In general relativity, the curvature of spacetime can be expressed mathematically in terms of a spacetime metric. In this section, we review a simple example of this: the metric for an Ellis wormhole; and then we discuss the metric for the Double Negative (Dneg) wormhole that we designed for Interstellar.

A. The Ellis wormhole

13 Ether flow through a drainhole: A particle model in general relativity ,” J. Math. Phys. 14, 104– 118 (1973). 13. Homer G. Ellis, “,” J. Math. Phys., 104–(1973). https://doi.org/10.1063/1.1666161 14 14. Fifteen years later, Morris and Thorne3 wrote down this same metric, among others, and being unaware of Ellis's paper, failed to attribute it to him, for which they apologize. Regretably, it is sometimes called the Morris-Thorne wormhole metric. d s 2 = − d t 2 + d ℓ 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , (1) where r is a function of the coordinate ℓ given by r ( ℓ ) = ρ 2 + ℓ 2 , (2) and ρ is a constant. In 1973 Homer Ellisintroduced the following metric for a hypothetical wormhole, which he called a “drainhole”whereis a function of the coordinategiven byandis a constant.

As always in general relativity, one does not need to be told anything about the coordinate system in order to figure out the spacetime geometry described by the metric; the metric by itself tells us everything. Deducing everything is a good exercise for students. Here is how we do so.

First, in −dt2 the minus sign tells us that t, at fixed ℓ, θ, ϕ , increases in a timelike direction; and the absence of any factor multiplying −dt2 tells us that t is, in fact, proper time (physical time) measured by somebody at rest in the spatial, { ℓ , θ , ϕ } , coordinate system.

Second, the expression r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) is the familiar metric for the surface of a sphere with circumference 2πr and surface area 4πr2, written in spherical polar coordinates { θ , ϕ } , so the Ellis wormhole must be spherically symmetric. As we would in flat space, we shall use the name “radius” for the sphere's circumference divided by 2π, i.e., for r. For the Ellis wormhole, this radius is r = ρ 2 + ℓ 2 .

Third, from the plus sign in front of dℓ2 we infer that ℓ is a spatial coordinate; and since there are no cross terms dℓdθ or d ℓ d ϕ , the coordinate lines of constant θ and ϕ , with increasing ℓ, must be radial lines; and since dℓ2 has no multiplying coefficient, ℓ must be the proper distance (physical distance) traveled in that radial direction.

Fourth, when ℓ is large and negative, the radii of spheres r = ρ 2 + ℓ 2 is large and approximately equal to | ℓ | . When ℓ increases to zero, r decreases to its minimum value ρ. And when ℓ increases onward to a very large (positive) value, r increases once again, becoming approximately ℓ. This tells us that the metric represents a wormhole with throat radius ρ, connecting two asymptotically flat regions of space, ℓ → −∞ and ℓ → +∞.

11 11. James B. Hartle, Gravity: An Introduction to Einstein's General Relativity ( Addison-Wesley , San Francisco , 2003). t and θ = π/2) look like when embedded in a flat 3-dimensional space, the embedding space. Hartle shows that equatorial surfaces have the form shown in Fig. 1 In Hartle's textbook,a number of illustrative calculations are carried out using Ellis's wormhole metric as an example. The most interesting is a computation, in Sec. VII , of what the two-dimensional equatorial surfaces (surfaces with constantand/2) look like when embedded in a flat 3-dimensional space, the. Hartle shows that equatorial surfaces have the form shown in Fig.—a form familiar from popular accounts of wormholes.

1 Figureis called an “embedding diagram” for the wormhole. We discuss embedding diagrams further in Sec. II B 3 , in the context of our Dneg wormhole.