by Thorne, Kip
The out-back dimension plays a major role in Interstellar, though the Professor and others don’t use the phrase “out-back,” but instead just refer to “the fifth dimension.” Out-back is central to the next two chapters, and to Chapters 25, 29, and 30.
* * *
34 But the relativistic laws do offer the possibility of backward time travel by a circuitous route: going outward in space and returning before we left. To this I return in Chapter 30.
35 See Chapter 3 for a brief description of this quest.
22
Bulk Beings
2D Brane and 3D Bulk
In 1844 Edwin Abbott wrote a satirical novella titled Flatland: A Romance of Many Dimensions (Figure 22.1).36 Though its satire on Victorian culture seems quaint today and its attitude toward women outrageous, the novella’s venue is highly relevant to Interstellar. I recommend it to you.
Fig. 22.1. The cover of the first edition of Flatland.
It describes the adventures of a square-shaped being who lives in a two-dimensional universe called Flatland. The square visits a one-dimensional universe called Lineland, a zero-dimensional universe called Pointland, and most amazing of all to him, a three-dimensional universe called Spaceland. And, while living in Flatland, he is visited by a spherical being from Spaceland.
In my first meeting with Christopher Nolan, we were both delighted to find the other had read Abbott’s novella and loved it.
In the spirit of Abbott’s novella, imagine that you are a two-dimensional being, like the square, who lives in a two-dimensional universe like Flatland. Your universe could be a tabletop, or a flat sheet of paper, or a rubber membrane. In the spirit of modern physics, I refer to it as a two-dimensional (2D) brane.
Being well educated, you suppose there is a 3D bulk, in which your brane is embedded, but you’re not certain. Imagine your excitement when one day you are visited by a sphere from the 3D bulk. A “bulk being,” you might call him.
At first you don’t realize it’s a bulk being, but after much observation and thought, you see no other explanation. What you observe is this: Suddenly, with no warning and no apparent source, a blue point appears in your brane (top left of Figure 22.2). It expands to become a blue circle that grows to a maximum diameter (middle left), and then gradually shrinks to a point (bottom left) and disappears completely.
Fig. 22.2. A three-dimensional sphere passes through a two-dimensional brane.
You believe in conservation of matter. No object can ever be created from nothing, yet this object was. The only explanation you can find is shown in the right half of Figure 22.2. A three-dimensional bulk being—a sphere—passed through your brane. As it passed through, you saw in your brane its changing two-dimensional cross section. The cross section began with a point at the sphere’s south pole (top right). It expanded to a maximal circle, the sphere’s equatorial plane (middle right). It then shrank to a point, the sphere’s north pole, and disappeared (bottom right).
Imagine what would happen if a 3D human being, living in the 3D bulk, passed through your 2D brane. What would you see?
Bulk Beings from the Fifth Dimension, Passing Through Our 3D Brane
Suppose that our universe, with its three space and one time dimensions, really does live in a five-dimensional bulk (four space and one time). And suppose there are “hyperspherical beings” who live in the bulk. Such a being would have a center and a surface. Its surface would consist of all points, in four space dimensions, that are some fixed distance from the center, for example, 30 centimeters. The bulk being’s surface would have three dimensions and its interior would have four.
Suppose that this hyperspherical bulk being, traveling in the bulk’s out direction or back direction, were to pass through our brane. What would we see? The obvious guess is correct. We would see spherical cross sections of the hypersphere (Figure 22.3).
Fig. 22.3. A hyperspherical bulk being passing through our brane, as seen in our brane.
A point would appear from nothing (1). It would expand to become a three-dimensional sphere (2). The sphere would expand to a maximum diameter (3), then contract (4), shrink to a point (5), and disappear.
Can you guess what we would see if a four-dimensional human being living in the bulk were to pass through our brane? To speculate about this, you need to imagine what a four-dimensional human being—with two legs, a torso, two arms, and a head—must “look like” in the bulk, with its four space dimensions. And what its cross sections must look like.
The Nature of Bulk Beings, and Their Gravity
&
If there are bulk beings, what are they made of? Certainly not atom-based matter like us. Atoms have three space dimensions. They can only exist in three space dimensions, not four. And this is true of subatomic particles as well. And it is true also of electric fields and magnetic fields (Chapter 2) and the forces that hold atomic nuclei together.
Some of the world’s most brilliant physicists have struggled to understand how matter and fields and forces behave if our universe really is a brane in a higher-dimensional bulk. Those struggles have pointed rather firmly to the conclusion that all the particles and all the forces and all the fields known to humans are confined to our brane, with one exception: gravity, and the warping of spacetime associated with gravity.
There might be other kinds of matter and fields and forces that have four space dimensions and reside in the bulk. But if there are, we are ignorant of their nature. We can speculate. Physicists do speculate. But we have no observational or experimental evidence to guide our speculations. In Interstellar, on Professor Brand’s blackboard, we see him speculating (Chapter 25).
It’s a reasonable, half-educated guess that, if bulk forces and fields and particles do exist, we will never be able to feel them or see them. When a bulk being passes through our brane, we will not see the stuff of which the being is made. The being’s cross sections will be transparent.
On the other hand, we will feel and see the being’s gravity and its warping of space and time. For example, if a hyperspherical bulk being appears in my stomach and has a strong enough gravitational pull, my stomach may begin to cramp as my muscles tighten, trying to resist getting sucked to the center of the being’s spherical cross section.
If the bulk being’s cross section appears and then disappears in front of a checkerboard of paint swatches, its space warp might lens the swatches, bending the image I see, as in the top half of Figure 22.4.
Fig. 22.4. A bulk being, passing through our brane, bends and swirls our view of a paint-swatch wall.
And if the bulk being is spinning, it might drag space into a whirling motion that I can feel and see, as in the bottom of Figure 22.4.
Interstellar’s Bulk Beings
All the characters in Interstellar are convinced that bulk beings exist, though they use that name only rarely. Usually, the characters call the bulk beings “They.” A reverential They. Early in the movie, Amelia Brand says to Cooper, “And whoever They are, They appear to be looking out for us. That wormhole lets us travel to other stars. It came along right as we needed it.”
One of Christopher Nolan’s clever and intriguing ideas is to imagine that They are actually our descendants: humans who, in the far future, evolve to acquire an additional space dimension and live in the bulk. Late in the movie, Cooper says to TARS, “Don’t you get it yet, TARS? They aren’t beings. They’re us, trying to help, just like I tried to help Murph.” TARS responds, “People didn’t build this tesseract” (in which Cooper is riding; Chapter 29). “Not yet,” Cooper says, “but one day. Not you and me but people, people who’ve evolved beyond the four dimensions we know.”
Cooper, Brand, and the crew of the Endurance never actually feel or see our bulk descendants’ gravity or their space warps and whirls. That, if it ever occurs, is left for a sequel to Interstellar.
But older Cooper himself, riding through the bulk in the closing tesseract of Chapter 30, reaches out to the Endurance’s crew and his younger self, reaches out through the bulk, reaches out gravitationally. Brand feels and sees his presence, and thinks he is They.
* * *
36 Widely available on the web. See, for example, the end of the article “Flatland” on Wikipedia.
23
Confining Gravity
The Trouble with Gravity in Five Dimensions
If the bulk does exist, then its space must be warped. If it were not warped, then gravity would obey an inverse cube law instead of inverse square, our Sun could not hang onto its planets, and the solar system would fly apart.
OK. I’ll slow down and explain this more carefully.
Recall (Chapter 2) that the Sun’s gravitational force lines, like those of the Earth and any other spherical body, point radially toward its center and pull objects along themselves toward the Sun (Figure 23.1). The strength of the Sun’s gravitational pull is proportional to the density of the force lines (the number of lines passing through a fixed area). And since the transverse areas (spheres) through which the lines pass have two dimensions, the lines’ density goes down with increasing radius r as 1/r2, and so does gravity’s strength. This is Newton’s inverse square law for gravity.
Fig. 23.1. The gravitational force lines around the Sun.
String theory insists that gravity in the bulk is also described by force lines. If space in the bulk is not warped, then the Sun’s gravitational force lines will spread radially outward into the bulk (Figure 23.2). Because of the bulk’s extra dimension (just one in Interstellar), there are three transverse dimensions into which gravity can spread instead of just two. Therefore, if the bulk exists and is not warped, then the density of force lines and thence gravity’s strength should decrease as 1/r2 when we move away from the Sun, rather than as 1/r3. The sun’s pull on the Earth would be two hundred times weaker, and on Saturn 2000 times weaker. With gravity weakening so rapidly, the Sun couldn’t hold onto its planets; they would fly away into interstellar space.
Fig. 23.2. Gravitational force lines spread radially into the bulk, if the bulk is not warped. The dotted circles are solely to guide your eyes. [Patterned on a figure in Lisa Randall’s Warped Passages (Randall 2006).]
But they don’t fly away. And their measured motions reveal unequivocally that the Sun’s gravity weakens as the inverse square of the distance. The conclusion is inescapable: if there is a bulk, it must be warped in some manner that prevents gravity from spreading into the fifth dimension, the out-back dimension.
Is Out-Back Curled Up?
If the bulk’s out-back dimension were curled up into a tight roll, then gravity could not spread far into the bulk, and the inverse square law would be restored.
Fig. 23.3. If the out-back dimension (yellow) is curled up, then outside the blue circle a particle’s gravitational force lines (red) are parallel to our brane.
Figure 23.3 depicts this for the gravity of a tiny particle that resides at the center of the blue disk. In this picture, two space dimensions are suppressed, so we see only one of our brane’s dimensions (call it north-south) along with the bulk’s out-back dimension. Near the particle, inside the blue disk, the force lines spread in the out-back dimension as well as north-south, so (with the missing dimensions restored) gravity’s strength obeys an inverse cube law. However, outside the blue disk the curl-up makes the force lines lie parallel to our brane. They spread no further into out-back, and Newton’s inverse square law is restored.
Physicists who struggle to understand quantum gravity think this is the fate of all the extra dimensions except possibly one or two: they are curled up on microscopic scales, preventing gravity from spreading too fast. In Interstellar, Christopher Nolan ignores these curled-up dimensions and focuses on just one bulk dimension that’s not curled up. This becomes his out-back, fifth dimension.
Why should out-back not be curled up? For Chris the answer is simple: A curled-up bulk has very little volume—nowhere near enough volume to be an arena for interesting science fiction. For Cooper to travel into the bulk riding in the tesseract, as he does in the movie, the tesseract needs far more volume than a curled-up dimension would provide.
Out-Back: The Anti-DeSitter Warp
In 1999, Lisa Randall at Princeton University and MIT and Raman Sundrum at Boston University (Figure 23.4) conceived another way to stop gravitational force lines from spreading into the bulk: the bulk could suffer what is called “Anti-deSitter warping.” This warping might be produced by what are called “quantum fluctuations of bulk fields”—but that’s irrelevant to my story so I do not explain it here.37 Suffice it to say that this mechanism to produce the warping is very natural. By contrast, the Anti-deSitter (AdS) warping itself does not look natural at all. It looks downright weird.
Fig. 23.4. Lisa Randall (1962– , right) and Raman Sundrum (1964– , left).
Suppose you’re a microbe, and you live in a face of a microscopic tesseract (Chapter 29). You travel, in your tesseract, out from our brane; perpendicularly out (straight up in Figure 23.5). And suppose you have a microbial pal, who also travels perpendicularly out from our brane. When you and your pal depart our brane, you are 1 kilometer apart (1000 meters; about 0.6 miles). Although you both travel precisely outward, perpendicular to our brane, your separation plummets precipitously due to AdS warping. When you have traveled a tenth of a millimeter (the thickness of a human hair), your separation has decreased tenfold: from 1 kilometer to 100 meters. The next 0.1 millimeter of travel reduces your separation by another factor of ten, to 10 meters; the next 0.1 millimeter reduces it to 1 meter; and so forth.
Fig. 23.5. AdS warping of the bulk.
This shrinkage of distances parallel to our brane is hard to imagine. I don’t know a good way to draw it, no better way than Figure 23.5. But it has marvelous consequences.
It has the potential to explain a mystery called the “hierarchy problem in the laws of physics”—but that’s outside the scope of this book.38 And because of the shrinkage, there is very little volume, above or below our brane, into which gravitational force lines can spread (Figure 23.6). Closer to our brane than 0.1 millimeter, the force lines spread into three transverse dimensions with impunity, so gravity obeys an inverse cube law. Farther than 0.1 millimeter, the force lines are bent parallel to our brane and so spread into just two transverse dimensions, whence gravity obeys the observed inverse square law.39
Fig. 23.6. If the bulk experiences AdS warping, then gravitational force lines bend parallel to our brane, because far from the brane there is very little volume in which to spread. [Patterned on a figure in Lisa Randall’s Warped Passages (Randall 2006).]
The AdS Sandwich: Plenty of Room in the Bulk
Sadly, the precipitous shrinkage of distances parallel to our brane, as you move outward, makes the bulk’s volume above and below our brane too small for Cooper and his tesseract, and too small for any other human activity in the bulk. I recognized this problem way back in 2006, when Interstellar was in its infancy, and I quickly conceived a solution for my science interpretation of the movie: Confine the AdS warping to a thin layer around our brane, a “sandwich.” Do so by placing two other branes, confining branes, alongside ours (Figure 23.7). In the sandwich between these branes, the bulk suffers AdS warping. Outside the sandwich, the bulk is totally unwarped. So there is all the volume any sci-fi writer could want, outside the sandwich, for bulk-based adventures.
How thick must the sandwich be? Thick enough to bend gravitational force lines—emerging from our brane—parallel to our brane and hold them there, so we in our brane see gravity obey an inverse square law. But no thicker, because added thickness means greater total transverse shrinkage, which may cause trouble for bulk-based adventures. (Suppose our whole universe, a
s seen from outside the AdS layer, were shrunk to the size of a pin head!) The required thickness turns out to be about 3 centimeters (roughly an inch), so as you travel from our brane to a confining brane, distances parallel to our brane shrink by fifteen powers of ten: a thousand trillion.
Fig. 23.7. The AdS sandwich between two confining branes. The AdS layer between the branes is lightly grayed.
In my interpretation of Interstellar, Gargantua is in the far reaches of the observable universe: roughly 10 billion light-years from Earth. Cooper, in the tesseract, rises through the AdS layer, from Gargantua’s core into the bulk. There the distance to Earth is 10 billion light-years divided by a thousand trillion, which is about the same as the distance between the Sun and the Earth, one “astronomical unit” (1 AU; Figure 23.7). Cooper then travels that 1 AU distance through the bulk, parallel to our brane, to reach the Earth and visit Murph; see Figure 29.4.
DANGER: The Sandwich Is Unstable
In 2006, I used Einstein’s relativistic laws to work out a mathematical description of the AdS layer and its confining branes. Because I had never before worked with relativity in five dimensions, I asked Lisa Randall to critique my analysis. Lisa browsed it quickly, and then told me some good news and some bad news.