Can we test this? Well, suppose there is a ladder leading from a point on the floor to a hatch at the centre of the top of the cylinder. If Alice is standing on the floor, time is moving more slowly for her than it is for Bob because of her rapid circular motion. If she climbs up a few rungs of the ladder (moving against the “pull of gravity”), her speed of motion will be reduced (notice that this speed is zero at the top of the ladder). So the slowing of time relative to Bob will likewise be reduced (at the top of the ladder her time will be running at the same rate as Bob’s). In this way Einstein reasoned that time might be expected to move at different rates depending on where a person is in a gravitational field. In particular, time should move faster (i.e., less slowly) the higher up we are a gravitational field (i.e., the higher we climb against the “pull of gravity”).

Let’s apply this lesson to real gravity on the surface of the Earth. If Alice and Bob are standing side by side, their respective time is obviously moving at the same rate. If Alice now climbs to the top of a tower (moving against the “pull of gravity”), time should be moving more quickly for her than it is for Bob at the bottom of the tower. If she spends the day at the top of the tower and then climbs back down, she will have aged more than Bob by virtue of the fact that she spent some time in a part of the universe where time moves more quickly. The effect is quite small in Earth’s relatively weak gravitational field, but easily measured using accurate atomic clocks. In fact, the Global Positioning System, which relies on very accurate measurements of time intervals made by atomic clocks both here on the Earth and high above in orbiting satellites, must account for this gravitational warping of time in order to work properly. There is no doubt that this effect is real. It has been measured to great precision. What is amazing is that this effect was predicted by Einstein simply through the power of careful thought and imagination—through a “thought experiment”!

The Warping of Space

Let’s now move on to our second example: the warping of space that accompanies a gravitational field. Imagine that Alice is standing on a platform with wheels that is rolling swiftly past Bob (to the right in the figure). Bob wishes to measure the width of Alice’s shoulders while she is in motion. How will he do this? At the instant she is passing in front of him he thrusts both hands forward such that, in that instant, he is touching both of her shoulders at the same time, one hand on one shoulder, the other hand on the other shoulder. Of course since Alice is moving, he can touch her shoulders in this way only for a split second before she moves on past his hands. He then carefully withdraws his hands, walks over to a metre stick, and measures the distance between his hands, which is the distance between her shoulders as far as he is concerned. The surprise is that the distance he measures in this way gets smaller and smaller the faster she is moving. The effect is difficult to measure for speeds we experience in everyday life, but becomes dramatic as Alice approaches the speed of light. Close to the speed of light, Alice will be paper thin according to Bob. However, it is important to realize that—just as in the case of time dilation—Alice herself experiences no such contraction. As far as she is concerned, she maintains the same ratio of width to height. No distortions. This effect is called length contraction, and has been verified experimentally to extremely high precision.

Before we go on to see the consequences of this effect for gravity, let’s try to understand a bit better what is going on here, simply because it’s really quite fascinating. Both length contraction and time dilation can be understood as simple consequences of something called “relativity of simultaneity”. It is a quirky fact about our universe that if Bob snaps his fingers on both hands simultaneously, the snapping of his fingers will not be seen as simultaneous to someone in motion relative to Bob. In our previous example, if Bob snaps his fingers simultaneously according to him, then according to Alice she will see the fingers of his right hand snap first, followed by the fingers of his left hand. Why? I won’t try to answer this here. So what? I will answer this: According to Bob, when he is measuring the width of Alice’s shoulders, his right hand is touching Alice’s left shoulder at the same time that his left hand is touching her right shoulder. These events, like the snapping of his fingers, are simultaneous according to Bob. But Alice does not experience them as simultaneous. What she sees is first Bob’s right hand thrusting forward to touch her left shoulder, followed shortly afterwards by his left hand thrusting forward to touch her right shoulder. It’s no surprise to her that Bob is measuring a shorter width. After all, between the times that Bob touches her left shoulder and touches her right shoulder, she has moved some distance to the right. Her right shoulder has moved closer to where her left shoulder used to be, so there is nothing really strange about Bob’s shorter measurement. (The situation is a bit more complicated that I have described, but this is the main effect going on.)

Notice that what’s happening here is delightfully bizarre: when Bob “reaches into” Alice’s “moving frame”, he thinks he is touching her shoulders at the same time (and in fact he is, according to his sense of simultaneity), but in fact his hands are touching two shoulders, one of which is in the future of the other, according to Alice’s sense of simultaneity! It is important to stress, however, that although we can understand why Bob’s measurement comes out smaller in terms of this relativity of simultaneity, it doesn’t remove the physically real fact that Alice is contracted in Bob’s frame—she really does occupy a lesser volume of (Bob’s) space!

So what are the consequences for gravity? Imagine that, when the space station is not rotating, exactly 100 Alices can be fit—shoulder to shoulder—around the inside circumference of the cylinder. And suppose we place 100 Bobs around the outside circumference of the space station, one Bob for each Alice. After the space station has begun to rotate, at one instant of time (simultaneous for all of the Bobs) each measures the width of the Alice immediately in front of them at that moment. Just as in the previous case of Alice gliding past Bob on a moving platform, each Bob will determine that his respective Alice is contracted in width. We still have 100 Alices, but each Alice is now thinner, so that 100 of them strung together no longer span the circumference. What actually happens as the space station begins to spin faster is that gaps begin to appear between the shoulders of the Alices. If the space station is spinning fast enough, each of these gaps will be wide enough to fit another Alice, so that we could now have 200 Alices standing shoulder to shoulder inside the space station!