## Relativity, Special

## Relativity, Special

# Relativity, Special

German-American physicist Albert Einstein’s (1879–1955) theory of relativity consists of two major parts, the special theory of relativity and the general theory of relativity. Special relativity deals with phenomena that become noticeable when traveling near the speed of light at a constant straight-line velocity. Such objects are said to exist in inertial reference frames. General relativity deals with noninertial reference frames, namely, those that are accelerating, and with the phenomena that occur in strong gravitational fields. General relativity also uses the curvature of space to explain gravity.

## History

In the seventeenth century, English physicist and mathematician Sir Isaac Newton’s (1642–1727) Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) accomplished a grand synthesis of physics that used three laws of motion and the law of gravity to explain motions we observe both on the Earth and in the heavens. These laws worked very well, and continue to be used in modern day engineering.

In eighteenth and nineteenth centuries, philosophical and religious thought led many scientists to accept the argument that seemingly separate forces of nature shared an absolute reference frame. Against this backdrop, nineteenth century experimental work in electricity and magnetism resulted in James Clerk Maxwell’s (1831–1879) unification of concepts regarding electricity, magnetism, and light in his four famous equations describing electromagnetism. Maxwell and others scientists were, however, convinced that even electromagnetic waves needed propagating medium, just as waves in air require air and waves in water require water; thus, he and other scientists tried to establish the existence and properties of an “ether” or transmission medium for electromagnetic waves.

The nonexistence of ether was subsequently demonstrated by ingenious experiments of Albert Michelson (1852–1931) and Edward Morley (1838–1923). The importance and implications of the Michelson-Morley experiment was lost to much of the scientific world. Many scientists thought the Michelson-Morley result was simply a problem of experimental design or accuracy. In contrast, Einstein, then a clerk in the Swiss patent office, developed a theory of light that incorporated implications of Maxwell’s equations and of the nonexistence of ether.

Important implications of Einstein’s special theory—“special” because restricted to inertial reference frames—were length contraction and time dilation for bodies moving near the speed of light. In separate papers published in 1889, Irish physicist George Francis FitzGerald (1851–1901) and Dutch physicist Hendrik Antoon Lorentz (1853–1928) had pointed out that if the ether did exist, then the length of an object would change as it moved through the ether, the amount of contraction related to the square of the ratio of the object’s velocity to the speed of light. Subsequently this proposed effect was know as the FitzGerald-Lorentz contraction. Near the same time, French mathematician Jules-Henri Poincare´ (1854–1912) pointed out problems with concepts of simultaneity—the idea of a single, universal time-flow—and, just a year before Einstein published the special theory of relativity, Poincaré pointed out that observers in different reference frames would measure time differently. These anomalies led to the development of both relativity and quantum mechanics in the early part of the twentieth century.

In formulating his special theory of relativity, Einstein made the simple fundamental assumption that the laws of physics must be the same in all inertial (moving) reference frames. Since the speed of light arises directly out of the properties of magnetic and electric fields, as described by Maxwell, this meant that the speed of light must be constant regardless of its direction of propagation and independent of the velocity of the observer. The speed of light arises from physics, and physics must be the same, Einstein assumed, for all observers in inertial frames.

Key to the development of special relativity was Einstein’s confidence in the results of the Michelson-Morley experiment. To understand this experiment, imagine a bored brother and sister on a long train ride. (Einstein liked thought experiments using trains.) To pass the time, they get up and start throwing a baseball up and down the aisle of the train. The boy is in the front and the girl in the back. The train is traveling at 60 mph, and they can each throw the ball at 30 mph. As seen by an observer standing on the bank outside the train, the ball appears to be traveling 30 mph (60– 30) when the boy throws the ball to the girl and 90 mph (60+30) when the girl throws it back. The Michelson-Morley experiment was designed to look for similar behavior in light. Earth orbiting the Sun takes the place of the train, and the measured speed of light (like the baseball’s speed) should vary by Earth’s orbital speed depending on the direction the light is traveling. The experiment did not work as expected; the speed of light did not vary. Because Einstein took this result as the basic assumption that led to the special theory of relativity, the Michelson-Morley experiment is sometimes referred to as the most significant negative experiment in the history of science.

The orbit of the planet Mercury around the sun has some peculiarities that can not by explained by Newton’s classical laws of physics. The general theory of relativity can explain these peculiarities; they are described in the article on general relativity.

## Special relativity

To understand many concepts in relativity one first needs to understand the concept of a reference frame. A reference frame is a system for locating an object’s (or event’s) position in both space and time. It consists of both a set of coordinate axes and a clock. An object’s position and motion will vary in different reference frames. Go back to the example above of the boy and girl tossing the ball back and forth in a train. The boy and girl are in the reference frame of the train; the observer on the bank is in the reference frame of Earth. The reference frames are moving relative to each other, but there is no absolute reference frame. Either reference frame is as valid as the other.

For his special theory of relativity, published in 1905, Einstein assumed the result of the Michelson-Morley experiment. The speed of light will be the same for any observer in any inertial reference frame, regardless of how fast the observer’s reference frame is moving. Einstein also assumed that the laws of physics are the same in all reference frames. In the special theory, Einstein limited himself to the case of nonaccelerating, nonrotating reference frames (moving at a constant velocity), which are called inertial reference frames.

From these assumptions, Einstein was able to find several interesting consequences that are noticeable at speeds close to the speed of light (usually taken as greater than one tenth the speed of light). These consequences may violate our everyday common sense, which is based on the sum total of our experiences. Because we have never traveled close to the speed of light we have never experienced these effects. We can, however, accelerate atomic particles to speeds near the speed of light, and they behave as special relativity predicts.

## Space-time

Special relativity unified our concepts of space and time into the unified concept of space-time. In essence time is a fourth dimension and must be included with the three space dimensions when we talk about the location of an object or event. As a consequence of this unification of space and time the concept of simultaneous events has no absolute meaning. Whether or not two events occur simultaneously and the order in which different events occurs depends on the reference frame of the observer.

If, for example, you want to meet a friend for lunch, you have to decide both which restaurant to eat at and when to eat lunch. If you get either the time or the restaurant wrong you are not able to have lunch with your friend. You are in essence specifying the space-time coordinates of an event, a shared lunch. Note that both the space and time coordinates are needed, so space and time are unified into the single concept of space-time.

## Unusual effects of motion

Imagine a rocket ship traveling close to the speed of light. A number of unusual effects occur: Lorentz contraction, time dilation, and mass increase. These effects are as seen by an outside observer at rest. To the pilot in the reference frame of the rocket ship all appears normal. These effects will occur for objects other than rocket ships and do not depend on there being someone inside the moving object. Additionally, they are not the result of faulty measuring devices (clocks or rulers); they result from the fundamental properties of space-time.

A rocket moving close to the speed of light will appear shorter as seen by an outside observer at rest. All will appear normal to an observer such as the pilot moving close to the speed of light inside the rocket. As the speed gets closer to the speed of light, this effect increases. If the speed of light were attainable the object would appear to have a length of zero to an observer at rest. The length of the rocket (or other moving object) measured by an observer at rest in the reference frame of the rocket, such as the pilot riding in the rocket, is called the proper length. This apparent contraction of a moving object as seen by an outside observer is called the Lorentz contraction.

A similar effect, time dilation, occurs for time. As seen by an outside observer at rest, a clock inside a rocket moving close to the speed of light will move more slowly. The same clock appears normal to the pilot moving along with the rocket. The clock is not defective; the rate at which time flows changes. Observers in different reference frames will measure different time intervals between events. The time interval between events measured both at rest in the reference frame of the events and with the events happening at the same place is called the proper time. This time dilation effect increases as the rocket gets closer to the speed of light. Traveling at the speed of light or faster is not possible according to special relativity, but if it were, time would appear to the outside observer to stop for an object moving at the speed of light and to flow backward for an object moving faster than light. (The idea of time dilation is amusingly exaggerated in a famous limerick by Arthur Reginald Buller, 1912: “There was a young lady named Bright, Whose speed was much faster than light. She set out one day, In a relative way, And returned on the previous night.”) As seen by an outside observer, the mass of the rocket moving close to the speed of light increases. This effect increases as the speed increases so that if the rocket could reach the speed of light it would have an infinite mass. As for the previous two effects to an observer in the rocket, all is normal. The mass of an object measured by an observer in the reference frame in which the object is at rest is called the rest mass of the object.

These three effects are usually thought of in terms of an object, such as a rocket, moving near the speed of light with an outside observer who is at rest. But it is important to remember that according to relativity there is no preferred or absolute reference frame. Therefore the viewpoint of the pilot in the reference frame of the rocket is equally valid. To the pilot, the rocket is at rest and the outside observer is moving near the speed of light in the opposite direction. The pilot therefore sees these effects for the outside observer. Who is right? Both are.

## Speed of light limit

Think about accelerating the rocket in the above example. To accelerate the rocket (or anything else) an outside force must push on it. As the speed increases, the mass appears to increase as seen by outside observers including the one supplying the force (the one doing the pushing). As the mass increases, the force required to accelerate the rocket also increases. (It takes more force to accelerate a refrigerator than a feather.) As the speed approaches the speed of light the mass and hence the force required to accelerate that mass approaches infinity. It would take an infinite force to accelerate the object to the speed of light. Because there are no infinite forces no object can travel at the speed of light. An object can be accelerated arbitrarily close to the speed of light, but the speed of light can not be reached. Light can travel at the speed of light only because it has no mass. The speed of light is the ultimate speed limit in the universe.

*E* =*mc* ^{2}

This famous equation means that matter and energy are interchangeable. Matter can be directly converted to energy, and energy can be converted to matter. The equation, *E* =*mc* ^{2}, is then a formula for the amount of energy corresponding to a certain amount of matter. E represents the amount of energy, m the mass, and c the speed of light. Because the speed of light is very large a small amount of matter can be converted to a large amount of energy. This change from matter into energy takes place in nuclear reactions such as those occurring in the sun, nuclear reactors, and nuclear weapons. Nuclear reactions release so much energy and nuclear weapons are so devastating because only a small amout of mass produces a large amount of energy.

## A pair of paradoxes

A paradox is an apparent contradiction that upon closer examination has a noncontradictory explanation. Several paradoxes arise from the special theory of relativity. The paradoxes are interesting puzzles, but more importantly, help illustrate some of the concepts of special relativity.

Perhaps the most famous is the twin paradox. Two twins are initially the same age. One of the twins becomes an astronaut and joins the first interstellar expedition, while the other twin stays home. The astronaut travels at nearly the speed of light to another star, stops for a visit, and returns home at nearly the speed of light. From the point of view of the twin who stayed home, the astronaut was traveling at nearly the speed of light. Because of time dilation the homebound twin sees time as moving more slowly for the astronaut, and is therefore much older than the astronaut when they meet after the trip. The exact age difference depends on the distance to the star and the exact speed the astronaut travels. Now think about the astronaut’s reference frame. The astronaut is at rest in this frame. Earth moved away and the star approached at nearly the speed of light. Then Earth and star returned to their original position at nearly the speed of light. So, the astronaut expects to be old and reunite with a much younger twin after the trip. The resolution to this paradox lies in the fact that for the twins to reunite, one of them must accelerate by slowing down, turning around and speeding up. This acceleration violates the limitation of special relativity to inertial (nonaccelerating) reference frames. The astronaut, who is in the noninertial frame, is therefore the younger twin when they reunite after the trip. Unlike much science fiction in which star ships go into a fictional warp drive, real interstellar travel will have to deal with the realities of the twin paradox and the speed of light limit.

The garage paradox involves a very fast car and a garage with both a front and back door. When they are both at rest, the car is slightly longer than the garage, so it is not possible to park the car in the garage with both doors closed. Now imagine a reckless driver and a doorman who can open and close both garage doors as fast as he wants but wants only one door open at a time. The driver drives up the driveway at nearly the speed of light. The doorman sees the car as shorter than the garage, opens the front door, allows the car to drive in, closes the front door, opens the back door, allows the car to drive out without crashing, and closes the back door. The driver on the other hand, sees the garage as moving and the car as at rest. Hence, to the driver the garage is shorter than the car. How was it possible, in the driver’s reference frame, to drive through the garage without a crash? The driver sees the same events but in a different order. The front door opens, the car drives in, the back door opens, the car drives through, the front door closes, and finally the back door closes. The key lies in the fact that the order in which events appear to occur depends on the reference frame of the observer. (See the section on space-time.)

## Experimental verification

Like any scientific theory, the theory of relativity must be confirmed by experiment. So far, relativity has passed all its experimental tests. The special theory predicts unusual behavior for objects traveling near the speed of light. So far no human has traveled near the speed of light. Physicists do, however, regularly accelerate subatomic particles with large particle accelerators like the recently canceled Superconducting Super Collider (SSC). Physicists also observe cosmic rays which are particles traveling near the speed of light coming from space. When these physicists try to predict the behavior of rapidly moving particles using classical Newtonian physics, the predictions are wrong. When they use the corrections for Lorentz contraction, time dilation, and mass increase required by special relativity, it works. For example, muons are very short lived subatomic particles with an average lifetime of about two millionths of a second. However when they are traveling near the speed of light physicists observe much longer apparent lifetimes for

### KEY TERMS

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

**Lorentz contraction—** An effect that occurs in special relativity; to an outside observer the length appears shorter for an object traveling near the speed of light.

**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.

**Time dilation—** An effect that occurs in special relativity; to an outside observer time appears to slow down for an object traveling near the speed of light.

muons. Time dilation is occurring for the muons. As seen by the observer in the lab time moves more slowly for the muons traveling near the speed of light.

Time dilation and other relativistic effects are normally too small to measure at ordinary velocities. But what if we had sufficiently accurate clocks? In 1971, two physicists, J.C. Hafele and R.E. Keating, used atomic clocks accurate to about one billionth of a second (one nanosecond) to measure the small time dilation that occurs while flying in a jet plane. They flew atomic clocks in a jet for 45 hours then compared the clock readings to a clock at rest in the laboratory. To within the accuracy of the clocks they used, time dilation occurred for the clocks in the jet as predicted by relativity. Relativistic effects occur at ordinary velocities, but they are too small to measure without very precise instruments.

The formula *E* =*mc* ^{2} predicts that matter can be converted directly to energy. Nuclear reactions that occur in the Sun, in nuclear reactors, and in nuclear weapons confirm this prediction experimentally.

Albert Einstein’s special theory of relativity fundamentally changed the way scientists characterize time and space. So far it has passed all experimental tests. It does not however mean that Newton’s law of physics is wrong. Newton’s laws are an approximation of relativity. In the approximation of small velocities, special relativity reduces to Newton’s laws. Relativity, in turn is an approximation to some other, even more comprehensive law that will include the facts of quantum physics (which relativity ignores).

*See also* Cosmology; Quantum mechanics.

## Resources

### BOOKS

Cheng, Ta-Pei. *Relativity, Gravitation, and Cosmology: A Basic Introduction.* New York: Oxford University Press, USA, 2005.

Einstein, Albert. *Relativity.* New York: Crown, 1961. Mould, R.A. Basic Relativity. Springer Verlag, 2001.

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

Schroödinger, *Edwin. Space-Time Structure, Reprint Edition.* Cambridge University Press, 2002.

### PERIODICALS

Cho, Adrian. “Special Relativity Reconsidered.” *Science.* 307 (2005): 866-868.

Ashby, Neil. “General Relativity: Frame-Dragging Confirmed.” *Nature.* 431 (2004): 918-919.

Paul A. Heckert

K. Lee Lerner