The Essence of General Relativity, Part I: Einstein’s Rotating Disk Thought Experiment

Introduction

Albert Einstein had a powerful imagination and was famous for “thought experiments”: if “A” is true and “B” is true, then a bit of careful thought reveals that “C” must also be true, where “C” often turned out to be some amazing, deep new insight into the hidden “gears and wheels” at work in our universe. Perhaps the most beautiful of these is his “rotating disk” thought experiment (circa 1907), which played a crucial role in his development of the ideas that eventually led to general relativity — his revolutionary theory of space, time and gravity (1915).

As most people know by now, general relativity has superseded Isaac Newton’s model of gravity. Gravity is no longer thought of as some sort of “force” acting on objects. Following Einstein, physicists today understand gravity as a beautiful manifestation of the warping of the geometry of space and time. But wasn’t Newton’s model of gravity brilliantly successful for 250 years? Yes. So why do we need another model? There were basically two facts that knocked Newtonian gravity off its pedestal:

1. Experimental: Newtonian gravity fails to correctly predict the observed orbits of the planets, in particular the planet Mercury. The predictions are quite good, but careful observations in the 19th century revealed slight discrepancies that no one could find a way to resolve. General relativity resolved these discrepancies. (For more detail follow this link.)
2. Theoretical: Newtonian gravity is not compatible with Einstein’s idea that our universe has a speed limit: no material object or information of any kind can travel faster than 299,792, 458 m/s (about 1 billion km/hr). For instance, at this speed it takes about 8 minutes for light to travel from the Sun to the Earth. So if the Sun were to suddenly disappear, the universal speed limit forbids us any means of knowing of this catastrophe until 8 minutes later. Light itself respects this required time delay: the Sun would continue to blaze in our sky during the whole 8 minutes it takes the last ray of light from the Sun to make its journey to the Earth. Only then would the Earth be plunged into darkness. Newton’s model of gravity does not respect this required time delay: it predicts that at the same instant the Sun disappears, the “gravitational force” it exerts on the Earth would also disappear, causing the Earth to immediately break out of its usual orbit. Observers on night side of the Earth could be immediately aware of this catastrophe by noticing a change in the apparent motion of the distant stars produced by the change in Earth’s motion. General relativity resolved this problem.

Einstein was aware of these problems and the need for an improved model of gravity. But when he began his quest he did not realize how radically different from Newton’s model it would turn out to be! At its conceptual foundation, Einstein’s geometrical model of gravity is diametrically opposite to Newton’s force model. Understanding the basic idea is very much like having first seen the beautiful young woman in this famous Gestalt picture and then later, suddenly seeing the old woman, who was there all along but you just weren’t looking at it in the right way. Can you see both?

We will begin this essay by introducing the gravitational analogue of this Gestalt picture, called Einstein’s “equivalence principle”. Taking this equivalence principle seriously, together with some earlier ideas about the nature of space and time he had learned from his work on the theory of special relativity (1905), Einstein devised his ingenious rotating disk thought experiment. The result was strong motivation for the idea that, whatever gravity is, it is very likely to be intimately connected with a warping of space and time (or spacetime, for short). This was not an entirely new idea: in a landmark lecture delivered on the 10th of June, 1854, the great mathematician Georg Friedrich Bernhard Riemann (see picture) asked the question: might the space we live in be warped? The question was half a century ahead of its time. Unaware of Riemann’s question, Einstein asked if spacetime (not just space) might be warped. And unlike Riemann, Einstein provided also a strong physical motivation for the idea. At this time Einstein was not particularly mathematically inclined, but armed with this new insight Einstein learned of Riemann’s ideas and laboured to understand the mathematics he had developed. Out of this, general relativity was born. It is a rich and fascinating story, but in this essay (Part I) I will focus only on how Einstein arrived at this tantalizing potential connection between gravity and warped spacetime.

Weightlessness

Let’s start by imagining that NASA has constructed a space station in the shape of a giant soup can, and this soup can is freely floating in an orbit around the Earth (I know, a soup can is not as cool-looking as the International Space Station, but work with me…). We fill up this giant tin can with air and imagine our intrepid astronaut Alice floating around inside, weightless just like astronauts floating around inside the space shuttle. Let’s first remind ourselves what “weightless” means.

Newton came up with a beautiful argument to imagine how an object such as the Moon orbits the Earth. The illustration at right is from his book, the Principia (1687), one of the most important and influential books ever written. Imagine standing at the top of a mountain and throwing a rock straight out, in a direction horizontal to the Earth’s surface. The rock will describe a curved trajectory that takes it some distance away from the base of the mountain before it hits the ground. If we throw the rock harder, it will travel a further distance before it hits the ground. We can easily imagine throwing it hard enough to make it go all the way around and hit the Earth at the base of the mountain. Throwing it harder still, we could even imagine it circling the Earth several times before hitting the ground (we are ignoring air resistance!). Of course if we throw it too hard (faster than about 40,000 km/hr), it will actually escape Earth’s gravitational field never to return—what goes up does not always come down! So there must be some speed less than this (it turns out to be about 28,000 km/hr) for which the rock will circle the Earth and return to exactly its starting point (and hit us in the back of the head if we’re not careful), whizzing by us with exactly the same speed it had when it was first thrown. (At this speed it takes about 84 minutes to travel around the Earth). So what will it do now? It will simply go around again, exactly as before, and again, and again: the rock will be in orbit around the Earth, at an altitude equal to the height of the mountain. The European Space Agency Web site has a nice animation of this.