In a famous series of stories in the 1940s, physicist George Gamow related the adventures of one Mr. C.G.H. Tompkins, a humble bank clerk who had vivid dreams of worlds where strange physical phenomena intruded into everyday life. In one of these worlds, for instance, the speed of light was 15 kilometers per hour, putting the weird effects of Einstein's theory of special relativity on display if you so much as rode a bicycle.

Not long ago I figuratively encountered one of Mr. Tompkins's great grandsons, Mr. E. M. Everard, a philosopher and engineer who is carrying on his ancestor's tradition. He told me of an amazing experience he had involving some recently discovered aspects of Einstein's theory of general relativity, which I will share with you. His remarkable story is replete with curved spacetime, cats twisting in midair, an imperiled astronaut dog paddling through a vacuum to safety—and Isaac Newton perhaps spinning in his grave.

**Dangerous Curves Ahead**

In a far-off region of the cosmos, Mr. Everard had gone outside his spaceship to repair an errant antenna. He noticed that the beautiful lights of the distant stars looked distorted, as though he were viewing them through a thick lens. He felt, too, something gently stretching his body. Suspecting he knew what was afoot, he took a laser pointer and a can of shaving cream from his utility belt and turned on his jet pack to test his idea.

With the laser beam serving as a guide, he jetted straight out 100 meters, turned left to travel several dozen meters in that direction and finally returned to his starting point, drawing a triangle of foam like a cosmic skywriter. Then he measured his triangle’s vertex angles with a protractor and added them up. The result was more than 180 degrees.

Far from being nonplussed by this apparent violation of the rules of geometry, Mr. Everard fondly remembered a mischievous non-Euclidean incident in his childhood, when he drew triangles on the globe in his parents’ study. There, too, the angles added up to more than 180 degrees. He concluded that the space around him also must be curved much like the surface of that globe, so many years and light-years away. The curvature would account for the distorted starlight and the slightly unpleasant feeling of being stretched.

Thus, Mr. Everard understood he was experiencing textbook effects of general relativity. Experiments of a rather more refined nature than his jaunting about with shaving cream had confirmed these effects long ago: matter and energy cause space and time to curve, and the curvature of spacetime causes matter and energy (such as his laser beam and the light from the stars) to follow curved trajectories. His feet and his head “wanted” to follow slightly different curves, and the discrepancy produced the stretching sensation.

Musing on these facts, Mr. Everard pressed the button to engage his jet pack again to return to his spaceship—and nothing happened. Alarmed, he saw his fuel gauge was at zero, and he was a good (or rather, bad) 100 meters from the safety of his air lock. In fact, he and his triangle of foam were drifting away from his spacecraft at a constant velocity.

Acting quickly, he flung his protractor, laser, can of foam and all the other items on his utility belt directly away from his spacecraft. In accord with the principle of momentum conservation, with each throw he recoiled a little in the opposite direction—toward his ship. He even unharnessed his jet pack and shoved that dead weight away as forcefully as he could. Alas, when he had nothing left to hurl, he found he had done only enough to counteract his initial motion away from the ship. He was now floating motionless with respect to his ship but still far away from it. His situation may have seemed hopeless: his high school physics teacher had impressed on him that it is not possible to accelerate a body without an external force or some kind of mass ejection.

Fortunately for our adrift friend, he had earlier established that he was in a curved space, and he was wise enough to know that some conservation laws in physics work differently in a curved space than in the flat (uncurved), Newtonian space of his school years. In particular, he remembered reading a 2003 physics paper in which planetary scientist Jack Wisdom of the Massachusetts Institute of Technology showed that an astronaut could move through curved space in ways that would be impossible according to Newton’s laws of motion—simply by making the right movements with his arms and legs. In other words, he could swim. It did not require any fluid to push against; he could dog-paddle through the vacuum.

Wisdom’s trick is rather like how a cat, dropped upside down, can twist its body and retract and extend its legs so that it flips over and lands on its feet. The laws of Newtonian mechanics permit the cat to change its orientation, but not its velocity, without needing to push on anything or be pushed by anything.

Astronauts such as those onboard the International Space Station use a version of the cat-twisting trick to turn around in weightlessness without needing to grab onto a handhold. In the curved spacetime of general relativity, a cat or an astronaut can pull off more impressive stunts. Our hero covered the distance back to his spacecraft in somewhat more than an hour—no Olympic record but certainly quick enough to ensure that he would live to undertake more adventures.

**Swimming Lessons**

How exactly does Wisdom’s phenomenon work? How is it an adventurer such as Mr. Everard can swim in space? In a flat space—the kind assumed by Newtonian mechanics and also special relativity—the center of mass of an isolated system (for example, astronaut plus dead jet pack) never accelerates. Suppose Mr. Everard had tied a long cord to his jet pack before he shoved it away and then reeled it back in. Throughout the entire exercise, as the jet pack and astronaut first moved farther apart and later came together again, the center of mass of the two would be unchanged. At the end, he and his jet pack would be back at their initial position. More generally, Mr. Everard cannot move merely by cyclically changing his shape or structure and then restoring it again.

In curved space, the situation is different. To understand why, imagine an alien creature with two arms and a tail, all of which it can extend and retract. To simplify the discussion, imagine that virtually all of the alien’s mass is concentrated at the ends of its three limbs, a quarter of it in each hand and the other half at the tip of the tail. Floating in flat space, this alien is helpless. If it extends its tail by, say, two meters, the hands move forward one meter and the tail tip moves back one meter, maintaining the center of mass. Retracting the tail again brings the whole alien back to its starting position, just as with Mr. Everard and his inert jet pack. Similar things happen if the alien tries extending its arms. Whatever combination or sequence of limb extensions and retractions the alien carries out, its center of mass stays the same. The best it can do is use the cat trick (extend limbs, swing them around, retract them, swing them back again) to change the direction it is pointing.

But now imagine that this alien lives in a curved space, one shaped like the surface of a sphere. To help you picture it, I will use geographical terms to describe positions and directions on the sphere. The alien starts on the sphere’s equator, its head pointing west and its arms and tail all retracted. It extends both arms, one to the north and one to the south. It then lengthens its tail while keeping its arms extended at right angles to its body. As in flat space, if the mass-laden tail tip moves one meter to the east, the hands move one meter west. Here is the crucial difference on the sphere: the alien keeps its arms aligned with the sphere’s lines of longitude and the distance between those lines is greatest at the equator. Thus, when the alien’s hands (nearer the north and south poles of the sphere) move one meter westward, its shoulders (on the equator) move more than one meter. Now when the alien retracts its arms, along the lines of longitude, it ends up with its hands more than one meter west. When it retracts its tail, restoring its original body configuration, it finds itself a short distance westward along the equator from its original position!

*[Watch an animation of swimming on a sphere and another of swimming on a saddle shape.]*

By cyclically repeating these movements, the alien crawls along the equator. The unusually heavy tail tip and hands are not essential to the swimming; it is just easier to see how far the arms move in response to the tail stretching if all the mass is concentrated at those three points. And, as it happens, if the alien species depended on the curved-space swimming for its survival, it might evolve heavy knobs at its extremities to improve the efficiency of its swimming. After all, mass located at its elbows will not reach so far around the curvature of the sphere as its hands do and therefore will not produce as much extra movement of the body.

A sphere is a two-dimensional surface, but the same principle works in curved four-dimensional spacetime. Cyclic changes in the configuration of a system can lead to a net displacement. Wisdom’s proposed swimmer was a tripod with telescoping legs. The legs can be retracted or extended in length, and the angle between them can be widened or narrowed. The tripod swims by extending its legs, spreading them, retracting them and closing them. The greater the curvature of spacetime where the tripod is, the farther it gets displaced by this sequence of moves.

**Moving Violations?**

Though surprising at first, swimming is a direct consequence of basic conservation laws, not a violation of them. Swimming works because the very concept of a center of mass is not well defined in a curved space. Suppose we have three one-kilogram balls located at the vertices of an equilateral triangle. On a flat surface, their center of mass is the geometric center of the triangle. You can calculate where the center of mass is located in a number of different ways, and each method gives the same result. You can find the point that is an equal distance from all three balls. Or you can replace two of the balls with a single two-kilogram ball located halfway between them and then calculate the center of mass of that ball and the third ball (the point one third of the way along the line to the third ball). The result will be the same. This geometric fact carries over into the dynamics of the system: the center of mass of an isolated system never accelerates.

On a curved surface, however, different computations may not give the same result. Consider a triangle formed by three equal-mass balls in Singapore, Dakar and Tahiti—all near the equator. A point equidistant to the three balls is near the North Pole. But if you replace the balls in Singapore and Dakar with a heavier one in between them and then calculate the position that is one third of the way along the great circle from that ball to the one in Tahiti, your answer will lie close to the equator. Thus, the “center of mass” on a curved surface is ambiguous. This geometric fact ensures that a system in a curved space can move even when it is isolated from any outside influences.

Other subtleties also arise. A standard physics homework assignment involves adding up the forces on a body to determine the net force. Physics students express forces as vectors, which are drawn as arrows. To add two vectors, they slide the arrows around so that the base of one arrow meets the tip of the other. In a curved space, this procedure has pitfalls: the direction of a vector can change when you slide it around a closed path. The procedure for calculating the total force on a body in curved space is therefore considerably more complicated and can result in oddities such as swimming.

Some effects in Newtonian gravitation may seem similar to spacetime swimming at first glance. For instance, an astronaut orbiting Earth could alter his orbit by stretching tall and curling into a ball at different stages. But these Newtonian effects are distinct from spacetime swimming—they occur because the gravitational field varies from place to place. The astronaut must time his actions, like a person on a swing does to swing faster. He cannot change his Newtonian orbit by rapidly repeating very small motions, but he *can* swim through curved spacetime that way.

That the possibility of spacetime swimming went unnoticed for nearly 90 years reminds us that Einstein’s theories are still incompletely understood. Although we are unlikely to construct a swimming rocket anytime soon, Nobel laureate physicist Frank Wilczek, also at M.I.T., has argued that Wisdom’s work raises profound questions about the nature of space and time.

In particular, Wisdom’s findings bear on the age-old question of whether space is a material object in its own right (a position known as substantivalism) or merely a convenient conceptual device to express the relations among bodies (a position known as relationalism) [see “A Hole at the Heart of Physics,” by George Musser; Scientific American, September 2002].

To illustrate these viewpoints, imagine that Mr. Everard is floating in an otherwise empty universe. He would have no stars or galaxies to serve as reference points to judge his motion. Physicist and philosopher Ernst Mach, a relationalist, argued in 1893 that motion would be meaningless in this situation. Yet even a completely empty space can be curved, in which case Mr. Everard could swim through it. It therefore seems that spacetime acts as a virtual fluid against which the motion of an isolated body can be defined. Even completely empty space has a specific geometric structure—another point in favor of substantivalism. At the same time, though, matter (or any other form of energy) is what gives spacetime its geometric structure, so spacetime is not independent of its contents—a point in favor of relationalism. This debate, which crops up in the attempts to develop a unified theory of physics, remains unresolved.

**On the Wings of Time**

Worn out by the effort to swim back to his spaceship, Mr. Everard was resting inside the cabin and letting the autopilot plot a course back home. Suddenly, the alarm went off and the red lights started flashing, indicating that the spaceship was falling onto a massive planet. Mr. Everard was delighted by this opportunity for new and interesting discoveries, but landing on this planet would be a challenge. The ship had too little fuel for a powered descent, and the planet lacked an atmosphere, making a parachute useless.

Fortunately, he remembered the 2007 paper that my colleague, mathematical physicist Ricardo A. Mosna of the State University of Campinas in Brazil and I wrote. Inspired by Wisdom’s example, we came up with another way to exploit general relativity to control motion. Our analysis indicates that an object can slow its descent toward, say, a planet by repeatedly stretching and contracting in an asymmetric fashion—meaning the extending motion is faster than the retracting. A ship equipped with a device moving in that fashion could act as a glider even in the absence of air.

In this case, the effect has to do with the temporal rather than spatial qualities of the motion, which brings to light one of the deepest aspects of Einstein’s theories: the connection between space and time. In Newtonian mechanics, physicists can specify the location of events using three coordinates for spatial position and one for the time, but the concepts of space and time are still distinct. In special relativity, they are inextricably intertwined. Two observers with different velocities may not agree on their measurements of the distance or time interval between two events, but they do agree on a certain amalgam of space and time. Thus, the observers see time and space, considered separately, differently—yet see the same spacetime.

In general relativity, the structure of spacetime becomes distorted (that is, curved), producing what we perceive as the force of gravity. Whereas Newtonian gravity involves only space, relativistic gravity also involves time. This distortion of both space and time leads to effects such as one known as frame dragging: a rotating body (such as Earth) exerts a slight force in the direction of its rotation on other nearby objects (such as orbiting satellites). Loosely speaking, the spinning Earth drags spacetime itself around slightly. More generally, the velocity of motion of a mass influences the gravitational field it produces. Frame dragging and the glider are both examples of this phenomenon.

The swimming effect arises from non-Euclidean geometry, and the relativistic glider is a consequence of indissolubility of space and time. Other such phenomena may remain to be recognized and understood within the inscrutable equations of general relativity. Mr. Everard and other disciples surely have more adventures in store.

*Note: This article was originally printed with the title, "Adventures in Curved Spacetime."*