generalrelativityGPS

Science

by Pat Murphy & Paul Doherty

General Relativity at Home

In Pat Murphy's latest novel, There and Back Again, Bailey Beldon, a reluctant adventurer from the Asteroid Belt, finds himself swept along on an adventurous journey to the center of the galaxy. Accelerating to near light speed, Bailey finds himself far from home in both space and time. As any well-read science fiction reader knows, Bailey, traveling at close to the speed of light, ages much more slowly than his friends who are puttering about happily in the Asteroid Belt. During the course of his travels, Bailey must deal with the consequences of relativity and its effects on time.

If you read this column regularly, you know that we generally focus our attention on everyday phenomena, science that you can see at work in the world around you. So you may be puzzled at this point. How, you ask, can your readers experience relativity at home? Is this column just a excuse to plug Pat's new book? How can you tie Bailey's predicament to anything that your readers might experience in their everyday lives?

Well, it does provide us with an excuse to mention Pat's novel (currently available from Tor Books), but we really will tie relativity to something that you can use at home. We'll start by talking a little bit about relativity&emdash;both special relativity and general relativity--including a discussion of how your speed and your position in a gravity field affect the passage of time. We'll describe how scientists have tested the predictions of relativity, using some ridiculously precise clocks. We'll visit the Andes, where the predictions made by general relativity were instrumental in helping Paul escape a dangerous situation. And finally, we'll come back home and discuss how you personally can use general relativity to your advantage, especially if (like Pat) you have no sense of direction whatsoever.

WHAT'S SO SPECIAL ABOUT RELATIVITY?

In many a science fiction novel, characters like Pat's pal Bailey have had to dealwith the consequences of special relativity. Special relativity predicts that a clock moving with respect to you will be seen to run slower. If you could watch the hands of a clock in a spaceship speeding past you, you'd see that the faster the clock travels, the slower time passes on the ship; the clock seems to slow toward a stop as the spaceship approaches the speed of light. The clock can never reach the speed of light--the physicists are adamant about that--so it never comes to a complete stop.

If you like, you can calculate the effect of special relativity on moving clocks. (If math makes you crazy, you can skip this paragraph. But this is simple math--just arithmetic, really. So be bold and give it a try.) Calculating how movement affects time is easiest if you express the speed of the clock as a fraction of the speed of light. Let's call that fraction "f." When Pat's character

Bailey is traveling at 99.5% the speed of light, f = 0.995. To calculate how much time passes on Bailey's spaceship, you just plug f into an equation. For every hour that passes on a stationary clock, the time that passes on Bailey's clock will be the square root of (1-f^2) hours. When one hour passes on a stationary clock, one tenth of an hour passes on Bailey's clock.

While traveling at 99.5% light speed, Bailey experiences only a year's time for every ten years that pass back home. Relativity affects all clocks including biological clocks that control aging, so while Bailey's friends age ten years, Bailey only ages one.

Here on the earth. people travel at a tiny fraction of the speed of light. A jetliner travels at 300 meters per second, just a little slower than the speed of sound. Light travels 300 million meters per second! So the fractional speed of light of an airliner is f = 10^-6. That's one millionth the speed of light. Clocks onboard

the airliner run slower than clocks on the ground, but not by much. When one second passes for a clock on the ground, then a clock on the plane ticks off one second minus half a picosecond. (A picosecond is a trillionth of a second, 10^-12 seconds.) Scientists say that the plane clock runs slow by about one part in 10^12 or one part in a trillion. You would have to wait 100,000 years for the airplane clock to lose a second! (To do this last calculation, Paul got to use one of his favorite memory tricks: the number of seconds in a year is about pi times 10^7.)

MEET ME AT THE BLACK HOLE

The effects of special relativity have been used in a number of science fiction stories. Science fiction writers haven't used the time-shifting effects of general relativity nearly as often (though Pat is now giving it some serious thought). General relativity predicts that clocks deep in a gravity field (close to the surface of the earth, for example, or at the event horizon of a black hole) will run slower than clocks far from the gravitational effects of masses. The rate at which time passes in a given place depends on something physicists define as the gravitational potential. Now we're going to use a few equations to show you just how physicists use gravitational potential in calculations. You can skip this if you want to, but it's really pretty easy if you go slow. (That's what Pat says anyway. Paul thinks it's safe at any speed.)

If you've ever taken a basic physics class, you've probably heard of the gravitational potential energy. Physicists call it "U." Near the surface of the earth where the acceleration of gravity (known as "g") is constant, the gravitational potential energy is equal to the object's mass (m) times the acceleration of gravity (g) times the height (h). Written in standard physics shorthand, that's U = mgh. (The result is in joules, a unit for measuring energy.)

The gravitational potential (V) is the gravitational potential energy per unit mass, U/m. The gravitational potential is then V = gh, (joules per kg). The potential depends on the acceleration of gravity (g) and the height (h). Near the surface of the earth g = 10 meters/second2. Using that number, you can calculate the gravitational potential difference between the ground and a point 1 meter higher. It's just V = 10 x 1 = 10 J/kg.

Using V, a physicist can calculate how the distance from the earth's surface affects the passage of time. When a clock on your mantelpiece measures 1 second, a clock a meter below it will measure 1 second x (1-V/c^2). (That "c" represents the speed of light.) So the lower clock runs slower by (1- 10/10^17) = 1- 10^-16. That means it's slow by one part in 10^16. If both your clocks were exceedingly accurate, you'd have to wait nearly a billion years before they differed by one second.

That calculation is for a simple case, where the gravitational potential is constant. If you were traveling through space, where the force of gravity changes as you travel, you have to substitute the correct expression for the gravitational potential difference between the clocks at each point in your trip. For example, if you were to approach the event horizon of a black hole, the gravitational potential energy would approach a huge negative value and the clocks on your ship would slow down nearly to a stop.

As clocks (and time) slow down, so do the vibrating electrons that produce the radiation we call light. The rate of an electron's vibration determines the frequency of the light. As the electrons' vibration slows, the frequency of the light shifts from the blue end of the spectrum (high-frequency visible light) to the red end of the spectrum (low frequency visible light). As time shifts, so does

the frequency of the light, shifting from blue toward red. Because of that shift, physicists call this effect the "gravitational red shift."

VERY, VERY, VERY, VERY PRECISE TIME

All the above calculations were worked out by Einstein, after many "thought experiments" back before 1915. Since then, the predictions made by Einstein have been tested with actual experiments involving real clocks. Since scientists can't currently accelerate clocks to near light speed, they did the next best thing.

They made clocks so accurate that they can measure the tiny time shifts caused by relativistic effects of slower speeds.

These very, very, very, very precise clocks are called atomic clocks. Rather than relying on the swinging of a pendulum or the vibration of a quartz crystal, an atomic clock bases its time-keeping on the oscillations of electrons between atomic energy levels.

The most common atomic clock uses atoms of cesium. Cesium atomic clocks are so good that they are used to define the second itself---one second is 9,192,631,770 oscillations of one particular microwave spectral line, called a hyperfine transition, emitted by an electron in an atom of cesium isotope 133. This very oscillation is the one used in cesium atomic clocks.

Today's cesium clocks easily measure time to within 2 parts in 10^14. That's 1 second in 1.4 million years. (For comparison, Paul's quartz crystal wristwatch is good to within one second a day, about 1 part in 10^5.) Cesium clocks can be used for many things, one of which is to test relativity theory.

In 1971, scientists J. C. Hafele and Richard Keating did the most direct test of relativity possible. They flew one set of atomic clocks around the world on a commercial jet air liner and then compared them to a reference set left behind on the ground. The clocks flew strapped to the front bulkhead in coach class. The plane flew around the world to the east taking 41 hours to fly the entire circuit. The experimenters recorded the altitude and speed of the plane, which flew at an altitude of 10 km (30,000 feet) above sea level, and a speed of 800 km/hr (500 mph). The atomic clocks on the plane lost 59 nanoseconds overall. They lost 184 nanoseconds because of their speed of travel relative to the earth surface clocks, but they also gained 125 nanoseconds due to the gravitational red shift.

To check their results, the scientists then flew the clocks around the world again the other way. Both flights confirmed the predictions of relativity to within the experimental measurement accuracy of 10%.

MEANWHILE, UP IN THE ANDES

So how could the theory of relativity and these clocks that lose just a few billionths of a second per day affect us in our everyday lives? That's where Paul's trip to the Andes comes in. In November 1998, while traveling in the Andes, Paul benefited from Hafele's and Keating's test of relativity.

Paul had set out to climb a mountain, that being the sort of thing he likes to do for fun. (Unlike Pat's hero, Bailey, Paul actively seeks out strange adventures.) The mountain Paul chose to climb was Cerro Guillatiri, on the border between Chile and Bolivia, high up above the Atacama desert. He was climbing with an equally adventurous friend, Bob Ayers.

"In the morning, we parked our rental truck in a deep arroyo and started hiking toward the cloud-shrouded mountain," Paul explains. "There were no trails up this remote summit. As we climbed in the thin air above 15,000 feet, the day began to clear. What we saw was beautiful, and terrifying. The mountain was erupting! Steam plumes rose from several vents and rocks tumbled down the summit slopes. We climbed until we got a good view and snapped some photographs, then we decided that this mountain was a little too exciting. We turned around and headed down."

"As we dropped down onto the flat terrain of the desert, every arroyo began to look the same. Bob and I had each carefully noted the way back to the truck, but each of us remembered a different gully as the correct one. If we chose the wrong gully, we could be in big trouble."

"To find the correct gully, I pulled out my global positioning system (GPS) receiver. I had marked a waypoint at the location of the truck as we started our climb. The GPS unit pointed out the correct route back to that point. A few hours later, we were standing at the truck again, guzzling water, eating potato chips, and congratulating ourselves on having cheated death again."

The global positioning system that helped Paul cheat death uses an array of 24 satellites which orbit the earth every 12 hours. The orbits are arranged so that you can almost always receive radio signals from 4 or more satellites at the same time from every place on earth.

Each satellite carries a cesium atomic clock. The Walkman-sized device Paul carried received signals from 4 (or more) GPS satellites at once. The signals told the unit where each satellite was located in space and the exact atomic time at which each signal was sent. The receiver used the satellite signals to figure out the exact time at its location to atomic clock accuracy. By knowing the exact time each signal was received and sent, the receiver could tell how long the radio signal took to travel from each satellite. Since the signal traveled at the speed of light, the receiver then knew the distance to each satellite. Since the location of each satellite in orbit was transmitted along with the time signal, the receiver could easily calculate the position of the GPS unit in Paul's hand on the side of Mt. Guillatiri.

All of this however depended on knowing the exact time. Without corrections for relativity the global positioning system would not work!

The satellites orbit 10 times faster and a thousand times higher than a plane. (To be more specific, they orbit with an orbital radius of 20,000 kilometers at a speed of 10,000 kilometers/hour. That's 6000 miles/hour or 3 kilometer/second.) At these speeds and heights, the relativistic effects are orders of magnitude greater than the effects measured on the round-the-world airliners.

Because the clocks on the satellites are traveling faster than earth surface clocks, they run slow---as predicted by special relativity. Because they have higher gravitational potential than earth surface clocks, they also run fast as predicted by general relativity. In this case, the gravitational time shift wins and the satellite clocks run faster than clocks on the earth. They run fast by 4 parts in 10^10. (That's four parts in ten billion.)

When the first GPS satellite was launched, its relativistic correction program was not turned on for twenty days. During that time, its atomic clock provided a great test for relativity, gaining the predicted 4 parts in 10^10. The clock gained 38,000 nanoseconds per day. The effect of this error in calculating position was significant, adding an error of 38,000 feet per day. That's over 7 miles per day adding up day after day!

WHAT RELATIVITY MEANS TO ME

The predictions of relativity allowed Paul to pinpoint his pickup truck on the slopes of an exploding mountain. For Pat and Paul and science fiction writers and readers, the consequences of these predictions also points out one of the interesting difficulties about predicting the future.

Back in 1915, Einstein's theory of general relativity seemed to have little practical use. Even today, its primary importance is in astronomy, in examining black holes, gravity waves, and the big bang. But this theory has also proven essential to the development of the GPS system, which allows users to determine latitude, longitude, and altitude to within a few hundred meters and local time within a few ten billionths of a second.

Paul, who owns a GPS receiver, and Pat, who is often lost and plans to acquire one, are both very grateful to Albert Einstein. Paul advises people to learn to use a GPS receiver in conjunction with other navigation skills. (See page 00.) He advises against depending solely on a GPS receiver, since you need a backup plan if you drop the GPS down a crevasse (the sort of thing that is quite likely on Paul's expeditions) or the batteries in the receiver die (the sort of thing that is more likely to happen in Pat's household).

A GPS receiver--a navigation tool used by hikers, sailors, pilots, and drivers everyday--depends on atomic clocks and the theory of relativity. Truly it is a way to hold the effects of general relativity in your hands.

Note: For more about Pat Murphy's and Paul Doherty's work, check out their web sites at: www.brazenhussies.net/murphy and www.exo.net/~pauld.

Finding Your Way

Since we couldn't figure out a way to have you experiment at near-light speed, we decided to provide a few tips to help you navigate at speeds you are more likely to find yourself traveling. Here are a few tricks Paul uses to minimize how lost he gets while hiking--or when wandering around an unfamiliar city.

1) Always look back. As you are walking turn around and look at where you came from. It's important to remember what the landmarks look like from this perspective. After all, you might have to retrace your steps. (Although Paul admits that his y chromosome pushes him to make loop trips and never retrace his steps, he often finds that circumstances force him to go back, as they did in the Andes.)

2) Watch the stars.

Keep track of the sun during the day, and the moon and pole star at night. In the northern hemisphere, the sun and moon cross the southern half of the sky. In the southern hemisphere, they cross the northern half of the sky, something people from the northern hemisphere find most disconcerting. Keeping track of the sun can keep you from going in circles.

3) Check your instruments.

If you have a compass, check it occasionally. Paul has an emergency compass attached to the zipper of his storm jacket. He remembers several times when he was lost in a cloud on a mountain top. His instincts told him which way was north, and the compass gave him a different answer. When the second compass in his backpack agreed with the one on his jacket, he accepted their answers and made his way safely down.

Suggested World Wide Web Sites

To find out more about atomic clocks, we suggest you search through the US Naval Observatory and the National Institute of Standards and Technology homepages:

http://www.usno.navy.mil/

http://www.nist.gov

To find out about global positioning system, check out the Trimble tutorial:

http://www.trimble.com/gps/index.htm.

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