Albert Einstein (response to Ernst Reichenbaecher)

William Shakespeare (Julius Caesar)

The ancients believed that celestial patterns steered the individual fortunes of human beings and the collective destinies of civilizations. For example, if a particular constellation, or stellar grouping, was high in the sky on the night a certain king was born, he would be blessed with the fiery gifts of a warrior. Another heavenly configuration and he was doomed to die in battle. If Venus kissed Jupiter in the chapel of lights, then a royal marriage was brewing. But if the stars were all wrong on the night of marital bliss, the bride would alas be barren.

Is it lunacy to believe that there is a deep connection between earthly and celestial events? Not if one takes the concept literally.

The term “lunacy” itself derives from beliefs in periodic influences of the Moon. As the shining beacon of the nocturnal sky, Earth’s satellite was thought to exert quite a pull on terrestrial affairs.

There is little evidence that the Moon has driven anyone mad, or induced anyone to sprout extra facial hair. Yet, especially for those attuned to the rhythms of the sea, it clearly exerts a pull on many lives. For those who earn their living hauling in lobsters from the Bay of Fundy off the coast of maritime Canada, each working day is shaped by lunar forces. Amid some of the highest tides in the world, one could not help but concede that the sandstone of human destiny is carved by heavenly guided waters.

Today we distinguish between scientific forces and spiritual influences. To the ancients this distinction was not so clear. Early astronomers did double duty, serving both to record the positions of the celestial spheres and to apply this information for astrological forecasts. Their expertise in predicting eclipses, planetary conjunctions, and other celestial events, as well as offering critical navigational knowledge, earned them the mantel of exalted prophets.

Even as late as the 16th century, many scientific researchers, such as the German mathematician Johannes Kepler, sold horoscopes on the side for extra income. Kepler, in his first astrological calendar, proudly predicted a cold spell and a Turkish invasion of Styria (now Austria). Not only did he peddle forecasts, he deeply believed that they offered special insight into the determinants of human character. He once wrote that his father was “vicious, inflexible, quarrelsome and doomed to a bad end” because of the clashing influences of Venus and Mars.

How did the heavenly orbs set the pace of their own motions and influence the course of earthly events? Kepler originally thought this happened because the planets somehow possessed minds of their own. However, after he developed a clearer understanding of celestial mechanics, he realized this could not be the case. “Once I firmly believed that the motive force of a planet was a soul,” he wrote. “Yet as I reflected, just as the light of the Sun diminishes in proportion to

distance from the Sun, I came to the conclusion that this force must be something substantial.”

Thus, what ultimately changed Kepler’s opinions on these matters was the realization that planetary motion could be explained through simple mathematical rules. This revelation came through a systematic study of the orbital behavior of the planet Mars and generalizing these results to other bodies in the solar system. Kepler discovered that each planet travels along an elliptical path around the Sun, sweeping out equal areas (of the region inside the ellipse) in equal times. He also found relationships between each planet’s orbital period and its average distance from the Sun. These discoveries, known as Kepler’s laws, led him to conclude that rock-solid mathematical principles, not ethereal spiritual influences, govern celestial mechanics.

Picking up where Kepler left off, Isaac Newton brilliantly revealed the mainspring of this clockwork cosmos. He discovered that the same force that guides acorns down from oak trees and cannonballs down to their targets similarly steers the Moon around Earth and the planets around the Sun. Calling this force gravitation, from the Latin gravitas or heavy, he showed that it exerts an attractive pull between any pair of massive objects in the universe. The Moon, for instance, is pulled toward Earth just like a ripe fruit from its branch. Earth is similarly drawn toward the Moon—which explains the movements of the tides.

Newton further demonstrated that the strength of gravity varies in inverse proportion to the square of the distance between any two masses. That is, if two objects are flung twice as far away from each other, their gravitational attraction drops by a factor of 4. This reduction in strength with distance explains why the Moon, rather than any of the stars (as massive as they are by comparison), lifts and lowers Earth’s ocean waters.

Some contemporary believers in astrology have asserted that the marching parade of constellations exerts a changing gravitational influence on the temperaments of children born under these signs.

When disaster strikes they like to think that the fault lies not in ourselves but in our stars. (Indeed, the word “disaster” derives from the Latin for “the unfavorable aspects of stars.”) However, as the late astronomer Carl Sagan was fond of pointing out, the gravitational attraction of the delivering obstetrician, hands cupping the baby’s head, outweighs the pull of any distant star. Though the stars are far more massive, the obstetrician is much, much closer. Besides, it’s unclear how any gravitational force could affect thought processes, unless one hangs upside down to permit greater blood flow to the brain.

When applied to the proper venue, material objects in space, Newtonian theory is remarkably successful. Yet it harbors an essential mystery. Why do bodies orbit their gravitational attractors, rather than moving directly toward them? Why doesn’t the Moon, for instance, immediately plunge toward Earth and destroy all civilization?

Newton’s answer was to propose a property called inertia that keeps still objects at rest and moving objects traveling in a straight line at a constant speed. Like a universal hypnotist, inertia places each object in a trance to continue doing what it is already doing. The only thing that can break inertia’s spell is the application of an external force (or unbalanced set of forces). Still, the magic is lifted only provisionally, allowing the body to change paths only during the interval in which the force is applied. Then it resumes its straight-line motion, until perhaps another force takes hold.

Now consider the case of the Moon. Inertia compels the Moon to keep going in a straight line, but gravity continuously pulls it toward Earth. The compromise is a curving motion, resulting in an essentially circular path.

Though gravity is a force, inertia is not. Rather, inertia represents the state of nature in the absence of all forces. As strange as it might seem, according to Newtonian theory, if all the forces in the universe suddenly “turned off,” every object would continue moving forever uniformly. This state of affairs would result from no specific cause but rather from a lack of causes.

In trying to fathom the underlying reason for inertia, one is reminded of the Taoist paradox that, in trying to pin down something’s definition, its true meaning slips away. The machinery of inertia is remarkable in that there is no machinery. Nevertheless, from Newton’s time onward, physicists and philosophers have sought a deeper understanding of why constant linear motion constitutes nature’s default mode.

To make matters even trickier, all motion is relative. This principle was put forth by Galileo and firmed up by Newton, well before the time of Einstein. The speed of any object depends on the frame (point of view) in which it is observed. For instance, if two truck drivers, traveling at the same speed but in opposite directions, wave to each other on a highway, each will observe the other to be moving twice as fast as their speedometers would indicate. If, on the other hand, each is traveling at identical speeds in the same direction, each will appear to the other to be at rest—presuming they ignore all background scenery.

You would think that the property of inertia would similarly be relative. If, according to one reference frame, inertial motion appears unblemished by extra forces, why not in all frames? Strangely, though, while this is true for observers moving at constant velocities with respect to each other, it is emphatically not true for accelerating observers. Newton cleverly demonstrated this principle by means of a simple thought experiment involving a spinning bucket of water.

Sometimes the most ordinary household objects can offer deep insights about the physical universe. If we concoct the right experiment, there is no need for an expensive particle accelerator to probe the mysteries of force, nor a high-powered telescope to reveal the enigmas of the cosmos. A visit to our basement or backyard might well provide all the materials required.

Take, for example, an ordinary pail. Fill it to the brim with plain tap water. Suspend the bucket from a rope, attached to the limb of a tree. If the bucket is still, the water should appear completely level.

Not the stuff of Nobel prizes, so far, but here’s where things get strange. Spin the bucket. Twirl it around gently but resolutely. As the bucket turns, you notice several things. First, the water remains in the pail. Thanks to inertial tendencies, it pushes against the walls of the bucket but doesn’t spill out.

Yet something does change about the fluid. Its surface begins to hollow out, as if sculpted by a potter. In short order, the once-level top has become as curved as a soup bowl.

The principle of inertia can explain this concavity, but only if you adopt the right perspective. From your point of view, the reason is simple: The water is building up against the sides because, despite the spinning of its container, it wants to travel in a straight line. This lowers the central part of the fluid, hollowing it out.

Consider, however, the perspective of a tiny observer (a savvy ant, perhaps) perched on the side of the pail. If he ignores the world beyond the bucket, he might well believe that the bucket isn’t spinning at all. For him, therefore, inertia should keep everything inside the bucket at rest. Then, imagine his surprise if he looks down at the water and sees it change shape. What bizarre supernatural effect, he might wonder, could deliver such a targeted punch?

Newton used his bucket argument to make the point that, while the principle of inertia does not depend on the relative velocity of two reference frames, it clearly does depend on the relative acceleration of the frames. In physics, acceleration refers not just to alterations in speed but also to changes in direction. Therefore, a spinning bucket is accelerating because the motion of any point within it keeps changing direction. But indeed that is true about Earth itself—rotating about its own axis as it revolves around the Sun. Therefore, given all these gyrating vantage points, how can we uniquely define inertia’s unmistakable action? Where in this whirling cosmic carnival can we find solid ground?

Newton’s answer was to define a fixed, universal reference frame, called “absolute space,” placed in an exalted position above any other framework. “Absolute space,” he wrote, “in its own nature, without relation to anything else, remains always similar and immovable.”

To the concept of absolute space, Newton added another expression, called “absolute time.” Absolute time represents the uniform ticking of an ideal universal clock. Together, absolute space and time serve to define absolute motion—an inviolate description of movement through the cosmos.

Common parlance, Newton pointed out, fails to distinguish between relative motion (measured with respect to any fleeting frame) and absolute motion (defined with regard to the steel scaffolds of absolute space and time). The bucket example, however, demonstrates why such confusion of terms won’t do. To understand dynamics properly, he emphasized, one must reject the ephemeral and take a firm universal perspective. “Relative quantities,” he wrote, “are not the quantities themselves whose names they bear….” Those who mistake transient measures for true quantities, he continued, “violate the accuracy of language, which ought to be kept precise….”

Despite Newton’s admonition, in the centuries after his death a growing community of scholars came to find his distinction rather artificial. With everything in the cosmos in ceaseless motion, why should any one reference frame stand still? By the 19th century, a number of scientists replaced Newton’s artifice with an all-pervading invisible substance, known as the aether. Absolute motion could thereby be defined with respect to the aether stream. Nobody, however, could detect the aether; it seemed as elusive as a ghost.

Viennese physicist Ernst Mach took a different approach. In his popular book on mechanics, he dismissed the notion of an absolute frame. Rather, he argued that it is the combined pull of distant stars that keeps inertia’s hammock aloft. “Instead of referring a moving body to [absolute] space,” he wrote, “let us view directly its relation to the bodies of the universe, by which alone such a system of coordinates can be determined.” Hence, objects resist acceleration

because they are in some mysterious way “connected” to the myriad other bodies in the cosmos. Even a lowly pail on Earth responds to mammoth energies trillions of miles away. This far-reaching concept has come to be known as Mach’s principle.

It’s strange to think of remote stars steering the water in a bucket. Yet the idea that the Moon guides the tides of Fundy seems perfectly normal. Mach’s principle just stretches such cosmic connections much, much further—until they engulf the entirety of space itself. When Mach published his treatise, he freely admitted that he had no experimental proof for his hypothesis. Yet because it was based on the actions of real celestial bodies, he asserted that his explanation was heads above Newton’s abstract design. This argument stirred the youthful imagination of Albert Einstein, who dreamed of putting Mach’s ideals into practice.

Albert Einstein, the greatest physicist of modern times, was born in Ulm, Germany, on March 14, 1879. As a child he had a keen curiosity about the principles underlying the way things work. In an autobiographical essay, he recalled his wonder at the age of 4 or 5 when he was lying ill in bed and his father presented him with his first compass.

“The fact that the needle behaved in such a definite manner did not fit at all into the pattern of occurrences which had established itself in my subconscious conceptual world (effects being associated with ‘contact’). I remember to this day—or I think I remember— the deep and lasting impression this experience made on me. There had to be something behind the objects, something that was hidden.”

Then, at the age of 12, a family friend gave him a book on Euclidean geometry. The young thinker marveled at the crisp certainty of the mathematical arguments presented. Soon he learned how to construct his own proofs, creating geometric rules from simple propositions.

Like many philosophers before him, Einstein was intrigued by the contrast between the imperfect arena of sensory experiences and the ideal realm of abstract concepts. He wondered which aspects of the world required hands-on experimentation and which could be deduced through pure thought. His life’s journey stepped carefully between these two positions. Ultimately, the latter would win out, and his mathematical side would overtake his more practical side. He would become obsessed by the idea of finding inviolable mathematical principles, elegant and beautiful in their simplicity of expression, that could explain all of nature.

One of Einstein’s first “thought experiments” involved a seeming contradiction between Newtonian physics and the known properties of light. At the age of 16 he imagined chasing a light wave and trying to catch up with it. He pictured himself running faster and faster until he precisely matched the speed of the flash. Then, he wondered, would the signal seem still to him, like two trains keeping pace?

Newtonian physics would suggest the affirmative. Any two objects moving at the exact same velocity should observe each other to be at rest—that is, their relative velocity should be zero. However, by Einstein’s time, physicists knew that light was an electromagnetic wave. James Clerk Maxwell’s well-known equations of electro-magnetism made no reference to the velocities of observers. Anyone recording the speed of light (in a vacuum) must measure the same value. Hence, two of the giants of physics, Newton and Maxwell, appeared locked in a conceptual battle.

Others tried to find a way out of this dilemma by proposing effects due to the invisible aether (which by that time had experimentally been discredited), but it was Einstein who developed the definitive solution. In a breakthrough known as the special theory of relativity, he demonstrated that Newtonian mechanics and Maxwellian electrodynamics could be reconciled by abandoning the notions of absolute space and time. By asserting that measured distances and durations depend on the relative velocities of the observer and the observed, Einstein developed dynamical equations that preserve the constancy of the speed of light.

Let’s see how special relativity works. Suppose a runner tries to catch up to a light wave. As he moves faster and faster, approaching light speed, his personal clock (as measured by his thoughts, his metabolism, and any timepieces he is wearing) would slow down relative to that of someone standing still. Compared to the tortoise-like ticking of his own pace, light would still seem to be whizzing by at its gazelle-like speed. He wouldn’t know that his own time is moving slower, unless he later compares his findings with a stationary observer. Then he would realize that he had experienced fewer minutes while running than he would have just by standing. All this ensures that any observer, whether moving or still, records exactly the same value for light speed.

This phenomenon, of clocks ticking more slowly if they move close to the speed of light, is called time dilation. Time dilation is nature’s hedge against anyone catching up with its fastest sprinter. Nature would rather slow down stopwatches than allow runners carrying them to violate its sacred speed limit.

A related mechanism, called length contraction (or sometimes Lorentz-Fitzgerald contraction), involves the shortening of relativistic objects along the direction of their motion. This is a clear consequence of time dilation and the constancy of light. If one uses a light signal to measure a length (by recording how long it takes to go from one end to the other) but one’s clock is slower, one would naturally find the object to be shorter.

In 1907, two years after Einstein published his special theory of relativity, the Russian-German mathematician Hermann Minkowski proposed an extraordinary way to render it through pure geometry. Minkowski suggested that Einstein’s theory could be expressed more eloquently within a four-dimensional framework. With a thunderous speech, he proclaimed the very end to space and time as separate entities, replacing them with unified four-dimensional space-time.

Within this framework, known as Minkowski space-time, anything that happens in the universe is called an “event.” Spilling a

morning cup of coffee in a Ganymede café could be one event; shipping out emergency supplies of Venusian organically grown house blend on a sweltering afternoon could be another. The “distance” between these two occurrences, called the space-time interval, involves combining the differences in time and space between the two events.

How, you might ask, can time be “added to” space? The answer involves using nature’s universal constant velocity, the speed of light. Multiplying velocity (in miles per hour, for example) by time (in hours) yields length (in miles). The time units cancel out, leaving only length units. Multiplying all time values by the speed of light converts them into length values in a consistent way. Then we can employ a modified form of the Pythagorean theorem—the geometric relationship that relates the hypotenuse and sides of a right triangle— to find the space-time interval.

Technically, the procedure is as follows: Take the spatial distances in all three directions and square them. Next, take the time difference, multiply by the speed of light, and square the result. Finally, subtract that value from the sum of the squares of the spatial distances, yielding the square of the space-time interval.

Notice that the spatial distances are added, but the time difference (multiplied by light speed) is subtracted. The procedure that governs which terms to add and which to subtract is called the signature. In standard Euclidean geometry, of the sort Einstein studied as a child, all distances are additive. Hence, the signature is fully positive. In Minkowski space-time, on the other hand, the temporal “distance” is subtracted, yielding a mixed signature of three “plusses” (for space) and one “minus” (for time).

The mechanism, in general, to determine the space-time interval for any given set of events and region of the universe is called the metric. For Minkowski space-time, the metric is relatively simple: Add the squares of the spatial terms and subtract the square of the temporal term (multiplied by the speed of light). It is known as a

“flat” metric—“flat” indicating that the shortest distance between two points is a straight line, not a curve. However, as we’ll see, other metrics are decidedly more complex.

The opposite signs of the space and time parts of the Minkowski metric indicate that the space-time interval can be positive, negative, or zero. These have three distinct meanings. In the first case, positive, the spatial terms dominate, and we call the interval “spacelike.” A spacelike interval means that causal communication is impossible because there is simply too much spatial separation for a signal to travel in the given interval of time. If, in contrast, the interval is negative, it is called “timelike.” In that case, the temporal dominates the spatial, and signals have more than enough time to make the journey. Finally, a third case, known as a “zero” or “lightlike” interval, refers to the exact amount of space crossed by a light signal in a given time. This does not, of course, imply zero separation in three-dimensional space. Rather, it enshrines the special status of light rays as the quickest connections through space-time.

So in our futuristic scenario, if a café on Ganymede orders fresh supplies of coffee beans specially grown on Venus, almost half a billion miles away, there must be a minimum time delay between the order and shipment of roughly three-quarters of an hour for these events to be causally connected. This allows enough time for communication (by radio or other means) to occur. Therefore, if the order is placed at 11:45 a.m. and the shipment goes out at noon, the time delay would be too brief for a signal to have traveled from one place to the other. The interval would be spacelike, and we’d have to chalk up the sequence to pure coincidence. If, on the other hand, the shipment leaves the following day, the time delay would be sufficient for us to conclude that it was in direct response to the order. The interval, in that case, would be comfortably timelike.

Special relativity is a highly successful theory. A young boy’s day-dreams, polished by years of painstaking calculations, have delivered an extraordinarily accurate description of near-light-speed dynamics.

Its astonishing predictions have proven correct in numerous applications. For example, when placed on high-speed aircraft, ultraprecise cesium clocks lag by the precise amount Einstein predicted. Mammoth accelerators boost elementary particles to near light speeds by faithfully timing their actions to relativity’s rhythms. Nuclear reactions generate energetic offspring that—invigorated by time dilation—live longer lives than their slower cousins.

Given such a fantastic achievement, why didn’t Einstein stop there? Why did he wrestle with nature’s laws for another decade, until he could mold special relativity into a far more mathematically intricate theory, known as general relativity? The reason stems from two critical omissions: acceleration and gravity.

An avid reader of Mach, Einstein knew that special relativity failed to answer Mach’s question, “How does it come about that inertial systems are physically distinguished above all other coordinate systems?” That is, what makes constant velocity the favorite type of motion in nature?

Einstein also realized that this question was deeply linked to the mysteries of gravitational attraction. Why, without air resistance, do light feathers and heavy stones drop toward Earth at the same rate? Clearly, he surmised, gravity’s pull cannot just depend on the bodies in question but must be seated in space-time itself.

In similar fashion to his earlier theory, gravity came to Einstein’s attention in the form of a thought experiment. He imagined someone falling off the roof of a house while simultaneously dropping an object (say a box of tools). As the unfortunate man descends, he notices that it remains right next to him. Although the dropped object is falling independently, he can reach out and grab it whenever he wants. Except when he hazards to look down, his situation

seems to him exactly as if he had remained at rest. That is, of course, until he and the object simultaneously hit the ground with a thud.

These musings led Einstein to posit a fundamental property of nature, called the “equivalence principle,” that governs what happens when objects fall freely due to gravity. It states that no physical experiment can distinguish between free-falling motion and the state of rest. For example, astronauts plummeting toward Earth in a windowless spaceship could well imagine that they are floating in deepest space. Unless they fired the ship’s braking rockets, they would notice no extra forces that could distinguish the two situations. For all intents and purposes, they would be resting in an inertial frame.

Hurling a mighty rock through Newton’s stain-glassed vision of eternity, Einstein’s brilliant proposition shattered its unified concept of inertia into myriad parts. No longer could science consider the state of constant linear motion to be a global property. Rather, it would depend on the gravitational field at any point in space. (A field is a point-by-point description of how forces act on objects.) Like a patchwork quilt, the fabric of the universe would henceforth consist of local free-falling frames sewn together. Each segment would represent inertia according to the immediate gravitational conditions. A piece near Earth would describe free motion in terrestrial gravity, for example; a piece near Jupiter would describe Jovian gravity. All that remained, Einstein realized, would be finding the thread to stitch these fragments together. But that would not be easy.

For a number of years, while working in Zurich, Prague, and Berlin, Einstein wrestled with the difficult issue of connecting the coordinate systems of disparate parts of space-time. Realizing that solving this problem would require potent mathematical machinery, he turned to his close friend, mathematician Marcel Grossmann, who guided him through the nuances of higher geometry. Finally, in 1916, Einstein completed and published his general theory.

Like its antecedent theory, general relativity is four-dimensional. However, it is more flexible and far-reaching than special relativity.

It deals with the varying types of motion caused by the attraction of any kind of matter or energy. Representing Einstein’s comprehensive theory of gravitation, it describes how materials produce and respond to changes in space-time geometry.

To achieve this more general theory, Einstein had to move from flat Minkowski space-time to what are known as Riemannian manifolds. These are named after Bernhard Riemann, a German mathematician who died in 1866 at the early age of 39. Manifolds are multi-dimensional geometric representations that can twist and turn like flags in the wind. Unlike Minkowski space-time, Riemannian manifolds can bend at any given point, turning straight lines into curved paths.

Mathematicians express this curvature using several related expressions, technically known as “tensors.” A tensor is an object that undergoes specific predictable changes whenever a manifold’s coordinate system is transformed (rotated, for example). Therefore, like perfect lenses, they provide consistent images no matter which way they are turned. Because of this regularity, tensors offer an ideal way of characterizing geometries.

Einstein’s formalism refers to a number of tensors. The most complete measure of curvature is called the Riemann tensor. It can be reduced into another object called the Ricci tensor, named for Italian mathematician Gregorio Ricci-Curbastro, the founder of tensor calculus. Add another term, and this becomes the Einstein tensor. Finally, all these expressions are related to the metric tensor— the generalization of Minkowski’s space-time interval to curved manifolds.

Einstein hoped that one of these expressions for curvature could be directly linked to the matter and energy in any particular region of the universe. Indeed, the Einstein tensor does this job quite nicely. Einstein’s general relativistic relations equate it to yet another tensor, called the stress-energy tensor, which characterizes the material and energy content of each part of space. In other words, these tensors

connect the bending of space-time with the nature of the substances within it. Generally, the more mass a region contains, the greater its warping, like a field of snow trampled deeper by heavier boots.

From the matter comes the curvature. This, in turn, affects the metric. Unlike Minkowski’s metric, of simple plusses and minuses, in the general case each distance or time term is multiplied by an independent factor. These factors can vary from place to place and from moment to moment. They respond to the conditions in a particular locale. Thus, in short, the changing distribution of material in a region alters its web of connections between points, leading to new avenues of motion.

We can imagine special relativity, described by Minkowski’s metric, as a staid rectangular building, constructed of uniform rows of identical steel girders. Each girder joins the other in perfect perpendicular fashion, maintaining the same shapes, sizes, and relationships forever. If the universe were like this, it would be as homogeneous as a 1960s public housing project. Moreover, it would have no gravity, since all paths through space would be endless straight corridors.

A more general Riemannian manifold, in contrast, has a far more flexible structure, echoing the complexity of the actual universe. Depending on the coefficients set forth in its metric, each of its girders can vary in size from point to point. Over time they can shrink or expand, becoming indefinitely small or unimaginably large. The result is an elastic architecture more akin to the lithe, flowing creations of Spanish designer Antonio Gaudi than to conventional buildings. Indeed, it is an architecture malleable enough to model the evolving dynamics of an intricate cosmos.

Just as tourists weaving through La Sagrada Familia, Gaudi’s sinuous church in Barcelona, must take more convoluted paths than if they were traipsing down a flat, straight sidewalk, objects in Riemannian space are often forced through circumstance into curved trajectories. This is true for the planets of the solar system, as they

follow elliptical paths around the Sun. While, according to Newton’s theory, the gravitational pull of the Sun breaks the planets’ natural inertial states, in Einstein’s theory they are in their natural states. The mass of the Sun warps space-time, changing, in turn, the motion of the planets. Therefore a “straight line”—or more properly a “geodesic” (most direct path)—in Riemannian space-time may not look straight at all in ordinary space.

Although the real world is one of curved space, for sufficiently small regions of the universe (such as laboratories on Earth), Newtonian and Einsteinian theory barely differ in their predictions. Experiments done on particle accelerators such as CERN (European Organization for Nuclear Research) in Switzerland do not need to take general relativity into account. Because tiny regions of space-time are essentially flat, they can be modeled well by Newtonian mechanics for low speeds and special relativity for near light speeds. Nevertheless, Einstein proposed several key tests of general relativity that could distinguish it from other theories for sufficiently curved regions. These tests involve the most warped part of the solar system: the region closest to the Sun.

Einstein’s first prediction concerned the orbital precession of the planet Mercury. It was well known that planetary orbits don’t stay in place forever; rather they advance slightly each time, like the minute hand of a clock. The gravitational theories of Newton and Einstein offer somewhat different values for this rate. Therefore, Einstein was pleased when he discovered that his prediction was more accurate.

Despite this success, Einstein realized that his theory required a stronger test to distinguish it from other possible theories of gravity. Determining that the Sun’s gravitational well would be sufficient to bend starlight, he hoped that astronomers would find a means to measure this effect. Such efforts were delayed, unfortunately, because of the poor state of international cooperation during the First World War.

In 1919, with the war finally over, British astronomer Arthur

Eddington organized two expeditions to the southern hemisphere to record effects on starlight during a total solar eclipse. At totality the Sun’s rays would be completely occluded for several minutes, giving an observer enough time to examine the Sun’s warping of the space around it by measuring the bending of light rays from stars near the edge of the Sun’s disk on their way to Earth. Eddington himself led one of the teams down to the island of Principe off the coast of Africa. The other group, serving as backup, went to Sobral in Brazil. The backup plan proved most fortunate when Eddington’s voyage turned out to be literally a wash. A deluge of rain drenched Eddington and his team members as they tried to make out the stars through the clouds. They did manage to take some photos, but the ones from Brazil were generally much clearer. Merging these results, Eddington calculated the bending. It agreed reasonably well with general relativistic predictions.

A third test of general relativity, called gravitational redshift, involved the light emitted by the Sun itself. Resembling Doppler shifts, the predicted effect pertained to the reddening of light escaping a deep gravitational well. According to Einstein’s theory, the strong gravitational field near the surface of the Sun should slow down the rate of clocks there, resulting in the lowering of luminous frequencies. This postulate can be tested by looking at the spectral lines of atoms—which act like tiny timepieces. The same process should occur near the surfaces of other stars—especially compact ones like Sirius B, the white dwarf companion to Sirius A (the Dog Star). Unfortunately, while the Sun and other bright stars are easy to observe, the physics of their churning surfaces is hard to decipher. So this test was less clear-cut than either Mercury’s precession or the Sun’s light bending. Sirius B’s redshift would not be measured until the mid-1920s.

However, by 1919 the weight of the data for the other two tests was clearly in favor of Einstein’s theory. Eddington, who at that time was one of the few people in the world who properly understood it,

announced that general relativity was right. Headlines around the world proclaimed the death knell of the Newtonian age and heralded the debut of Einsteinian physics.

Today, the scientific community considers general relativity the most accurate and elegant description of the workings of gravity. Nevertheless, many theorists take issue with some of its profound limitations. The foremost of these quandaries concerns its lack of any obvious connection to quantum mechanics—the other physical revolution in the early 20th century.

Of the four fundamental forces of nature, three have been well interpreted through quantum principles. Physicists have combined electromagnetism and the weak interaction (the force that precipitates nuclear decay) into a unified quantum field theory, called electroweak theory. Researchers have similarly modeled the strong interaction (the force that binds protons and neutrons in atomic nuclei) through a theory known as quantum chromodynamics. Yet gravity, the fourth force, remains the odd man out.

Given the vastly different scopes and methodologies that separate quantum mechanics from general relativity, it is no wonder that the search for a full quantum treatment of gravitation has proved elusive. While quantum theory deals with the lilliputian domain of elementary particles, general relativity concerns itself with stellar and galactic behemoths, as well as the vast cosmos itself. Quantum mechanics proclaims, through its famous uncertainly principle, the impossibility of knowing the exact positions and velocities of any object at the same time. This won’t go for relativity, which requires such information to render predictions. Moreover, while the quantum world generally relies on a fixed background in space and time, general relativity incorporates space and time into its very dynamics. Thus, while quantum physics conducts its mysterious

drama on the space-time stage, Einsteinian gravitational theory is pulling the carpet out from under its feet.

Early attempts to fashion a quantum theory of gravity were further stymied by the presence of mathematical monstrosities, called “infinities,” in the basic equations. These anomalies stem from trying to consider tinier and tinier regions, eventually homing in on exact geometric points. By dividing such infinitesimal distances, one is left with indeterminate expressions. For the other forces, physicists have found ways of canceling such problematic items, but not so for gravity. Gravity, considered on its most miniscule scale, is plagued with unavoidable infinite terms that render attempts at calculation meaningless.

A clever way of handling this situation derives from modern string and membrane theories, which posit that point particles do not even exist. Rather, they theorize, the smallest units in nature are vibrating strings and sheets of energy. By excluding mathematical points, these theories abolish the infinities from quantum calculations. For this reason many theorists believe they offer the correct pathway to quantum gravity.

Such theories predict that for minute distances probed at ultra-high energies, gravitational behavior would begin to deviate from standard general relativity. Einstein’s equations (relating space-time geometry to its material content) would accrue extra terms, leading to measurably different results. Thus, gravity would have two different faces, its familiar visage seen in the ordinary motions of stars and planets and an exotic countenance discernible only under extreme circumstances.

Where might such a hidden face be found? Perhaps in the fiery first instants of the universe, gravity could scarcely be distinguished from the other natural interactions. Maybe, as physicist John Wheeler once proposed, the early cosmos was a space-time foam—a jumble of free-flowing geometry leaping from one quantum configuration to another. In those turbulent moments, gravity and the other forces could have continuously exchanged properties and iden-

tities, energetically exploring myriad characteristics. Within this swirling amorphous billow of inconceivably intricate connections, even the number of spatial dimensions could have varied wildly. Reality, if we could somehow perceive its earliest state, would have been a maddening labyrinth.

Then, as the universe cooled down, each natural interaction might have locked into place. One by one, like crystals slowly assembling on a watery surface, each force would assume its permanent form. As space-time’s froth turned more solid, gravitational behavior would settle into its current profile. Finally, like an icy lake in winter, the ripples and eddies of sultrier times would be completely frozen over.

Obviously, we cannot travel back in time and experience the nascent cosmic conditions ourselves. But perhaps sifting through current astronomical data could somehow reveal aspects of this embryonic development—much like a doctor surmising from a child’s health what his fetal environment may have been like. Or maybe powerful particle accelerators, such as CERN’s Large Hadron Collider scheduled to go on line in 2007, will replicate the high temperatures of the early universe and produce discernable effects.

Alternatively, we could hunt for regions in space where matter is dense enough that conventional general relativity could break down. Astronomers believe such conditions might be present in the massive remnants of collapsed stellar cores—ultracompact objects known as neutron stars and black holes. Within their shrouded interiors, the lexicon of ordinary physics could give way to an unknown language so bizarre as to be barely comprehensible.

When Einstein developed the equations of relativity, he hoped they would resolve the dilemmas posed by Newtonian physics without generating new problems of their own. Ideally, he envisioned an airtight description of the cosmos without any open ends. A strict

determinist in the tradition of philosopher Baruch Spinoza, Einstein expected that a full accounting of nature would prove unambiguous and unique. A deity, the esteemed physicist argued, would have no reason to create an imperfect universe with any aspect subject to chance or interpretation. “God does not play dice,” he famously remarked.

Ironically, however, loose threads began unraveling from Einstein’s supposedly seamless garment almost as soon as he had fashioned it. One of the first general relativistic solutions, calculated by German astrophysicist Karl Schwarzschild in 1916, possessed a curious open end, called a singularity, that seemed impossible to remove or explain. A singularity is a point or region where certain parameters (such as density or pressure) zoom to infinity, creating a breach in the fabric of space-time. Einstein deplored singularities because they rendered theories mathematically incomplete.

Schwarzschild, officially the director of the Potsdam Astrophysical Observatory but then serving on the Russian front as an artillery expert, developed his solution to describe the relativistic properties of stars. He modeled stars as spheres of particular sizes and masses. Churning these variables through Einstein’s equations, he obtained a metric describing the warping of geodesics (“straightest” paths) near such bodies.

Physicists often like to test-drive solutions by exposing them to extreme conditions. In the case of the Schwarzschild metric, this involved imagining what would happen if the star’s mass was high but its radius extremely small. Strangely, this changed its character from a simple dent in space-time to a bottomless pit. Beyond a certain point, called the “event horizon,” geodesics entering this region would no longer be able to escape. Hence, light rays—traveling along geodesics—could enter but never leave. Today, we call this situation a “black hole”—so dubbed by John Wheeler for its light-trapping properties.

Since the 1960s, when Wheeler introduced the expression,

astronomers have identified a number of black hole candidates. One might wonder how they can detect such coal black objects against the backdrop of darkest space. Like a ghost sitting on a seesaw and lifting a startled child resting on the other end, astronomers have sensed these unseen bodies through the reactions of those around them. Many black hole candidates have been found in binary star systems by noting their actions on visible stars. Black holes are thought to victimize their companions by absorbing their material in a process called “accretion.” As such captured matter plunges into the black hole’s bottomless gravitational well, it reaches ultrahigh temperatures, causing it to emit highly energetic radiation, mainly in the form of X-rays. Astronomers have recorded such characteristic signals, leading them to conclude that black holes likely exist.

Black holes, according to current thinking, comprise one of three possible end points for stellar evolution. When a star’s primary source of energy—its nuclear fuel—becomes exhausted, its central core collapses and its outer envelope expands. The peripheral material exudes into space—either in a gradual dissipation (for lighter stars) or in a catastrophic supernova explosion (for heavier stars). In the former case, the remaining core settles down into a hot, tiny beacon, called a white dwarf. Such will be the fate of the Sun.

A star between 1.4 and 3 times the mass of the Sun, however, suffers a far more turbulent fate. Its core implodes so suddenly and energetically that the very atoms inside it are completely pulverized. Throughout the collapsing body, positive protons and negative electrons fuse into neutrons. This happens simultaneously with the supernova explosion of the outer shell—similar to the pulling back of the undertow when ocean waves are building up. The core—an ultradense amalgamation of neutrons known as a neutron star— remains as a relic of the catastrophe.

Finally, if a star is more than three times heavier than the Sun, its violent transformation is even more powerful. Not just the core’s atoms but also its elementary particle constituents are utterly

destroyed. Nothing remains of matter as we know it. What’s left is an infinitely dense singularity cupped by a deep, light-trapping gravitational well—in other words, a black hole.

Despite promising candidates and sensible formation theories, scientists can only speculate about a black hole’s shrouded interior. The region between a black hole’s central singularity and its event horizon constitutes perhaps the most enigmatic frontier in modern astronomy. General relativity advises us that a series of extraordinary, but ultimately deadly, events would transpire for any brave or foolish soul who dares to venture within its ghastly domain.

A black hole would be a most insidious snare to anybody entering it, for sure, as it would give little warning of the perils in store. At first, astronauts on such a doomed spacecraft would feel nary a jangling of their silverware as they approached the dark, frozen object. Looking at their watches, they’d notice nothing of particular interest, little knowing that their timelines were rapidly diverging from those on Earth. The reason for such a discrepancy is that the time axis tilts in regions of gravitational distortion. This variation in the direction of time’s axis resembles the twisting of the quills of a porcupine, pointing in different ways on various parts of its curved body. The tilting of time’s axis near the black hole contrasts with its “upright” direction in relatively flat regions far away from it, leading to a comparative dilation of time—similar to special relativistic effects but due to gravity rather than high speeds. So as time passed normally for the unfortunate crew, those following their travails from a safe distance (we imagine here a remarkably powerful telescope observing the ship) would be horrified to see them moving more and more slowly. Like characters from a George Romero flick, they would seem like languid automata inching their way across the deck of an increasingly dormant vessel.

Eventually, as far as the outside world is concerned, the ship and its occupants would grind to a virtual halt at the brink of the black hole’s event horizon. Their clocks would be moving so slowly, rela-

tive to Earth’s, that for all intents and purposes they’d be statues. Not so, however, from the astronauts’ perspective. Time would continue for them unabated as they sailed through the invisible barrier. From that point on, there would be no turning back. To escape, they’d have to reverse course at a rate faster than light—an impossibility.

What would transpire next for the fated passengers depends on the size and nature of their captor. Physicists have generalized Schwarzschild’s simple model to encompass more elaborate possibilities. Additional black hole solutions have been found, representing spinning and electrically charged varieties. The complete description of a black hole state, named the Kerr-Newman solution for theorists Roy Kerr and Ted Newman, delimits all possible masses, sizes, rotational rates, and charges.

A curious expression coined by Wheeler, “black holes have no hair,” designates physicists’ opinion that these are the only parameters that have meaning for such bodies. Everything else notable about them (such as the specifics of their origins) would be shorn off by relativity’s meticulous barber. Cruelly, this would also be true for anything or anyone that happened to be ingested. There’d be no mark or tattoo on a black hole’s bald pate that would indicate its contents.

Still, given their wide range of possible masses and rates of spin around their axes (as well as whether they are electrically neutral or charged), not all black holes are the same. Candidates have been detected with an enormous variety of sizes—ranging from large stars to the central dynamos of galaxies. Primordial black holes, born from density fluctuations in the early universe, could have been as light as 1/100,000 gram.

More massive objects tend to form bigger black holes. For example, a black hole three times the mass of the Sun would have a Schwarzschild radius (distance from the singularity to the event horizon) of approximately 5 miles, the size of a small city. In the center of the Milky Way, by comparison, there may be a black hole

estimated to be more than 3 million times as massive as the Sun. Its Schwarzschild radius is thought to stretch out almost 5 million miles—or 11 times the radius of the Sun.

Variations in size would have major impact for our trapped astronauts. A small black hole would almost immediately crush them—offering them not even a moment’s respite to contemplate their fate. If, on the other hand, they were “lucky” enough to fall into a large black hole, they would have ample time to soak in their surroundings—a flood of lethal radiation—while taking a gut-wrenching plunge to its center. As they sank into the abyss, tidal forces would stretch them out along their path of motion while squeezing them like a tube of toothpaste in the other directions. In either case, quick or slow, the ultimate result would be a complete pulverization of every molecule in the astronauts’ bodies.

One is reminded of the scene in the film Arsenic and Old Lace, when mad Dr. Einstein (played by Peter Lorre) decries his cohort’s decision to apply slow torture instead of quick murder to the captured protagonist (played by Cary Grant). The trembling plastic surgeon begs his coconspirator to just get the killing over with. “Not the Melbourne method!” he pleads to no avail. “Two hours!” Nevertheless, the choice of a two-hour technique offers the leading character precious time to be rescued.

Given sufficient time, could astronauts find a way to escape a black hole’s crushing singularity? That would depend on whether a highly theoretical conjecture about such objects turns out to be true.

On the face of it, a black hole represents a one-way journey to a crushing death. But that’s just the classical picture. According to quantum notions, captured material does slowly leak out—in a trickle of energy, called Hawking radiation, that exudes from the event horizon over the course of trillions of years. Whether or not

such leaked energy could convey information about the original objects is still controversial. For decades, Cambridge physicist Stephen Hawking, the developer of the theory, argued that it does not. During a recent talk at a scientific conference, however, he indicated that he has changed his mind. Bits of information, he now believes, could be released in the trickle. Nevertheless, because it would be painstakingly slow and would not constitute information on the actual original bodies that were sucked into the black hole, this method of “escape” would hardly be comforting to trapped astronauts about to be pureed.

Of greater possible interest is the notion of “tunneling” intact through the black hole to another part of space-time through a type of interconnection called a “wormhole.” This hypothetical link between disparate segments of the universe appears when the Schwarzschild metric and other black hole solutions are plotted on special charts, called Kruskal diagrams, that convey their causal structures. These diagrams suggest that a black hole’s seemingly bottomless funnel might not be bottomless at all. Rather, it could be connected via a space-time “throat” to a second funnel. Just as matter would vanish without trace into the first funnel, it would materialize without sign of origin from the second. Theorists have deemed the all-emitting second funnel a “white hole,” to contrast it with its all-absorbing polar opposite.

Given their hypothetical nature, the greatest use of wormholes so far has been as a plot device in science fiction stories. Speculative writers had long sought a rapid transit system for conveying terrestrials and aliens from one sector of the cosmos to another—a kind of “subway to the stars.” With conventional space travel so slow, wormhole connections appeared a far superior solution. Readers or filmgoers loved to suspend disbelief and take wild rides through interspatial tunnels to worlds unknown.

Ironically, one such science fiction drama stimulated bonafide scientific discussion about wormholes. In the early 1980s, astronomer Carl Sagan was preparing to write Contact, a novel envisioning the

first human-alien encounter. Realizing that such a rendezvous would require quite a hop across space, he contemplated ways of doing so in a reasonable amount of time.

“That was my problem,” recalled Sagan. “To get [the female protagonist] to a great distance away from Earth in the Milky Way galaxy, to meet the extraterrestrials, come back and do all that within the lifetime of the people she has left behind.” Sagan knew that black hole tunnels had been discussed as possible gateways but didn’t think they’d be safe or feasible. He decided to ask his friend, Caltech astrophysicist Kip Thorne, for advice.

“In the early 1980s there was a common misconception that you might be able to travel from one place to another in the Galaxy, without covering the intervening distance, by plunging into a black hole,” continued Sagan. “But there was something about the whole idea that made me nervous. It was for that reason that I contacted Kip Thorne.”

When Sagan called him, Thorne confirmed that, although black holes theoretically offered the possibility of interspatial connections, space travelers would be strongly advised not to attempt them. Like a tunnel through an active, lava-filled volcano, such a shortcut would almost certainly prove lethal. Stretched out like taffy, bombarded like in a microwave oven, accelerated like on the most evil thrill ride imaginable, no sane person would wish to buy such a ticket—even if they could somehow get a chance to meet E.T. They might as well go to Universal Studios—which, unlike a black hole, is safety inspected.

Thorne wondered, though, if a more user-friendly wormhole could be developed. Along with graduate student Michael Morris, he examined how a black hole could be modified to eliminate its deadly features while preserving its potential to connect with other regions of space. Tinkering with various solutions to Einstein’s equations of general relativity, they managed to fashion a streamlined “traversable wormhole” that would permit safe passage between one

region of the universe and another. They sent the results to Sagan, who incorporated them into his novel. In 1987 they published these findings in the American Journal of Physics—hoisting the issue into the mainstream of theoretical discussion. Shortly thereafter, New Zealand physicist Matt Visser (then at Washington University in St. Louis) developed an alternative set of navigable wormhole models—proving that there were many ways to carve stable tunnels through space.

Before submitting any engineering bids just yet, any prospective wormhole entrepreneur should stop and consider the enormity of such an undertaking. Constructing a wormhole would require the technological know-how of a civilization far more advanced than ours. Gargantuan amounts of material—many times the mass of the Sun—would need to be compressed and molded into ultracompact configurations. Such a colossal enterprise—assuming it’s even possible—could easily be many centuries away.

Furthermore, in addition to the immense technical challenges, the traversable wormhole models all share one major catch: Stabilizing the wormhole’s throat would require a special kind of substance, dubbed “exotic matter,” with repulsive rather than attractive properties. Like scaffolding holding up a coal mine’s ceiling, exotic matter would keep the tunnel from caving in—allowing astronauts to pass through without being crushed. So why couldn’t scientists simply find or create such material? The tricky point is that, unlike any of the familiar substances around us, exotic matter would, under certain conditions, be observed as having negative mass. A ripe hanging apple tossed by the wind will eventually fall to the ground. But a negative mass apple would rise to the clouds instead. In other words, it would weigh less than zero.

How could something weigh less than zero pounds? Could such strange fruit exist? Are there watermelons somewhere in the universe

that would levitate from grocer’s scales? Are there negative mass boxes of chocolate-covered cherries that would actually remove poundage with each serving? After enough bites, could we float like Mary Poppins? Despite numerous experiments, scientists have yet to detect particles with negative mass. Even positrons, the oppositely charged antimatter counterparts to electrons, have positive mass. Experiments at the Stanford Linear Accelerator have confirmed that positrons indeed fall down, not up.

Curiously, the laws of gravitational physics—whether expressed in Newtonian or general relativistic form—don’t explicitly rule out the existence of negative mass. Therefore, following the dictum (attributed to physicist Murray Gell-Mann) “Whatever isn’t forbidden is compulsory,” surely it must lie somewhere. British astronomer Hermann Bondi once speculated that every positive mass particle could possess a negative mass companion, just as magnetic north poles must waltz with south poles. Then where are these sub-weightless creatures hiding? Could they be huddled in some remote corner of space—banished to the universe’s Siberia through sheer gravitational repulsion? Or could they reside closer to Earth, albeit in some dim attic of possibilities we have yet to explore?

As it turns out, you wouldn’t need all that much exotic matter to prop open a wormhole. In 2003, Visser and two colleagues calculated that the spatial vacuum—the fuzzy realm of fluctuating quantum fields where uncertainty reigns supreme—could well provide such material. As modern quantum theory has shown, no vacuum is truly empty. The Heisenberg uncertainty principle, a key element of quantum physics, permits particles to materialize from sheer nothingness, as long as they remain only for brief intervals. Conceivably, through this process, tiny amounts of negative mass could randomly emerge from the void. Normally, these bits of flotsam and jetsam would return to the great emptiness, but perhaps they could somehow be captured first. If just a smattering could be netted, Visser’s team showed that it would suffice to keep a

wormhole’s throat open. Like jalapeño sauce, just a few potent drops would be more than enough.

Another potential place to fish for negative mass would be in the deep space-time troughs of neutron stars and black holes. Near the packed centers of collapsed stars, where gravity wears titan’s boots, the conventional laws of physics might be well-enough trampled to permit small quantities of exotic matter to leak out. To detect such elusive material, we’d need to drop an enormous test object (like a planet) into a stellar relic and measure precisely what happens. As the test body plunged into the well, theoretically the negative mass would reveal its presence with a characteristic echo.

Some of the theoretical models permitting negative mass involve extending Einstein’s equations by an additional dimension, thus augmenting the four dimensions of space-time by one more. Dating back to an early 20th-century proposal by German mathematician Theodor Kaluza, extra dimensions have become a popular avenue for enlarging the scope of general relativity and encompassing electromagnetism and the other natural forces in a unified theory.

Traditional higher-dimensional theories, including Kaluza’s, are usually designed to forbid any influence of extra dimensions on the known laws of physics. For example, in a model proposed soon after by Swedish physicist Oskar Klein, the fifth dimension is curled up so tiny that it could never be observed. Like the minute stitches on a finely woven sheet, space-time would feel perfectly smooth to the touch—without indication of something extra.

However, more recent unified theories (such as particular versions of what is called M-theory) involve large extra dimensions— new directions that aren’t twisted up into miniscule knots. Rather, the additional dimensions lie along pathways that cannot be accessed by conventional matter but can still make their presence known. The theory allows for physical tests by indirect means.

Intriguingly, the possibility of negative mass could be used as one way of detecting extra dimensions. Particular solutions of gen-

eral relativity, extended by an extra dimension, display curious sensitivity to the sign (plus or minus) of a particle’s mass. They offer stark predictions for what would happen to objects under extreme gravitational circumstances—for example, near the event horizon of a black hole. The existence of negative mass would produce characteristic behavior that astronomers might be able to measure.

The same contemporary higher-dimensional theories offer another startling prediction. Not only do they distinguish between negative and positive mass, they also differentiate between two different uses of the term “mass” itself. They yield distinct values for “inertial mass,” a body’s resistance to forced changes in motion, and “gravitational mass,” which causes a gravitational field. Both originate in Newtonian physics, albeit in separate equations. The former enters into Newton’s second law of motion—force equals mass times acceleration—and the latter into his law of universal gravitation. Newtonian mechanics, though, treats these formulations of mass as if they always have the same values. It uses just one variable for these two concepts. Einsteinian general relativity goes even further. The equivalence principle on which it is based mandates that inertial mass and gravitational mass are identical. But what if they were slightly different?

Imagine that you often hear about someone named Moe. First, your next-door neighbor tells you that Moe trimmed some of the trees on your block. Then you learn from the couple around the corner that Moe plowed your street after a snowstorm. You might well conclude that the same handyman did all this work.

Then you find out that the guy who trimmed the trees is tall and has long blond hair and a stick-thin build. After also hearing that the snow plower is short and has curly dark hair and a paunch, you would realize there are two different Moes. Could there really be two different types of mass that do two different jobs?

General relativity describes gravity as a geometric effect in four dimensions. Particles move through space independently of their mass. Wispy neutrinos and bulky upsilon particles, acted on only by

gravity, must travel along identical paths because the inertial mass and gravitational mass, being precisely equal, do not enter into the equations of motion.

However, once a fifth, uncurled extra dimension supplements space-time’s ordinary four, standard general relativity undergoes a profound transformation. For solutions of Einstein’s equations in five dimensions, an extra force rears its head. This new force depends on the motion of ordinary space-time along the fifth dimension. Moreover, it accelerates particles as a function of their mass, clearly violating the equivalence principle. It would cause, for example, two asteroids plunging toward the Moon—one large, the other tiny—to fall at slightly different rates.

Questions about the absolute validity of the equivalence principle and other issues concerning the nature of gravitation have stimulated a number of experiments designed to test the fundamental assumptions underlying general relativity. Given that Einstein’s marvelous theory is now a proud nonagenarian, perhaps she could use some checkups to gauge her health. Will she continue to be the beloved grand dame of modern physics, or will one of her offspring assume her exalted position? Bets are on her continued survival, but it will be interesting to see what the prognoses reveal.

General relativity is, by its very nature, harder to test directly than other physical theories. Unlike laboratory-based disciplines such as biophysics or materials science, its focus is far more remote and less tangible. We can’t simply place the fabric of the universe under a microscope to see if it obeys certain geometric relationships. Unlike, say, an unknown metal, we can’t pound space-time with a hammer, press it with a die, or stretch it out on a roller to ascertain its tensile properties. Nevertheless, researchers have devised subtler methods of putting it through the wringer.

The results from the 1910s—the perihelion advancement of Mercury and the behavior of starlight near the Sun—were important early gauges of relativity’s overall viability—akin to making sure a patient has a reasonable heart rate and blood pressure. Another critical test, the existence of gravitational redshifts, similarly checked out fine. But by the 1950s many researchers expressed dismay that no more tests were available. For example, at a 1957 conference, physicist Bryce DeWitt threw a piece of chalk up in the air, caught it, and then remarked (slightly exaggerating): “We know almost nothing about gravitation. There is only one experiment which we do over and over again, and that is what I have just done.” Fortunately, a bevy of new probes now offer Einstein’s body of work an even more extensive physical examination. In assorted experiments, precise equipment has been scanning it from head to toe, seeking signs of even the slightest flaw.

Providing the very legs on which relativity stands, the equivalence principle must remain solid enough to support the theory. Accurate measurements of the equality of inertial and gravitational mass offer vitally important data. If they were to indicate even the slightest discrepancy, the implications would be monumental. Modifying Einstein’s theory would become a necessity, not just speculation.

One device for testing equivalence, called a torsion balance, dates further back than general relativity itself yet continues to be updated and refined. Torsion means twisting or turning. Through a balance device, such actions can reveal how forces affect materials. At the turn of the 20th century, Baron Roland von Eötvös of Hungary used such a sensitive instrument—a weight hanging from a rotating rod delicately balanced on a pivot—to measure minute differences in the accelerations of various substances. He devised it to record any subtle effects produced by small discrepancies between inertial mass and gravitational mass. Thanks to its meticulous design, the equipment was precise enough to rule out such a difference down to one part per hundred million.

Eötvös’s measurements stood as the benchmark for decades, offering a firm basis for Einstein’s assumptions. Then in the early 1960s astronomer Robert H. (Bob) Dicke of Princeton, along with colleagues G. Roll and R. Krotkov, suggested a clever way of substantially improving on Eötvös’s method. Realizing that the Sun exerts a periodic pull on terrestrial objects—due to Earth’s 24-hour rotation—they measured the accelerations of various objects with respect to the Sun rather than Earth. The device they used was a triangular array of weights: two made of aluminum and one made of gold. An electrical system served to keep the set balanced. If any of the weights felt an extra tug and the device started to tilt ever so slightly, an electrical signal would immediately rectify it. The amount of this signal was carefully recorded.

Now suppose the equivalence principle was false and acceleration depended on mass. Then aluminum would react slightly differently than gold to the Sun’s pull. As Earth turned around on its axis, the torsion balance would try to tilt slightly in different directions. The electrical system would thereby need to exert a periodic correction—with an unmistakable 24-hour cycle. Dicke and his co-workers found no such cycle. Within a difference of one part in 100 billion, they confirmed that aluminum and gold accelerate at the same rate under gravity.

Refining Eötvös’s concept even further, in the 1990s a group of researchers led by Eric Adelberger of the University of Washington constructed several torsion balances with even greater sensitivity. They designed each balance to test particular features of gravity on a variety of scales. Rather than just looking at the effects of Earth and the Sun, they fashioned their instruments to measure gravitational influences as close as the fly’s wings and as far away as the center of the Milky Way. To honor both Eötvös and their university, they named their collaboration the Eöt-Wash group—pointing out that “vös” in Hungarian is pronounced somewhat like “Wash” in English.

Since beginning its experiments, the Eöt-Wash group has delivered an impressive array of data indicating that light and heavy objects accelerate exactly the same way—with a maximum discrepancy of approximately one out of 10 trillion. Gravity, the team has found so far, behaves in an identical fashion, whether it is twirling stars around a galactic core or lowering a speck of dust toward the ground. With these successes in hand, the team is pushing its equipment to its absolute limit, hoping to map out every facet of gravity’s terrain.

Today, not all tests of the equivalence principle involve nimble balances twisting and turning in labs. Some of the newer experiments have forsaken Chubby Checker moves for Obi-Wan Kenobi maneuvers. With lasers and space probes now used to make ultraprecise measurements, tests of general relativity have entered the Star Wars age.

In the 1960s and 1970s, space agencies such as NASA (National Aeronautics and Space Administration) captivated the public through unprecedented manned missions, like the Apollo Moon landings. These days such centers have broadened their scope to include a wealth of scientific satellites and other instrumentation designed to investigate the nature of space itself. The Hubble Space Telescope, the most famous of these instruments, has been joined by numerous other devices probing the deep structure of the cosmos.

Witness a new APOLLO (Apache Point Observatory Lunar Laser-ranging Operation) mission, one that sends laser beams instead of people to the Moon. It makes use of five retro-reflectors—banks of special prisms left behind by the astronauts on the lunar surface. These mirrored surfaces reflect incident light back to Earth, enabling precise measurements of the distance to the Moon. By shining a laser from Earth onto one of these and timing how long it takes for the

beam to return, scientists have been able to pinpoint the Earth-Moon distance within a fraction of an inch. Led by astrophysicist Tom Murphy of the University of California at San Diego and including Adelberger as one of the team members, researchers hope to use this method to check for subtle differences between the motions of the Earth and Moon in the Sun’s gravitational field. If such discrepancies are found, they could point to minuscule violations of the equivalence principle.

To test the actions of gravity on varying masses, we might wonder why scientists don’t just drop two objects and see if they land simultaneously—as, legend would have it, Galileo did from the Leaning Tower of Pisa. The free fall would need to take place in a total vacuum to prevent air resistance from skewing the results, so the project planned by the ESA (European Space Agency), called MICROSCOPE (MICROSatellite à traînée Compensée pour l’Observation du Principe d’Equivalence), is designed to do just that. Targeted for launch in 2008, it will reconstitute the leaning tower as a floating satellite, orbiting almost 700 miles above Earth. This vehicle will shield two cylinders, made of platinum and titanium, which will be released simultaneously and allowed to move freely inside. Because both masses will be subject to the same gravitational field, namely Earth’s, the equivalence principle predicts that they should follow identical orbits. Each time they deviate from their uniform paths an electrical field will steer them back into place. Therefore, by measuring the electrical signals required to keep both objects moving along the same trajectory, researchers will gain precise information about whether or not the equivalence principle is violated.

In case MICROSCOPE doesn’t constitute proof enough of Einstein’s conjecture, yet another mission is planned after 2011. Known as STEP (Satellite Test of the Equivalence Principle), it is a joint project of NASA and the ESA. Housed within an Earth-orbiting satellite, hollow test cylinders of various masses will be

stacked inside each other like Russian dolls and then placed in a cryogenic (ultracold) vacuum flask. Superconducting shielding will protect the apparatus from external disturbances. (Superconductivity is a low-temperature quantum effect that allows certain materials to maintain electrical currents and magnetic fields indefinitely. It offers a buffer against external electromagnetic influences.) Highly sensitive SQUIDs (Superconducting Quantum Interference Devices) will measure the concentric cylinders’ relative motions as the satellite circles through Earth’s gravitational field. They’ll be able to detect motions as fine as 50-quadrillionths of an inch. Like an overzealous traffic cop, they’ll record even the slightest inkling of a violation.

The equivalence principle is not the only aspect of gravitational physics being tested in space. An orbiting satellite called GP-B (Gravity Probe B) is currently engaged in a far-reaching study of two general relativistic predictions: frame dragging and the geodetic effect. These properties, specific to Einstein’s theory, are quite subtle and have never before been tested.

Frame dragging involves the twisting of space-time due to the rotation of massive objects. Emanating from each body in the cosmos, like the streamers from a maypole, are manifold geodesics. These strands correspond to the shortest paths through that region of the universe—namely the routes traveled by light rays. When a body twirls around in its clockwork dance, it swings its streamers with it. Objects clinging to these streamers, like May Day revelers, must similarly whirl around, changing directions as they spin.

Although relativistic frame dragging was first postulated by Austrian physicists Joseph Lense and Hans Thirring in 1918, it wasn’t until 1959 that Leonard Shiff of Stanford proposed a direct way of testing it. Shiff calculated that a spinning gyroscope orbiting 400 miles above Earth would change its tilt by a fraction of a milliarcsecond (an extremely tiny angle, roughly 300-billionths of a degree) each time it orbits. Though minute, this precession could potentially be detected; it is the main impetus for GP-B.

A second source of tilting, the geodetic effect, arises from the denting of space-time by massive bodies. Dutch scientist Willem de Sitter discovered this property in 1916. When a car drives over bumps in the road, it may swing from side to side. Similarly, if a spinning gyroscope travels through warped space—near a planet, for instance—its axis of rotation tends to lean in various directions. This effect, approximately 6,600 milliarcseconds per year, is also minuscule but decidedly more pronounced than frame dragging.

Both effects are so tiny that we might be tempted to ignore them, or dispute whether they are worth spending our hard-earned tax dollars on testing. However, effects that are small in our solar system can have profound implications for the wider cosmos. For example, Einstein’s theory accounts for a tiny change in the orbit of the planet Mercury of 43 arcseconds per century. That minuscule result helped confirm the space-bending properties of general relativity—leading, for example, to predictions about massive black holes in the centers of galaxies. Similarly, precise measurements of the frame-dragging and geodetic effects would undoubtedly produce a wealth of new cosmological conjectures.

The GP-B apparatus is specially designed to accomplish this task. Inside an Earth-orbiting satellite is a dewar of superfluid helium, maintaining a temperature of 1.8 degrees above absolute zero. The dewar, in turn, houses a cigar-shaped quartz chamber. Within the chamber are four spinning spherical gyroscopes suspended in an electric field and encased by superconducting lead foil. The extreme cold, electrical levitation, and lead foil each cushion the gyroscopes from stray disturbances. Cold minimizes the random jostling of molecules; levitation minimizes the rocking due to the motion of the vessel; and the foil dampens the influence of Earth’s magnetic field. All this ensures that the gyroscopes are steered almost exclusively by gravitational effects.

As in the case of the STEP project, superconductivity plays a second, even more vital role. The gyroscopes are encircled with

superconducting loops hooked up to SQUID devices. As they turn ever so slightly, the SQUIDs are sensitive enough to record minute magnetic changes resulting from these reorientations. These devices provide the jeweler’s tools needed to examine the fine facets of relativity.

The gyroscopes themselves do not look like the archetypal toy spinning top (though mechanically they act in a similar fashion). Rather, each is a glassy sphere about the size of a Ping-Pong ball, machined to amazing smoothness. The surface of the ball does not depart from that of a perfect sphere by more than a millionth of an inch. To put this into perspective: If Earth were as spherical, its highest mountains and deepest oceans would represent deviations of only about 8 feet!

The GP-B satellite follows a polar orbit 400 miles above Earth. To provide a steady reference point, the orbital plane lines up with a star named HR8703, in the constellation Pegasus. This guide star offers an absolute background against which astronomers can take their measurements. As you can see, nothing about the mission has been left to chance.

The principal investigator of the GP-B experiment, who also happens to be the chief organizer and developer of the STEP project, is Stanford University physicist C. W. Francis Everitt. Born in 1934 in Sevenoaks, a town in the rolling Kentish countryside of south-eastern England, Everitt first learned about general relativity at an unusually young age. One day when he was 12 years old and was sitting at the family dinner table, his father fascinated him with compelling accounts of gravity’s actions in the universe. “My father,” relates Everitt, “who was an engineer/patent attorney with wide intellectual interests, talked to us about Einstein’s and Eddington’s popular books on the meaning of relativity. He contrasted these with quantum mechanics which, like Einstein, he found not entirely tasteful.”

Everitt did not pursue general relativity as his specialty, however, until 1961 when he assumed a position at the University of Pennsylvania. In that scholarly setting, Stanford physicist Bill Fairbank, a

pioneering researcher in gravitational and space science, gave a series of talks describing a number of “far out” experiments. “I found them and him fascinating,” recalls Everitt. “Since GP-B was the ‘farthest out’ of the lot, I volunteered to join his group to work on it. At a deeper level I was also much influenced by a remark [Nobel Prize– winning physicist Patrick] Blackett made to me in London: ‘If you can’t think of what physics to do next, invent some new technology; it’ll always lead to new physics’.”

Everitt has remarkable perseverance, given the decades taken for his major projects to reach fruition. From the time he began working on GP-B until the instant it blasted off into space, nine U.S. presidents took their oaths of office, musical tastes ran from Doo Wop to Hip Hop, and the world’s population more than doubled. Yet he persisted in his endeavors until he could set his creations free in space.

Convincing NASA to construct GP-B in times of tightening budgets required the skills of an expert salesman. Costing hundreds of millions of dollars, it was the most expensive and technologically ambitious science spacecraft ever commissioned by NASA, and its development became the subject of acrimonious debate in the science community. By and large, theoretical physicists wanted it, while astronomers thought it was unnecessary. Like many things in California, it came to represent a focal point of discontent between those in the north and those in the south. Stanford researchers, from the San Francisco area, wanted it built; while many Caltech researchers, from the Los Angeles area, wanted it scrubbed. An exception in the latter camp was Kip Thorne, who consistently supported the mission and was present at the launch. The experiment came perilously close to being closed down several times by NASA, whose critical visits to the Stanford campus were likened by one senior figure as akin to interrogations by the Spanish Inquisition.

Even the probe’s launch, from Vandenberg Air Force Base in south-central California, proved a nail-biting test of patience. From December 6, 2003, to April 17, 2004, the mission was held up

because of a revamping of its electronics. Then, a broken cord on the launch tower caused another delay. Finally, on April 19, things seemed ready to roll. The mission’s organizers bused several hundred people to the site—including various scientists and journalists keen to catch a glimpse of the historic event. Everything went well until the four-minute mark, when the launch was suddenly aborted due to unfavorable weather conditions. The wind profile at the time was not ideal, and nobody wanted to take any chances. Everybody then went back to their hotels, with some people drowning their sorrows at the local taverns. Everitt, however, did not seem downhearted.

Sure enough, the next day when the launch experts tried again, fate was much kinder. First, the epochal words: “Five, four, three, two, one….” A tense pause and then: “We have liftoff for Gravity-Probe B to test Einstein’s theory of relativity in space!”

A bright, unnatural light burst over the semiarid landscape of central California. The ball of radiation shot rapidly into the blue morning sky. Perceptibly later, a swath of ragged noise like an avalanche swept over the assembled onlookers. They appeared not to notice. Their eyes were fixed on the actinic light, now halfway up the sky, which marked the location of the spacecraft. It moved surprisingly quickly, heading out over the Pacific atop its Delta II rocket, accelerating to its final speed of about 17,000 miles per hour. In haste, cameras began to click, and a spontaneous cheer followed the probe upward. Heads craned back, hands sheltered narrowed eyes, and after little more than a minute, the sky was empty again, save for a few startled seagulls.

“We did it!” exclaimed one of the onlookers, Dimitri Kalligas. His tone mixed triumph and relief. Having worked on the mission at Stanford for several years in the late 1980s and early 1990s, he relished his dreams finally coming alive. He had traveled from his native Greece, together with his wife and two children to savor the moment. His eyes remained transfixed on the rapidly disappearing probe, even as his kids started pulling him impatiently toward the waiting shuttle bus. With a heartfelt gesture, he did the sign of

the cross and turned away; the spacecraft was in the heavens where it belonged. Perfectly aligned with its guide star, it has been in orbit ever since.

Another critical test of general relativity doesn’t involve space probes; it is taking place right here on Earth. The LIGO (Laser Interferometer Gravitational-Wave Observatory) project is attempting to detect gravitational waves, the elusive ripples in the space-time fabric first predicted by Einstein in 1916. A joint project of scientists from Caltech and MIT, the observatory’s detectors began operation in 2001 and have scanned for signals ever since.

Researchers believe cosmic catastrophes, such as supernova explosions or collisions between black holes, generate volleys of gravitational waves. These shock waves are thought to fan out in all directions from such disturbances, rattling any massive objects lying in their paths—in the same way that shops rumble when an elevated train passes overhead. Although they’ve yet to be found directly, astronomers Joseph Taylor and Russell Hulse have used binary pulsars (pairs of rapidly spinning neutron stars) to show that they are likely to exist. For this work they received the 1993 Nobel Prize for Physics.

The LIGO project was proposed by physicist Rainer (Rai) Weiss of MIT, along with Kip Thorne, Ronald Drever, Rochus Vogt, and other researchers at Caltech. Born in Berlin in 1932 to a politically active family, Weiss emigrated with them at a young age to the United States to escape the terrors of the Nazi regime. Like Everitt, he was not originally trained in general relativity but rather in another branch of physics. Weiss received his Ph.D. at MIT, in the field of atomic physics under the supervision of Jerrold Zacharias.

Zacharias had dedicated himself to building high-precision time-pieces based on the predictable rhythms of atoms, an extraordinarily important endeavor with broad implications for a variety of scien-

tific fields. As Weiss related, even Einstein in his final years, while engrossed in the search for a unified field theory, expressed interest in the MIT project to develop such clocks. If such devices could be perfected, one of their possible applications would be precise measurement of the effects of gravitation on time. This would help provide further confirmation of general relativity. Zacharias proudly introduced his project to Weiss.

“Jerrold said to me,” recalled Weiss, “that he had made himself a clock called the ‘fountain clock,’ which was a brand new idea involving tossing atoms high into the air and timing them. The idea was to get a long observation time on the atom. He kept telling me that if we could get the clock running, I would travel to the Jungfraujoch, a scientific observatory high in the Swiss Alps. He would be with his clock in the valley and we would measure the Einstein redshift. That’s what set the bee in my bonnet about relativity. But the clock didn’t work; it was a total failure.”

Nevertheless, Weiss’s interest in experimental tests of general relativity only grew. Obtaining a postdoctoral fellowship with Bob Dicke, he learned about attempts to measure gravitational radiation. “With Dicke I did something wacky,” continued Weiss. “I worked on a gravitometer to measure scalar waves [a hypothesized mode of radiation] hitting the Earth.”

Dicke, a master at cutting through thorny mechanical dilemmas, also instilled in Weiss the value of solid experimental design. Returning to MIT as a professor, Weiss embraced the teachings of his mentors and became one of the world’s leading experts in high-precision measurements of gravity.

The capstone of Weiss’s career is LIGO. Weiss developed the notion of using a special technique called laser interferometry to track minute movements of matter due to gravitational waves. Interferometry involves focused beams of light with well-defined frequencies (that is, laser beams) traveling along separate paths and then coming together again. The pattern created when the beams reunite provides precise information about the difference in path lengths.

Imagine the laser beam to be a troop of soldiers, marching down a road in perfect lockstep. At one point the band needs to cross a river, traversing two parallel bridges that at first glance appear to be identical. They split into two groups, continuing to march all the while. When they reunite, they realize that half of them are now marching out of cadence with the others. A member of the corps of engineers measures the bridges, and sure enough, one is 10 inches longer than the other. The extra length created the asynchrony. The same thing happens with light if it is forced to take several different trajectories. The results are characteristic interference patterns—bright and dark fringes that indicate where the beams are in or out of phase. The spacing of these fringes pertains to discrepancies between the routes.

Weiss and his collaborators realized that such hairbreadth measures would be needed if science had any chance of sensing the ghostly touch of gravitational pulses. Imagine two black holes colliding thousands of light-years from Earth. The resulting catastrophe would send shock waves through the fabric of space, with these rumbles eventually reaching Earth. Nevertheless, even the signal from such a cataclysmic event would offer only a feather touch on earthly objects. The end points of a yard-long iron rod, if it were completely free to move, would be displaced trillions of times less than the diameter of a speck of dust. Thus, the Caltech and MIT researchers planned LIGO to be miles long (for maximum effect) and calibrated as finely as state-of-the-art technology permitted.

The LIGO detectors, located in the states of Louisiana and Washington, are uniquely designed to record the murmurs of passing gravitational waves. Having two widely separated instruments helps rule out the effects of local terrestrial vibrations, such as miniearthquakes or other rumblings, that could masquerade as true signals. A team of planners selected each location to be as far away from urban noise as possible. No one would like a symphony of jackhammers, a band of tractor trailers, and an ensemble of landing jets to serenade the delicate equipment each day—not when it is listening for the subtler melodies of deep space.

Each detector is L shaped, with two perfectly straight vacuum pipes meeting at the corner. Like a colossal bowling alley, each pipe stretches out 2.5 miles long, with target masses on both ends. The idea is that gravitational waves would roll through the tubes, nudge the targets in each arm, and slightly alter their mutual separations. Along one arm, the masses would be pushed slightly closer, while along the other they’d be jostled slightly farther apart. Twin laser beams, meeting at the corner, would record these relative differences through the mechanism of light inference. The characteristic inference patterns would offer a telltale sign of gravitational disturbances (like from eons-old cosmic collisions) faintly touching Earth.

The direct detection of gravitational waves would be a crowning achievement for Einstein’s theory. It would well justify all the time and money spent on detectors and probes. As Weiss emphasized, “Observing gravitational waves would yield an enormous amount of information about the phenomenon of strong-field gravity. If we could detect black holes colliding that would be amazing.”

Such observations would offer a window into regions in which Einstein’s theory differs most greatly from Newton’s. General relativity is the foundation of modern-day astrophysics and cosmology. We cannot know if our theories of the cosmos are correct unless we can trust Einstein.

While future experiments may indicate a need for its modification, general relativity remains the gold standard. Yet despite its mathematical elegance and predictive success, some physicists are disappointed that it has never fully incorporated Mach’s principle. Einstein’s scheme never established a direct connection between local inertia and distant matter.

In the 1950s, British astrophysicist Dennis Sciama made a well-regarded attempt to bridge the gap. He wrote down equations

designed to make the locally measured mass of a particle depend on the rest of the matter in a continuously expanding universe. According to his calculations, the enormity of material in the cosmos would outweigh disparate regional influences and produce the uniform tendencies we know as inertia. Sciama never fully developed his model, however—he passed away in 1999 before completing his grand vision. Other physicists have launched similar efforts to encompass Machian notions, but none of their schemes have panned out so far. Perhaps their imaginations haven’t been properly nourished, say with cheap, wholesome cuisine.

Enter a trio of hungry cosmologists, famished for truth and a hearty meal. One of us (Paul Wesson), invited colleagues Sanjeev Seahra and Hongya Liu to a working dinner at a no-frills restaurant in Waterloo, Canada. Over heaping plates of seafood, the trio pondered ways of formulating Mach’s principle in terms of gravitational waves moving through an altered version of Einsteinian space-time. Through streams of relativistic calculations, hastily jotted down on available napkins, an intriguing picture emerged of a profoundly interconnected cosmos.

The modified theory involves expressing the space-time metric (which measures distances between space-time points) in complex numbers, instead of real (ordinary) numbers. Complex numbers, including terms such as the square root of negative one, play little role in traditional gravitational physics. However, they comprise an important part of quantum mechanics, helping to explain hidden connections between particles. In particular, they permit a complete description of particles in terms of “wave functions”: entities that can stretch out over vast regions of space.

By describing mass in terms of elongated waves rather than conventional clumps, the group found that it could express local inertia as a manifestation of the geometry of the universe as a whole. Thus, the combined effects of curvature throughout the entirety of space-time could exert a tug significant enough to affect the acceleration of

objects on Earth. The more matter in space-time (such as stars, galaxies, and quasars), the greater its fabric bends and the more pronounced the effect.

This approach to Mach is compatible with Einstein’s standard theory but goes considerably further. In a mathematical sense it extends general relativity to complex numbers, opening the way to all sorts of wavelike phenomena that were formerly the purview of quantum mechanics. In a physical sense the idea that a particle is a wave—whose behavior depends on the rest of the matter in the universe—links the local to the remote. This result came as a surprise to both quantum and classical physicists familiar with the approach, since it shows a way of bridging the two topics. More work is under way to see if the bridge represents a broad boulevard or just a catwalk.

Ancient mariners used to steer by the stars—relying on those distant beacons to help them sail across uncharted seas. If Mach’s principle is true, the stars guided their vessels in subtler ways than they ever could have imagined.