Remember, as far as each Alice is concerned, no contraction whatsoever is happening. As mentioned before, each Alice always maintains the same ratio of width to height as measured by herself or anyone rotating along with the Alices inside the space station. What must be happening is that the physical circumference of the space station, as reckoned by people inside it, is growing! What started as a 50 metre jog around the space station when it was not spinning is now a 100 metre jog. It is important to realize that, just as it is time itself that is warped inside the space station, here it is space itself that is warped, relative to space as measured by the Bobs at rest outside the space station.

With this expansion of the spatial circumference, what happens to the metal skin of the space station? Same thing that happens with the Alices. A strip of metal that used to stretch 50 metres around the circumference when the station is not spinning is now being asked to stretch 100 metres around, which of course it can’t do. The metal would rip. So we need to either make the hull of the space station out of a flexible material that can expand, or we need to keep on welding in new sections of metal as the spatial circumference expands.

Notice that if the circumference is increasing, the physical volume of space inside the space station must also be increasing. And this effect is real. Suppose that, when the space station is not rotating, 10 thousand buckets of water could be poured into it before it is filled up. As the space station (and the water inside) begins to rotate, the physical volume of the space inside increases so that now additional buckets of water could be poured in from the outside. Although nothing changes on the outside—the Bobs outside see a rotating cylinder that always occupies the same volume of space—the amount of space inside is increasing; there is simply more room inside. Suppose that the entire interior volume of our rotating space station is filled with water (which is rotating along with the space station). If the rotation of the space station (and the water inside) were suddenly brought to a halt, the volume of space inside the space station would suddenly shrink to its normal value and would no longer be able to contain the excess water, and the latter would explode into the surrounding space!

Using the same equivalence principle argument we discussed above for time warping, Einstein reasoned that a gravitational field might be expected to be accompanied by some sort of warping of space. And indeed, this is exactly what happens in Einstein’s theory of general relativity. Moreover, this effect has been observed in experiments. For example, consider a huge imaginary spherical surface in space, centred on the Sun, with a radius equal to the radius of the orbit of Venus, say. As an astronaut in a spaceship you could navigate the surface of this sphere and measure its surface area (tedious and time consuming, but doable). Knowing the surface area you could calculate what you might expect the volume of space inside to be (just like by measuring the surface area of a basketball you can figure out the volume inside by using elementary geometry). The difference from the basketball case is that the gravitational field of the sun warps the space around it in such a way that there is more volume inside the surface than you predict using elementary geometry. If instead of just hovering on the surface of this imaginary sphere you take your spaceship inside, and move around, you will find that there is more room in there than you expected. More stuff could be put inside. The space is warped. This effect has recently been measured very accurately using NASA’s Saturn-bound Cassini spacecraft.

So we have seen how Einstein was led to the idea that a warping of space and time might be somehow intimately connected with the phenomenon of gravity. This is, of course, only the beginning of a long and endlessly fascinating story. The next step is to understand how a warping of spacetime can resolve the puzzle of “apparently at rest and yet continually accelerating” that we mentioned earlier. This is discussed in Part II of this essay. But we have covered enough ground for now!

Richard Epp
Director of Outreach Programmes
Perimeter Institute for Theoretical Physics