What is warped spacetime?

Imagine walking down a street in your neighbourhood and coming upon an ordinary-looking house at the end of the street. You observe the house from the outside, walking all the way around it, and in doing so get a pretty good idea of its size, and thus how big you would expect it to be on the inside. You might picture a living room, a kitchen, two bedrooms and so on. Now imagine walking through the front door and finding that the house is much bigger on the inside than you expected. There is a cavernous living room, a restaurant-sized kitchen, fourteen spacious bedrooms. In short, what you thought would be an ordinary house actually contains a large mansion’s worth of rooms! Such an Alice in Wonderland experience is an example of “warped” (or “curved”) space. Surprisingly, this is not just the stuff of fiction: using his imagination and the power of mathematics, Albert Einstein discovered that our universe is actually like this, and used the phenomenon of warped space (and warped time) to explain what gravity is. This is the basis of Einstein’s theory of General Relativity.

To set the stage for Einstein’s ideas, let us begin by recalling what Sir Isaac Newton said about gravity. Imagine a planet, say the Earth, moving in empty space. Being completely empty, there is nothing around (like the Sun) to exert any forces on the Earth, so it moves in a straight line, coasting with whatever speed it had to start with. Remember, there is no friction in empty space to slow, or in any other way change the Earth’s motion — it will coast forever in the same direction and with the same speed, much like a puck sliding across a (nearly) frictionless “air-hockey” table. Now let us introduce the Sun, as shown in the figure below.

Newton believed that the Sun exerts a mysterious “gravitational force” on the Earth, causing it to veer off its otherwise straight-line trajectory. Supposing the Earth is initially moving “up” at point A, instead of coasting along the straight line from A towards P it experiences a “gravitational force”, labelled F_A and directed towards the centre of the Sun, pulling it off track so that it instead ends up at point B some time later. At point B the Earth would like to continue on a straight-line path towards Q, but Newton’s “gravitational force” (now labelled F_B) again causes it to veer towards the Sun. Due to this “gravitational force”, the Earth is continuously “falling towards the Sun” instead of moving in a straight line as it would like to. In this way, Newton explained how the Earth and the other planets orbit the Sun.

Newton’s picture of gravity has several serious problems. For one thing, Einstein realized that Newton’s theory was not compatible with his discovery that nothing can travel faster than the speed of light (part of his theory of Special Relativity). According to Newton’s theory, if the Sun were to suddenly disappear when the Earth was at point B, the force F_B would instantly disappear, allowing the Earth to break out of its circular orbit and continue on the straight line towards point Q. Einstein, on the other hand, argued that there is no way the Earth could know that the Sun had disappeared until at least eight minutes after point B, eight minutes (or thereabouts) being the time it takes light (or any other signal moving at the speed of light) to travel from the Sun to the Earth. In other words, Einstein reasoned that the Earth should continue on its circular orbit for at least another eight minutes after point B before breaking orbit. Newton’s theory had no way of accounting for this necessary time delay. Even worse, astronomers at the time knew of a clear cut example where Newton’s theory does not correctly predict what is actually observed in the motions of the planets, in particular the planet Mercury — click here for more details.

In response to these problems, Einstein provided a new and completely different picture of gravity. Rather than a mysterious “gravitational force”, he embraced the idea that space and time might be warped, and that this warping might account for the phenomenon we call gravity. It is interesting to note that the question of whether the space in our universe might be warped was actually first posed sixty years before this by the great mathematician Georg Friedrich Bernhard Riemann, who in 1854 provided the mathematical tools necessary to describe warped spaces. However, it took a series of spectacular “thought experiments” on the part of Einstein, within the context of his newly discovered theory of Special Relativity, to realize that space (and time) must, indeed, be warped, and he went on to pursue the detailed consequences of this fact for our understanding of how the universe works.

Let’s now turn to Einstein’s picture of gravity in detail, beginning again with our Earth-Sun example. Einstein’s idea was that the mass of the Sun warps the space around it, in a way very similar to our strange Alice in Wonderland house. To understand the nature of this warping, consider marking out a large, imaginary sphere enclosing the Sun and some of the space around it, as indicated by the dashed circle in the figure below.

Now imagine astronauts flying about on the surface of this imaginary sphere, making various measurements, say of the distance around its equator or its surface area. Just like looking at our Alice in Wonderland house from the outside, the astronauts can gauge the size of the sphere and from this information can infer the volume of space contained inside it. Or think of a basketball: measuring how big the basketball is on the outside we can infer exactly the volume of space inside the ball. But if we now let the astronauts fly around inside this volume they will find, according to Einstein, that there is more space inside the sphere than they expected. There’s simply more room to move about. Like our cavernous living room analogy, more could be fit into this “room” than they thought. In short, Einstein suggested that the mass of the Sun warps the space around it so that there is more volume inside any sphere we choose to mark out in space (and surrounding the Sun) than one would expect based on knowing just the size of the sphere and using ordinary geometry. And, very importantly, the amount of warping is greater nearer to the Sun and diminishes as we move further away.