What has this got to do with weightlessness? Instead of one rock, imagine throwing two rocks, side by side, both initially moving in the same direction. Both will travel along separate orbits, but they will always remain close to one other (actually, the orbits will cross each other twice before the rocks return—can you see why?). Standing on one of the rocks, we would see the other rock simply floating nearby—just “hanging in space”. Now imagine that one of the rocks is an astronaut and the other is the space shuttle. It should be clear now why astronauts float “weightless” inside the space shuttle! The same thing applies to Alice in our big tin can.

Artificial Gravity and the Equivalence Principle

Now, as a space engineer you have been given the task of creating an “artificial gravity field” in this space station so that Alice and her comrades can work in a normal gravity environment like on Earth. How will you do it? Simple: make the big tin can rotate on its axis! Just like in the notorious “spinning cylinder” amusement park ride in which you go round and round and the floor drops out from beneath you, Alice will be “stuck” to the outer wall of the rotating space station. It will feel to her as if there is some mysterious “force” pushing her against the wall. Moreover, rather than lying flat with her back against this wall, as in the amusement park ride, she can stand up so that the wall is now the floor! As far as gravitational effects are concerned, standing on this floor would feel no different to her than standing on the surface of the Earth. She could also walk around on this floor normally as if she were walking on the Earth.

The important question to ask now is: Does Alice’s experience of this “artificial” gravity effectively mimic the features of “real” gravity she experiences when standing on the Earth? Actually, yes. To help see this, imagine that Alice is holding an apple in her hand. If she were standing on the Earth she would feel the apple’s “weight”, which appears to be due to some sort of mysterious “gravitational force” pulling the apple to the ground. Since the idea of gravity as a force is in question here, let’s describe this situation without reference to “weight” or “gravitational force”. We can simply say: to keep the apple from falling to the ground Alice finds that she must exert an upward force on the apple. Certainly this much is true. Imagining, in addition to this, there is some sort of gravitational force at work is unnecessary and, in fact, wrong.

To see this, let’s now transplant Alice and her apple to the rotating space station. Taking a top view of the situation we can see that when Alice is holding the apple in her hand, the apple—like Alice—is moving in a circle. The law of inertia tells us that any object always wants to move in a straight line at a constant speed, and any deviation from this natural state of motion requires a force of some kind acting on the object. So, since the apple is moving in a circle (which is not a straight line) there must be some force acting on it! And a bit of thought tells us that this force must be directed towards the centre of the rotating cylinder (always perpendicular to the path; always pushing the apple sideways, off the straight line path it wants to go on). But where is this force coming from? Alice’s hand, of course! Alice finds that: in order to hold the apple in place she must exert an upward force on it (force F in the figure; “upward” in this case meaning towards the centre of the rotating cylinder). She could interpret this as a force required to counterbalance some mysterious “gravitational force” tugging on the apple, i.e. the apple’s “weight”, or she could see it for what it is: the force required to continuously push the apple off its straight line, inertial trajectory (line from her hand to point P), making it move instead in a circle.

To further clarify this, suppose she now pulls her hand aside and lets the apple “fall”. At the instant she removes her hand the force she was exerting on the apple disappears. With no force acting on it, the apple is no longer compelled to move in a circle and it instantly begins moving on an inertial trajectory: the apple continues its motion along a straight line path with the speed and direction it had at the instant it was released (line from her hand to point P). After a few seconds on this force-free trajectory, the apple runs into the wall of the rotating cylinder (at point P), i.e. it hits the floor at Alice’s feet. (Note that during the apple’s free-fall, Alice continues to rotate around so that her feet reach point P at essentially the same time the apple does.) To sum up: Alice finds that when she lets go of the apple it falls to the floor, in exactly the same way it would if she were standing on the Earth witnessing the effects of the Earth’s gravitational field.

Moreover, if you study the diagram carefully you will see that from Alice’s perspective the apple appears to be accelerating (moving ever faster towards the floor), just as an apple accelerates when it falls towards the Earth. To see this, observe that near the beginning of its trajectory, just after it is released, the apple is drifting through space in almost the same direction as Alice is moving, so from Alice’s perspective the apple will appear to be almost hovering in space, falling only very slowly. But later, near point P, Alice’s trajectory is no longer parallel to the apple’s trajectory. In particular, the trajectory of her feet (which follows the curve of the rotating cylinder) has curved around so as to now be on a rapid collision course with the apple. So although Alice sees what appears to be an accelerating apple, just as she would in a real gravitational field, what’s actually going on is quite the opposite: it is Alice who is accelerating, not the apple; she is the one moving on a curved trajectory that eventually intersects the apple’s straight, non-accelerated, force-free trajectory.