by Thorne, Kip
Chapter 14. Wormholes
For greater detail on wormholes, I especially recommend Lorentzian Wormholes: From Einstein to Hawking (Visser 1995), despite its being nearly twenty years old. I also recommend the last chapter of Black Holes & Time Warps (Thorne 1994), Chapter 9 of Time Travel and Warp Drives (Everett and Roman 2012), and Chapter 8 of Black Holes, Wormholes, and Time Machines (Al-Khalili 2012). For an up-to-date discussion of the exotic matter required to hold a wormhole open, see Chapter 11 of Time Travel and Warp Drives (Everett and Roman 2012).
Chapter 15. Visualizing Interstellar’s Wormhole
Paul Franklin’s team and I give much greater detail about our work on wormhole visualization in one or more articles that we plan to make available on the web at http://arxiv.org/find/gr-qc.
Chapter 16. Discovering the Wormhole: Gravitational Waves
For up-to-date information about LIGO and the search for gravitational waves, see the website of the LIGO Scientific Collaboration, http://www.ligo.org, especially the “News” and “Magazine” sections; also the LIGO Laboratory’s website http://www.ligo.caltech.edu, and also Kai Staats’s 2014 movie at http://www.space.com/25489-ligo-a-passion-for-understanding-complete-film.html. On the web you can also find a number of pedagogical lectures by me about gravitational waves and the warped side of the universe, for example my three “Pauli Lectures” at http://www.multimedia.ethz.ch/speakers/pauli/2011, which should be watched in the opposite order to their listing (that is, from the bottom, upward); and at a moderately technical level, http://www.youtube.com/watch?v=Lzrlr3b5aO8. For movies of black-hole collisions and the gravitational waves they emit, based on the SXS team’s simulations, see http://www.blackholes.org/explore2.html.
There are no up-to-date books about gravitational waves for the general reader, but I do recommend Einstein’s Unfinished Symphony: Listening to the Sounds of Space-Time (Bartusiak 2000), which is not extremely out of date. For the history of research on gravitational waves from Einstein onward, see Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves (Kennefick 2007).
Chapter 17. Miller’s Planet
In this chapter I make a large number of claims about Miller’s planet: its orbit, its rotation (it always keeps the same face toward Gargantua except for rocking), Gargantua’s tidal forces that deform it and make it rock; and Gargantua’s whirl of space that it experiences and how the whirl influences inertia, centrifugal forces, and the speed-of-light speed limit. These claims are all supported by Einstein’s relativistic laws of physics, his general relativity. I don’t know of any books or articles or lectures for nonspecialists that discuss and explain these things, for a planet orbiting close to a spinning black hole, except my Chapter 17. Readers at the advanced undergraduate level may try to check my claims using concepts and equations in Hartle’s textbook, Gravity: An Introduction to Einstein’s General Relativity (Hartle 2003).
The questions I raise in the section “Past History of Miller’s Planet” do not require much relativistic physics. They can be answered almost entirely with Newton’s laws of physics, and the best places to seek relevant information are books and websites that deal with geophysics or the physics of planets and their moons.
Chapter 18. Gargantua’s Vibrations
For a description of Bill Press’s discovery that black holes can vibrate and Saul Teukolsky’s deduction of the equations that govern those vibrations, see pp. 295–299 of Black Holes & Time Warps (Thorne 1994). The technical article about black-hole vibrations and their ringdown that underlies both Figure 18.1 and Romilly’s data set is Yang et al. (2013) by Huan Yang, Aaron Zimmerman, and their colleagues.
Chapter 21. The Fourth and Fifth Dimensions
For more detail on the unification of space and time, see pp. 73–79 of Black Holes & Time Warps (Thorne 1994). For the superstring breakthrough by John Schwarz and Michael Green and how that forced physicists to embrace a bulk with extra dimensions, see The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory (Greene 2003).
Chapter 22. Bulk Beings
For a highly rated, animated movie of Edwin A. Abbott’s Flatland (Abbott 1884), see Flatland: The Film (Ehlinger 2007). For extensive discussions of the mathematics underlying Flatland and the story’s connections to nineteenth-century English society, see The Annotated Flatland: A Romance of Many Dimensions (Stewart 2002). For visual insights into the fourth space dimension, see The Visual Guide to Extra Dimensions, Volume 1: Visualizing the Fourth Dimension, Higher-Dimensional Polytopes, and Curved Hypersurfaces (McMullen 2008).
Chapter 23. Confining Gravity
For much of the content of this chapter, I recommend Warped Passages: Unraveling the Mysteries of the Universe’s Hidden Dimensions (Randall 2006). This is a thorough discussion of modern physicists’ ideas and predictions about the bulk and its extra dimensions, written by Lisa Randall who, with Raman Sundrum, discovered that AdS warping can confine gravity near our brane (Figures 23.4 and 23.6). The idea of an AdS layer and sandwich, which I rediscovered, was first proposed and discussed in a technical paper by Ruth Gregory, Valery A. Rubakov, and Sergei M. Sibiryakov (Gregory, Rubakov, and Sibiryakov 2000), and the AdS sandwich was shown to be unstable in a technical paper by Edward Witten (Witten 2000).
Chapter 24. Gravitational Anomalies
For the history of the anomalous precession of Mercury’s orbit and the search for the planet Vulcan, I recommend a scholarly treatise by science historian N. T. Roseveare, Mercury’s Perihelion from Le Verriere to Einstein (Roseveare 1982), and also the more readable but less comprehensive account by astronomers Richard Baum and William Sheehan, In Search of the Planet Vulcan: The Ghost in Newton’s Clockwork Universe (Baum and Sheehan 1997).
For the discovery of evidence for dark matter in our universe and the current search for dark matter, I recommend a highly readable book, The Cosmic Cocktail: Three Parts Dark Matter (Freeze 2014), by one of the leading researchers in this quest, Katherine Freeze.
For the anomalous acceleration of the universe’s expansion and the dark energy that presumably causes it, I recommend the last chapter of The Cosmic Cocktail (Freeze 2014) and also The 4% Universe: Dark Matter, Dark Energy, and the Race to Discover the Rest of Reality (Panek 2011).
Chapter 25. The Professor’s Equation
The ideas that Newton’s gravitational constant G might change from place to place and time to time, and might be controlled by some sort of nongravitational field, were hot topics in the Princeton University physics department when I was a PhD student there in the early 1960s. These ideas had been proposed by Princeton’s Professor Robert H. Dicke and his graduate student Carl Brans in connection with their “Brans-Dicke theory of gravity” (Chapter 8 of Was Einstein Right? [Will 1993]), an interesting alternative to Einstein’s general relativity. For a brief personal memoir about this, see “Varying Newton’s Constant: A Personal History of Scalar-Tensor Theories” in Einstein Online (Brans 2010). The Brans-Dicke theory has motivated a number of experiments that searched for varying G, but no convincing variations were ever found; see, for example, Chapter 9 of Was Einstein Right? (Will 1993). These ideas and experiments motivated my interpretation of some of Interstellar’s gravitational anomalies and how to control them: bulk fields control the strength of G and make it vary.
The Professor’s equation, shown on his blackboard in Figure 25.6, builds on these ideas. It also incorporates Einstein’s relativistic laws (general relativity), extended into the bulk’s fifth dimension, which are laid out in a technical review article by Roy Maartens and Koyama Kazuya (Maartens and Kazuya 2010), and it incorporates a branch of mathematics called the “calculus of variations”; see, for example, http://en.wikipedia.org/wiki/Calculus_of_variations. For a few technical details about the Professor’s equation, see the appendix Some Technical Notes.
Chapter 26. Singularities and Quantum Gravity
For a first fora
y into quantum fluctuations and quantum physics more generally, I recommend The Ghost in the Atom: A Discussion of the Mysteries of Quantum Physics (Davies and Brown 1986). I don’t know any articles or books for nonphysicists about the quantum behavior of human-sized objects such as LIGO’s mirrors; at a technical level, I discuss this in the second half of my third Pauli lecture (the one listed first) at http://www.multimedia.ethz.ch/speakers/pauli/2011. In John Wheeler’s autobiography, he discusses how he came up with the idea of quantum foam (Chapter 11 of Geons, Black Holes and Quantum Foam: A Life in Physics [Wheeler and Ford 1998]).
In Chapter 11 of Black Holes & Time Warps (Thorne 1994) I discuss what was known in 1994 about the interiors of black holes, and how we came to know it—including the BKL singularity and its dynamics; quantum gravity’s control of the singularity’s core and its connection to quantum foam; and the infalling singularity (mass-inflation singularity), which had only recently been discovered by Erik Poisson and Werner Israel (Poisson and Israel 1990) and was not yet fully understood. The upflying singularity was discovered so recently that there is not yet any detailed discussion of it for nonphysicists; the technical discovery article is Marolf and Ori (2013) by Donald Marolf and Amos Ori. Matthew Choptuik’s discovery that tiny, transient naked singularities are possible was announced and explained in his technical article (Choptuik 1993).
Chapter 27. The Volcano’s Rim
The volcano-like surface that underlies much of this chapter (Figures 27.3, 27.5, and 27.9) can be described with elementary physics equations, as can the Endurance’s trajectory, the trajectory’s instability on the rim, and the Endurance’s launch toward Miller’s planet. See the appendix Some Technical Notes.
Chapter 28. Into Gargantua
In the Prologue of Black Holes & Time Warps (Thorne 1994), I describe, in much greater detail than here, what it would look like and feel like to fall through a black hole’s horizon, both as seen and felt by the infalling person and as seen by someone else outside the black hole. And I describe how the look and feel are influenced by the mass of the black hole and by its spin.
Andrew Hamilton has constructed a “Black Hole Flight Simulator” for computing what it looks like to fall into a nonspinning black hole. His computations are similar to those done for Interstellar by Paul Franklin’s team (Chapters 8, 9, and 15), but preceded Interstellar by many years. Andrew has used his simulator to produce a remarkable set of film clips that can be found on his website, http://jila.colorado.edu/~ajsh/insidebh, and in planetariums around the world (see http://www.spitzinc.com/fulldome_shows/show_blackholes).
Andrew’s film clips differ from what we see in Interstellar in several ways: First, for pedagogical purposes Andrew sometimes paints a grid of lines on the black hole’s horizon (there is no such grid for real black holes and none in Interstellar), and when he does so, he also replaces the star that imploded to form the black hole by a “past horizon.”59 Second, in his “Journey into a Realistic Black Hole,” http://jila.colorado.edu/~ajsh/insidebh/realistic.html, Andrew endows the hole with a jet and an accretion disk. Gas from the disk falls into and through the horizon, and that infalling gas dominates what the camera sees at and beneath the horizon. In Interstellar, by contrast, there is no jet, and the accretion disk is so anemic that it is not currently sending any of its gas into and through the horizon, so the hole’s interior looks rather dark. However, in Interstellar Cooper encounters a dim fog of light and white flakes from stuff that fell in before him. These are not the result of simulations, but instead were put in by hand by the Double Negative artists.
Chapter 29. The Tesseract
When Christopher Nolan told me he was going to use a tesseract in Interstellar, I was delighted. At age thirteen I read about tesseracts in Chapter 4 of George Gamow’s marvelous book One, Two, Three, . . . Infinity (Gamow 1947), and that had a major role in making me want to become a theoretical physicist. You can find a detailed discussion of tesseracts in The Visual Guide to Extra Dimensions (McMullen 2008). Christopher Nolan’s complexified tesseract is unique; there is not yet any public discussion of it anywhere, except in this book and others connected to the movie Interstellar.
In Madeleine L’Engle’s classic science fantasy novel for children, A Wrinkle in Time (L’Engle 1962), children travel via a tesseract—they “tesser”—to find their father. My own interpretation of this is a journey through the bulk, riding in the face of a tesseract, like my interpretation of Cooper’s trip from Gargantua’s core to Murph’s bedroom, Figure 29.4.
Chapter 30. Messaging the Past
For physicists’ current understanding of backward time travel in four spacetime dimensions without a bulk, see the last chapter of Black Holes & Time Warps (Thorne 1994), the chapters by Hawking, Novikov, and me in The Future of Spacetime (Hawking et al. 2002), and Time Travel and Warp Drives (Everett and Roman 2012). These are all by physicists who have contributed in major ways to the theory of time travel. For a historical account of modern research on time travel, see The New Time Travelers: A Journey to the Frontiers of Physics (Toomey 2007). For a comprehensive discussion of time travel in physics, in metaphysics, and in science fiction, see Time Machines: Time Travel in Physics, Metaphysics and Science Fiction (Nahin 1999). From Eternity to Here: The Quest for The Ultimate Theory of Time (Carroll 2011) is a wonderful discussion of almost everything physicists know, or speculate, about the nature of time.
I don’t know any good books or articles, for general readers, about time travel when our universe is a brane that lives in a higher dimensional bulk; but as I discuss in Chapter 30, Einstein’s laws extended to higher dimensions give basically the same predictions as without a bulk.
For some technical details of Cooper’s sending messages backward in time to Murph, see the appendix Some Technical Notes.
Chapter 31. Lifting Colonies off Earth
For Murph’s method (reducing G) for lifting the colonies off Earth, in my interpretation of Interstellar, see my remarks about Chapter 25, above.
In the early 1960s, when I was a PhD student at Princeton University, one of my physics professors, Gerard K. O’Neill, was embarking on an ambitious feasibility study for colonies in space, colonies somewhat like the one we see at the end of Interstellar. His study, augmented by a NASA study that he led, resulted in a remarkable book, The High Frontier: Human Colonies in Space (O’Neill 1978), which I highly recommend. But do pay attention to the book’s introduction by Freeman Dyson, which discusses why O’Neill’s dream of space colonies in his lifetime was shattered, but envisions them in the more distant future.
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59 Stated more precisely and more technically, he has his camera fall into the maximally extended Schwarzschild solution or Reissner-Nordstrom solution of Einstein’s equations instead of into a black hole.
SOME TECHNICAL NOTES
The laws of physics that govern our universe are expressed in the language of mathematics. For readers comfortable with math, I write down a few formulas that come from the physical laws and show how I used them to deduce some things in this book. Two numbers that appear frequently in my formulas are the speed of light, c = 3.00 × 108 meters/second, and Newton’s gravitational constant, G = 6.67 × 10–11 meters3/kilogram/second2. I use scientific notation so 108 means 1 with eight zeros after it, 100,000,000 or a hundred million, and 10–11 means 0.[ten zeros]1, that is, 0.00000000001. I don’t aspire to accuracy any higher than 1 percent, so I show only two or three digits in my numbers, and when a number is very poorly known, only one digit.
Chapter 4. Warped Time and Space, and Tidal Gravity
The simplest, quantitative form of Einstein’s law of time warps is this: Place two identical clocks near each other, and at rest with respect to each other, separated from each other along the direction of the gravitational pull that they feel. Denote by R the fractional difference in their ticking rates, by D the distance between them, and by g the acceleration of gravity tha
t they feel (which points from the one that ages the fastest to the one that ages the slowest). Then Einstein’s law says that g = Rc2/D. For the Pound-Rebca experiment in the Harvard tower, R was 210 picoseconds in one day, which is 2.43 × 10–15, and the tower height D was 73 feet (22.3 meters). Inserting these into Einstein’s law, we deduce g = 9.8 meters/second2, which indeed is the gravitational acceleration on Earth.
Chapter 6. Gargantua’s Anatomy
For a black hole such as Gargantua that spins extremely fast, the horizon’s circumference C in the hole’s equatorial plane is given by the formula C = 2πGM/c2 = 9.3 (M/Msun) kilometers. Here M is the hole’s mass, and Msun = 1.99 × 1030 kilograms is the Sun’s mass. For a very slowly spinning hole, the circumference is twice this size. The horizon’s radius is defined to be this circumference divided by 2π: R = GM/c2 = 1.48 × 108 kilometers for Gargantua, which is very nearly the same as the radius of the Earth’s orbit around the Sun.
The reasoning by which I deduce Gargantua’s mass is this: The mass m of Miller’s planet exerts an inward gravitational acceleration g on the planet’s surface given by Newton’s inverse square law: g = Gm/r2, where r is the planet’s radius. On the faces of the planet farthest from Gargantua and nearest it, Gargantua’s tidal gravity exerts a stretching acceleration (difference of Gargantua’s gravity between the planet’s surface and its center a distance r away) given by gtidal = (2GM/R3)r. Here R is the radius of the planet’s orbit around Gargantua, which is very nearly the same as the radius of Gargantua’s horizon. The planet will be torn apart if this stretching acceleration on its surface exceeds the planet’s own inward gravitational acceleration, so gtidal must be less than g: gtidal < g. Inserting the formulas above for g, gtidal , and R, and expressing the planet’s mass in terms of its density ρ as m = (4π/3)r3ρ, and performing some algebra, we obtain . I estimate the density of Miller’s planet to be ρ = 10,000 kilograms/meter3 (about that of compressed rock), from which I obtain M <3.4 × 1038 kilograms for Gargantua’s mass, which is about the same as 200 million suns—which in turn I approximate as 100 million suns.
