As part of the day-to-day activities in my line of work, I occasionally attend seminars on various aspects of computer science. Some of the seminars are called "distinguished lectures"; these are given by people at the forefront of their field, who have been working in that area for some time, so I make extra sure to attend those.
The advance notice for one of these distinguished lectures indicated that it was about a "computational model of sketching." Now, we all know what sketching is -- it's drawing this and that with stick figures and rough circles, and so on. There are lots of computer programs out there to do this stuff. And the obvious thing that comes to mind first when one hears about a "computational model of sketching" is a set of tools to turn rough circles into perfect circles, crooked lines into straight lines, cross-hatched areas into smooth shading, and so forth.
Totally boring. My first inclination was not to go. However, my second inclination was to reason as follows: it's a distinguished lecture; therefore, it must be interesting. If it's truly interesting, it can't be about what I think it's about; therefore, I must be wrong about what it's about. Ergo, I have to go and find out what it's really about. So I went, and I wasn't disappointed; it was about the many ways that people encode a lot of information on the back of an envelope during quick technical discussions, using not only drawings but also speech, gestures, and pictorial conventions (arrows, for example), and how to make use of those to preserve the content. We all know how something can seem so obvious when you're talking about it, and then when you wake up the next morning you can't make heads or tails of it -- it's completely lost! So it was interesting to hear about how other people try to prevent that from happening.
But what's also interesting is that I somehow took as more compelling the idea that the lecture was distinguished, and therefore interesting, rather than that it was about sketching, and therefore boring. It reminds me of how science works; normally, scientific progress is made in a bottom-up fashion: first you gather the data, then you analyze the data for patterns, and then, based on the patterns, you derive some sort of rule for the phenomenon you're studying. But every now and then -- in a flash of insight -- an overriding principle can be followed top-down to reach some startling conclusions.
Einstein worked top-down, for example, and he came up with some earth-shaking stuff. As I mentioned in my last essay, special relativity starts out with the overriding principle that the speed of light is a fundamental constant, no matter how you measure it, and ends up by predicting some pretty strange effects on space and time. For example, it asserts that just because you and I are moving by each other on passenger trains, I'll see your clock moving slow, and you'll see my clock moving slow.
Now, isn't that some sort of contradiction in terms? If you see my clock going slow compared to yours, and I already see yours slow compared to mine, shouldn't my own clock then run doubly slow compared to itself? What on earth is going on here?
The escape hatch from this paradox, which is sometimes called the twin paradox, is that you can only compare clocks when they are in the same place, or at rest with respect to one another. While we're relatively moving, we can't agree on what's simultaneous. But in order to compare the speeds of our clocks, we need to agree on the two times we'll check them; if we can't agree on that, we can't agree on their speeds! And in order to do that, we have to either meet in the same place, or at least head in the same direction at the same speed.
But -- and here's the rub -- to do that, at least one of us is going to have to change speeds, or change direction, or both. And that's an acceleration. To put it graphically, consider the path of two cars as shown in Figure 1.
Figure 1. Cars A and B meeting up after separating.
If two cars, A and B, start moving apart, then there's no way that they can continue moving in their respective directions and eventually meet up, or head in the same direction. One of the cars has to accelerate. If I'm the one in car A, who continues in the same speed in the same direction, and you're the one in car B who has to turn around and catch up with me, then I'm moving inertially and you're not, and it'll be my clock that runs normally, and your clock that runs slow.
The reason why this isn't just common sense is that ordinary trains and cars don't move fast enough for the special relativity effects to become noticeable. Even at typical airplane speeds, it takes atomic clocks to get the precision necessary to notice the difference. But the differences that are noticeable are always in full accordance with Einstein's predictions. In fact, within the precision allowed by today's experiments, none of the predictions made by Einstein's special theory of relativity has ever been invalidated. Not a single one! It is an outstandingly successful theory, and for that reason and others, it was accepted fairly quickly after Einstein published it.
All the same, Einstein was bothered by something. We decide that it's car A whose clock stays running at the same rate, and car B whose clock slows down, because car A is moving inertially. But what does that mean? It means whatever happens in car A satisfies Newton's first law of motion, namely:
Any object at rest or moving uniformly, remains at rest or moving uniformly, as long as no force acts upon it.
However, that's not strictly true in car A, because if you're sitting there in the car and you drop a tennis ball, it doesn't just hang in mid-air, it falls to the floor. And if you want to pass a drink to someone sitting in the back seat, you can't simply send it floating in the right direction, because what it will do instead is spill on the floor; instead, you have to carry the darned thing into the person's waiting hands.
This doesn't bother most people, because we all know what's behind this: the force of gravity. But Einstein made it a habit to let things that didn't bother anyone else bother him, and this time it led him to one of his greatest triumphs -- the general theory of relativity. This theory allowed Einstein to do away with gravity as a force, and made it instead a consequence of the way that space and time "want" objects to move, and the various ways in which objects "follow" space and time.
Einstein's view of gravity may sound pretty strange, but it was motivated by an odd coincidence in Newton's view of gravity. Since it will be easier to see how Einstein came up with general relativity if we understand this coincidence, let's look at Newton's conception first.
In 1665, at the age of 23, Newton had already formulated his three laws of motion. Just so we're clear on what they are, let's recap. We've already taken a quick glance at Newton's first law of motion. To put it in mathematical terms, if the net force F on an object is zero, then so is its acceleration a:
(1) F = 0 implies a = 0
Newton's second law of motion is possibly his most familiar one; it states that if there is a force acting upon an object, then the object accelerates in direct proportion to the size of the force. We can write "in direct proportion to" in mathematics by using a tilde sign, thus:
(2) F ~ a
What this means, in practical terms, is that if you throw a ball, and then throw it twice as hard, it ought to accelerate twice as much and end up with twice as great a speed on the second throw (neglecting friction, air resistance, and other effects). In other words, F and a are related by a ratio, which is inherent in the object -- it doesn't change if you move the object from the Earth to the Moon, or wherever. Newton called this quantity mass. Since this mass represents the tendency of the object to resist acceleration, or remain inert, it is often called inertial mass, but usually it is just called mass, and denoted by the letter m. We can then rewrite Equation 2 in its usual form, as:
(2a) F = ma
Newton's third and final law of motion is usually stated as the law of action and reaction: "If object A acts upon object B, then B acts equally, and in an opposite direction, upon A." It should be emphasized that there's no delay involved. It's not as if A shoves on B, and then B, in a response of justified anger, shoves back on A. They happen together. We can write this third law as
(3) FA on B = FB on A
As a simple example, consider the act of pushing down on a table with your hand. Your hand is exerting a downward force on the table. At the same time, the table exerts an upward force on your hand. The moment you lift your hand from the table, it no longer exerts a force on the table, and the table no longer exerts any force on your hand. That's all the third law is: it's a case of physics bookkeeping.
Back to Newton in 1665. He was at his childhood home, his school at Cambridge having been evacuated to minimize an outbreak of the plague. It's plausible that he was sitting under a tree one evening when an apple fell, although it probably didn't hit him on the head. Instead, he likely saw it fall from another tree, at a time when the Moon was visible in the sky. Newton wondered to himself why the apple fell to the Earth, and the Moon didn't. Was the Moon somehow exempt from the gravity of the Earth? Perhaps the Moon was far enough away such that the gravity of the Earth was insufficiently strong to affect it.
Then it occurred to him that perhaps the Moon was falling, but was simultaneously moving "sideways," and that that sideways motion was enough to keep the Moon in orbit around the Earth. For example, if you drop a ball, it falls straight to the ground. If you throw it so as to give it some horizontal motion, then it doesn't drop straight to the ground when you let go of it. Instead, it continues the horizontal motion of your throwing hand, as predicted by Newton's first law. Gravity pulls on the ball so that it does hit the ground eventually, but by the time that happens, the ball has moved a significant distance. What's more, the harder you throw, the faster the ball moves forward, and the further it goes before hitting the ground.
Now, if the Earth were flat and infinite in extent, then it wouldn't matter how hard you threw -- the ball would eventually fall to the ground, although you could get it to go as far as you wanted by throwing it harder and harder. But in fact, the Earth isn't flat and infinite -- this was known as far back as Eratosthenes (c. 284-192 B.C.). Instead, it's roughly a sphere with a radius of 6,400 km.
This makes a big difference! When you throw a ball, it curves back down to the ground, and the faster you throw it, the gentler the curve. What if you threw it so hard that the curve of the ball was just as gentle as the curve of the Earth? Then it would never fall down, and would instead orbit the Earth, just as the Moon does. And as long as nothing slowed the ball down (such as air resistance, but let's ignore that for the moment), the ball would continue to orbit the Earth, without you having to rethrow it every now and then.
Newton decided, therefore, that the fact that the Moon didn't come crashing into the Earth didn't mean that the Earth's gravity didn't pull on the Moon, as it did on tennis balls. There didn't need to be anything up there to hold the Moon up, or keep it moving in orbit against the pull of gravity; all that mattered was that it had enough speed to begin with. And if the Earth was indeed pulling on the Moon, why shouldn't any object pull on any other object? If Newton could uncover a law of gravity to cover the Earth and the Moon, the same law should also cover the Sun pulling on the Earth, or any of the other planets, or two bowling balls pulling on each other, or whatever. It would be a truly universal law of gravitation.
However, as great an insight as this is, it's not of much use scientifically until it's quantified and tested. It's no good to simply say, "The Sun pulls on the Earth with gravity, and that's that," because there's nothing you can observe to say that it doesn't. It's not falsifiable, in other words, and a proposition that isn't falsifiable isn't worth the paper it's written on. One might just as well say, "The Sun pulls the Earth with telekinesis, and that's that."
Newton therefore needed to come up with some formula that would predict the strength of gravity's pull, depending on some parameters of the objects pulling each other. But which parameters should he start with? Newton was aware of the experiments of the Italian physicist and astronomer Galileo Galilei (1564-1642), in which he dropped objects of different weights from the same height, and observed that they all fell in the same amount of time, with the same motions. (Actually, he rolled them down inclined planes, but as long as other factors such as the angular momentum of rotation are properly accounted for, you can translate rolling down an inclined plane into free falling from a height.)
In other words, if you drop an anvil with a mass of 10 kg and a basketball with a mass of 1 kg, they both experience identical accelerations due to gravity, despite the fact that one has a mass 10 times greater than the other. By Newton's second law, that means that the force of gravity on the anvil also has to be 10 times greater than the force of gravity on the basketball in order to overcome the extra inertia of the anvil. In general, the force of gravity upon any object is therefore proportional to the mass of that object, which we can write as:
(4) Fgrav ~ m
where Fgrav represents the force of gravity on the object. But, by Newton's third law, if the Earth is pulling on the anvil, or basketball, or whatever, then the anvil, or basketball, or whatever must be pulling on the Earth at the same time. The object falls down in response to the Earth's gravity, but the Earth also "falls up" in response to the object's gravity. The magnitude of the acceleration, and hence the fall, is inversely proportional to the mass (by Newton's second law), and since the Earth is so much more massive than anvils and basketballs, its fall is totally unnoticeable. Nevertheless, fall it does; so the force of gravity must also be proportional to the mass of the Earth, which we'll denote with a capital M:
(5) Fgrav ~ M
If a quantity is proportional to the first thing, and also proportional to the second thing, it must then be proportional to the product of the two things. We can therefore combine Equations 4 and 5 into:
(6) Fgrav ~ Mm
Since these masses help determine the strength of the gravity between the two objects, these are sometimes called the gravitational masses of the objects, although usually they are just referred to as mass, so long as there is no confusion about whether the mass is inertial or gravitational.
What else? The force of gravity might depend on the distance between the Earth and the object. In everyday experience, that doesn't appear to be the case. If you drop an anvil a distance of 1 m on the top floor of a skyscraper, it doesn't fall any differently than if you drop the anvil 1 m on a ground level sidewalk. However, gravity doesn't emanate from the ground, it emanates from the center of the Earth, which is some 6,400 km below the sidewalk. So it might not be true that gravity is independent of the separation between two objects; it might just be that the difference in height between the top and bottom of a skyscraper is simply too small, in relation to 6,400 km, to be perceptible.
Fortunately, Newton had other data to work with. The German astronomer and astrologer Johannes Kepler (1571-1630) had earlier worked out three laws of planetary motion, the last of which said that the period T of a planet's orbit -- the time it takes to go around the Sun, in years -- is related to its average distance r from the Sun. Specifically, after poring through detailed observations of the planets conducted by the Danish astronomer Tycho Brahe (1546-1601), Kepler discovered that:
(7) T2 ~ r3
How did this help Newton? Kepler's law not only worked for planets orbiting the Sun, but also worked for objects orbiting the Earth, such as the Moon. If the Earth had two moons, they would also obey Kepler's law. By combining Kepler's law with what was known of orbital mechanics, he was able to derive another relationship for Fgrav:
(8) Fgrav ~ 1 / r2
The force of gravity is inversely proportional to the square of the distance between the two attracting objects. If you increase the distance by a factor of 2, the square of the distance goes up by a factor of 2 squared, or 4, and the force goes down by that same factor of 4.
Is there anything else? If the Earth and the falling object were ideal point masses, with no length, no width, and no depth, then there are no other properties to speak of. Of course, that's not the case: the Earth is a big ball of rock and metal, and the other object could be any shape you want. Fortunately, Newton was able to show that these extra complications essentially didn't matter, and therefore the force of gravity depends on the two masses and the distance separating them, and nothing else. By combining Equations 6 and 8, then, we can get (as Newton did):
(9) Fgrav ~ Mm / r2
As with Newton's second law of motion, we can rewrite this by introducing a constant of proportionality. This time, however, the constant is not simply inherent in the falling object, or just inherent in the Earth -- it is inherent and constant for any two attracting objects in the universe. This constant is denoted by the capital letter G, to indicate that it's rather important. In addition, we write m1 and m2 to denote the masses of any two objects, rather than the Earth and some other object, and Newton's formula can now be written as it usually is:
(10) Fgrav = Gm1m2 / r2
G is often called the gravitational constant, and is experimentally determined to be about 6.67x10-11 m3/kg s2. Newton first tried out this equation -- without knowing the correct value of G -- by comparing the Moon's orbital motion with that of terrestrial ballistics. Because he started out with an incorrect value for the size of the Earth, the numbers didn't work out right at first, and Newton disappointedly set aside the theory. Fortunately, some years later, the English astronomer Edmond Halley (1656-1742) convinced Newton to give it another try, with newer and better data, and this time it worked out so well that Newton wrote out his theory of gravitation in detail in his magnum opus, the Principia Mathematica, published in 1687.
A few comments about Newton's law of universal gravitation. First of all, the constant G is a tiny number. If written out in ordinary figures, without scientific notation, it would be 0.0000000000667 m3/kg s2. That means that at least one of the objects has to be fairly massive in order for the force of gravity to be detectable using ordinary measures. The Earth is massive enough, at about 6x1024 kg, so objects do accelerate noticeably when you drop them here.
However, ordinary objects just aren't massive enough. In principle, a pair of 10 kg anvils, set some distance apart in a room, will attract each other in accordance with Newton's law, but in practice you can wait all you want, and the anvils will refuse to budge, because of friction. In order to get moving, the gravitational force has to be exceed the force of friction, and the size of the constant G insures that won't happen, even if the anvils happen to rest on marbles.
All the same, the force is there, and in 1798 the English physicist Henry Cavendish (1731-1810) was able to measure the gravitational force between two lead balls, one 8 inches across, the other 2 inches across, using a delicate instrument called a torsion balance. In fact, it was his experiment that yielded the first accurate estimate of G.
Another observation is related to Galileo's experiments with falling bodies. So long as we confine our experiments to the surface of the Earth, Equation 10 is highly constrained. The mass of the Earth M is, for all intents and purposes, constant, as is the distance r between the center of the Earth and the object. We can therefore rewrite Equation 10 as:
(11) Fgrav = m (GM / r2)
where everything in the parentheses is essentially constant near the surface of the Earth. The lone variable is the mass of the object, m. However, by comparing this with Newton's second law as written in Equation 2a, we can derive a value for the acceleration due to gravity. We could write this as agrav, but conventionally, this quantity is known simply as g:
(12) g = GM / r2
This gives us, correctly, the acceleration experienced by falling objects near the surface of the Earth: 9.8 m/s2.
Now, observe something peculiar (and this is what that odd coincidence is all about). In order to derive this figure, we had to equate the gravitational mass of an object, in Equation 10 or 11, with the inertial mass of the object, in Equation 2a. The undeniable fact that both masses are denoted by the same letter m must not cloud the equally undeniable fact that there is no inherent reason why those two should be the same at all. For example, the electromagnetic attraction between two charged particles is given by Coulomb's law:
(13) FEM = Kq1q2 / r2
You'll notice that this formula looks very much like Newton's formula for gravity; it states that the electromagnetic force is equal to a constant K, times the charge on both of the objects, divided by the square of their distance. The difference between Coulomb's formula and Newton's is that the property that generates the electromagnetic force is not mass but charge, while the property that determines the acceleration due to that electromagnetic force is still mass.
Therefore, if you have a basketball and an anvil with equal charges, and you let them move only in response to the attraction of another charged object, they won't experience the same amount of acceleration -- the anvil will move slower, because even though the attracting force is the same, the anvil is more massive and therefore harder to move. But if you let them move only in response to the gravitational attraction of another massive object, they always move with the same acceleration, to a very high level of precision.
Gravity thus appeared to be a privileged force, since only its generating property, gravitational mass, seemed to be equated with the property that governed an object's response to force, inertial mass. For over 200 years, scientists puzzled about what to do with this strange equivalence of gravitational and inertial mass.
Einstein's solution to the puzzle came, as it often did with him, in the form of a thought experiment. Suppose you're standing in an elevator on the ground floor of a building. When an elevator starts to rise, it speeds up a little at the beginning; then, through most of the ride, it remains at a constant speed, until near the end, when it slows down a little and comes to rest at your floor.
At those moments when the elevator is either speeding up or slowing down, you feel different than when the elevator is standing still or moving at a constant speed. In particular, when it's speeding up, you feel an extra downward pull on you, as if you suddenly weighed more. If you drop a ball during this time, it falls faster than it would if you dropped it in a stationary elevator or a uniformly moving one. In short, an elevator that's accelerating upward produces a sensation of increased gravity.
The opposite happens when the elevator slows down. Suddenly, you feel lighter than you normally do; dropped objects fall more slowly than they would in a stationary elevator. Another way to say the elevator is slowing down is that it's accelerating negatively, or downwards, so an elevator that's accelerating downward produces a sensation of decreased gravity.
If an accelerating elevator could change your sensation of gravity, Einstein reasoned, it could also create a sensation of gravity when there was none to begin with. But instead of an elevator, consider a spacecraft in empty space. As long as it's stationary, everything feels weightless. You don't feel any compression between your feet and the floor of the spacecraft, if that's where your feet currently are, and if you drop objects, they don't fall to the floor but remain floating where they are. Furthermore, if you gently toss a drink over to a companion, the drink doesn't go spilling onto the floor, but continues to float on a direct line to your companion. In short, it's not like a car on earth at all; it's a truly inertial frame of reference.
However, suppose the spacecraft begins accelerating upward (as measured with respect to the spacecraft's ceiling) at 1 g -- that is, 9.8 m/s2. Inside the spacecraft, it would feel as though you were experiencing a pull, toward the bottom of the spacecraft, of 1 g. If you dropped a ball, it would instantly begin to fall to the floor, just as it would on earth. Someone "at rest" outside the spacecraft would see the ball move uniformly, while the spacecraft accelerated upward to meet the ball, but aboard the spacecraft, you would see it the other way. In fact, with regard to everyday experiences, everything on board the spacecraft would behave precisely as it would in a stationary spacecraft on a launching pad on the Earth. The only way you could tell the difference would be if the spacecraft had a window you could look through.
Einstein proposed that not only would all everyday experiences be the same, but that every last physical property of the accelerating spacecraft would in fact be indistinguishable from those of a spacecraft sitting on the Earth. To put it more generally, Einstein asserted what he called the principle of equivalence:
There is no way to distinguish between a gravitational field, operating in a uniformly moving (or stationary) frame of reference, and an accelerating frame of reference.
The principle of equivalence neatly explains why all objects near the Earth fall with the same acceleration (and it therefore also explains the equivalence between gravitational and inertial mass). If you have an anvil and a basketball next to each other in a stationary spacecraft in empty space, they remain motionless (and therefore still next to each other) so long as no force is applied to either of them. If you then accelerate the spacecraft at 1 g, then of course they hit the floor at the same time, since nothing has changed to separate them. An outside observer would see the pair of objects sitting still, while the spacecraft accelerated upward to meet them together.
But by the principle of equivalence, this is precisely what happens in a gravitational field as well. In short, Einstein claimed that you could magic away gravity, simply by changing your perspective from a uniformly moving frame of reference to an accelerating one. The principle of equivalence is the central column of Einstein's general theory of relativity.
Now, just as with Newton, Einstein could make all the claims of equivalence he wanted, but in order to actually convince anyone, he had to make quantitative predictions. Furthermore, since Newton's laws were the accepted ones, Einstein had to make predictions that were different, under the proper conditions, from what was predicted using Newton's laws. It would do no good for Einstein to simply say, "If you drop a tennis ball in an accelerating spacecraft in outer space, it will fall just as it does on earth," because that's also what Newton's laws say, and since Newton's theory was already entrenched, there would be no reason for physicists to accept a more complex theory, if both theories agreed on everything. In order to supplant Newton's laws, general relativity would have to make some unusual predictions.
Fortunately, Einstein was able to reason his way to a number of those strange conclusions, much as he had done with special relativity, and all of those predictions have been tested and verified, to a level of precision equal to experimental error. Essentially, Einstein said that space and time operate in such a way to keep all objects moving in an inertial frame of reference, unless impeded by some other force (such as the electromagnetic interference of the atoms making up the Earth). In order to explain the acceleration of objects as seen by someone standing on the Earth, space and time had to be curved, or "warped," by massive objects, so that its geometry wasn't flat. The exact formulation of this in Einstein's paper involved some advanced mathematics, and can't be completely understood without that math.
One of the first people to understand it was the British astronomer and physicist, Arthur Eddington (1882-1944). (Eddington was once told by a reporter that he -- the reporter -- knew of only three people who claimed to understand general relativity. Eddington characteristically replied, "Three? Who's the other?") In 1919, during a total eclipse of the Sun, he carried out an experiment to test one of the predictions of general relativity. According to Einstein's theory, stars that appeared close to the edge of the Sun would have their light bent by the space-time warping created by the Sun's enormous mass. Your eyes don't know about the bending, so they just follow the last known heading of the light back to where the star appears to be. (See Figure 2.)
Figure 2. Bending of a star's light predicted by general relativity (not to scale).
Even with the Sun's mass (2x1030 kg), the amount of bending predicted by Einstein is very small -- only about 1.6 arcseconds, the width of a golf ball as seen from 4 km away. What made it tougher was that Newton's laws also predicted a bending of just half that size, so that Eddington had to distinguish an angle of only 0.8 arcseconds. Usually, stars that close to the Sun are washed out in its glare. However, during a total eclipse, the Sun's disc is blocked by the Moon, and stars quite close to the edge of the Sun are discernible. Even so, the experimental error was nearly as large as that angle, but the measurement came out generally in favor of Einstein. Eddington cautiously claimed victory, and all later measurements have vindicated general relativity. The recent Hipparcos mission was able to measure the deflection at very large angles from the Sun, so we're no longer restricted to testing this prediction during solar eclipses.
Most of the relevant predictions can only be understood quantitatively by specialists, but we can comprehend a few using only what we already know from special relativity. We'll look at just one case here -- the behavior of space and time in a gravitational field, such as that of the Earth.
Suppose you have two clocks, A and B. Clock A is on the ground, and clock B is up in the air, 1 km overhead. Each clock clearly thinks it's running at a uniform rate. But do they run at the same rate? In Newtonian physics, time is absolute, and they run at the same rate. The clocks are motionless with respect to each other, so special relativity also says they run at the same rate.
However, general relativity says differently. In general relativity, you can only compare clocks if they're in the same reference frame with respect to both motion and acceleration. They're not moving with respect to one another, but clock B is higher -- it would fall to clock B's level if unsupported. So let's have them meet. We drop clock B -- it's indestructible -- until it lands. At the moment it strikes the ground, it's moving at a high rate of speed, due to gravity. According to special relativity, each clock should see the other one running slowly. Which one is right?
Again, the deciding principle is that inertial clocks don't slow down --only non-inertial ones do. Clock B is moving inertially; if an observer moving with the falling clock dropped a ball, it would not fall faster than he does, but would continue to float alongside him. Clock A is not moving inertially, according to general relativity, since the force of gravity can be equated with a non-inertial frame of reference. Therefore it's clock B that runs at the normal rate, and clock A that runs slow. That is precisely the prediction that general relativity makes: Clocks lower down in a gravitational field run slower than clocks higher up.
Here's another way to put it. Suppose you're in a big elevator shaft, 1 km tall, in empty space, free from the effects of gravity. There's a clock at the top of the shaft -- that's clock B -- and a clock in the elevator car at the bottom -- that's clock A. As long as the elevator doesn't move, both clocks are in the same reference frame and run at the same rate.
If the elevator begins accelerating upward at 1 g, it goes faster and faster until it hits the top of the shaft and we can finally compare the clocks. Both clocks think they're running at a uniform rate, but because clock A is accelerating, it must be running slower than clock B. How much slower? If you accelerate at 1 g, then when you've gone 1 km, you're moving at approximately 140 m/s (about 310 mph). Special relativity says that a clock moving 140 m/s runs slow by about 1 part in 9 trillion.
By Einstein's principle of equivalence, there is no difference between this situation and the original one, so a clock on the ground runs about 0.00000000001% slower than one that's 1 km up in the air. Specialized atomic clocks can verify this prediction to better than one part in 1,000.
However, the same reasoning allows us to compare clocks at any height. Suppose clock A remains on the ground, but clock B is raised to an enormous height -- an infinite height, in fact. Let's run the same experiment, letting B free-fall in its inertial frame down to the ground. How fast is it moving when it hits the ground? The speed depends only on the mass of the Earth and its radius:
(14) v = sqrt(2GM / r)
Einstein's general relativity simply says that you plug that speed into the special relativity equations, and that tells you how slow clocks run at that height. If we perform the necessary substitutions, we get a time dilation effect of
(15) t/to = sqrt(1 - (2GM / rc2))
This means that clocks on the surface of the Earth run slow by about 1 part in 1.4 billion. Now that is large enough to be verified experimentally -- in fact, GPS devices, which depend on clocks aboard geostationary satellites in orbit far above the Earth, must take into account the relative speed-up of those clocks compared with those on Earth.
Incidentally, the same equivalence principle allows us to deduce that lengths are compressed in a gravitational field, just as they are in objects moving at a high speed. For example, if you take a perfectly rigid sphere (one that isn't deformed by tidal forces) from outer space and bring it back to the earth, its height is actually less than its width. Unfortunately, even if there were such a thing as a perfectly rigid sphere, you couldn't measure this change, because when you measure the width with a ruler, you're holding the ruler horizontally and it's uncompressed, whereas when you're measuring the height, you're holding the ruler vertically and it's compressed just as much as the object! So the measurements will come out just the same. As far as I know, there are no good experiments to test this particular prediction of relativity.
Adapted from Astronomical Games: May 2001.
Brian Tung is a computer scientist by day and avid amateur astronomer by night. He is an active member of the Los Angeles Astronomical Society and runs his own astronomy Web site. His previous publications in Strange Horizons can be found in our archives.