…if light takes the path with the least time between the two points, and light beams bend under the influence of gravity, then the shortest distance between two points is a curved line. Einstein was shocked by this conclusion: if light could be observed travelling in a curved line, it would mean that space itself is curved.
Space Warps
At the core of Einstein’s belief was the idea that “force” could be explained using pure geometry. For example, think of riding on a merry-go-round. Everyone knows that if we change horses on a merry-go-round we feel a “force” tugging at us as we walk across the platform. Because the outer rim of the merry-go-round moves faster than the centre, the outer rim of the merry-go-round must shrink according to special relativity. However, if the platform of the merry-go-round now has a shrunken rim or circumference, the platform as a whole must be curved. To someone on the platform, light no longer travels in a straight line, as though a “force” were pulling it toward the rim. The usual [Riemann’s] themes of geometry no longer hold. Thus the “force” we feel while walking between horses on a merry-go-round can be explained as the curving of space itself.
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Imagine, for example, a rock stretched on a spread bedsheet. Obviously the rock will sink into the sheet, creating a smooth depression. A small marble shot onto the bedsheet will then follow a circular or elliptical path around the rock. Someone looking from a distance at the marble orbiting around the rock may say that there is an “instantaneous force” emanating from the rock that alters the path of the marble. However, on close inspection it is easy to see what is really happening: The rock has warped the bedsheet, and hence the path of the marble.
By analogy, if the planets orbit around the sun, it is because they are moving in space that has been curved by the presence of the Sun. thus the reason we are standing on the earth, rather than being hurled into the vacuum of outer space, is that the earth is constantly warping the space around us.
This picture can be summarized by what is called Mach’s principle, the guide Einstein used to create his general theory of relativity. We recall that the warping of the bedsheet was determined by the presence of the rock. Einstein summarized this analogy by stating: The presence of matter – energy determines the curvature of the space – time surrounding it. This is the essence of the principle that Riemann failed to discover, that the bending of space is directly related to the amount of energy and matter contained within that space.
This, in turn, can be summarized by Einstein’s famous equation, which essentially states:
Matter – energy
where the arrow means “determines”.
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Living in a curved space
Another, more relevant example for our universe might be living in a curved space given by a hypersphere, a sphere in four dimensions. If we look ahead, light will circle completely around the small perimeter of the hypersphere and return to our eyes. Thus we will see someone standing in front of us, with his back facing us, a person who is wearing the same clothes as we are.
Is this person a fake image created by mirrors? To find out, we stretch out our hand and put it on his shoulder. We find that the person in front of us is a real person, not just a fake. If we look into the distance, in fact, we see an infinite number of identical people, each facing forward, each with his hand on the shoulder of the person, in front.
But what is most shocking is that we feel someone’s hand sneaking up from behind, which then grabs our shoulder. Alarmed, we look back, and see another infinite sequence of identical people behind us, with their faces turned the other way.
What is really happening? We, of course, are the only people living on the hypersphere. The person in front of us is really ourself. We are staring at the back of our own head. By placing our hand out in front of us, we are really stretching out our hand around the hypersphere, until we place our hand on our own shoulder.
The counterintuitive stunts that are possible in a hypershere are practically interesting because many cosmologists believe that out universe is actually a large hypersphere. There are also other equally strange topologies, like hyperdoughnuts and möbius strips.
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No longer a passive arena, space becomes an active player in the drama unfolding in our universe.
In 1921, astronomers had verified that starlight indeed bends as it travels around the sun, precisely as Einstein had predicted.
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