Black Holes, Ants, and Roller Coasters

Many physicists regarded centrifugal force less as a force than a fiction. Then a Polish astrophysicist discovered that black holes won't work without it--without it working in reverse, that is.

By Hans Christian Von Baeyer|Saturday, July 01, 1995
Sitting upside down in a roller coaster car as it hurtles across the top of a loop-the-loop is not the most suitable position for reflecting on the meaning of physics. Nevertheless, a couple of summers ago I found myself doing just that when my eight-year-old daughter, Lili, dared me to accompany her on such a ride. And at the appropriate moment, I observed that since gravity always points downward, and since our seats were above us and also pushing us downward, the only thing holding us up was the centrifugal force. What if it stopped operating, I wondered? Or worse, what if it reversed direction? Then what?

My fellow physicists would have dismissed my concern as idle nonsense, or at least they would have until recently. But, as it happens, we have learned that there are indeed circumstances--exceptional circumstances, to be sure--under which the centrifugal force can vanish or even reverse. And this astonishing discovery has profound consequences for the way we understand all motion.

The idea of the centrifugal force has long made physicists a bit uncomfortable. Although the force appears whenever a moving object strays from a straight path, its existence depends on the point of view of the observer. Thus when a car rounds a curve, the passengers feel pressed outward by an unseen agency they call the centrifugal force, which increases with the speed of the car and disappears when the car stops moving or the road straightens out. However, people watching the scene from the side of the road, though they too would notice the outward push, would not ascribe its cause to the centrifugal force. Instead, they would explain the effect by pointing to the inertia of the passengers--their natural tendency to keep traveling in a straight line tangent to the curve of the road. Only the riders themselves, whose frame of reference is itself tracing out a curve, would invoke the centrifugal force.

Because the centrifugal force exists in some reference frames (such as moving automobiles and roller coaster cars) but not in others (like the road or the grounds of the amusement park), it has acquired a bad name. College texts disdainfully call it a pseudoforce. My high school physics teacher went even further, vehemently insisting that the force was fictitious and banning mention of it in his classroom. Yet, years later, on the roller coaster, it saved me from crashing to the ground. So what is it then--fictitious or real? The squabble is purely semantic: everyone agrees on the actual effect, the outward push; only the name is at issue. As a compromise, physicists have agreed to call it an inertial force, a term that implies the centrifugal force is as fundamental as the related property of inertia. Inertia is an expression of nature’s laziness--of its inherent resistance to change--and the centrifugal force is just one particular form of that tendency.

This interpretation of the centrifugal force has remained unchallenged since the seventeenth century, so the community of physicists expressed understandable skepticism toward the recent announcement that there is still more to be learned about a matter as trivial as rounding a curve.

Twenty years ago, while working on the theoretical swirl of gas and dust that forms a black hole, Polish astrophysicist Marek Abramowicz noticed a peculiar phenomenon. He knew that in an ordinary star like the sun, gravity tends to increase internal pressure by compressing the gas, whereas rotation decreases that pressure by causing a centrifugal force that counteracts gravity. But in Abramowicz’s computer simulation, rotation of the material had just the opposite effect: it increased the internal pressure. Instead of becoming progressively flattened at the poles with increased rotation, the way Earth did soon after it was formed and the way normal stars do, a cloud of gas collapsing in on itself to form a black hole behaves in the opposite manner: as the speed of rotation increases, the flattening tends to diminish.

Abramowicz, who now chairs the astronomy department at Chalmers University of Technology in Göteborg, Sweden, mulled over these and a number of related puzzles for many years before he finally hit upon a simple explanation. In 1990 it occurred to him that these mysteries could be cleared up at once if near a black hole centrifugal force was pointed in the wrong direction--inward. At first his intuition, molded by decades of experience, recoiled from the thought. After all, the word centrifugal comes from the Latin for fleeing from the center, making an inward- directed centrifugal force a logical contradiction. But Abramowicz overcame his initial prejudice by arguing that centrifugal force happens to have certain properties in one place and contradictory ones in another--a bit like the planet Venus, which was sometimes called the morning star and sometimes the evening star before its true nature was understood.

Working with a number of colleagues, Abramowicz eventually succeeded in constructing a mathematical proof of his daring conjecture. Unfortunately, the proof is couched in the language of Einstein’s general theory of relativity. This explanation of gravity makes use of a four- dimensional framework called curved space-time, which is impossible to imagine, so for a nonmathematical discussion, analogies must suffice. The following simple representation of the strange landscape in the neighborhood of a black hole combines Abramowicz’s ideas with a metaphor invented by Einstein himself.

Imagine an ant crawling around in the desert. It is a two- dimensional creature, which means that it understands only the surface of the desert floor and has no concept of up and down or above and below. Its universe is a two-dimensional surface embedded in the three dimensions of the real world--just as our own three-dimensional space is embedded in Einstein’s unimaginable four-dimensional one. If a rock in the middle of the desert attracts the ant by means of a gravitational force, it does so because it distorts the desert floor as though it were a flexible rubber sheet. In effect, the rock creates an indentation in the sheet. The ant, believing it is crawling along a straight line, naturally follows the curved contours of the desert and is thus led willy-nilly toward the rock. General relativity can be summarized by saying that massive objects curve space-time, and curved space-time in turn shapes the orbits of all the objects that move around in it.

Imagine that the ant crawls in a circle around the rock (see the illustrations opposite). Of course, as it moves it feels a centrifugal force, though at the walking speed of an ant that force is quite feeble. Also imagine that our ant is obsessively methodical, and to keep track of the direction of the force, the ant draws arrows in the sand. As they should, all the arrows point outward, away from the rock.

Now suppose that the rock is replaced by a heavier one, and the hole becomes deeper and steeper. If the ant happens to crawl in a circle along the vertical wall of the hole, it will be surprised to realize that it feels no centrifugal force at all! The reason for this strange illusion is that the direction of the force at any point along this circle happens to be perpendicular to the surface--a concept that the ant cannot even imagine. Only forward, backward, left, and right have meaning for the two- dimensional insect. Downward doesn’t exist.

Finally, imagine that the rock is really massive: it now sinks deep into the sand, creating a cave shaped like a gopher hole, with a short neck and a balloonlike cavern. Once the ant gets past the neck of the hole, where the walls are also vertical, and crawls around in a circle on the ceiling of the cavern, it begins to feel a centrifugal force again. But now, amazingly, the direction of the force is roughly opposite that of the arrows in the sand. Indeed, since the flat ant can recognize only the surface, along which it can crawl, and not the air, through which it cannot fly, it perceives the push of the centrifugal force to be directed along the shortest route toward the rock. If the ant were to crawl around the waist of the gopher hole, where the walls become vertical once more, it would find that the centrifugal force vanishes again. In the lower hemisphere, however, the ant would find that the force returns to its normal direction.

Of course, a gopher hole is not a realistic representation of a black hole. But it furnishes an example of a simple arrangement in ordinary space where the centrifugal force seems to vanish, and even point in the wrong direction--just as it does outside a real black hole.

But why does it do so? The secrets of the behavior of matter in strong gravitational fields are buried deep inside the inaccessible mathematical formalism of general relativity. To enhance his own understanding, Abramowicz continued searching for plausible physical explanations that could be expressed in words rather than equations yet that reached beyond mere analogies. In particular, he wondered what was special about the circle where the centrifugal force vanishes--that place in the gopher hole where the walls are vertical.

The answer turned out to be simple and revealing. The special orbit around a black hole where the centrifugal force is inoperative is located precisely where gravity is strong enough to bend light around into a complete circle. That light is deflected by gravity was Einstein’s most revolutionary prediction in 1916, when he advanced his general theory of relativity, and has been amply verified since then. Once that basic fact is accepted, it is not difficult to imagine a gravitational field powerful enough to turn a light beam clear around so it closes in on itself. Abramowicz found that whenever an object, such as a spaceship, travels along the circular path of a light beam, it experiences no centrifugal force.

At this point two clarifications are in order. A spaceship traveling along the circular path of a light beam outside a black hole does not move at the speed of light--no material object can reach that limit. Indeed, its speed may be as modest as an ant’s. Nor is the spaceship in a free orbit, like the moon or a communications satellite circling Earth. To keep from falling into the black hole, the spaceship must exert a powerful rocket thrust aimed straight downward, as it were, a thrust sufficient to balance and neutralize the strong gravitational pull. But as long as the orbit through space is the same one that a light beam would follow, and as long as gravity has been overcome by means of a constant counterthrust, any spaceship, regardless of weight or speed, will coast in a circle without experiencing a centrifugal force.

Once Abramowicz got this far, he decided to go for broke. He generalized the lessons he learned from black-hole research and proceeded to demonstrate that the centrifugal force always vanishes when an object follows the path of a light beam--regardless of whether that path is straight, circular, or curved by gravity in any manner whatsoever. With that discovery he found himself back in the footsteps of Galileo. In the conventional view, the centrifugal force vanishes when an object travels in a straight line--but how do you define a straight line? Galileo might have responded to that question by appealing to the mathematical definition: a straight line is the shortest distance between two points as measured by a rigid ruler. To surveyors and experimental physicists, though, the straight line of geometry is a mere abstraction. They give a more practical, operational answer: for them a real straight line is defined by a laser beam. Abramowicz’s theorem states that even in a gravitational field the centrifugal force vanishes along straight lines, provided they are defined operationally instead of mathematically.

Abramowicz calls this appealing idea the Principle of Seeing Is Believing, and he illustrates it graphically. Imagine that you are in outer space, near a black hole or some other strong gravitational source. You build yourself a neat rectangular laboratory, using light beams to align the edges true and square. Your ship may require a motor to keep it from crashing downward, and gravity may be pretty strong inside your lab, but otherwise everything seems normal. Now suppose you perform an experiment to find out whether an object moving in a predetermined straight line is subject to a centrifugal force. You find out that it isn’t, as predicted by Galileo, but a critic asks you how you know that your line is straight.

Seeing is believing, answers Abramowicz. If it looks straight, you must believe that it is straight. His point is that from the perspective of a distant observer, the walls, floor, and ceiling of the lab, and even the straight line, are as bent out of shape as a room in a fun house--but as long as they follow the trajectories of light beams, they are, for the purposes of checking the laws of mechanics, straight.

The seeing is believing principle rescues Abramowicz’s findings from remaining curiosities of exclusive interest to the tiny community of black-hole physicists. In fact, it is so general that it has implications for the way we think about motion even here on Earth, where gravity is extremely weak. In particular, the new principle boils down to a radically new way of specifying what is meant by the ancient term natural motion.

Aristotle and other Greek philosophers found it useful to distinguish between natural and forced motion. In the heavens they singled out circular motion as natural and attempted to fit all planetary orbits into patterns composed of circles. Their ideas about common, everyday motion were more pragmatic. Understandably they considered the downward motion of a falling object natural, but in a horizontal direction, they felt, objects naturally stop moving. Carts not pulled by horses quickly come to a halt, and so do ships with furled sails. The motion of a stone flung horizontally off a cliff was explained as follows: the stone gradually loses the artificial forward impetus it received from the thrower, and its natural downward tendency takes over and eventually dominates.

By an ingenious argument, Galileo radically redefined natural motion. He pointed out that if a marble rolls along a smooth trough tilted downward, it accelerates. If the trough is tilted upward instead, the marble decelerates. It follows that when the trough is perfectly level, the marble neither accelerates nor decelerates and thus retains its original speed. Natural motion, according to Galileo’s definition, which later served as the anchor for Newton’s mechanics, is motion in a straight line with constant velocity. In this view the downward deflection of a stone thrown forward is unnatural and must be ascribed to the effect of an external agency, namely gravity.

Einstein revolutionized physics once more by incorporating gravity into the fabric of space and time and thus ending the practice of regarding it as an external, unnatural force. According to general relativity, the actual, parabolic path of the stone (neglecting air resistance) is also its natural trajectory. Once it has been flung into space, the stone just follows the gently curved contours of space and time until it hits the ground. Whatever is, is natural, Einstein seemed to say.

The definition of natural motion implied by the Principle of Seeing Is Believing falls somewhere between the definitions suggested by Galileo and Einstein. A laser beam shining horizontally from the top of a cliff will bend downward almost imperceptibly. So small is the deviation from a mathematical straight line, in fact, that it is practically impossible to observe. While crossing an average room, for example, a light beam drops by less than the diameter of an atomic nucleus--a distance that is too small to measure. The bending of light is observable only in much stronger gravitational fields, such as the sun’s, which deflects starlight. Nevertheless, in principle at least, every light beam (except a vertical one) droops a little.

The natural motion of a stone projected horizontally should now be defined as motion along the gently curved trajectory of a light beam. This definition is suggested by several considerations:

First, by Abramowicz’s theorem, the centrifugal force--the hallmark of unnatural motion--vanishes along this trajectory.

Second, according to the physical definition of a straight line, the natural trajectory is in fact straight.

And finally, an even more physical way to arrive at a definition of a straight line is to hurl the stone harder and harder. According to Newtonian physics, its trajectory will approach a straight line as its speed becomes infinite. But according to the special theory of relativity, the stone’s speed can only approach the speed of light, so its path cannot become straighter than that of a light beam.

In contrast to Einstein’s conception of natural motion, which includes the effects of gravity, the new definition requires gravity to be neutralized by some external force. The same was true for Galileo’s formulation: his marble rolling in a horizontal straight line was supported by a trough whose sole purpose was to cancel the vertical force of gravity. The new definition updates this image by taking into account the fact that the manufacturer of the trough had to check its straightness by reference to a light beam--which, according to Einstein, happens to droop a little.

In a sense, then, the new definition of natural motion represents a step backward. It adds nothing to the predictions of general relativity, but it revives the discarded idea that gravity is an external, unnatural force. For these reasons some experts dismiss it as being fundamentally alien to general relativity. But Abramowicz rejects such criticism. He argues that the advantage of his approach lies precisely in the separation of gravitational effects from inertial ones--a separation that can lead to valuable insight into the physics hidden in a thicket of formulas. In the case of the puzzling rotational flattening of a gas cloud, for example, Abramowicz and a colleague found that their deeper understanding of the physical mechanisms allowed them to reproduce with pencil and paper the results of a complicated computer program.

The most curious consequence of the Principle of Seeing Is Believing is the new wrinkle it adds to our understanding of the familiar concept of the centrifugal force. Consider a toy rocket that keeps a constant speed and follows a gently curved path inside the unimaginably thin wedge that exists between straight lines drawn by Galileo and Abramowicz (see the illustration on page 59), all the while providing a constant counterthrust against gravity. According to the conventional view, the rocket experiences a centrifugal force pushing it back toward the Galilean straight line, outward and upward--just what I felt on the roller coaster. But according to Abramowicz, the centrifugal force actually pushes the rocket back toward its real natural path--the line defined by a light beam. The centrifugal force, in other words, points downward! (Outside the wedge, the two predictions about the direction of the centrifugal force agree.)

The ant we left wandering inside the gopher hole would have no trouble understanding the strange tug on that little rocket. Just imagine that the insect crawls out of its hole and wanders off toward a smaller stone--one that makes only a mild indentation in the desert sand and represents, say, the planet Earth (see the illustration opposite). The ant walks past that stone along a path that is straight as seen by a balloonist high above the desert. The balloonist, unaware of the dip in the desert floor, and accustomed to Newtonian thinking, believes that the ant feels no centrifugal force.

But the ant itself, on the ground, and versed in the subtleties of general relativity, disagrees. As it passes the stone, its trajectory takes a little dip, which causes a centrifugal force directed downward. If this force were perpendicular to the ground, the two-dimensional ant would be unable to feel it. However, because of the slope of the land, the force is not perpendicular but inclined a bit toward the side--in the direction of the stone. The ant therefore feels a slight centrifugal force, directed inward, even though its path seems straight from the balloonist’s point of view, as well as from its own. This paradox illustrates the slender difference between Galileo’s concept of natural motion and the one implied by Abramowicz. The ant feels an inertial force despite hewing to a path that is straight and, to Galileo’s thinking, natural.

If, on the other hand, the ant were to curve its path slightly toward the stone, it would incur an additional centrifugal force that points horizontally outward, away from the stone. The ant could pick this path in such a way that this new outward force is precisely enough to cancel the inward-directed force due to the dip. The path would look slightly bent both to the balloonist and to the ant, but it would also correspond to the path of natural motion according to Abramowicz. Like the rocket ship following the path of light around a black hole, the ant would feel no centrifugal force at all.

This metaphor, of course, is not perfect. The balloonist, looking down in three dimensions on the ant’s two-dimensional world, has no precise analogue in our world of four-dimensional space-time. And whereas light plays a crucial role in our world as a three-dimensional yardstick, the poor ant can rely on no similarly useful tool in its world of two dimensions. Nevertheless, the essential difference between Galileo’s and Abramowicz’s ideas of natural motion holds. The curved path that the two- dimensional ant follows past the stone is roughly analogous to the path that a light beam would take when passing by Earth.

So my panic on the roller coaster turns out not to have been altogether unfounded. Centrifugal force does not behave quite the way we thought it should. Unless the track curves downward sufficiently, the centrifugal force will vanish, or reverse, and Lili and I will plunge to our death. Fortunately it doesn’t take much of a curve to prevent that disaster. As long as the track dips below a light beam shining out horizontally from its apex, I have no cause for worry.

But I will never again think of natural motion and the centrifugal force in the way Galileo and Newton taught us.
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