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Special Relativity in a Nutshell

  • By Brian Greene
  • Posted 09.15.11
  • NOVA

The relativity of space and of time is a startling conclusion. I have known about it for more than 25 years, but even so, whenever I quietly sit and think it through, I am amazed. From the well-worn statement that the speed of light is constant, we conclude that space and time are in the eye of the beholder.

London traffic at night

Einstein showed that time and space are not fixed entities, as they seem to us, but are actually changeable. As Brian Greene explains, it’s all based on an observer’s state of motion compared to that of another observer moving at a different speed. Enlarge Photo credit: © Tiffany Chan/iStockphoto

Time and space are relative

Each of us carries our own clock, our own monitor of the passage of time. Each clock is equally precise, yet when we move relative to one another, these clocks do not agree. They fall out of synchronization; they measure different amounts of elapsed time between two chosen events. The same is true of distance. Each of us carries our own yardstick, our own monitor of distance in space. Each yardstick is equally precise, yet when we move relative to one another, these yardsticks do not agree; they measure different distances between the locations of two specified events.

If space and time did not behave this way, the speed of light would not be constant and would depend on the observer’s state of motion. But it is constant; space and time do behave this way. Space and time adjust themselves in an exactly compensating manner so that observations of light’s speed yield the same result, regardless of the observer’s velocity.

Getting the quantitative details of precisely how the measurements of space and time differ is more involved, but requires only high school algebra. It is not the depth of mathematics that makes Einstein’s special relativity challenging. It is the degree to which the ideas are foreign and apparently inconsistent with our everyday experiences. But once Einstein had the key insight—the realization that he needed to break with the more than 200-year-old Newtonian perspective on space and time—it was not hard to fill in the details. He was able to show precisely how one person’s measurements of distances and durations must differ from those of another in order to ensure that each measures an identical value for the speed of light.

man running by clocks

Einstein taught us that we must consider not only motion through space but motion through time. The two are inextricably intertwined. Enlarge Photo credit: © Mark Evans/iStockphoto

An extremely simple idea

To get a fuller sense of what Einstein found, imagine that Bart has a skateboard with a maximum speed of 65 miles per hour. If he heads due north at top speed—reading, whistling, yawning, and occasionally glancing at the road—and then merges onto a highway pointing in a northeasterly direction, his speed in the northward direction will be less than 65 miles per hour. The reason is clear. Initially, all his speed was devoted to northward motion, but when he shifted direction some of that speed was diverted into eastward motion, leaving a little less for heading north.

The combined speed of any object’s motion through space and its motion through time is always precisely equal to the speed of light.

This extremely simple idea actually allows us to capture the core insight of special relativity. Here’s how:

We are used to the fact that objects can move through space, but there is another kind of motion that is equally important: Objects also move through time. Right now, the watch on your wrist and the clock on the wall are ticking away, showing that you and everything around you are relentlessly moving through time, relentlessly moving from one second to the next and the next. Newton thought that motion through time was totally separate from motion through space—he thought these two kinds of motion had nothing to do with each other. But Einstein found that they are intimately linked.

Motion through time

In fact, the revolutionary discovery of special relativity is this: When you look at something like a parked car, which from your viewpoint is stationary—not moving through space, that is—all of its motion is through time. The car, its driver, the street, you, your clothes are all moving through time in perfect synch: second followed by second, ticking away uniformly.

speeding car

All the "motion" of a motionless car is through time, but as soon as it speeds away, some of that car’s motion through time is diverted into motion through space. Enlarge Photo credit: © David Birkbeck/iStockphoto

But if the car speeds away, some of its motion through time is diverted into motion through space. And just as Bart’s speed in the northward direction slowed down when he diverted some of his northward motion into eastward motion, the speed of the car through time slows down when it diverts some of its motion through time into motion through space. This means that the car’s progress through time slows down, and therefore time elapses more slowly for the moving car and its driver than it elapses for you and everything else that remains stationary.

That, in a nutshell, is special relativity.

Complementary motions

In fact, we can be a bit more precise and take the description one step further. Bart had no choice but to limit his top speed to 65 miles per hour. This is important to the story, because if he sped up enough when he angled northeast, he could have compensated for the speed diversion and thereby maintained the same net speed toward the north. But no matter how hard he revved the skateboard’s engine, his total speed—the combination of his speed toward the north and his speed toward the east—remained fixed at the maximum of 65 miles per hour. And so when he shifted his direction a bit toward the east, he necessarily caused a decreased northward speed.

Special relativity declares a similar law for all motion: The combined speed of any object’s motion through space and its motion through time is always precisely equal to the speed of light. At first, you may instinctively recoil from this statement since we are all used to the idea that nothing but light can travel at light speed. But that familiar idea refers solely to motion through space. We are now talking about something related, yet richer: an object’s combined motion through space and time.

The key fact, Einstein discovered, is that these two kinds of motion are always complementary. When the parked car you were looking at speeds away, what really happens is that some of its light-speed motion is diverted from motion through time into motion through space, keeping their combined total unchanged. Such diversion unassailably means that the car’s motion through time slows down.

melting clock

Because motion through both time and space must always add up to the speed of light, when an object (such as a photon) moves through space at light speed, Einstein reasoned, there’s no "room" for motion through time, and time, consequently, stops. Enlarge Photo credit: © Barbara Henry/iStockphoto

Stopping time

As an example, imagine Bart could go 500 million mph on his skateboard—that’s about three-fourths the speed of light, which is about 670 million mph—and his sister Lisa, who is standing still, could see his watch as he sped past her. She would see that it was ticking about two-thirds as fast as her own. For every three hours that passed on Lisa’s watch, she would see that only two had passed on Bart’s. His rapid motion through space would have proved a significant drain on his speed through time.

This is not dexterous wordplay, sleight of hand, or psychological illusion. This is how the universe works.

Moreover, the maximum speed through space is reached when all light-speed motion through time is fully diverted into light-speed motion through space—one way of understanding why it is impossible to go through space at greater than light speed. Light, which always travels at light speed through space, is special in that it always achieves such total diversion.

And just as driving due east leaves no motion for traveling north, moving at light speed through space leaves no motion for traveling through time! Time stops when traveling at the speed of light through space. A watch worn by a particle of light would not tick at all. Light realizes the dreams of Ponce de León and the cosmetics industry: It doesn’t age.

The nature of nature

As this description makes clear, the effects of special relativity are most pronounced when speeds (through space) are a significant fraction of light speed. But the unfamiliar, complementary nature of motion through space and time always applies. The lesser the speed, the smaller the deviation from prerelativity physics—from common sense, that is—but the deviation is still there, to be sure.

streaking comet

We may not, indeed we cannot, intuitively grasp the relativity of space and time. Nevertheless, that is the way the universe works, Greene says. Here, Comet Hale-Bopp. Enlarge Photo credit: © tpuerzer/iStockphoto

Truly. This is not dexterous wordplay, sleight of hand, or psychological illusion. This is how the universe works.

In 1971, Joseph Hafele and Richard Keating flew state-of-the-art cesium-beam atomic clocks around the world on a commercial Pan Am jet. When they compared the clocks flown on the plane with identical clocks left stationary on the ground, they found that less time had elapsed on the moving clocks. The difference was tiny—a few hundred billionths of a second—but it was precisely in accord with Einstein’s discoveries. You can’t get much more nuts-and-bolts than that.

Brian Green – photo credit: © Lark ElliottBrian Greene is a professor of physics and mathematics at Columbia University. He is the author of The Fabric of the Cosmos, from which this essay was excerpted and on which the four-part NOVA series premiering in fall 2011 is based. Greene is also the author of The Elegant Universe, the subject of a three-part NOVA series that aired in 2003, and The Hidden Reality.

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