The Science of Interstellar

by Thorne, Kip

  Using Einstein’s relativistic equations, I have deduced a formula that connects the slowing of time on Miller’s planet, S = one hour/(seven years) = 1.63 × 10–5 to the fraction α by which Gargantua’s spin rate is less than its maximum possible spin: . This formula is correct only for very fast spins. Inserting the value of S, we obtain α = 1.3 × 10–14; that is, Gargantua’s actual spin is less than its maximum possible spin by about one part in a hundred trillion.

  Chapter 8. Imaging Gargantua

  The equations that I gave to Oliver James at Double Negative, for the orbital motion of light rays around Gargantua, are a variant of those in Appendix A of Levin and Perez-Giz (2008). Our equations for the evolution of bundles of rays are a variant of those in Pineult and Roeder (1977a) and Pineult and Roder (1977b). In several papers that we’ll make available at http://arxiv.org/find/gr-qc, Paul Franklin’s team and I give the specific forms of our equations and discuss details of their implementation and the simulations that resulted.

  Chapter 12. Gasping for Oxygen

  Here are the calculations that underlie my statements in Chapter 13. They are a nice example of how a scientist makes estimates. These numbers are very approximate; I quote them accurate to only one digit.

  The mass of the Earth’s atmosphere is 5 × 1018 kilograms, of which about 80 percent is nitrogen and 20 percent is molecular oxygen, O2—that is, 1 × 1018 kilograms of O2. The amount of carbon in undecayed plant life (called “organic carbon” by geophysicists) is about 3 × 1015 kilograms, with roughly half in the oceans’ surface layers and half on land (Table 1 of Hedges and Keil [1995]). Both forms get oxidized (converted to CO2) in about thirty years on average. Since CO2 has two oxygen atoms (that come from the atmosphere) and just one carbon atom, and the mass of each oxygen atom is 16/12 that of a carbon atom, the oxidization of all this carbon, after all plants die, would eat up 2 × 16/12 × (3 × 1015 kilograms) = 1 × 1016 kilograms of O2, which is 1 percent of the atmosphere’s oxygen.

  For evidence of sudden overturns of the Earth’s oceans and the theory of how they might be produced, see Adkins, Ingersoll, and Pasquero (2005). The standard estimate of the amount of organic carbon in sediments on the ocean bottoms that might be brought to the surface by such an overturn focuses on an upper sedimentary layer that is mixed by ocean currents and animal activity. This mixed layer’s carbon content is the product of an estimated rate of deposit of carbon into the sediments (about 1011 kilograms per year) and the average time it takes for its carbon to be oxidized by oxygen from ocean water (1000 years), giving 1.5 × 1014 kilograms, one-twentieth of that on land and in ocean surface layers (Emerson and Hedges 1988, Hedges and Keil 1995). However: (i) The estimated deposition rate could be wrong by a huge amount; for example, Baumgart et al. (2009), relying on extensive measurements, estimate a deposition rate in the Indian Ocean off Java and Sumatra that is uncertain by a factor of fifty and, extrapolated to the whole ocean could give as much as 3 × 1015 kilograms of carbon in the mixed layer (the same as on land and in the ocean’s surface layers). (ii) A substantial fraction of the deposited carbon could sink into a lower layer of sediment that does not get mixed into contact with seawater and oxidized except possibly during sudden ocean overturns. The last overturn is thought to have been during the most recent ice age, about 20,000 years ago—twenty times longer than the oxidation time in the mixed layer. So the unmixed layer could have twenty times more organic carbon than the mixed layer, and as much as twenty times that on land and in the ocean’s surface. If brought to the ocean surface by a new overturn and there oxidized, this is nearly enough to leave everyone gasping for oxygen and dying of CO2 poisoning; see the end of Chapter 12. Thus such a scenario is conceivable, though highly unlikely.

  Chapter 15. Visualizing Interstellar’s Wormhole

  Christopher Nolan chose several kilometers for the diameter of Interstellar’s wormhole. The wormhole’s angular diameter as seen from Earth, in radians, is this diameter divided by its distance from Earth, which is about 9 astronomical units or 1.4 × 109 kilometers (the radius of Saturn’s orbit). Therefore, the wormhole’s angular diameter is about (2 kilometers)/(1.4 × 109 kilometers) = 1.4 × 10–9 radians, which is 0.0003 arc-seconds. Radio telescopes routinely achieve this angular resolution using transworld interferometry. Optical telescopes on the ground using a technique called “adaptive optics,” and the Hubble space telescope in space, achieve angular resolutions a hundred times worse than this in 2014. Interferometry between twin Keck telescopes in Hawaii in 2014 can achieve a resolution ten times worse than the wormhole’s angular diameter, and it is very plausible that in the era of Interstellar optical interferometry between more widely spaced optical telescopes will make possible resolutions better than the wormhole’s 0.0003 arc-seconds.

  Chapter 17. Miller’s Planet

  If you are familiar with Newton’s gravitational laws in mathematical form, then you may find it interesting to explore a modification of them by the astrophysicists Bohdan Paczynski and Paul Wiita (Paczynski and Wiita 1980). In this modification, the gravitational acceleration of a nonspinning black hole is changed from Newton’s inverse square law, g = GM/r2 to g = GM/(r – rh)2. Here M is the hole’s mass, r is the radius outside the hole at which the acceleration g is felt, and rh = 2GM/c2 is the radius of the nonspinning hole’s horizon. This modification is a surprisingly good approximation to the gravitational acceleration predicted by general relativity.60 Using this modified gravity, can you give a quantitative version of Figure 17.261 and deduce the radius of the orbit of Miller’s planet? Your result will be only roughly correct, because the Paczynski-Wiita description of Gargantua’s gravity fails to take account of the dragging of space into a whirling motion by the black hole’s spin.

  Chapter 25. The Professor’s Equation

  The meaning of the various mathematical symbols that appear in the Professor’s equation (Figure 25.6) is explained on his other fifteen blackboards, which can be found on the web at this book’s page at Interstellar.withgoogle.com. His equation expresses an “Action” S (the classical limit of a “quantum effective action”) as an integral over “Lagrangian” functions L. These Lagrangians involve the spacetime geometries (“metrics”) of the five-dimensional bulk and our four-dimensional brane, and also involve a set of fields that live in the bulk (denoted Q, σ, λ, ξ, and φi), and also “standard model fields” that live in our brane (including the electric and magnetic fields). The fields and spacetime metrics are to be varied, seeking an extremum (maximum or minimum or saddle point) of the Action S. The conditions that produce an extremum are a set of “Euler-Lagrange” equations that control the evolutions of the fields. This is a standard procedure in the calculus of variations. The Professor and Murph make guesses for a list of unknown bulk fields φi and unknown functions U(Q), Hij(Q2), M(standard model fields), and unknown constants Wij that appear in the Lagrangian. In Figure 25.9 you see me writing a list of their guesses on the blackboard. Then for each set of guesses, they vary the fields and spacetime geometries, deduce the Euler-Lagrange equations, and then explore in computer simulations those equations’ predictions for the gravitational anomalies.

  Chapter 27. The Volcano’s Rim

  This note is for readers who are familiar with the mathematical description of Newton’s laws of gravity and the conservation of energy and angular momentum. I challenge you to deduce the following formula for the volcano-like surface from (i) the Paczynski-Wiita approximate formula for Gargantua’s gravitational acceleration, g = GM/(r – rh)2 (see the technical notes for Chapter 17, above) and (ii) the conservation laws for energy and angular momentum. The formula, using the notation of the technical notes for Chapter 17 plus L for the Endurance’s angular momentum (per unit mass), is

  .

  The first term is the Endurance’s gravitational energy (per unit mass), the second is its circumferential kinetic energy, and the sum of V(r) and the radial kinetic energy v2/2 (with v its radial veloci
ty) is equal to the Endurance’s conserved total energy (per unit mass). The rim of the volcano is at the radius r where V(r) is a maximum. I challenge you, using these equations and ideas, to prove my claims, in Chapter 27, about the Endurance’s trajectory, the trajectory’s instability on the rim of the volcano, and its launch toward Edmunds’ planet.

  Chapter 30. Messaging the Past

  In the bulk as well as in our brane, the locations in spacetime, to which messages and other things can travel, are controlled by the law that nothing can travel faster than light. We physicists use spacetime diagrams to explore the consequences of this law. We draw spacetime diagrams in which, at each event, there is a “future light cone.” Light travels outward from that event along the light cone; everything else, moving slower than light, travels from that event either along or inside the cone. See, for example, Gravity: An Introduction to Einstein’s General Relativity (Hartle 2003).

  Figure TN.1 shows the pattern of future light cones inside and on faces of the tesseract, in my interpretation of Interstellar. (It is the mathematical description of spacetime warping that I refer to in footnote 1 of Chapter 30. Physicists call this pattern of light cones “the causal structure of spacetime” inside the tesseract.) Figure TN.1 also shows the world line (violet curve) of the gravitational-wave message (force) sent by Cooper through the tesseract’s interior to Murph’s bedroom; and the world line (red dashed line) of the light ray from the bedroom through tesseract faces, by which Cooper sees the bedroom. This is a spacetime version of the purely spatial diagram in Figure 30.5.

  Figure TN.1. The causal structure of spacetime inside the tesseract with one space dimension omitted.

  Can you understand from this diagram how it is that the gravitational-wave message travels at the speed of light, yet moves backward relative to bedroom time and Cooper’s time? And can you understand how, by contrast, the light ray travels at the speed of light and moves forward relative to bedroom time and Cooper’s time? Compare with our discussion of Escher’s drawing, Figure 30.6.

  * * *

  60 This Paczynski-Wiita modification of gravity was used in developing the black hole’s influence on spacecraft orbits for a gravitational-slingshot video game associated with Interstellar; see Game.InterstellarMovie.com.

  61 For a related calculation, see the technical notes for Chapter 27, below.

  ACKNOWLEDGMENTS

  For welcoming me into Hollywood and teaching me so much about that remarkable world, I thank, first and foremost, my partner Lynda Obst; and also Christopher Nolan, Emma Thomas, Jonathan Nolan, Paul Franklin, and Steven Spielberg.

  I thank Lynda for the friendship and collaboration that gave birth to the treatment from which Interstellar sprang, and for guiding Interstellar through its trials and tribulations until it landed in the remarkable hands of Christopher Nolan, who transformed it so greatly.

  For welcoming me into the visual-effects world and giving me the opportunity to lay foundations for visualizing Interstellar’s wormhole and the black hole Gargantua and its accretion disk, I thank Paul Franklin, Oliver James, and Eugénie Von Tunzelmann; and for collaborating tightly with me on those foundations, I thank Oliver and Eugénie.

  For wise comments and suggestions on the manuscript of this book, I’m grateful to Lynda Obst, Jeff Shreve, Emma Thomas, Christopher Nolan, Jordan Goldberg, Paul Franklin, Oliver James, Eugénie Von Tunzelmann, and Carol Rose. For their dogged commitment to accuracy and consistency in every line of the manuscript, I thank Leslie Huang and Don Rifkin. For crucial assistance and/or advice about figures, I thank Jordan Goldberg, Eric Lewy, Jeff Shreve, Julia Druskin, Joe Lops, Lia Halloran, and Andy Thompson. For crucial assistance in getting permission for use of figures, I thank Pat Holl. And for making the book a reality, I’m grateful to Drake McFeely, Jeff Shreve, Amy Cherry, and my Hollywood attorneys Eric Sherman and Ken Ziffren (yes, most everyone who works in Hollywood has to have an attorney or agent; even a scientist on the fringes).

  And for her patience and support throughout this adventure, I’m grateful to my wife and life partner, Carolee Winstein.

  FIGURE CREDITS

  Following figures © Warner Bros. Entertainment Inc.: 1.2, 1.3, 3.4, 3.6, 5.6, 8.1, 8.5, 8.6, 9.7, 9.9, 9.10, 9.11, 11.1, 14.9, 15.2, 15.4, 15.5, 17.5, 17.9, 18.1, 19.2, 19.3, 20.1, 20.2, 24.5, 25.1, 25.7, 25.8, 25.9, 27.8, 28.3, 29.8, 29.14, 30.1, 31.1

  Following figures © Kip Thorne: 2.4, 2.5, 3.2, 3.5, 4.3, 4.4, 4.8, 4.9, 5.1, 5.2, 5.3, 5.4, 6.1, 6.2, 6.3, 6.4, 6.5, 7.1, 7.2, 7.3, 7.5, 8.2, 8.7, 9.8, 13.4, 13.5, 13.6, 14.5, 15.1, 15.3, 16.2, 16.5, 16.8, 17.1, 17.2, 17.3, 17.4, 17.6, 19.1, 21.3, 22.2, 22.3, 22.4, 23.2, 23.5, 23.6, 23.7, 23.8, 24.1, 24.4, 24.6, 24.7, 25.2, 25.3, 25.5, 25.6, 26.5, 26.10, 26.11, 26.12, 26.13, 27.1, 27.3, 27.6, 27.10, 28.2, 29.1, 29.2, 29.12, 30.2, 30.3, 30.4, TN.1

  1.1:Carolee Winstein

  1.2:Melinda Sue Gordon. © Warner Bros.

  1.3:Tyler Ott

  1.4:Rosie Draper

  2.1:NASA, N. Benitez (JHU), T. Broadhurst (Racah Institute of Physics/The Hebrew University), H. Ford (JHU), M. Clampin (STScI), G. Hartig (STScI), G. Illingworth (UCO/Lick Observatory), the ACS Science Team, and ESA

  2.2:Adam Evans, www.sky-candy.ca

  2.3:Courtesy of NASA/SDO and the AIA, EVE, and HMI science teams

  2.6:Property of the estate of Matthew H. Zimet. Courtesy Eva Zimet

  2.7:© Best View Stock/Alamy

  2.8:Image of Earth: NASA

  2.9:© Picture Press/Alamy

  2.10:© Russell Kightley/Science Source

  2.11:Image of Earth: NASA

  3.1:Waldseemuller map: map image courtesy of the Norman B. Leventhal Map Center at the Boston Public Library/Sidney R. Knafel Collection at Phillips Academy, Andover, MA. Ortelius map: from Library of Congress Geography and Map Division Washington, D.C. Bowen map: Geographicus Rare Antique Maps

  3.3:Double Negative Visual Effects: Eugénie von Tunzelmann and Oliver James.

  3.6:Kip Thorne. © Warner Bros.

  4.2:United States Government, adapted by Kip Thorne

  4.5:© Lia Halloran, www.liahalloran.com

  4.6:© Lia Halloran and Kip Thorne

  4.7:© Lia Halloran and Kip Thorne

  5.5:Courtesy NASA/JPL-Caltech

  5.6:Double Negative Visual Effects: Eugénie von Tunzelmann and Oliver James.

  5.7:Karl Schwarzschild: photograph by Robert Bein, courtesy AIP Emilio Segrè Visual Archives. Roy Kerr: Roy P. Kerr. Stephen Hawking: © Richard M. Diaz. Robert Oppenheimer: en.wikipedia.org—J._Robert_Oppenheimer. Andrea Ghez: Mary Watkins (UCLA)

  5.8:Keck/UCLA Galactic Center Group; Andrea Ghez

  5.9:Akira Fujii/ESA/Hubble

  6.5:© Kip Thorne, patterned after illustrations by Edward Teo (2003)

  7.4:Left: © Robert Gendler (robgendlerastropics.com). Right: © Kip Thorne

  7.6:Steve Drasco, assistant professor of physics, California Polytechnic State University, San Luis Obispo

  7.7:Courtesy NASA/JPL-Caltech

  8.1:Double Negative Visual Effects: Eugénie von Tunzelmann and Oliver James.

  8.3:Star field image: Alain Riazuelo, IAP/UPMC/CNRS; drawing: Kip Thorne. Movie camera icon courtesy basarugur

  8.4:Star field image: Alain Riazuelo, IAP/UPMC/CNRS; drawing: Kip Thorne

  8.5:Image: Double Negative Visual Effects: Eugénie von Tunzelmann and Oliver James; drawing: Kip Thorne. © Warner Bros. and Kip Thorne

  8.6:Image: Double Negative Visual Effects: Eugénie von Tunzelmann and Oliver James; drawing: Kip Thorne. © Warner Bros. and Kip Thorne. Movie camera icon courtesy basarugur

  9.1:Photo: NASA/STScI. Spectrum: Maarten Schmidt

  9.2:Property of the estate of Matthew H. Zimet. Courtesy Eva Zimet

  9.3:Property of the estate of Matthew H. Zimet. Courtesy Eva Zimet

  9.4:Property of the estate of Matthew H. Zi
met. Courtesy Eva Zimet

  9.5:James Guillochon

  9.6:James Guillochon

  9.7:Double Negative Visual Effects: Eugénie von Tunzelmann and Oliver James.

  9.8:Movie camera icon courtesy basarugur. © Kip Thorne

  9.9:Double Negative Visual Effects: Eugénie von Tunzelmann and Oliver James.

  10.1:Steve Drasco, assistant professor of physics, California Polytechnic State University, San Luis Obispo

  11.2:NASA Goddard Earth Sciences Data and Information Services Center/Giovanni. Special thanks to James G. Acker.

  13.1:© Kip Thorne and Richard Powell, www.atlasoftheuniverse.com

  13.2:Freeman Dyson

  13.3:Copyright © American Institute of Aeronautics and Astronomics, Inc. 1983. All rights reserved.

  14.1:Apple © Preto Perola/Shutterstock.com. Ant: © Katarzyna Cielecka/Fotalia.com

  14.2:Wormhole drawing on left previously published in Misner, Thorne, and Wheeler (1973); remainder of figure: Kip Thorne

  14.3:Property of the estate of Matthew H. Zimet. Courtesy Eva Zimet

  14.4:Property of the estate of Matthew H. Zimet. Courtesy Eva Zimet

  14.6:Left: © Catherine MacBride. Right: © Mark Interrante

  14.7:Property of the estate of Matthew H. Zimet. Courtesy Eva Zimet

  14.8:Property of the estate of Matthew H. Zimet. Courtesy Eva Zimet