Relativity, General

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Relativity, General

The theory of relativity was developed by the German physicist Albert Einstein (1879–1955) in the early twentieth century and quickly became one of the basic organizing ideas of physics. Relativity actually consists of two theories, the special theory (announced in 1905) and the general (1915). Special relativity describes the effects of straight-line, constant-velocity motion on the mass and size of objects and on the passage of time. It also states that mass and energy can be transformed into each other and that movement faster than the speed of light is impossible. General relativity describes the effects of accelerated (non-constant or curving) motion and of gravitational fields on mass, size, and time. It also states that matter and “empty” space influence each other in a complex fashion and that the universe is finite in size.

According to relativity, our commonsense notions of space and time are only approximately true at best and are completely unreliable in many situations (e.g., in intense gravitational fields or for relative speeds approaching that of light). Relativity’s predictions have been extensively tested by experiment and found to be highly accurate; relativity is thus a “theory” in the scientific sense that it is a well-tested structure of ideas that explains a specific aspect of nature. The theory may be—will almost certainly be—changed and improved in the future, but as far as it goes it is not doubtful or speculative.

History

In the seventeenth century, Isaac Newton (1642–1727) proposed three laws of motion and one of gravitation to describe and predict the motions of objects on the Earth and in the heavens. Newton’s laws worked very well, and became the centerpiece of the system of laws known as Newtonian physics. In the late nineteenth century, however, physicists began to be troubled by certain experiments that did not obey the predictions of the physics they knew. The need to account for these anomalies led to the development of both relativity and quantum mechanics in the early twentieth century.

One such anomaly was the Michelson-Morley experiment, which disproved the hypothesis that light waves propagate through an intangible, universe-filling substance called the ether as ripples propagate in water. The puzzle for nineteenth-century physicists was this: light was guaranteed by Maxwell’s well-tested electromagnetic theory to have a specific velocity (186,282 miles per second [299, 972 km/sec], usually designated c ). The Michelson-Morley experiment showed that light does not travel at c with respect to the ether, while certain astronomical tests had also ruled out the emitter theory of light, according to which every beam of light travels at c with respect to its source, like a bullet fired from a gun. Yet if a beam of light does not move at c with respect either to ether or to its source, what does it move at c with respect to? Einstein’s answer was simple yet radical: light moves at c with respect to everything, all the time. From this single bold hypothesis Einstein unfolded the entire theory of special relativity.

Special relativity is called “special” because its equations are valid only for one special set of cases, namely, systems of phenomena moving in straight lines at constant velocities (inertial reference systems).

Einstein at once sought to extend his equations to describe reference systems undergoing acceleration, not just inertial reference systems. This required him to account for gravity, which accelerates all objects. Since Newton, physicists had conceived of gravity as a “force,” a mysterious attraction exerted instantaneously by every bit of matter on every other. Newton himself had been uncomfortable with this notion, but could not think of an alternative; Einstein did. As part of his general theory of relativity, he proposed that gravity is a manifestation of the geometry of space itself, and that this geometry is imposed on space by the matter in space. This implies that space is a thing having specific, changeable properties, not a featureless void—“absolute space.” The proposal that space is not absolute was one of Einstein’s most daring ideas.

On the basis of his general theory, Einstein made a number of interesting claims, such as that gravity propagates at the speed of light, that light itself must be diverted by gravity, that time passes more slowly in a stronger gravitational field than in a weaker, and that the universe is finite in size. Scientists at once began looking for ways to test some of these claims. In 1919, a total eclipse of the sun allowed British astronomers to photograph stars whose light had, at just that time, to graze the sun on its way to Earth. Ordinarily, the sun’s glare prevents such observations; during an eclipse, they are possible because the sun is blocked by the moon. Einstein predicted that the stars’ light would be bent a certain, measurable amount by the sun’s gravity, changing the stars’ apparent position. The observations were made, and Einstein’s prediction was confirmed precisely.

Another early check on general relativity was its ability to solve the long-standing puzzle of the orbit of the planet Mercury, which has peculiarities that cannot by explained by Newton’s laws. Each time Mercury orbits the sun the point at which it comes closest to the sun, its perihelion, shifts by a small amount. This small but measurable motion—termed precession—had first been measured by the French astronomer Urbain Leverrier (1811–1877) in 1859. The rate of precession of Mercury’s orbit is 43 seconds of arc per century, meaning that about three million years are required for a complete cycle of Mercury’s perihelion around the sun. All planets, including Earth, precess, but the effect was only measurable for Mercury because Mercury is closest to the sun, where the gravitational field is strongest. Newton’s theory of gravity could not account for Mercury’s precession, but general relativity could—a strong argument in its favor. In later years, many experiments have confirmed the predictions of general relativity to high precision. Such confirmations are still regularly announced by physicists seeking to determine the bounds of the theory’s accuracy.

Basic concepts of the theory

To understand relativity one needs to first understand the concept of a reference frame. A reference frame is a conceptual system for locating objects and events in space and time. It consists of a “frame” or set of spatial coordinate axes (for example, north-south, east-west, and up-down) and of a “clock” (that is, any means of measuring time). Such a system is called a reference frame because any object or event’s position and state of motion can, in principle, be described by referring to points on the axes of the frame and to readings given by clocks. A description of any object or event’s location and speed depends on the reference frame on which the description is based. If, for example, you are riding in a car, you are at rest in a reference frame that is rigidly attached to the car but are moving in a reference frame that is rigidly attached to the road. The two frames are moving relative to each other. A key insight of Einstein’s was that there is no absolute reference frame. That is, the reference frame attached to the car is precisely as valid as the frame attached to the road. A reference frame that is moving at a constant velocity in a straight line is termed an inertial reference frame. A reference frame that is accelerating or rotating is termed noninertial. As mentioned above, the theory of general relativity expands the theory of special relativity from inertial to noninertial reference frames.

One of the effects of special relativity was to combine our concepts of space and time into a unified concept: space-time. According to the space-time concept, space and time are not independent as they are in everyday experience and in Newtonian physics: that is, in relativistic space-time an observer’s state of motion through space (velocity) has a real effect on how quickly time passes in their frame of reference (that is, a frame of reference moving with them) relative to observers in different frames of reference. General relativity allows that space-time to be “curved” by the matter contained in it, and explains gravity as a manifestation of curved space-time.

General relativity

Principle of equivalence

Einstein’s general theory of relativity, announced in 1915, uses the principle of equivalence to explain the force of gravity. There are two logically equivalent statements of this principle. First, consider an enclosed room on Earth. In it, one feels a downward gravitational force. This force is what we call weight; it causes unsupported objects to accelerate downward at a rate of 32 ft/s² (9.8 m/s²). Now imagine an identical room located in space, far from any masses. There will be no gravitational forces in the room, but if the room is accelerated “upward” (in the direction of its ceiling) at 9.8 m/s²—say, by a rocket attacked to its base— then unsupported objects in the room will accelerate toward its floor at rate of 9.8 m/s², and a person standing in the room will feel normal Earth weight. We experience a similar effect when we are pushed back into the seat of a rapidly accelerating car. This type of force is termed an inertial force and is a result of being located in an accelerating (noninertial) reference frame. The inertial force acts in the opposite direction of the acceleration producing it (i.e., the room accelerates toward its ceiling, objects in the room “fall” toward its floor). Is it possible to tell, solely by means of observations made inside the room, whether the room is on Earth or not? No; the conclusion is that gravitational forces are indistinguishable from inertial forces in an accelerating reference frame.

What if the room in space is not accelerating? Then there will be no inertial forces, so objects in the room will not fall and the occupants will be weightless. Now imagine that the room is magically transported back to Earth, but by a slight error it appears in the air 100 ft (30.5 m) above the ground rather than on the surface. Earth’s gravity will at once begin to accelerate the room downward at 9.8 m/s². Just as when the room is accelerating in space, this acceleration will produce an inertial force that is indistinguishable from a gravitational force. In this case, however, the inertial force is upward and the gravitational force (Earth’s pull) is downward. These forces cancel out exactly, rendering the occupants of the room weightless—for as long as it takes the room to fall 100 feet, at least. In general, then, objects that are in free fall—that is, falling freely in a gravitational field—will be weightless. Astronauts in orbit around Earth are weightless not because there is no gravity there, but because they are in free fall. You can test the claim that freely falling objects will be weightless by putting a small hole in the bottom of an empty plastic milk jug and filling the jug with water. Drop the jug, uncovering the hole at the moment of release. While the jug is falling no water will come out the hole, proving that freely falling water is weightless.

The second statement of the principle of equivalence involves the concept of mass. Mass appears in two distinct ways in Newton’s laws. In Newton’s second law of motion, the amount of force required to accelerate an object increases as its mass increases. That is, it takes twice as much force to accelerate two kilograms of mass at a given rate as it takes to accelerate one kilogram of mass. The sort of mass that appears in Newton’s second law is termed the inertial mass. Meanwhile, in Newton’s law of gravity the gravitational force between two objects increases as the mass of the objects increases. The mass in the law of gravity is termed the gravitational mass. The inertial mass and gravitational mass of an object are expressed using the same units and are always equal, but there is no obvious reason, in Newtonian physics, why they should be. Newtonian physicists were obliged to accept the identity of inertial and gravitational mass as a sort of perfect coincidence. Einstein, however, declared that they are exactly the same thing. This is the second statement of the principle of equivalence.

These two statements of the principle of equivalence are logically equivalent, meaning that it is possible to use either statement to prove the other. The principle of equivalence is the basic assumption behind the general theory of relativity.

Geometrical nature of gravity

From the principle of equivalence, Einstein was able to derive the general theory of relativity. General relativity explains the force of gravity as a result of the geometry of space-time. To see how it does so, consider the example given above of the enclosed room being accelerated in space far from any masses. A person in the room throws a ball perpendicular to the direction of acceleration—that is, across the room. Because the ball is not being pushed directly by whatever is accelerating the room, it follows a path that is curved as seen by the person in the room. (You would see the same effect if you rolled a marble on a tray in an accelerating car. The marble’s path would curve toward the back of the car.) Now imagine that the ball is replaced by a beam of light shining sideways in the enclosed, accelerating room. The person in the room sees the light beam follow a curved path, just as the ball does and for the same reason. The only difference is that the deflection of the light beam—how much it drops as it crosses the room—is smaller than the deflection of the ball, because the light is moving so fast it gets to the wall of the room before the room can move very far.

Now consider the same enclosed room at rest on the surface of earth. A ball thrown sideways will follow a downward curved path because of Earth’s gravitational field. What will a light beam do? The principle of equivalence states that it is not possible to distinguish between gravitational forces and inertial forces; therefore, any experiment must have the same result in the room at rest on earth as in the room accelerated in space. The equivalence principle thus predicts that a light beam will therefore be deflected downward in the room on earth just as it would in the accelerated room in space. In other words, light falls.

The question is, why? Light has no mass. According to Newton’s law of gravity, only mass is affected by gravity. Light, therefore, which is weightless, should move in a straight line. Einstein proposed that in a sense it does move in a straight line; that, in fact, the nature of straight lines is changed by the presence of mass, and this geometrical change is what gravity is. Another way of saying this is that space-time is “curved.” (The physical meaning of this statement is far from obvious, and this description is not meant to offer a complete explanation of the concept of curved space-time.)

Prior to Einstein, people thought of space and time as being independent of each other, and of space as being absolute (unaffected by matter and energy in it) and flat (Euclidean in geometry). Euclidean geometry is the set of rules that describes the geometry of flat surfaces and is studied in high-school geometry classes. In general relativity, however, space-time is not necessarily Euclidean; the presence of a mass curves or warps space-time. This warping is similar to the curvature in a horizontal sheet of rubber that is stretched downward by a weight placed in the center. The curvature of space-time is impossible to visualize, because it is the curvature of a four-dimensional space rather than of than a two-dimensional surface, but can be described mathematically and is quite real. The curvature of space-time produces the effects we call gravity. When we travel long distances on the surface of Earth, we must follow a curved path because the surface of Earth is curved; similarly, an object traveling in curved space-time follows a curved path. For example, Earth orbits the sun because space-time near the sun is curved. Earth’s nearly circular path around the sun is analogous to the path of marble circling the upper part of a curved funnel, refusing to fall in; an object falling straight toward the sun is like a marble rolling straight down into the funnel.

One consequence of the curvature of space-time by matter is that the universe is finite in size. This does not mean that space comes to an end, as the space inside a balloon comes to an end at the inner surface of the balloon; space is finite but unbounded. Physicists often compare our situation to that of imaginary two-dimensional (perfectly flat) beings living on the surface of a sphere, who can make measurements only on the surface of the sphere and cannot see or even visualize the three-dimensional space in which their sphere is embedded. If they explore the whole surface of their universe they will find that it has only so many square inches of surface (is finite) but has no edges (is unbounded). Our universe is analogous. Furthermore, according to general relativity, the size of the universe depends directly on the amount of matter and energy in it.

Experimental verification

Bending of light

The first experimental confirmation of general relativity occurred in 1919, shortly after the theory was published. Newton’s law of gravity predicts that gravity will not deflect light, which is massless; however, the principle of equivalence, on which general relativity is founded, predicts that gravity will bend light rays. The nearest mass large enough to have a noticeable effect on light is the sun. The apparent position of a star almost blocked by the sun should be measurably shifted as the light from the star is bent by the sun’s gravity. As described above, observations made during the total eclipse of 1919 found the predicted shift.

More recently, this effect has been observed in the form of gravitational lenses. If a galaxy is located directly between us and a more distant object, say a quasar, the mass of the galaxy bends the light coming almost straight towards us (but passing around the galaxy) from the more distant object. If the amount of bending is just right, light from the quasar that would otherwise have missed us is focused on us by the galaxy’s gravity. When this occurs we may see two or more images of the quasar, dotted around the image of the intervening galaxy. A number of gravitational lenses have been observed.

Binary pulsar

The 1993 Nobel Prize in physics was awarded to U.S. physicists Joseph Taylor (1941–) and Russell Hulse (1950–) for their 1974 discovery of a binary pulsar. A pulsar, or rapidly rotating neutron star, is the final state of some stars; a star become a neutron star if, once its nuclear fuel has burnt out, its gravity is strong enough to collapse it to about the size of a small city. A binary pulsar is two pulsars orbiting each other. Because pulsars are extremely dense they have extremely strong enough gravitational fields. Binary pulsars therefore provide an excellent experimental test of general relativity’s predictions, which vary most from the predictions of Newtonian theory for strong fields. General relativity predicts that some systems of objects—including binary pulsars—should emit gravity waves that travel at the speed of light, and that these waves should remove energy from such systems. This energy loss should slowly brake the speed of rotation of a binary pulsar. Taylor and Hulse were able to measure a binary pulsar’s rate of slowing, and showed that it agreed with the predictions of general relativity.

Consequences of general relativity

The German astronomer Karl Schwarzschild (1873–1916) first used general relativity to predict the existence of black holes, which are stars that are so dense that not even light can escape from their gravitational field. Because the gravitational field around a black hole is so strong, we must use general relativity to understand the properties of black holes; indeed, most of what we know about black holes comes from theoretical studies based on general relativity. Ordinarily we think of black holes as having been formed from the collapse of a massive star, but American physicist Stephen Hawking (1942–) has combined general relativity with quantum mechanics to predict the existence of primordial quantum black holes. These primordial black holes (if they exist) were formed by the extreme turbulence of the big bang during the formation of the Universe. Hawking predicts that over sufficiently long time these small, quantum black holes—and larger black holes, too—can evaporate, that is, lose their mass to surrounding space despite their intense gravity, like drops of water evaporating into dry air. This view has replaced the earlier, too-simple belief that nothing can escape from a black hole.

General relativity also has important implications for cosmology, the study of the structure of the Universe. The equations of general relativity state not only that the universe is finite but that it may be contracting or expanding. Einstein noticed this result of his theory, but assumed that the universe must be stable in size, neither contracting nor expanding, and therefore added to his equations a numerical term called the “cosmological constant.” This constant was basically a fudge factor that Einstein used to adjust his equations so that they predicted a static universe. Later, American astronomer Edwin Hubble (1889–1953), after whom the Hubble Space Telescope is named, discovered that the universe is expanding. Einstein visited Hubble, examined his data, and admitted that Hubble was right. Einstein later called his cosmological constant the biggest blunder of his life; however, modern cosmologists have found that Einstein may have been right after all about the need for a cosmological constant in the equations of general relativity. Recent observations show that the universe’s rate of expansion is probably accelerating. This means that some force resembling negative gravity—a “force” that originates in matter but that pushes other matter away rather than attracting it—may exist. If it does, a nonzero value for Einstein’s cosmological constant may be required to describe the structure of the universe. Astronomers are debating and researching this question intensively.

Albert Einstein’s general theory of relativity fundamentally changed the way we understand gravity and the universe in general. So far, it has passed all experimental tests. This, however, does not mean that Newton’s law of gravity is wrong. Newton’s law is an approximation of general relativity; that is, in the approximation of small gravitational fields, general

KEY TERMS

General relativity— The part of Einstein’s theory of relativity that deals with accelerating (noninertial) reference frames.

Principle of equivalence— The basic assumption of general relativity: gravitational forces are indistinguishable from apparent forces caused by accelerating reference frames, or alternatively, gravitational mass is identical to inertial mass.

Reference frames— A system, consisting of both a set of coordinate axes and a clock, for locating an object’s (or event’s) position in both space and time.

Space-time— Space and time combined as one unified concept.

Special relativity— The part of Einstein’s theory of relativity that deals only with nonaccelerating (inertial) reference frames.

relativity reduces to Newton’s law of gravity. General relativity, too, is only an approximate description of certain aspects of nature. This is known because general relativity does not agree with the predictions of quantum mechanics (the other great organizing idea of modern physics) in describing extremely small phenomena. Quantum mechanics, similarly, makes excellent predictions in its own domain of application (the extremely small) but erroneous predictions at the cosmic scale. Physicists are striving to discover an even more general or unified theory that will yield both general relativity and quantum mechanics as special cases.

Resources

BOOKS

Clark, Ronald W. Einstein: The Life and Times. New York: World Publishing, 1971.

Einstein, Albert. Relativity. New York: Crown, 1961. Plebanski, Jerzy and Andrzej Krasinski. An Introduction to General Relativity and Cosmology. Cambridge, UK: Cambridge University Press, 2006.

Rindler, Wolfgang. Relativity: Special, General, and Cosmological. New York: Oxford University Press USA, 2006.