xray radiation—with black holes of a few solar masses, and in the centers of numerous galaxies—with black holes of between one million and several billion solar masses. ↩
 For a review see Marek Abramowicz and P. Chris Fragile, “Foundations of Black Hole Accretion Disk Theory,” Living Reviews in Relativity 16, no. 1 (2013): 1–88. ↩
 Shawn Poindexter, Nicholas Morgan, and Christopher Kochanek, “The Spatial Structure of an Accretion Disk,” Astrophysical Journal 673 (2008): 34–38. ↩
 Lee Billings, “Parsing the Science of Interstellar with Physicist Kip Thorne,” Scientific American, November 28, 2014. ↩
 Adam Rogers, “How Building a Black Hole for Interstellar Led to an Amazing Scientific Discovery,” Wired, October 22, 2014. ↩
 The best simulations currently available showing the gravitational lensing of background stars are those developed by Alain Riazuelo, at the Institut d’Astrophysique in Paris. Riazuelo has calculated the silhouette of black holes that spin very fast, like Gargantua, in front of a celestial background comprising several thousands of stars. ↩
 The goal of imaging accretion disks telescopically using very long baseline interferometry is nearing reality thanks to the Event Horizon Telescope Collaboration. ↩
 JeanPierre Luminet, “Image of a Spherical Black Hole with Thin Accretion Disk,” Astronomy and Astrophysics 75 (1979): 228–35. ↩
 In theory, there is a tertiary image which provides an extremely distorted view of the top of the accretion disk after the light rays have completed three halfturns, then a quaternary image which provides a view of the bottom of the disk that is even more squashed, and so on to infinity. ↩
 To describe the complete image I obtained, no caption could fit better than the following verses by the French poet Gérard de Nerval, written in 1854:
In seeking the eye of God, I saw nought but an orbit
Gérard de Nerval, “Le Christ aux Oliviers” in Les chimères, poésie et théâtre (Paris: Le Divan, 1928): 39. Translation by JeanPierre Luminet. ↩
Vast, black, and bottomless, from which the night which there lives
Shines on the world and continually thickens
A strange rainbow surrounds this somber well,
Threshold of the ancient chaos whose offspring is shadow,
A spiral engulfing Worlds and Days!  Jun Fukue and Takushi Yokoyama, “Color Photographs of an Accretion Disk around a Black Hole,” Publications of the Astronomical Society of Japan 40 (1988): 15–24. ↩
 S. U. Viergutz, “Image Generation in Kerr Geometry. I. Analytical Investigations on the Stationary EmitterObserver Problem,” Astronomy and Astrophysics 272 (1993): 355–77. ↩
 JeanAlain Marck, “ShortCut Method of Solution of Geodesic Equations for Schwarzschild Black Hole,” Classical and Quantum Gravity 13, no. 3 (1996) 393–402. See also JeanAlain Marck and JeanPierre Luminet, “Plongeon dans un trou noir (Diving into a Black Hole),” Pour la Science HorsSérie “Les trous noirs” (July 1997) 50–56. ↩
 JeanAlain Marck, “Color Animation of a Black Hole with Accretion Disk.” YouTube video. January 13, 2014. Conversion into a movie first appeared in the documentary Infinitely Curved by Laure Delesalle, Marc LachièzeRey, and JeanPierre Luminet (France: CNRS/Arte, 1994). ↩
 For simulations involving the galactic black hole Sagittarius A* see ChiKwan Chan et al., “The Power of Imaging: Constraining the Plasma Properties of GRMHD Simulations Using EHT Observations of SgrA*,” Astrophysical Journal 799, no. 1 (2014), doi: 10.1088/0004637X/799/1/1. ↩
 Oliver James et al., “Gravitational Lensing by Spinning Black Holes in Astrophysics and in the Movie Interstellar,” Classical and Quantum Gravity 32 no. 6 (2014): 1–41. ↩
 Phil Plait, “Interstellar Science,” Slate, November 6, 2014. ↩
 JeanPierre Luminet and Brandon Carter, “Dynamics of an Affine Star Model in a Black Hole Tidal Field,” The Astrophysical Journal Supplement Series 61 (1986): 219–48. ↩
 Brandon Carter and JeanPierre Luminet, “Pancake Detonation of Stars by Black Holes in Galactic Nuclei,” Nature 296 (1982): 211–14. ↩
 The Roche limit is named after Édouard Roche, a French mathematician and astronomer, who studied the problem of tidal forces in the context of planets and their satellites. Roche calculated this figure in 1847. ↩
 For black holes larger than 10^{8} solar masses, such as those suspected to lie in the centers of quasars, the Roche limit becomes significantly smaller than the gravitational radius. Planets or stars can be broken up by tidal forces only once they are inside a black hole of this magnitude. ↩
 Kip Thorne, The Science of Interstellar (New York: W. W. Norton & Company, 2014), 154. ↩
 This process is referred to as the Lense–Thirring effect. ↩
 Roger Penrose, “Gravitational Collapse: The Role of General Relativity,” Rivista del Nuovo Cimento, Numero Speziale 1 (1969): 252–76. ↩
 Bernard Carr and Stephen Hawking, “Black Holes in the Early Universe,” Monthly Notices of the Royal Astronomical Society 168 (1974): 319–416. ↩
 Philippe Brax and Carsten van de Bruck, “Cosmology and Brane Worlds: A Review,” Classical and Quantum Gravity 20, no. 9 (2003), doi:10.1088/02649381/20/9/202. ↩
 Wikipedia, “Brane Cosmology.” ↩
 Lisa Randall and Raman Sundrum, “Large Mass Hierarchy from a Small Extra Dimension,” Physical Review Letters 83, no. 17 (1999): 3,370–73. ↩
 See Carlo Rovelli, “Loop Quantum Gravity,” Living Reviews in Relativity 1 (1998); Michael Green, John Schwarz, and Edward Witten, Superstring Theory I (Cambridge: Cambridge University Press, 1987); Alain Connes, Noncommutative Geometry (Boston: Academic Press, 1994). ↩
 Peter Woit, Not Even Wrong: The Failure of String Theory And the Search for Unity in Physical Law (New York: Basic Books, 2006). ↩
 For an interesting discussion regarding Interstellar’s visualization of the tesseract, see Mike Seymour, “Interstellar: Inside the Black Art,” fxguide, November 18, 2014. ↩
 Despite the exciting visualization of the tesseract in Interstellar, for my own part, artistically speaking, I find the painting Crucifixion (Corpus Hypercubus) by Salvador Dali more deeply moving. Dali depicted the crucifixion of Jesus with a tesseract in place of the cross to signify that, just as God could exist in a space incomprehensible to humans, the hypercube exists in a 4dimensional space that is equally inaccessible to ordinary minds. See Salvador Dali, Crucifixion (Corpus Hypercubus), 1954, oiloncanvas, 94.3cm x 123.8cm. Metropolitan Museum of Art, New York. ↩
 Gravity, directed by Alfonso Cuarón, Warner Bros. Pictures (2013) film. ↩
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