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.
Even with the Sun’s mass (2×1030 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?