by Thorne, Kip
The good news: My idea of an AdS sandwich had been invented six years earlier by Ruth Gregory (University of Durham, UK), together with Valery Rubakov and Sergei Sibiryakov (Institute for Nuclear Research in Moscow, Russia). This showed I was not being stupid in my first mathematical foray into the bulk. I had rediscovered something worth discovering.
The bad news: Edward Witten (Princeton) and others had shown that the AdS sandwich is unstable! The confining branes are under pressure, rather like a playing card that you squeeze end to end between your finger and thumb (Figure 23.8). The card bends, and with further squeezing, it buckles. Similarly, the confining branes will bend and crash into our brane (our universe), destroying it. The entire universe destroyed! That’s the worst news ever!!
But I can think of several ways to save our universe, if it really does live in an AdS sandwich (which I very much doubt it does); several ways to “stabilize the confining branes,” in the jargon of physicists.
In my science interpretation of Interstellar, Professor Brand, working with Einstein’s relativity equations, rediscovers the AdS sandwich, as I did; see the photograph of his blackboard in Figure 3.6. How the confining branes are stabilized then gets intertwined with the Professor’s struggle to understand and control gravitational anomalies. In the movie, that struggle is spelled out mathematically on the sixteen blackboards in Professor Brand’s office; Chapter 25.
Fig. 23.8. A playing card, compressed end to end, bends and then buckles.
Traveling Through the AdS Layer
In the AdS layer, the AdS warpage of space produces tidal forces that are enormous by human standards. Any bulk being traveling through the layer to reach our brane must deal with those forces. Because we know nothing about the matter of which a bulk being is made—matter with four space dimensions—we have no idea whether this is an issue. In science fiction it can be left in the hands of the writers.
Not so for Cooper, riding in the tesseract (Chapter 29). In my interpretation of the movie, he has to cross the AdS layer. The tesseract must either protect him from the layer’s enormous tidal forces or clear the AdS layer away from his path. Otherwise he’ll be spaghettified.
By confining gravity, the AdS layer regulates its strength. In Interstellar we see gravity’s strength fluctuate, perhaps due to fluctuations in the AdS layer. These fluctuations—gravitational anomalies—play a huge role in Interstellar. To them we now turn.
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37 I discuss quantum fluctuations in Chapter 26 and bulk fields in Chapter 25.
38 For details see Lisa Randall’s Warped Passages (HarperCollins, 2006).
39 Why is the magic distance, at which the inverse square law begins, 0.1 millimeter instead of, say, 1 kilometer or 1 picometer? I have chosen 0.1 millimeter quite arbitrarily. Experiments have proved that gravity obeys the inverse square law down to about 0.1 millimeter, so that is an upper limit on the magic distance. It could perfectly well be smaller.
24
Gravitational Anomalies
A gravitational anomaly is something about gravity that doesn’t fit our understanding of the universe, or our understanding of the physical laws that control the universe—for example the falling books, in Interstellar, that Murph attributes to a ghost.
Since 1850, physicists have put a lot of effort into searching for gravitational anomalies and understanding those few that were found. Why? Because any true anomaly is likely to produce a scientific revolution; a major change in what we think is True . This, in fact, has happened three times since 1850.
In Interstellar, Professor Brand’s struggle to understand gravitational anomalies is very much in the spirit of these previous revolutions; so I describe the previous ones, briefly.
The Anomalous Precession of Mercury’s Orbit
Newton’s inverse square law for gravity (Chapters 2 and 23) forces the orbits of the planets around the Sun to be ellipses. Each planet feels small gravitational tugs from the other planets, and these tugs cause its ellipse to gradually change orientation, that is, to gradually precess.
In 1859, the astronomer Urbain Le Verrier at the Observatoire de Paris (France) announced he had discovered an anomaly in the orbit of the planet Mercury. When he computed the total precession of Mercury’s orbit caused by all the other planets, he got the wrong answer. The measured precession is larger than the planets could produce by about 0.1 arc second each time Mercury traverses its orbit (Figure 24.1).
Now 0.1 arc second is a tiny angle, just one ten-millionth of a circle. But Newton’s inverse square law insists there can be no anomaly whatsoever.
Le Verrier convinced himself that this anomaly is produced by the gravitational tug of an undiscovered planet closer to the Sun than Mercury; “Vulcan” he called it.
Astronomers searched in vain for Vulcan. They could not find it, nor could they find any other explanation for the anomaly. By 1890 the conclusion seemed clear: Newton’s inverse square law must be very slightly wrong.
Wrong in what way? A revolutionary way, it turned out. The way discovered by Einstein twenty-five years later. The warping of time and space endow the Sun with a gravitational force that obeys Newton’s inverse square law, but only nearly. Not precisely.
Upon realizing that his new relativistic laws explain the observed anomaly, Einstein was so excited that he suffered heart palpitations and felt like something snapped inside himself. “For a few days I was beside myself with joyous excitement.”
Fig. 24.1. The anomalous precession of Mercury’s orbit. In this picture, I exaggerate the orbit’s ellipticity (its elongated shape) and the magnitude of its precession.
Today the measured anomalous precession and the predictions by Einstein’s laws agree to within one part in a thousand (one-thousandth of the anomalous precession), which is the accuracy of the observations—a great triumph for Einstein!
The Anomalous Orbits of Galaxies Around Each Other
In 1933 the Caltech astrophysicist Fritz Zwicky announced he had discovered a huge anomaly in the orbits of galaxies around each other. The galaxies were in the Coma cluster (Figure 24.2), a collection of about a thousand galaxies, 300 million light-years from Earth, in the constellation Coma Berenices.
From the Doppler shifts of the galaxies’ spectral lines, Zwicky could estimate how fast they were moving relative to each other. And from the brightness of each galaxy, he could estimate its mass and thence its gravitational pull on the other galaxies. The galaxies’ motions were so fast that there was no way their gravitational pulls could hold the cluster together. Our best understanding of the universe and of gravity insisted that the cluster must be flying apart, and would soon be completely destroyed. If so, then the cluster must have formed by random motions of all those galaxies and would disrupt in a veritable blink of an eye compared to other astronomical phenomena.
Fig. 24.2. The Coma cluster of galaxies as seen through a large telescope.
This conclusion was totally implausible to Zwicky. Something was wrong with our conventional wisdom. Zwicky made an educated guess: The Coma cluster must be filled with some sort of “dark matter” whose gravity is strong enough to hold the cluster together.
Now, many anomalies that astronomers and physicists think they have discovered go away when observations improve. This one did not. Instead, it spread. By the 1970s it was clear that so-called dark matter permeates most all clusters of galaxies and even individual galaxies. By the 2000s, it was clear that the dark matter gravitationally lenses light from more distant galaxies (Figure 24.3), just as Gargantua gravitationally lenses light from stars (Chapter 8). Today that lensing is being used to map the dark matter in our universe.
Fig. 24.3. Dark matter in the galaxy cluster Abell 2218 gravitationally lenses more distant galaxies. The images of the lensed galaxies are arc-shaped (e.g., those I circled in purple), analogous to ar
c structures seen in Gargantua’s gravitational lensing, Chapter 8.
And today physicists are fairly sure that the dark matter is truly revolutionary, that it consists of fundamental particles of a type never before seen, but a type predicted by our best current understanding of the quantum laws of physics. Physicists have embarked on a holy-grail mission: a quest to detect these particles of dark matter, shooting through the Earth with near impunity, and measure their properties.
The Anomalous Acceleration of the Universe’s Expansion
In 1998 two research groups independently discovered an astounding anomaly in the expansion of our universe. For this discovery, the groups’ leaders (Saul Perlmutter and Adam Reiss at the University of California, Berkeley, and Brian Schmidt at the Australian National University) won the 2011 Nobel Prize in Physics.
Both groups were observing supernova explosions: explosions triggered when a massive star exhausts its nuclear fuel and implodes to form a neutron star, and the implosion energy blows off the star’s outer layers. They discovered that distant supernovae are dimmer than expected, and therefore farther away than expected. Farther enough away that the universe’s expansion must have been slower in the past than today. The expansion is accelerating. See Figure 24.4.
Fig. 24.4. The distance to the star at the time of explosion (the time that the light we see was emitted), under two assumptions: that the universe’s expansion is decelerating (red) or accelerating (blue). The explosion was dimmer than expected, so farther away. The universe must be accelerating.
Now, our best understanding of gravity and the universe required, unequivocally, that all things in the universe (stars, galaxies, galaxy clusters, dark matter, etc.) must pull on each other gravitationally. And by that pull they must slow the universe’s expansion. The universe’s expansion must slow down over time, not speed up.
For this reason, I, personally, didn’t believe the claimed acceleration, nor did many of my astronomer and physicist colleagues. We didn’t believe until other observations, by completely different methods, confirmed it. Then we caved.
So what’s going on? There are two possibilities: Something is wrong with Einstein’s relativistic laws of gravity. Or something else is filling the universe, in addition to ordinary matter and dark matter. Something that repels gravitationally.
Most physicists love Einstein’s relativistic laws and are loathe to give them up, and so lean toward repulsion. The hypothetical material that repels has been given the name “dark energy.”
The final verdict is not in. But if the cause of the anomaly is, indeed, dark energy (whatever that may be), then gravitational observations now tell us that 68 percent of the universe’s mass is in dark energy, 27 percent is in dark matter, and only 5 percent is in the kind of ordinary matter of which you, I, planets, stars, and galaxies are made.
So physicists today have another holy grail: to understand whether the universe’s accelerated expansion is caused by a breakdown of Einstein’s relativistic laws (and if so, what is the nature of the correct laws?), or is caused by repulsive dark energy (and if so, what is the nature of the dark energy?).
Gravitational Anomalies in Interstellar
The gravitational anomalies in Interstellar are seen on Earth, by contrast with the three anomalies that I described.
Physicists have put great effort into searching for such anomalies on Earth, beginning with Isaac Newton himself in the late 1600s. Those searches have produced many claimed anomalies, but all claims, upon deeper scrutiny, have collapsed.
The anomalies in Interstellar are startling for their weirdness and strength, and the way they change as time passes. If anything like them had occurred in the twentieth century or early twenty-first, physicists would surely have noticed them and explored them with great fervor. Somehow, gravity on Earth has been altered in the era of Interstellar.
And, indeed, Romilly tells Cooper so in the movie: “We started detecting gravitational anomalies [on Earth] almost fifty years ago,” and also, around that same time, the most signficant anomaly of all: the sudden appearance of a wormhole near Saturn, where before there was none.
In the movie’s opening scene, Cooper experiences an anomaly himself, while trying to land a Ranger spacecraft. “Over the Straights something tripped my fly-by-wire,” he tells Romilly.
The GPS system that Cooper has adapted to control harvesting machines, as they roam through corn fields, has also gone haywire, and a bunch of harvesters have converged on his farmhouse. He attributes this to gravitational anomalies that screwed up the gravity corrections that any GPS system relies on (Figure 4.2).
Early in the movie, we see Murph watch, transfixed, as dust falls unnaturally fast to the floor of her bedroom, collecting in a bar-code-like pattern of thick lines. And then we see Cooper stare at the lines (Figure 24.5) and toss a coin across one. The coin shoots to the floor.
Fig. 24.5. Cooper stares at the dust pattern on the floor of Murph’s bedroom. [From Interstellar, used courtesy of Warner Bros. Entertainment Inc.]
In my science interpretation of Interstellar, I presume that Professor Brand’s team has collected a large trove of data on the anomalies. The most interesting data to me as a physicist, and to Professor Brand in my movie interpretation, is new and changing patterns of tidal gravity.
We first met tidal gravity in Chapter 4: the tidal gravity produced by a black hole, and tidal gravity on Earth produced by the Moon and Sun. In Chapter 17 we saw Gargantua’s tidal gravity in action on Miller’s planet, triggering gigantic “Millerquakes,” tsunamis, and tidal bores. In Chapter 16 we met the tiny stretching and squeezing of tidal gravity in a gravitational wave.
Tidal gravity is produced not only by black holes, the Sun, the Moon, and gravitational waves but also, in fact, by all gravitating objects. For example, regions of the Earth’s crust that contain oil are less dense than regions containing only rock, so their gravitational pull is weaker. This leads to a peculiar pattern of tidal gravitational forces.
In Figure 24.6, I use tendex lines to illustrate that tidal-force pattern. (See Chapter 4 for a discussion of tendex lines.) Squeezing tendex lines (drawn blue) stick out of the oil-bearing region, while stretching tendex lines (drawn red) stick out of the denser, oil-free region. As always, the two families of tendex lines are perpendicular to each other.
Fig. 24.6. Tendex lines above a portion of the Earth’s crust. The red lines produce a tidal stretch along themselves. The blue lines produce a tidal squeeze.
An instrument called a gravity gradiometer can measure these tidal patterns (Figure 24.7). It consists of two crossed, solid rods attached to a torsional spring. On the ends of each rod are masses that feel gravity. The rods are normally perpendicular to each other, but in the figure the blue tendex lines squeeze the top two masses together and squeeze the bottom two together, while the red tendex lines stretch the right pair of masses apart and stretch the left pair apart. As a result, the angle between the rods decreases until the spring counterbalances the tidal forces. This is the gradiometer’s readout, its “readout angle.”
Fig. 24.7. A simple version of a gravity gradiometer, designed and built by Robert Forward at Hughes Research Laboratories in 1970.
If this gradiometer is flown rightward through the tidal pattern of Figure 24.6, its readout angle opens up above the oil-bearing region, and then closes down over the oil-free region. Gradiometers like this, but more sophisticated, are used by geologists to search for oil and also for mineral deposits.
NASA has flown a more sophisticated gradiometer called GRACE40 (Figure 24.8) to map tidal fields everywhere above the Earth, and watch slow changes of tidal gravity produced, for example, by the melting of ice sheets.
Fig. 24.8. GRACE: Two satellites, which track each other with a beam of microwaves, are pushed together by blue tendex lines and stretched apart by red tende
x lines. The tendex lines, from the Earth below, are not shown.
In my interpretation of Interstellar, most of the gravitational anomalies that Professor Brand’s team measures are sudden and unexpected changes in the patterns of tendex lines above the Earth’s surface, changes that occur for no obvious reason. The rocks and oil in the Earth’s crust are not moving. The melting of ice sheets is much too slow to produce these quick changes. People see no new gravitating masses coming near the gradiometers. Nevertheless, the gradiometers report changing tidal patterns. Falling dust accumulates in radial lines. Cooper sees the coin plunge to the floor.
The members of Professor Brand’s team monitor these changing patterns and eagerly record Cooper’s observations. Their trove of data becomes grist for the Professor’s quest to understand gravity, a quest that centers on the Professor’s equation.
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40 The Gravity Recovery and Climate Experiment, a joint US/German space mission launched in May 2002 and still collecting data in 2014.
25
The Professor’s Equation
In Interstellar, the gravitational anomalies excite Professor Brand for two reasons. If he can discover their cause, that may trigger a revolution in our understanding of gravity, a revolution as great as Einstein’s relativistic laws. More important: If he can figure out how to control the anomalies, that could enable NASA to lift large colonies of people off the dying Earth, and launch them toward a new home elsewhere in the universe.
