Consider a large and deep swimming pool with a very large drain in its center. Water flows down this drain, producing a whirlpool at the surface of the kind we commonly see in bathtubs. To keep the level of the pool constant, let us assume that the water lost down the drain is continuously being replaced through inlets at the sides of the pool. Imagine now three Stanford professors floating in the pool. Professor Schiff is floating on an air mattress between the whirlpool and the edge of the pool, with his feet closest to the drain. Professor Fairbank is floating on a similar air mattress but is straddling the whirlpool, well above the drain of the deep pool. Professor Cannon is treading water. For simplicity’s sake, let us also assume that each professor is anchored to the bottom of the pool by a tether attached to his waist. This is to prevent them from circling the drain, which would complicate the effect we are looking for. With this arrangement, the behavior of these professors is very similar to that of three gyroscopes in a spacetime being dragged by a rotating body. As with all the analogies for relativistic effects that we have used in this book, one must be careful not to push the analogy too far. Gyroscopes in spacetime are not the same as air mattresses in water, but if the analogy helps us remember the qualitative effects, it is a useful one.
First consider Professor Schiff. Because the water closer to the whirlpool moves around more quickly than the water farther away, the foot of his air mattress is dragged more quickly than the head, and so while the whirlpool rotates, say, counterclockwise as seen from above, Schiff’s air mattress rotates or precesses in a clockwise direction. This is precisely the behavior of the axis of a gyroscope on the equatorial plane in a dragged spacetime, with its axis pointing outward (Figure 4.3). Contrast this behavior with that of Professor Fairbank, whose air mattress straddles the whirlpool. The head and foot of his mattress are also pulled by the water, but because they are on opposite sides of the whirlpool the mattress is pulled in the same sense as the whirlpool, in other words counterclockwise. This is just what happens to a gyroscope on the rotation axis of the dragged spacetime, with its own axis perpendicular to the rotation axis. Finally, we see that Professor Cannon, who is treading water, doesn’t do much of anything. The direction of his body remains vertical, no matter where he goes in the pool. The same is true for a gyroscope whose axis is parallel to the rotation axis of the central body; the dragging of spacetime has no effect on it.
Of course, there is another key difference between the air mattress precession and the gyroscope precession due to the dragging of inertial frames: size. The predicted precession for a gyroscope on the equator of the Earth is only one-tenth of an arcsecond per year. Unlike the geodetic precession, the dragging of inertial frames does not depend on whether or not the gyroscope is moving through spacetime (the air mattresses precessed even though they were stationary in the pool), and so there is little difference in this case between the precession of a gyroscope on the Earth and that of a gyroscope in orbit. For a low Earth orbit it is between 0.1 and 0.05 arcseconds per year, depending on the tilt of the orbit relative to the Earth’s equatorial plane and on the initial direction of the gyroscope axis relative to the Earth’s rotation axis.
What makes this effect so interesting and important is that while the other effects that we have described in this book, including geodetic precession, have to do with such concepts as gravitational fields, curved spacetime, and nonlinear gravity, this effect tells us something about the inertial properties of spacetime. If you ask yourself “Am I rotating?” and you wish an answer with more accuracy than you can get simply by seeing if you are getting dizzy, you usually turn to a gyroscope, for the axis of a gyroscope is assumed to be non-rotating relative to inertial space. If you were to build a laboratory whose walls were constructed to be lined up with the axes of three gyroscopes arranged to be perpendicular to each other, you would conclude that your laboratory was truly inertial (and if the laboratory were in free fall, that would be even better). However, if your laboratory happened to be situated outside a rotating body, the gyroscopes would rotate relative to the distant stars because of the dragging effect just described. Therefore, your laboratory can be non-rotating relative to gyroscopes, yet still rotate relative to the stars. Relativists make a careful distinction between a laboratory that is locally non-rotating, that is tied to gyroscopes, and one that might rotate relative to distant stars. In this way, general relativity rejects the idea of absolute rotation or absolute non-rotation, just as special relativity rejected the idea of an absolute state of rest.
To understand this more clearly, contrast it with Newtonian theory. True, Newtonian theory proposed that all inertial frames are equivalent, regardless of their state of motion, but it still had to allow an absolute concept when it came to rotation. A simple example, known as Newton’s bucket, will illustrate this idea (see Figure 4.4). Fill a bucket with water, place it on a turntable, and start the turntable spinning. As the bucket starts to spin, the water doesn’t do much of anything at first, but eventually the friction between the water and the walls of the bucket causes the water to spin along with the bucket. As a consequence of this, the surface of the water becomes concave and the water begins to climb up the sides of the bucket; a depression forms in the center. Quite naturally, we attribute this behavior to centrifugal forces pushing the water away from the rotation axis. When the turntable is stopped, eventually the water slows down and returns to its initial state with a flat surface.
Figure 4.4 Newton’s bucket. Left: The bucket is not rotating and the surface of the water is flat. Right: The bucket is rotating and the surface of the water is concave. Would the surface be concave if the bucket were “not rotating” and the universe rotated around it?
As mundane and commonplace as this simple observation is, it has led to some of the most intriguing and vexing philosophical questions. Newton himself wrestled with them. One question is, how does the water know that it is rotating and should have a concave surface instead of a flat one? If we truly abhor the concept of absolute space, as relativity in either its Newtonian or Einsteinian forms teaches us, we cannot answer that the water knows that it is rotating relative to absolute, non-rotating space. With respect to what then? The best we can do is to answer that somehow the water knows that it is rotating relative to the distant stars and galaxies. As reasonable as this sounds, it does beg two questions. Suppose we performed this bucket experiment in an otherwise completely empty universe. With nothing to which to refer its state of motion, would the water know what to do as the turntable spun? Would its surface become concave or stay flat? That is the first question, to which there is no satisfactory answer based on physics. Up to a point, of course, this question is irrelevant, because we don’t live in an empty universe anyway.
The second question is somewhat more meaningful: Suppose we leave the bucket at rest, and let the entire universe rotate around it with the same rotation rate as the bucket had in the previous experiment, but in the opposite sense. Would the water become concave as before? If only the rotation of the bucket relative to the distant matter in the universe is important, then the two experiments should give the same concave shape for the water’s surface. In other words, it should not matter whether we say that the universe is non-rotating and the bucket is rotating, or that the bucket is non-rotating and the universe is rotating. Only the rotation of one relative to the other is relevant.
Unfortunately, Newtonian gravity predicted that the rotating universe would have no effect on the bucket, and therefore you had to invoke an absolute space to understand rotation. But in general relativity, the dragging of inertial frames provides the way out of this absolutism. As early as 1923, Eddington suggested as much in his beautiful textbook on general relativity. However, it wasn’t until the mid 1960s that theorists could show that the dragging effect provides a good accounting of how rotation is indeed relative. The demonstration consisted of a simple model calculation of the following situation: Imagine a spherical shell of matter, like a balloon, that i
s rotating about some axis (for the purposes of this discussion we can ignore the flattening of the balloon caused by centrifugal forces). At the center of the shell is a gyroscope with its spin axis perpendicular to the axis of rotation of the balloon. According to Newtonian gravitation, the interior of the balloon is absolutely free of gravitational fields. The gyroscope feels no force whatsoever. To a first approximation, the same is true in general relativity, except for the dragging of inertial frames effect, which produces forces in the interior of a rotating shell just as it would in the exterior. The effect of these forces is to cause the gyroscope to precess in the same direction as the rotation of the shell, but as you might imagine from our previous discussion, for a shell of planetary dimensions, say of the radius of a typical planet and containing the mass of a typical planet, the rate of precession is very small, much smaller than the rate of rotation of the shell.
But now imagine increasing the mass of the shell and increasing its radius (keeping its rate of rotation the same), and consider the limit in which the mass tends toward the mass of the visible universe and the radius tends toward the radius of the visible universe. The remarkable result is that as you increase these values, the rate of precession of the gyroscope in the center grows and, in the limit, tends toward the rate of rotation of the shell. In other words, inside a rotating universe the axes of gyroscopes rotate in step with the rotation of the universe; their axes are tied to the directions of distant bodies in that universe. Therefore, a laboratory tied to the gyroscopes, which we would define to be non-rotating, would indeed be non-rotating relative to the distant galaxies. Imagine now that we placed a bucket inside the shell instead of a gyroscope, and imagine that we kept the bucket fixed, or non-rotating, with the shell rotating around it. As you expand the size of the rotating shell, residual frame dragging forces would cause the water to climb the sides of the bucket. Therefore, an observer in this scenario would see exactly the same physical phenomenon as would an observer looking at a rotating bucket inside a non-rotating universe. The existence of the dragging of inertial frames then guarantees that rotation must be defined relative to distant matter, not relative to some absolute space. This is what makes the detection of this effect so vital.
In addition to resolving this conceptual problem, the relativistic frame dragging effect has important astrophysical implications beyond the solar system. Astronomers have found that some of the incredible outpouring of energy from quasars is directed along narrow jets of matter that stream at nearly the speed of light in opposite directions from a compact central region. The leading model for this phenomenon involves a vast disk of hot gas spiraling inward around a spinning supermassive black hole (we will return to these beasts, including the one in our own Milky Way, in Chapter 6). The combination of the magnetic fields generated by the charged particles in the gas and the extreme dragging of spacetime in the vicinity of the black hole generates strong electric fields that can accelerate particles away from the black hole along its spin axis. The particles can reach speeds close to that of light, and upon interacting with the twisted magnetic field lines they can emit radio waves and other forms of electromagnetic radiation. These radio jets can be seen extending millions of light years away from the central quasar. Rotating black holes represent extreme examples of the effect of the dragging of inertial frames, and so it would be very desirable to verify that this effect exists.
All well and good. But still, the effects on gyroscopes on and near the Earth are horribly small. What would possess anyone to actually try to measure them? It is here that the three Stanford professors return to the story.
At Stanford University in the 1950s, back before the days of coeducational athletic facilities, the Encina gymnasium and its walled-in, open-air swimming pool was restricted to males only (the women’s gym was on the other side of the campus). As such, it was customary for users to swim in the nude. Schiff had a virtually unshakable routine of going to the Encina pool every day at noon, swimming 400 yards, and eating a bag lunch afterwards while sunbathing. Even though he was chairman of the physics department, he would try his best to schedule meetings and appointments so as not to conflict with his noon swim. Fairbank knew about Schiff’s daily routine, and when he bumped into Cannon on campus one day in late 1959 and they began to talk about gyroscopes, Fairbank suggested that they go see Schiff at the swimming pool.
Each of these men had had gyroscopes on his mind for a while. Schiff had been thinking about gyroscopes ever since he opened his December 1959 issue of the professional physicists’ magazine Physics Today and saw the advertisement on page 29. There, hovering in an artist’s conception of interstellar space, was a perfect sphere girdled by a coil of electrical wires, captioned “The Cryogenic Gyro.” The advertisement announced the development at the Jet Propulsion Laboratory in Pasadena of a super new gyroscope consisting of a superconducting sphere supported by a magnetic field (from the coils), all designed to operate at 4 degrees above absolute zero. Schiff had taken a strong interest in tests of general relativity lately, and so he asked himself whether such a device could detect interesting relativistic effects. During the first two weeks of December he carried out the calculations, finding both the well-known geodetic effect, as well as the dragging of inertial frames effect. The latter discovery was entirely new, at least as applied to gyroscopes. Back in 1918, two German theorists, Josef Lense and Hans Thirring, had shown that the rotation of a central body such as the Sun would produce frame dragging effects on planetary orbits that were unfortunately utterly unmeasurable at the time, but no one had apparently looked at the effect of the rotation of a central body on gyroscopes.
Fairbank’s field was low-temperature physics, the properties of liquid helium, and the phenomenon of superconductivity, the disappearance of electrical resistance in many materials at low temperatures. He had also been thinking about the potential for a superconducting gyroscope that could be built in the new laboratory that he was setting up at Stanford, and he and Schiff had begun to talk about how these relativistic effects could be detected. Fairbank suggested measuring the precessions using gyroscopes in a laboratory on the equator, but this did not look promising. The reason was gravity. The best gyroscopes of the day had as their main element a spinning sphere, just as in the JPL advertisement. But the sphere had to be supported against the force of gravity, and the standard method of doing this was by electric fields or by air jets. Unfortunately, the forces required to offset gravity were so large that they introduced spurious forces or torques on the spinning ball that gave its spin axis a precession thousands of times larger than the relativistic effect being sought after, though easily small enough to permit accurate navigation and other commercial uses. This problem would effectively go away if the gyroscope were in orbit, where the gravitational forces are zero to high accuracy, and essentially no support is required. But remember, this was only two years after the Soviet Union’s launch of Sputnik, the first orbiting satellite, and Schiff and Fairbank could not imagine realistically being able to do this.
This was where Cannon came in. Cannon knew gyroscopes. He had helped develop gyroscopes used to navigate nuclear submarines under the Arctic icecap. He also knew aeronautics, and he was active in the fledgling space race that Sputnik had started. Before coming to Stanford from MIT, Cannon had already begun to consider the improvements in spacecraft performance that would come with orbiting gyroscopes.
Finally, the three were together (in their birthday suits) at the Stanford pool. When Schiff and Fairbank told Cannon about the proposed experiment, Cannon’s first response was astonishment. To pull it off, they would need a gyroscope a million times better than anything that existed at that time. His next response was: Forget about doing it on Earth, put it into space! An orbiting laboratory is not at all farfetched, and in fact NASA was already laying plans for an orbiting astronomical observatory. Furthermore, Cannon knew the right people at NASA whom they could contact. With that, a five-decade adventure had begun. Only
Cannon would live to witness the end of the story.
It is one of those strange twists of scientific history that, almost simultaneously with Schiff, Fairbank and Cannon, someone else was thinking about gyroscopes and relativity. Completely independently of the Stanford group, George E. Pugh at the US Pentagon was doing the same calculations. Pugh worked for a section of the Pentagon known as the Weapons Systems Evaluation Group, and for him, toying with gyroscopes was a perfectly reasonable activity because gyroscopes have obvious military applications in the guidance of aircraft and missiles. In a remarkable memorandum dated 12 November 1959, Pugh outlined the nature of the two relativistic effects, although he had the frame dragging effect wrong by a factor of two, and described the requirements for detecting them using an orbiting satellite. Some of Pugh’s ideas, such as a technique for compensating for the atmospheric drag felt by the satellite, ultimately became important ingredients in the Stanford experiment. It is highly unlikely, however, that the Pentagon actually incorporated relativistic gyroscope effects into its military guidance systems. Pugh’s classified work could not be published in the open scientific literature, and so Schiff was initially given credit for the idea of a gyroscope test. Only later was Pugh’s work declassified and recognized for equal credit.
Is Einstein Still Right? Page 11