The Science of Interstellar
Page 11
Fig. 13.2. Freeman Dyson’s bomb-powered propulsion system. [From Dyson (1968).]
Thermonuclear bombs (“hydrogen bombs”) are detonated just behind a hemispherical shock absorber that is 20 kilometers in diameter (Figure 13.2). The bomb debris pushes the ship forward, achieving, in Dyson’s most optimistic estimate, a speed one-thirtieth that of light. A less crude design could do somewhat better. In 1968 Dyson estimated that such a propulsion system would not be practical any sooner than the late twenty-second century, 150 years from now. I think that’s overly optimistic.
Laser Beam and Light Sail
In 1962 Robert Forward, another physicist whom I respect, wrote a short article in a popular magazine about a spacecraft with a sail, pushed by a distant, focused laser beam (Forward 1962). In a 1984 technical article, he made this concept more sophisticated and precise (Figure 13.3.)
An array of solar-powered lasers in space or on the Moon generates a laser beam with 7.2 terawatts of power (about twice the total power consumption of the United States in 2014!). This beam is focused, by a Fresnel lens 1000 kilometers in diameter. It is focused onto a distant sail, 100 kilometers in diameter and weighing about 1000 metric tons, that is attached to a less massive spacecraft. (The beam direction must be accurate to about a millionth of an arcsecond.) The beam’s light pressure pushes the sail and spacecraft up to about a fifth the speed of light halfway through a forty-year trip to Proxima Centauri. A modification of this scheme then slows the ship down during the second half of the trip, so it arrives at its destination with a speed low enough to rendezvous with a planet. (Can you figure out how the slow down is achieved?)
Fig. 13.3. Robert Forward’s laser beam and light sail propulsion system. [From Forward (1984).]
Forward, like Dyson, imagined his scheme practical in the twenty-second century. When I look at the technical challenges, I think longer.
Gravitational Slingshots in a Black-Hole Binary
My third example is my own wild—very wild!—variant of an idea due to Dyson (1963).
Suppose you want to fly across much of the universe (not just interstellar travel, but intergalactic travel) at near light speed in a few years of your own life. You can do so with the aid of two black holes that are orbiting each other, a black-hole binary. They must be in a highly elliptical orbit and must be large enough that their tidal forces do not destroy your ship.
Using chemical or nuclear fuel, you navigate your ship into an orbit that comes close to one of the black holes: a so-called zoom-whirl orbit (Figure 13.4). Your ship zooms close to the hole, whirls around it a few times, and then, when the hole is traveling nearly directly toward its companion, the ship zooms out, crosses over to the companion hole, and slides into a whirl around it. If the two holes are still headed toward each other, the whirl is brief: you zoom back toward the first hole. If the holes are no longer headed toward each other, the whirl is much longer; you must park yourself in orbit around the second hole until the holes are again headed toward each other, and then launch back toward the first hole. In this way, always traveling between holes only when the holes are approaching each other, your ship gets boosted to higher and higher speeds, approaching as close as you wish to the speed of light if the binary is sufficiently elliptical.
It is a remarkable fact that you only need a small amount of rocket fuel to control how long you linger around each hole. The key is to navigate onto the hole’s critical orbit, and there perform your controlled whirl. I discuss the critical orbit in Chapter 27. For now, suffice it to say that this is a highly unstable orbit. It is rather like riding a motorcycle around a very smooth volcano rim. If you balance delicately, you can stay on the rim as long as you want. When you wish to leave, a slight turn of the bike’s front wheel will send you careening off the rim. When you want to leave the critical orbit, a slight rocket thrust will enable centrifugal forces to take over and send your ship careening toward the other black hole.
Once you are as close to the speed of light as you wish, you can launch yourself off a critical orbit toward your target galaxy in the distant universe (Figure 13.5).
Fig. 13.4. Zoom-whirl orbit brings a spacecraft up to near light speed.
Fig. 13.5. Launching off a critical orbit toward a distant galaxy.
The trip may be long; as much as 10 billion light-years’ distance. But when you move at near light speed, your time flows far more slowly than on Earth. If you are close enough to light speed, you can make it to your target in a few years or less, as measured by you—slowing down with the aid of a highly elliptical black-hole binary at your target, if you can find one! See Figure 13.6.
You can return home by the same method. But your homecoming may not be pleasant. Billions of years will have passed at home, while you have aged only a few years. Imagine what you find.
These types of slingshots could provide a means for spreading a civilization across the great reaches of intergalactic space. The principal obstacle (perhaps insurmountable!) is finding, or making, the needed black hole binaries. The launch binary might not be a problem if you are a sufficiently advanced civilization, but the slow-down binary is another matter.
What happens to you if there is no slow-down binary, or there is one, but your aim is bad and you miss it? This is a tricky question because of the expansion of the universe. Think about it.
Fig. 13.6. Slowing down by slingshots in a target black-hole binary.
As exciting as these three far-future propulsion systems may seem, they truly are far future. Using twenty-first-century technology, we are stuck with thousands of years to reach other solar systems. The only hope (an exceedingly faint hope) for faster interstellar travel, in the event of an earthly disaster, is a wormhole like that in Interstellar, or some other extreme form of spacetime warp.
* * *
26 The kinetic energy is Mv2/2, where M is the helium atom’s mass and v is its speed. Equate this to the energy released, 0.0064 Mc2, where c is the speed of light. (I used Einstein’s famous result that when you convert mass into energy, the energy you get out is the mass multiplied by the square of the speed of light.) The result from equating these two formulas is v2 = 2 × 0.0064c2, which means v is close to c/10.
IV
THE WORMHOLE
14
Wormholes
How Wormholes Got Their Name
My mentor, John Wheeler, gave astrophysical wormholes their name. He based it on wormholes in apples (Figure 14.1). For an ant walking on an apple, the apple’s surface is the entire universe. If the apple is threaded by a wormhole, the ant has two ways to get from the top to the bottom: around the outside (through the ant’s universe) or down the wormhole. The wormhole route is shorter; it’s a shortcut from one side of the ant’s universe to the other.
Fig. 14.1. An ant explores a wormhole-endowed apple.
The apple’s delicious interior, through which the wormhole passes, is not part of the ant’s universe. It is a three-dimensional bulk or hyperspace (Chapter 4). The wormhole’s wall can be thought of as part of the ant’s universe. It has the same dimensionality as the universe (two dimensions) and it joins onto the universe (the apple’s surface) at the wormhole’s entrance. From another viewpoint, the wormhole’s wall is not part of the ant’s universe; it is just a shortcut by which the ant can travel across the bulk, from one point in its universe to another.
Flamm’s Wormhole
In 1916, just one year after Einstein formulated his general relativistic laws of physics, Ludwig Flamm in Vienna discovered a solution of Einstein’s equations that describes a wormhole (though he did not call it that). We now know that Einstein’s equations allow many kinds of wormholes (wormholes with many different shapes and behaviors), but Flamm’s is the only one that is precisely spherical and contains no gravitating matter. When we take an equatorial slice through Flamm’s
wormhole, so it and our universe (our brane) have just two dimensions rather than three, and when we then view our universe and the wormhole from the bulk, they look like the left part of Figure 14.2.
With one of our universe’s dimensions lost from the picture, you must think of yourself as a two-dimensional creature confined to move on the bent sheet or on the wormhole’s two-dimensional wall. There are two routes for travel from location A in our universe to location B: the short route (dashed blue curve) down the wormhole’s wall, or the long route (dashed red curve) along the bent sheet, our universe.
Of course, our universe is really three dimensional. The concentric circles in the left part of Figure 14.2 are really the nested green spheres shown to the right. As you enter the wormhole along the blue path from location A, you pass through spheres that get smaller and smaller. Then the spheres, though nested inside each other, cease changing circumference. And then, as you exit the wormhole toward location B, the spheres get larger and larger.
For nineteen years, physicists paid little attention to Flamm’s outrageous solution of Einstein’s equations, his wormhole. Then in 1935 Einstein himself and fellow physicist Nathan Rosen, unaware of Flamm’s work, rediscovered Flamm’s solution, explored its properties, and speculated about its significance in the real world. Other physicists, also unaware of Flamm’s work, began to call his wormhole the “Einstein-Rosen bridge.”
Fig. 14.2. Flamm’s wormhole.
Wormhole Collapse
It is often difficult to extract, from the mathematics of Einstein’s equations, a full understanding of their predictions. Flamm’s wormhole is a remarkable example. From 1916 until 1962, nearly a half century, physicists thought that the wormhole is static, forever unchanging. Then John Wheeler and his student Robert Fuller discovered otherwise. Looking much more closely at the mathematics, they discovered that the wormhole is born, expands, contracts, and dies, as shown in Figure 14.3.
Initially, in picture (a), our universe has two singularities. As time passes, the singularities reach out to each other through the bulk and meet to create the wormhole (b). The wormhole expands in circumference, (c) and (d), then shrinks and pinches off (e), leaving behind the two singularities (f). The birth, expansion, shrinkage, and pinch-off happen so quickly that nothing, not even light, has time to travel through the wormhole from one side to the other. Anything or anyone that attempts the trip will get destroyed in the pinch-off!
Fig. 14.3. Dynamics of Flamm’s wormhole (the Einstein-Rosen bridge). [Drawing by Matt Zimet based on a sketch by me; from my book Black Holes & Time Warps: Einstein’s Outrageous Legacy.]
This prediction is inescapable. If the universe were ever, somehow, to develop a spherical wormhole that contains no gravitating matter, this is how the wormhole would behave. Einstein’s relativistic laws dictate it.
Wheeler was not dismayed by this conclusion. On the contrary, he was pleased. He regarded singularities (places where space and time are infinitely warped) as a “crisis” for the laws of physics. And crises are wonderful tutors. By probing wisely, we can get great insights into the physical laws. To this I return in Chapter 26.
Contact
Fast-forward a quarter century, to May 1985: a phone call from Carl Sagan asking me to critique the relativistic science in his forthcoming novel Contact. I happily agreed. We were close friends, I thought it would be fun, and, besides, I still owed him one for introducing me to Lynda Obst.
Carl sent me his manuscript. I read it and I loved it. But there was one problem. He sent his heroine, Dr. Eleanor Arroway, through a black hole from our solar system to the star Vega. But I knew that a black-hole interior cannot be a route from here to Vega or to anywhere else in our universe. After plunging through the black hole’s horizon, Dr. Arroway would get killed by its singularity. To reach Vega fast, she needed a wormhole, not a black hole. But a wormhole that does not pinch off. A traversable wormhole.
So I asked myself, What do I have to do to Flamm’s wormhole to save it from pinching off; to hold it open, so it can be traversed? A simple thought experiment gave me the answer.
Suppose you have a wormhole that is spherical like Flamm’s, but unlike Flamm’s it does not pinch off. Send a light beam into the wormhole, radially. Since all the beam’s light rays travel radially, the beam must have the shape shown in Figure 14.4. It is converging (its cross-sectional area is decreasing) as it enters the wormhole, and it is diverging (its area is increasing) as it leaves the wormhole. The wormhole has bent the light rays outward, as would a diverging lens.
Fig. 14.4. A radial light beam traveling through a spherical, traversable wormhole. Left: As seen from the bulk with one space dimension removed. Right: As seen in our universe. [Adapted from a drawing by Matt Zimet based on a sketch by me; from my book Black Holes & Time Warps: Einstein’s Outrageous Legacy.]
Now, gravitating bodies such as the Sun or a black hole bend rays inward (Figure 14.5). They can’t bend rays outward. To bend light rays outward, a body must have negative mass (or equivalently, negative energy; recall Einstein’s equivalence of mass and energy). From this fundamental fact, I concluded that any traversable, spherical wormhole must be threaded by some sort of material that has negative energy. At least the material’s energy must be negative as seen by the light beam, or by anything or anyone else that travels through the wormhole at nearly the speed of light.27 I call such material “exotic matter.” (I later learned that, according to Einstein’s relativistic laws, any wormhole, spherical or not, is traversable only if it is threaded by exotic matter. This follows from a theorem proved in 1975 by Dennis Gannon at the University of California at Davis. Being somewhat illiterate, I was unaware of Gannon’s theorem.)
Now, it is an amazing fact that exotic matter can exist, thanks to weirdnesses in the laws of quantum physics. Exotic matter has even been made in physicists’ laboratories, in tiny amounts, between two closely spaced electrically conducting plates. This is called the Casimir effect. However, it was very unclear to me in 1985 whether a wormhole can contain enough exotic matter to hold it open. So I did two things.
Fig. 14.5. The Sun or a black hole bends a beam of light inward.
First, I wrote a letter to my friend Carl suggesting that he send Eleanor Arroway to Vega through a wormhole rather than a black hole, and I enclosed a copy of the calculations by which I had shown that the wormhole must be threaded by exotic matter. Carl embraced my suggestion (and wrote about my equations in the acknowledgment of his novel). And that is how wormholes entered modern science fiction—novels, films, and television.
Second, with two of my students, Mark Morris and Ulvi Yurtsever, I published two technical articles about traversable wormholes. In our articles, we challenged our physicist colleagues to figure out whether the combined quantum laws and relativistic laws permit a very advanced civilization to collect enough exotic matter inside a wormhole to hold it open. This triggered a lot of research by a lot of physicists; but today, nearly thirty years later, the answer is still unknown. The preponderance of the evidence suggests that the answer may be NO, so traversable wormholes are impossible. But we are still far from a final answer. For details, check out Time Travel and Warp Drives by my physicist colleagues Allen Everett and Thomas Roman (Everett and Roman 2012).
What Does a Traversable Wormhole Look Like?
What does a traversable wormhole look like to people like us who live in our universe? I can’t answer definitively. If a wormhole can be held open, the precise details of how remain a mystery, so the precise details of the wormhole’s shape are unknown. For black holes, by contrast, Roy Kerr has given us the precise details, so I can make the firm predictions described in Chapter 8.
So for wormholes, I can make only an educated guess, but one in which I have considerable confidence. Hence the symbol on this section’s header.
Mouth in Calif
ornia Desert
Mouth in Dublin
Fig. 14.6. The images seen through a wormhole’s two mouths. [Left photo by Catherine MacBride; right photo by Mark Interrante.]
Imagine we have a wormhole here on Earth, stretching through the bulk from Grafton Street in Dublin, Ireland, to the desert in Southern California. The distance through the wormhole might be only a few meters.
The entrances to the wormhole are called “mouths.” You are sitting in a sidewalk cafe alongside the Dublin mouth. I am standing in the desert beside the California mouth. Both mouths look rather like crystal balls. When I look into my California mouth, I see a distorted image of Grafton Street, Dublin (Figure 14.6). That image is brought to me by light that travels through the wormhole from Dublin to California, rather like light traveling through an optical fiber. When you look into your Dublin mouth, you see a distorted image of Joshua trees (cactus trees) in the California desert.
Can Wormholes Exist Naturally, as Astrophysical Objects?
In Interstellar, Cooper says, “A wormhole isn’t a naturally occurring phenomenon.” I agree with him completely! If traversable wormholes are allowed by the laws of physics, I think it extremely unlikely they can exist naturally, in the real universe. I must confess, though, that this is little more than a speculation, not even an educated guess. Maybe a highly educated speculation, but speculation nonetheless, so I labeled this section .