The Science of Interstellar

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.

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 that 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/M_sun) kilometers. Here M is the hole’s mass, and M_sun = 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 g_tidal = (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 g_tidal must be less than g: g_tidal < g. Inserting the formulas above for g, g_tidal, 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.

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