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.
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.
This stretching and squeezing is just a different way of thinking about the influence of the warping of space and time. From one viewpoint, the paths are stretched apart or squeezed together due to the planetary paths moving along the straightest routes possible in the warped space and time. From another viewpoint it is the tendex lines that do the stretching and squeezing. Therefore, the tendex lines must, in some very deep way, represent the warping of space and time. And indeed they do, as the mathematics of the Riemann tensor taught us.
Black holes are not the only objects that produce stretching and squeezing forces. Stars and planets and moons also produce them. In 1687 Isaac Newton discovered them in his own theory of gravity and used them to explain ocean tides.
The Moon’s gravity pulls more strongly on the near face of the Earth than on the far face, Newton reasoned. And the direction of pull on the Earth’s sides is slightly inward, because it is toward the Moon’s center, a slightly different direction on the Earth’s two sides. This is the usual viewpoint about the Moon’s gravity depicted in Figure 4.8.
Now, the Earth does not feel the average of these gravitational pulls, because it is falling freely along its orbit.[11] (This is like the Endurance’s crew not feeling Gargantua’s gravitational pull when they are in the Endurance, in its parking orbit above the black hole. They only feel centrifugal forces due to the Endurance’s rotation.) What the Earth does feel is the red-arrowed lunar pulls in the left half of Figure 4.8, with their average subtracted away; that is, it feels a stretch toward and away from the Moon, and a squeeze on its lateral sides (right half of Figure 4.8). This is qualitatively the same as around a black hole (Figure 4.7).
These felt forces stretch the ocean away from the Earth’s surface on the faces toward and away from the Moon, producing high tides there. And the felt forces squeeze the oceans toward the Earth’s surface on the Earth’s lateral sides, producing low tides there. As the Earth turns on its axis, one full turn each twenty-four hours, we see two high tides and two low tides. This was Newton’s explanation of ocean tides, aside from a slight complication: The Sun’s tidal gravity also contributes to the tides. Its stretch and squeeze get added to the Moon’s stretch and squeeze.
Because of their role in ocean tides, these gravitational squeezing and stretching forces—the forces the Earth feels—are called tidal forces. To extremely high accuracy, these tidal forces, computed using Newton’s laws of gravity, are the same as we compute using Einstein’s relativistic laws. They must be the same, since the relativistic laws and the Newtonian laws always make the same predictions when gravity is weak and objects move at speeds much slower than light.
In the relativistic description of the Moon’s tides (Figure 4.9), the tidal forces are produced by blue tendex lines that squeeze the Earth’s lateral sides and red tendex lines that stretch toward and away from the Moon. This is just like a black hole’s tendex lines (Figure 4.7). The Moon’s tendex lines are visual embodiments of the Moon’s warping of space and time. It is remarkable that a warping so tiny can produce forces big enough to cause the ocean tides!
On Miller’s planet (Chapter 17) the tidal forces are enormously larger and are key to the huge waves that Cooper and his crew encounter.
We now have three points of view on tidal forces:
• Newton’s viewpoint (Figure 4.8): The Earth does not feel the Moon’s full gravitational pull, but rather the full pull (which varies over the Earth) minus the average pull.
• The tendex viewpoint (Figure 4.9): The Moon’s tendex lines stretch and squeeze the Earth’s oceans; also (Figure 4.7) a black hole’s tendex lines stretch and squeeze the paths of planets and stars around the black hole.
• The straightest-route viewpoint (Figure 4.6): The paths of stars and planets around a black hole are the straightest routes possible in the hole’s warped space and time.
Having three different viewpoints on the same phenomenon can be extremely valuable. Scientists and engineers spend most of their lives trying to solve puzzles. The puzzle may be how to design a spacecraft. Or it may be figuring out how black holes behave. Whatever the puzzle may be, if one viewpoint doesn’t yield progress, another viewpoint may. Peering at the puzzle first from one viewpoint and then from another can often trigger new ideas. This is what Professor Brand does, in Interstellar, when trying to understand and harness gravitational anomalies (Chapters 24 and 25). This is what I’ve spent most of my adult life doing.
5
Black Holes
The black hole Gargantua plays a major role in Interstellar. Let’s look at the basic facts about black holes in this chapter and then focus on Gargantua in the next.
First, a weird claim: Black holes are made from warped space and warped time. Nothing else—no matter whatsoever.
Now some explanation.
Ant on a Trampoline: A Black Hole’s Warped Space
Imagine you’re an ant and you live on a child’s trampoline—a sheet of rubber stretched between tall poles. A heavy rock bends the rubber downward, as shown in Figure 5.1. You’re a blind ant, so you can’t see the poles or the rock or the bent rubber sheet. But you’re a smart ant. The rubber sheet is your entire universe, and you suspect it’s warped. To determine its shape, you walk around a circle in the upper region measuring its circumference, and then walk through the center from one side of the circle to the other, measuring its diameter. If your universe were flat, then the circumference would be π = 3.14159… times the diameter. But the circumference, you discover, is far smaller than the diameter. Your universe, you conclude, is highly warped!
11
In 1907, Einstein realized that if he were to fall, off the roof of his house for example, then as he fell he would feel no gravity. He called this the “happiest thought of my life,” because it got him started on his quest to understand gravity, the quest that led to his concepts of warped time and space and the laws that govern the warping.