Celebrating Einstein

"The General Theory of Relativity"

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C.  Gravity and Acceleration

One of the more significant clues that Einstein noted early in his investigation involved a simple way to produce gravity where it might not be expected to exist.

Imagine being inside an elevator and tossing a ball from one hand to the other.  The ball goes up, the ball comes down, over and over-nothing special there.  But now imagine being in the same elevator floating somewhere in outer space, away from the earth or any other large celestial bodies.  Tossing the ball there produces quite a different result:  instead of rising in an arc, getting slower near the top, and then speeding up as it falls, the ball moves in a straight line without slowing down at all.  Instead of falling into your other hand, the ball just moves straight toward the top of a wall or the ceiling, and bounces around up there in the corner a few times and then comes back toward you in another straight line.  We see in this a demonstration of Isaac Newton's first law of motion-the law of inertia:  things at rest tend to remain at rest, while things in motion tend to continue moving at a constant speed and direction, unless some force acts on them to change their rest or motion.

Now imagine this "space elevator" beginning to move, as shown in the top row of Figure 2.  Suppose it accelerates "upward" at a steady rate, getting faster and faster each second.  Now when you toss the ball, from the perspective of someone outside the elevator who isn't accelerating, the ball still moves in a straight line, but you and the elevator floor begin to overtake it as the elevator continues to gather speed.  From your perspective, it turns out that the ball rises at first but slows down, then falls toward your other hand.  The steady acceleration produces the same sort of effect as a gravitational field.  In fact, the similarity is so great that, if you couldn't see outside the elevator and had no idea where you were, and the acceleration was one g (9.8 meters per second per second), you'd have no way to tell that you weren't on earth.  Any experiment you could do on the elevator would give results consistent with your being in an ordinary gravitational field.

Figure 2

Figure 2.  Top row:  "elevator" accelerating "upward" in zero gravity, seen at three different times.  Tossed ball, as seen from outside the elevator, moves at constant velocity, as indicated by the straight line.  Bottom row:  view from within the elevator.  From this perspective, the tossed ball undergoing the same motion is seen as rising and falling, just as it would in a uniform gravitational field.

As the elevator accelerates upward, all objects floating around inside will appear to fall with the same downward acceleration, no matter what masses the objects have, provided they have enough mass that the air in the elevator doesn't noticeably inhibit their motion.  The action of gravity is exactly the same:  as long as gravity is the only thing influencing their motion, light objects will fall just as quickly as heavy ones.  (When they don't, it's because the buoyancy and viscosity of the air affect light objects more.  Gravity by itself doesn't treat light and heavy objects differently, as we learn from experiments with objects falling in vacuum.  One such experiment, using a feather and a hammer, was even done once on the moon.  In the absence of air, the feather fell right alongside the hammer when both were dropped together.)

What we in the elevator would see as objects falling under the influence of gravity, someone who wasn't accelerating like the elevator would see as objects moving at constant speeds in straight lines, just as the law of inertia describes.  Put another way, the spacetime path of each "falling" object is a straight line, even though the paths look curved from our perspective in the elevator; in spacetime, our own path, accelerating as we are, is the curved one.

A lesson Einstein grasped from contemplating such "thought-experiments" is that objects acting under the influence of gravity alone might be following the straightest possible paths in spacetime.  If spacetime is curved, the straightest possible path for an object may be curved itself, just as the straightest possible connection between two points on the earth's equator is a segment of the curved equator.  This suggests a way to generalize the law of inertia:  while the paths of undisturbed objects are straight lines in uncurved spacetime, the paths in curved spacetime are the straightest possible lines, with the only bends being those necessary to follow the curvature of spacetime itself.

Einstein saw that the curvature of spacetime might provide an explanation for the "ordinary" gravity seen with all objects.  If the spacetime near a massive object were curved, any other object passing by would move along whichever curved spacetime path was the straightest one in front of it, as long as gravity was the only influence on the motion.

Once Einstein understood this, it remained for him to figure out what determined the spacetime's exact curvature.  Different curvatures would result in different paths for objects in spacetime.  What curvature would result in the paths objects actually follow due to gravity?     (.....continued)

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Date Modified: 02/15/2006