The Science of Interstellar

Let’s make that more precise. In Figure 4.4 the Sun’s equatorial plane is a two-dimensional surface that bends downward in a three-dimensional bulk. This motivates the way we physicists think about our full universe. Our universe has three space dimensions (east-west, north-south, up-down), and we think of it as a three-dimensional membrane or brane for short that is warped in a higher-dimensional bulk. How many dimensions does the bulk have? I discuss this carefully in Chapter 21, but for the purposes of Interstellar, the bulk has just one extra space dimension: four space dimensions in all.

Now, it’s very hard for humans to visualize our three-dimensional universe, our full brane, living and bending in a four-dimensional bulk. So throughout this book I draw pictures of our brane and bulk with one dimension removed, as I did in Figure 4.4.

^{Fig. 4.4. Paths of Viking radio signals through the Sun’s warped equatorial plane.}

In Interstellar, the characters often refer to five dimensions. Three are the space dimensions of our own universe or brane (east-west, north-south, up-down). The fourth is time, and the fifth is the bulk’s extra space dimension.

Does the bulk really exist? Is there truly a fifth dimension, and maybe even more, that humans have never experienced? Very likely yes. We’ll explore this in Chapter 21.

The warping of space (warping of our brane) plays a huge role in Interstellar. For example, it is crucial to the very existence of the wormhole connecting our solar system to the far reaches of the universe, where Gargantua lives. And it distorts the sky around the wormhole and around the black hole Gargantua; this is the gravitational lensing we met in Figure 3.3.

Figure 4.5 is an extreme example of space warps. It is a fanciful drawing by my artist friend Lia Halloran, depicting a hypothetical region of our universe that contains large numbers of wormholes (Chapter 14) and black holes (Chapter 5) extending outward from our brane into and through the bulk. The black holes terminate in sharp points called “singularities.” The wormholes connect one region of our brane to another. As usual, I suppress one of our brane’s three dimensions, so the brane looks like a two-dimensional surface.

^{Fig. 4.5. Black holes and wormholes extending out of our brane into and through the bulk. One space dimension is removed from both our brane and the bulk. [Drawing by the artist Lia Halloran.]}

Tidal Gravity

Einstein’s relativistic laws dictate that planets, stars, and unpowered spacecraft near a black hole move along the straightest paths permitted by the hole’s warped space and time. Figure 4.6 shows examples of four such paths. The two purple paths headed into the black hole begin parallel to each other. As each path tries to remain straight, the two paths get driven toward each other. The warping of space and time drives them together. The green paths, traveling circumferentially around the hole, also begin parallel. But in this case, the warping drives them apart.

^{Fig. 4.6. Four paths for planetary motion in the vicinity of a black hole. The picture of the hole is extracted from Lia Halloran’s drawing, Figure 4.5.}

Several years ago, my students and I discovered a new point of view about these planetary paths. In Einstein’s relativity theory there is a mathematical quantity called the Riemann tensor. It describes the details of the warping of space and time. We found, hidden in the mathematics of this Riemann tensor, lines of force that squeeze some planetary paths together and stretch others apart. “Tendex lines,” my student David Nichols dubbed them, from the Latin word tendere meaning “to stretch.”

Figure 4.7 shows several of these tendex lines around the black hole of Figure 4.6. The green paths begin, on their right ends, parallel to each other, and then the red tendex lines stretch them apart. I draw a woman lying on a red tendex line. It stretches her, too; she feels a stretching force between her head and her feet, exerted by the red tendex line.

^{Fig. 4.7. Tendex lines around a black hole. The picture of the hole is extracted from Lia Halloran’s drawing, Figure 4.5.}

The purple paths begin, at their top ends, running parallel to each other. They are then squeezed together by the blue tendex lines, and the woman whose body lies along a blue tendex line is also squeezed.