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Weirder Than You Think
IN 1905, Albert Einstein published a paper that described his special theory of relativity for the first time. That theory, condensed to its essence, is this: Space and time are not two separate things. They are two parts of the same thing: spacetime.
One consequence of special relativity is that your motion through space is linked to your motion through time. The faster your spaceship moves, the more slowly the clock on your spaceship moves compared to a clock on Earth. This apparent slowing of a fast-moving clock is known as time dilation.
In the hundred years that have passed since Einstein published his theory, science fiction writers have had a great time exploring ways in which time dilation might affect space travelers. Maybe you've read Robert Heinlein's novel, Time for the Stars, a classic case of what physicists call the "twin paradox." In Time for the Stars, one twin travels to distant planets at near-light speed. He returns to Earth to find that his twin has become old (and rather cranky) while he was gone. Or perhaps you've read Joe Haldeman's Forever War, in which soldiers repeatedly travel long distances at near-light speed, returning from each interstellar journey to a human society that is transformed, centuries having passed on Earth while the soldiers were gone. Or you may have read one of the many other works of science fiction that include relativistic effects.
These works explore possible consequences of time dilation. But they never explain why the folks on the spaceship age more slowly than the folks on Earth. That's where we come in.
In this column, we will get into the nitty gritty details of time dilation. And, since we know you won't take our word for it, we'll offer actual proof of Einstein's theory. We will stretch your mind with some thought experiments and show you how to derive the equation that lets you calculate how much time shifts with speed. We won't have space to explore all the other weird consequences of special relativity. We'll have to save that for the two-hundredth anniversary.
At the Exploratorium, we always like to start with an experiment. So if you have Internet access, we suggest you start by taking a little virtual trip. Visit http://www.exploratorium.edu/relativity/index.html, the Web site that the Exploratorium created to celebrate the hundredth anniversary of Einstein's paper. At this Web site, you can blast off to the stars—to Epsilon Eridani, to be precise, a star that's ten light-years from Earth.
Since you're a science fiction reader, we probably don't need to remind you that a light-year is the distance that light travels in a year. And we also don't have to tell you that light travels at about 186,000 miles or 300,000 kilometers in a second. So a light-year is a long, long way (about 1016 meters).
Here's the story: Twenty years from now, the Exploratorium will be hosting a party on a planet orbiting Epsilon Eridani. The goal of your virtual trip is to get to the party on time. You can choose your speed of travel—and note the relativistic effects while you travel. If you've ever wanted to blast off to a distant planet, here's your chance! It's a virtual trip, but given the limitations of our budget, we think that's still pretty good.
Blast off to the stars, then come back and read on. If you don't have Internet access, just read on. You can read and enjoy the article even if you never visit the Web site.
THINKING LIKE EINSTEIN,
Before we start detailing the weirdness, we want to give you a better handle on the theory itself. Einstein started with a couple of basic assumptions. First, Einstein assumed that Galileo was right. Back in the 1600s, Galileo proposed that the laws of physics don't depend on the speed you are traveling. As long as you are traveling at a constant speed, the laws of physics are the same.
Suppose you're in a vehicle that's closed up—the doors are shut, the windows are blacked out, and you have no way to see the outside world. How can you tell whether you're sitting still or hurtling along in a straight line at a steady 100 kilometers an hour?
According to physicists, you can't.
In a closed vehicle, there is no experiment you can do to figure out whether you're standing still or moving at a constant velocity (that is, in a straight line and at a constant speed). Pour water into a glass, bounce a ball, swing a pendulum—do any experiment you can imagine. The laws of physics that dictate how liquids pour, balls bounce, and pendulums swing are the same at all constant velocities, whether you're moving at zero kilometers per second, 100 kilometers per second, or nearly the speed of light.
If the vehicle you're in speeds up, slows down, or turns, you can detect the change. But as long as the vehicle is moving in a straight line at a steady velocity, there's no way you can tell.
Physicists call the place from which motion is measured a "frame of reference." As a physicist might put it, "all laws of physics are the same in all uniformly moving frames of reference." This assumption is called Galilean relativity, and it's fundamental to Einstein's special theory of relativity.
THINKING LIKE EINSTEIN,
Before we get to the weirdness of interstellar travel, there's one more assumption we need to consider. Einstein assumed that the speed of light is the same for all observers.
Ordinarily, when you talk about how fast something's moving, you have to take note of where you're standing as you watch that motion. Suppose you're on a train traveling 100 kilometers per hour. You decide to get out of your seat and walk toward the dining car at the front of the train, so you amble up the aisle at a speed of two kilometers per hour. How fast are you moving?
Before you can answer that question, you need to ask another question: How fast am I moving relative to what? Relative to your seat on the train, you're moving 2 kilometers per hour. But relative to the track under the train, you're moving 102 kilometers per hour—the speed of the train plus your speed on the train.
That's the way it works for most things. But that's not how it works for light.
Back in 1887, physicists A. A. Michelson and E. W. Morley did an experiment that showed the speed of light does not depend on the movement of the observer. Michelson and Morley were trying to measure the speed of the Earth as it passed through the aether.
The aether, for those of you who have forgotten, is the undetectable stuff that nineteenth-century scientists supposed carried light through space. Back then, they figured that light had to travel through some kind of substance. After all, ocean waves travel in water; sound waves travel in air or water or some other substance. It seemed natural that light waves needed something to propagate through. Scientists subsequently discovered that light is weirder than they thought. There is no aether, and light travels through empty space unaided by any aether.
Anyway, Michelson and Morley came up with a way to measure the speed of light in the direction of Earth's motion and at right angles to Earth's motion. If light behaved like most things, those two speeds would have been different. If you're in a sailboat traveling into the waves, the wave speed relative to you is greater than if you are in the same boat traveling in the same direction as the waves or traveling at right angles to the wave's direction of travel.
Michelson and Morley wanted to calculate the difference between the speeds of light in these two directions. They planned to use the difference to calculate the speed of the Earth. But they were surprised to discover that there was no difference! The speed of light—measured in the direction of Earth's motion and at right angles to Earth's motion—was the same. The bottom line is: No matter how fast or slow you move, light always moves at the same speed.
Imagine, for instance, that Marco is on a spaceship traveling toward the sun at 99.99 percent of the speed of light, and Sophia is on a spaceship heading away from the sun at 99.99 percent of the speed of light. Even though they're heading in different directions, sunlight is rushing past them both. How fast is that light traveling relative to Marco? It's traveling at the speed of light—about 300,000 kilometers per second. How fast is that light traveling relative to Sophia? It's traveling at the speed of light—about 300,000 kilometers per second.
Or suppose you are driving a fast car at just a meter per minute below the speed of light. (It's a very fast car.) You turn on your headlights. Do you see the light moving slowly away from you at just a meter a minute?
Nope. The speed of light is the same for all observers, no matter how fast they are traveling. The light from your headlights always rushes away from you at 300,000 kilometers a second.
You're skeptical? Experiments have shown that the speed of light is independent of the source of the light. Back in 1964, T. Alvager measured the speed of a pulse of light emitted by a neutral pion (that's a subatomic particle) that was traveling at 99.975 percent of the speed of light. He found out that light traveling in the same direction as the pion was traveling at the speed of light.
IT'S ABOUT TIME!
Now that we've dealt with Einstein's basic assumptions, we can get to what you've all been waiting for: the really weird stuff. Let's talk about what happens when you blast off at near light speed. Suppose, for example, you travel a distance of ten light-years at ninety percent the speed of light.
As we noted up front, the faster your ship moves, the more slowly your ship's clock moves compared to a clock on Earth. And the more slowly you age, compared to your twin back on Earth.
Here's one way to think about it: You are always traveling through spacetime. Even when you're standing still, you're traveling through time—at a rate of twenty-four hours a day. When you also travel through space, you travel through time a little bit slower. Thomas Humphrey, one of the Exploratorium's physicists, explains this as a trade-off: In order to travel through space, you have to give up a bit of your travel through time.
If you think about it, you'll realize that this sort of trade-off happens all the time in your travel through space. Suppose you leave San Francisco's Ferry building on a ferry that travels at ten knots. (We're assuming that this ferry always travels at exactly ten knots.) You could travel east at ten knots to Oakland or you could travel north at ten knots to Sausalito. When you travel due north, your speed eastward is zero. When you travel due east, your speed northward is zero. When you travel due northeast, your eastward rate of speed is reduced to allow some northward progress and your northward rate of speed is reduced to allow some eastward progress. Your speed will be 7 knots east and 7 knots north. If you graphed the distance you traveled in an hour, you will always end up on a circle of radius ten nautical miles, but where you are on that circle varies depending on your direction. (Take a look at figure 1 to see see what we mean.)
A similar rule applies to space and time! When you travel through space, your passage through time slows down. At ordinary speeds, this slowing is imperceptible (except to a superaccurate clock). But as you approach the speed of light, your clocks slow down to a stop. Just as when the ferry goes almost due north, its eastward speed approaches zero.
The left graph shows the situation when you are traveling in a direction between north and east on a ferry that moves at 10 knots. Your speed is always on the arc. Traveling faster to the north means giving up some speed in the eastward direction.
The right graph shows the trade-off between travel through space and travel through time. Your travel through space is measured in fractions of the speed of light, or c. When your speed is 0, you travel through time at the usual rate: One second for you equals one second measured by someone standing still. At 1c, the speed of light, your travel through time stops. Zero seconds pass for you for each second that passes for someone standing still.
WHERE'S THE PROOF?
Maybe you're skeptical about all this. You want proof? Well, it just so happens we've got proof. Evidence from two different experiments support the existence of time dilation.
The first experiment is Pat's favorite. How can you prove that a fast-moving clock runs more slowly than a stationary one? You could send a very accurate clock on a long journey.
In October 1971, two physicists flew four atomic clocks on commercial airliners twice around the world, once in an easterly direction, and once in a westerly direction. These clocks were accurate to within a few nanoseconds. (That's mighty accurate since there are 1,000,000,000 nanoseconds in a second.) Using calculations from the special theory of relativity, the physicists predicted how time measured by the clocks on the planes would differ from time measured by clocks on the ground. What do you know! The experimental results matched the predictions, confirming that time dilation actually does take place with real clocks.
In 1975, physicists tried another experiment. They sent an atomic clock up in a U.S. Navy plane that traveled for 15 hours at 270 knots, or 140 meters per second—about 0.47 millionths the speed of light. As the plane flew back and forth over Chesapeake Bay, the onboard clock was compared to a clock on the ground by laser signal. During the flight, the onboard clock lost 5.6 nanoseconds, exactly the amount predicted by special relativity.
Those who read this column regularly are not surprised. Those faithful readers remember that we mentioned in an earlier column that special relativity is essential to the functioning of the GPS satellite system. The atomic clocks on the satellites run slow due to special relativity because the satellites are whizzing around the Earth in orbit. (They also run fast due to general relativity, but we're not dealing with that theory here. One thing at a time!)
So whenever you fly in a jet, you age a tiny bit more slowly than you would have if you'd stayed put. Check out the Exploratorium's "frequent flier seconds" calculator at http://www.exploratorium.edu/relativity/time_games.html. Enter your lifetime frequent flier mileage, and the calculator will tell you how many nanoseconds younger you are as a result. If you've flown 500,000 miles, you will have aged 1100 nanoseconds less than Earthbound observers.
Other evidence for special relativity comes from the subatomic particle called the muon, a particle that lives for just 2.2 microseconds. Muons are created when cosmic rays traveling through space strike molecules in the upper reaches of the atmosphere, some ten kilometers above Earth's surface. Even moving at nearly the speed of light, a muon should only be able to travel about 700 meters before it decays. So you might think no muons could ever reach Earth.
Not so! Many muons make the entire ten-kilometer trip. From the perspective of the Earth, these high-speed particles live ten times longer than they would if they were stationary, and consequently, they can travel ten times farther.
In 1966, physicists at CERN—the world's largest particle physics facility—repeated this natural occurrence in a laboratory. They created muons and sent them zooming around a ring at 0.997 times the speed of light. Those high-speed muons survived twelve times longer than did muons at rest.
(By the way, if you have a classic thermos bottle with a silvery inside you can "see" these muons. Fill the thermos with water, go into a dark place and let your eyes become dark-adapted, which takes about half an hour. Dark-adapted human eyes are very sensitive to light. Now gaze into the bottle. You will see dim flashes of light. These flashes come from muons moving through the water.)
CLOCKS AND CALCULATIONS
But why, you ask, does this happen? Why does moving faster through space make you move slower through time?
Here's a thought experiment that will help you understand—though we warn you that understanding this will twist your brain a bit. We thank Exploratorium physicist Ron Hipschman (whose brain is quite twisted) for his explanation of this mental experiment.
Suppose you have a spaceship that has a very simple clock. This clock is made of two mirrors, one above the other, exactly parallel to each other. Between these mirrors a pulse of light bounces up and down, making the clock tick every time it hits a mirror. (See the picture below.)
When the spaceship is on Earth, waiting to take off, both a person on the ship and an observer outside the ship agree that it takes a certain time (let's call it t0) for the clock to tick.
What happens when the ship flies past an Earthbound observer? From the point of view of the person on the ship, nothing has changed: The pulse of light bounces up and down, just as it did before, with the time t0 between ticks. But the Earth-based observer sees something like the picture below.
From the Earth-based observer's point of view, the light pulse travels on a diagonal path between the mirrors as the ship moves forward. This diagonal path is longer than the straight up-and-down path observed by the person on the ship.
Now we'll remind you of Einstein's two assumptions. The first assumption said that all laws of physics are the same in all uniformly moving frames of reference. The spaceship is moving at a steady speed, so the moving clock has to behave in the same way that the clock did when it was stationary.
The second assumption said that the speed of light is the same for all observers. That gets us into tricky territory. The person on the spaceship and the person on Earth both observe light traveling at about 300,000 kilometers per second. But the Earthbound person observes the light traveling a longer distance for each tick of the clock. Since the speed of light is always the same, it must take a longer time to travel this greater distance. Therefore the moving clock ticks slower for an Earthbound observer than it does for an observer on the ship. Voilą! We have time dilation!
Does that make your head hurt? We sympathize, but we won't stop there. We'll tell you that you can use this thought experiment and a bit of high school geometry and algebra to calculate exactly how much time slows aboard the moving ship. You'll end up with the same equation that Einstein did back in 1905!
To calculate the relationship between t0 and t, you'll use the Pythagorean theorem (which states that for a right triangle with sides a and b and hypotenuse d, a2 + b2 = d2). If you look at the illustration of the moving spaceship, you'll see that you could draw a right triangle. Side a is the distance between the two mirrors, the distance that the pulse of light travels in one tick of the clock when the ship is standing still. Side b is the distance the ship travels in one clock tick. And hypotenuse d is the distance the light travels on its diagonal path.
If you know the velocity of the ship and the speed of light, you can figure out the length of each line in the triangle by converting time and speed to distance. You can then create an equation that compares the time it takes the clock to tick in the stationary ship and in the moving ship. (We won't take you through the equations here, but you can find the full discussion on the Exploratorium web site at http://www.exploratorium.edu/relativity/time_math.html.)
When all is said and done, you end up with an equation that lets you convert t (the passage of time measured by the Earth observer) to t0, the passage of time measured by the observer on the spaceship. Here's that equation:
You can separate the right-hand side of this equation into two parts—time and a quantity that physicists call "gamma" (γ).
BUT THAT'S NOT ALLWe'd love to talk about how travel at near light speeds affects length and mass, but we're running out of space and time, which (as we pointed out) are really part of the same thing. We'll just say that gamma shows up in other calculations related to special relativity. Paul says, "Once you know the value of gamma, the math for time dilation and other aspects of special relativity becomes very simple." Pat says, "Yeah. Right." But both of us suggest that you visit the Exploratorium's web site, where you get the rare opportunity to experiment with relativity. For those who want to do the math, it's there. For those who just want to blast off to the stars and see what happens, that's an option, too.
The Exploratorium is San Francisco's museum of science, art, and human perception—where science and science fiction meet. Pat Murphy and Paul Doherty both work there. We'd like to thank the Exploratorium for the use of figures 2 and 3, created by David Barker. We'd like to thank all the members of the Exploratorium's Relativity team (Thomas Humphrey, Ron Hipschman, Dave Barker, Noah Wittman, Judith Brand, Sarah Reiwitch, Jenny Villagran, and Ruth Brown) for their inspiration and their contributions to the ideas in this column. To learn more about Pat Murphy's science fiction writing, visit her web site at www.brazenhussies.net/murphy. For more on Paul Doherty's work and his latest adventures, visit www.exo.net/~pauld.
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Copyright © 1998–2013 Fantasy & Science Fiction All Rights Reserved Worldwide