This article by Paul Doherty and Pat Murphy appeared in the Magazine of Fantasy and Science Fiction as the Science Fact Column.
General Relativity at home
In Pat Murphy's latest novel, Bailey Beldon, a reluctant adventurer from the asteroid belt, finds himself swept along on an adventurous journey to the center of the galaxy. Traveling near light speed and at high acceleration, Bailey finds himself far from home in both space and time.
As any well-read science fiction reader knows, Bailey, when he is traveling near 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 affects on time.
If you read this column regularly, you know that we generally focus our attention on everyday phenomena, strange science that you can see at work in the world around you. So you may be puzzled at this point. How, you say, can we experience relativity at home? We've inserted a totally shameless plug for Pat's book. Now how can we tie Bailey's predicament to anything that our readers can experience in their everyday lives?
Well, we'll start by talking a little bit about time and how it passes and how scientists have gotten downright picky about measuring its passage. And we'll talk about how those impossibly picky measurements of time have allowed scientists to test the predictions of both special and general relativity. We'll spend some time talking about how when relates to where you are in a gravity field and to how fast you are going. Then we'll head for the Andes, where corrections for general relativity allowed Paul to escape a potentially dangerous situation.
Finally, we'll come a little closer to home and discuss how you personally can use general relativity to your advantage, especially if (like Pat) you have no sense of direction whatsoever.
Since we are unlikely to be as "lucky" as Bailey and be swept at relativistic velocities across the galaxy we need to be able to measure small time differences caused by the tiny relativistic effects at the slow speeds we can achieve. For this we need a very stable clock. Luckily there have been excellent atomic clocks for 5 decades now. Atomic clocks are based on oscillations of electrons between atomic energy levels. Inside the atom electrons are free from friction that plagues pendulum clocks and even quartz crystal clocks. This allows them to keep very exact time.
The most common atomic clock uses atoms of cesium. In fact cesium atomic clocks are so good that they are used to define the second itself &emdash; one second is 9,192,631,770 oscillations of one particular microwave spectral line emitted by an electron in an atom of cesium 133. This very oscillation is the one used in cesium atomic clocks. Today's cesium clocks measure time to within 2 parts in 1014, to make that more real, that is 2 nanoseconds per day or 1 second in 1.4 million years. A clock that accurate can be used for many things, one of which is to test relativity theory.
Special relativity predicts that a clock moving with respect to you will be seen to run slower, and as it approaches the speed of light the clock slows toward a stop. (The clock can never reach the speed of light nor totally stop either.) General relativity predicts that clocks deep in a gravity field, for example close to the surface of the earth, or the event horizon of a black hole, will run slower than clocks far from the gravitational effects of masses.
The fractional amount of time dilation due to special relativity can be approximated for speeds much less than (<1 %) the speed of light by a simple rule, moving clocks run at 1-0.5 (v/c)2 compared to clocks at rest. Since the speed of light is 3 x 105 km/s and the speed of the plane is 0.2 km/s the clocks run slower by about 0.5 * (.2/ 3 * 105)2 = 0.5 * 10-10 or 2 parts in 1010 well within the measurement ability of 2 parts in 1014 of atomic clocks. Clocks in a gravitational field also run slower. Their rate depends on the gravitational potential. You've probably heard of the gravitational potential energy equal to mgh where m is the mass g is the acceleration of gravity and h is the height, for example newtonburger which contains a patty weighing 1 newton or 1/4 pound the potential energy when it is raised 4 meters goes up 1/4 * 10*4 = 10 Joules. The potential is the potential energy per unit mass. Potential energy near the surface of the earth is mgh, while potential is gh. A clock which is a height h above another clock will run slower by a fraction gh/c2 so a clock 1 meter above the floor of your room will run one part in (10 x 1/1017 = one part in 1016 faster than one on the floor. This is too little a change to be measured by an average cesium atomic clock. A clock on an airplane flying at 10 km above sea level will run one part in1012 faster and a clock in orbit 20000 km high will run two parts in 109 faster. These are easily measurable with an atomic clock.
end of mathematical digression.
Two scientists, Hafele and Keating, did the most direct test of relativity possible, in 1971 they flew one set of atomic clocks around the world on a commercial jet liner and then compared them to a reference set left behind on the ground. The scientists flew the clocks around the world twice, once east to west and then west to east. (The experiment is complicated by the fact that the clocks on the surface of the earth are moving relative to the distant fixed stars too because of the rotation of the earth. The 24,000 mile long equator of the earth rotates once in 24 hours which allows one to remember the speed of the equator in miles per hour, 24,000 miles in 24 hours or 1000 miles per hour.
The planes flew at an elevation of 10 km (30,000 feet) and at 800 km/hr (500 mph). The plane flying to the east was thus traveling at the speed of the earth's surface plus 500 mph while that flying to the west was flying at the speed of the earths surface minus 500 miles per hour. (If experimenters could afford a flight on a Concorde jet they could fly at 1000 mph opposite the rotation of the earth and remained at rest relative to the fixed stars while the earth clock rotated beneath them!)
(Experimental aside, Paul once flew back from Sweden to San Francisco flying a great circle route over the arctic, he enjoyed the show as the sun set then rose again moving west to east for a while as the plane sped across the surface of the earth faster than the rotation speed of the earth in the arctic! Watch the suns motion as you fly on polar routes!)
The atomic clocks on the planes flying east lost 184 nanoseconds (a nanosecond is 10-9 s) because of their speed of travel relative to the earth surface clocks. They gained 125 nanoseconds due to the gravitational red shift. The planes flying west gained 96 nanoseconds due to their motion and gained 177 nanoseconds due to gravity. The measured effects were within 10% of the predicted effects which was within the 20% error in the experimental technique. (The effect of gravitational redshift has now been confirmed to better than 1%)
By doing this experiment the scientists were actually doing a version of the twin paradox experiment. In the twin paradox twins are separated and one flies to a distant star then turns around and returns. The twin who turned around is reunited with his stay at home twin and is found to be younger. This is a paradox because the motions of the twins is relative. The twin who rode the rocket ship sees his twin accelerate away from him, then turn around and come back. This is exactly what the stay at home twin sees, so how can there be a difference in their ages? The resolution of the paradox comes because only one of the twins actually feels the forces of acceleration: speeding up away from the earth, turning around, and then returning and stopping. The twin on the rocket ship sees his brother move away and return but the twin on the rocket ship knows that he is the one that is really moving because he feels the accelerations. There is thus a real difference between the twins and one can age less than the other. The paradox is resolved.
Now let's return to earth and have Paul tell the story of his recent trip to the Andes and how general relativity played an important role. Unlike Pat's hero Bailey, Paul actively seeks out these adventures.
" Together with my friend Bob Ayers I wanted to climb Cerro Guillatiri on the Border between Chile and Bolivia, high up above the Atacama desert. In the morning, we parked our rental truck in a deep arroyo and started hiking toward the cloud shrouded mountain. 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, Guillatiri was erupting! steam plumes rose from several vents and rocks tumbled down the summit slopes. We climbed until we got a good view and some photographs, then we turned around and headed down. Guillatiri was a little too exciting! As we dropped down onto the flat terrain of the desert, every arroyo began to look the same. Each of us had carefully noted the way back to the truck, but now we couldn't agree on which gully was the correct one. To answer the question, 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 the way to our truck. A few hours later we were standing at the truck again, guzzling water and eating potato chips."
Without corrections for relativity the global positioning system would not work! The system uses an array of 24 satellites which orbit the earth every 12 hours. The orbits are arranged so that from every place on earth you can almost always receive radio signals from 4 or more satellites at the same time. Each satellite carries a cesium atomic clock. The walkman-sized device Paul carried received signals from 4 (or more) GPS satellites. 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 signals from up to 12 satellites simultaneously and used these signals to calculate its position on the earth and also the exact time. You see, by knowing the exact time the receiver could tell how long the radio signal took to reach it from the satellite, since the signal traveled at the speed of light it then knew the distance to each satellite and, since the location of the satellite was transmitted along with the time signal the position of the GPS unit in Paul's hand on the side of Mt. Guillatiri was easily calculated. All of this however depends on knowing the time.
The spacecraft orbit 10 times faster than the airliner, and a thousand times higher. ( They orbit with an orbital radius of 20,000 km at a speed of 10,000 km/hr, (6000 mi/hr) or 3 km/s,) the relativistic effects are thus orders of magnitude greater than the effects measured on the round-the-world airliners. In fact errors accumulate at 4 parts per 1010. When the first GPS satellite was launched its relativistic correction program was not turned on for 20 days, during that time its atomic clock provided a great test for relativity losing the predicted 4 parts in 1010 this meant losing 38,000 nanoseconds per day, meaning an error of 38,000 feet per day, over 7 miles per day of error adding up day after day! That's what relativity means. No one ever predicted that the basic theoretical physics discovered by Einstein would have an effect on nearly every serious navigator on earth within one century! (wupa.wustl.edu/nai/opeds/opeds98/WillMay98.html)
The seed of light is a foot per nanosecond (0.3m) a timing error of 1 nanosecond would mean an error in position of 1 foot. However the relativistic corrections are cumulative and as the days pass the error in earth location would grow larger and larger. Soon rendering the system useless.
So if you want to find the correct time a good source is a GPS unit, the time it displays comes from an array of orbiting atomic clocks. Corrected for general relativity. In addition the unit will tell you your latitude and longitude and altitude to help you find your way. Paul advises to learn to use a GPS receiver in conjunction with other navigation skills. This provides a backup in case you drop the GPS down a crevasse or the batteries in the receiver die.
So todays 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. What Bailey and indeed the galactic human population of Pat's latest novel need is the galactic equivalent of the GPS... but to find out what Bailey actually finds you'll have to read the novel.
Scientific Explorations with Paul Doherty
8 November 2004