The Science of Interstellar

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 – r_h)². Here M is the hole’s mass, r is the radius outside the hole at which the acceleration g is felt, and r_h = 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 17.2[61] 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), H_ij(Q²), M(standard model fields), and unknown constants W_ij 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 – r_h)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 velocity) 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).