Fear of Physics (20 page)

If Galileo was along with you in the elevator, he would swear he was back on Earth. Everything he spent his career proving about the way objects behave at the Earth’s surface would be the same for objects in the elevator. So, while Galileo realized that the laws of physics should be identical for all observers moving with a constant velocity, Einstein realized that the laws of physics are identical for observers moving at a constant acceleration as those in a constant gravitational field. In this way, he argued that even acceleration is relative. One person’s acceleration is another’s gravity.

Again, Einstein peered beyond the cave. If gravity can be “created” in an elevator, maybe we are
all
just in a metaphorical elevator. Maybe what we call gravity is really related to our particular vantage point. But what is particular about our vantage point? We are on the Earth, a large mass. Perhaps what we view as a force between us and the Earth’s mass can be viewed instead
as resulting from something that the presence of this mass does to our surroundings, to space and time.

To resolve the puzzle, Einstein went back to light. He had just shown that it is the constancy of light that determines how space and time are woven together. What would a light ray do in the elevator accelerating in space? Well, to an outside observer, it would go in a straight line at a constant velocity. But inside the elevator, which is accelerating upward during this time, the path of the light ray would appear to look like this:

In the frame of the elevator, the light would appear to bend downward, because the elevator is accelerating upward away from it! In other words, it would appear to fall. So, if our accelerating elevator must be the same as an elevator at rest in a gravitational field, light must also bend in a gravitational field! This is actually not so surprising. Einstein had already shown that mass and energy were equivalent and interchangeable. The energy of a light ray will increase the mass of an object that absorbs it. Similarly, the mass of an object that emits a light ray decreases by an amount proportional to the energy of the radiation. Thus, if light
can carry energy, it can act as though it has mass. And all massive objects fall in a gravitational field.

There is a fundamental problem with this notion, however. A ball that is falling, speeds up! Its velocity changes with position. However, the constancy of the speed of light is the pedestal on which special relativity is built. It is a fundamental tenet of special relativity that light travels at a constant velocity for all observers, regardless of their velocity relative to the light ray as seen by someone else. Thus, an observer at the top left-hand side of the elevator observing the light enter would be expected to measure the speed of light as
c,
but so would an observer at the lower right, measuring the light as it exits the elevator. It does not matter that the bottom observer is moving faster at the time he views the light than the first observer was when he measured it. How can we reconcile this result with the fact that light globally bends and therefore must be “falling”? Moreover, since Einstein suggested that if I am in a gravitational field I am supposed to see the same thing as if I were in the accelerating elevator, then, if I am
at rest,
but in a gravitational field, light will also fall. This can occur only if the velocity of light globally varies with position.

There is only one way in which light can globally bend and accelerate but locally travel in straight lines and always be measured by observers at each point to travel at speed
c.
The rulers and clocks of different observers,
now in a single frame
—that of the accelerating elevator, or one at rest in a gravitational field—must vary with position!

What happens to the global meaning of space and time if such a thing happens? Well, we can get a good idea by returning to our cave. Consider the following picture, showing the trajectory of an
airplane from New York to Bombay as projected on the flat wall of the cave:

How can we make this curved trajectory look locally like a straight line, with the airplane moving at a constant velocity? One way would be to allow the length of rulers to vary as one moved across the surface, so that Greenland, which looks much larger in this picture than all of Europe, is in fact measured to be smaller by an observer who first measures the size of Greenland while he is there and then goes and measures the size of Europe with the same ruler when he is there.

It may seem crazy to propose such a solution, at least to the person in the cave. But we know better. This solution is really equivalent to recognizing that the underlying surface on which the picture is drawn is, in fact,
curved.
The surface is really a sphere, which has been projected onto a flat plane. In this way, distances near the pole are stretched out in the projection compared to the actual distances measured on the Earth’s surface. Seeing it as a sphere, which we can do by benefit of our three-dimensional
perspective, we are liberated. The trajectory shown above is really that of a line of longitude, which is a straight line drawn on a sphere and is the shortest distance between two points. An airplane traveling at a constant velocity in a straight path between these points would trace out such a curve.

What conclusion can we draw from this? If we are to be consistent, we must recognize that the rules we have found for an accelerating frame, or for one in which a gravitational field exists, are not unreasonable but are equivalent to requiring the underlying spacetime to be
curved!
Why can’t we sense this curvature directly if it exists? Because we always get a local view of space from the perspective of one point. It is just as if we were a bug living in Kansas. His world, which consists of the two-dimensional surface he crawls on, seems flat as a board. It is only by allowing ourselves the luxury of embedding this surface in a three-dimensional framework that we can directly picture our orb. Similarly, if we wanted to picture directly the curvature of a
three
-dimensional space, we would have to embed it in a four-dimensional framework, which is as impossible for us to picture as it would be for the bug whose life is tied to the Earth’s surface, and for whom three-dimensional space is beyond his direct experience.

In this sense, Einstein was the Christopher Columbus of the twentieth century. Columbus argued that the Earth was a sphere. In order to grasp that hidden reality, he argued that he could set sail to the west and return from the east. Einstein, on the other hand, argued that to see that our three-dimensional space could be curved, one merely had to follow through on the behavior of a light ray in a gravitational field. This allowed him to propose three classic tests of his hypothesis. First, light should bend when it travels near the sun twice as much as if it were merely “falling” in
a flat space. Second, the elliptical orbit around the sun of the planet Mercury would shift in orientation, or precess, by a very small amount each year, due to the small curvature of space near the sun. Third, clocks would run slower at the bottom of a tall building than at the top.

The orbit of Mercury had long been known to precess, and it turned out that this rate was exactly that calculated by Einstein. Nevertheless, explaining something that is already seen is not as exciting as predicting something completely new. Einstein’s two other predictions of general relativity fell in this latter category. In 1919, an expedition led by Sir Arthur Stanley Eddington to South America to observe a total solar eclipse reported that the stars near the sun that could be observed during the darkness were shifted from where they were otherwise expected to be by exactly the amount Einstein had predicted. Light rays thus appeared to follow curved trajectories near the sun, and Einstein became a household name! It was not until forty years later that the third classical test proposed by Einstein was performed, in the basement of the Harvard physics laboratory of all places. Robert Pound and George Rebka demonstrated that the frequency of light produced in the basement was measured to be different when this light beam was received on top of the building. The frequency shift, though extremely small, was again precisely that by which Einstein had predicted a clock ticking at such a frequency would be shifted at the two different heights.

From the point of view of general relativity, the curved and accelerated trajectories followed by objects in a gravitational field, including light, can be thought of as artifacts of the underlying curvature of space. Again, a two-dimensional analogy is useful. Consider the two-dimensional projection of the motion of an object spiraling in toward a larger object, as seen on the cave wall:

We could postulate a force between the objects to explain this. Or we could imagine that the actual surface on which this motion occurs is curved, as we can picture by embedding it in three dimensions. The object feels no force exerted by the larger ball, but rather follows a straight trajectory on this curved surface:

In just such a way, Einstein argued that the force of gravity that we measure between massive objects can be thought of instead as a consequence of the fact that the presence of mass produces a curvature of space nearby, and that objects that travel in this space merely follow straight lines in curved space-time, producing curved trajectories. There is thus a remarkable feedback between the presence of matter and the curvature of space-time, which is
again reminiscent of the Ouroboros, the snake that eats itself. The curvature of space governs the motion of the matter, whose subsequent configuration in turn governs the curvature of space. It is this feedback between matter and curvature that makes general relativity so much more complicated to deal with than Newtonian gravity, where the background in which objects move is fixed.

Normally the curvature of space is so small that its effects are imperceptible, which is one reason why the notion of curved space seems foreign to us. In traveling from New York to Los Angeles, a light ray bends only about 1 millimeter due to the curvature of space induced by the Earth’s mass. There are times, however, when even small effects can add up. For example, the 1987 Supernova I referred to previously was one of the most exciting astronomical observations of this century. However, one can easily calculate—and, in fact, a colleague and I were so surprised by the result that we wrote a research paper on its importance—that the small curvature through which the light from the 1987 Supernova passed as it traveled from one end of our galaxy to the other to reach us was sufficient to delay its arrival by up to nine months! Had it not been for general relativity and the curvature of space, the 1987 Supernova would have been witnessed in 1986!