Empty out the drawer: Following Einstein’s path to General Relativity

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This month, we celebrate the centenary of Einstein’s discovery of a new theory of gravity – general relativity. Einstein’s achievement required perseverance and enormous creativity, as he struggled over a rough and winding road for eight years to formulate the theory. His path was guided by thought experiments and a philosophical approach to physics. His intellectual journey was also strikingly solitary.

Most scientific theories are the product of many minds working together, but general relativity was almost entirely the product of Einstein’s insights.

The story of Einstein’s discovery of general relativity also ends with a dramatic race to the prize, as Einstein’s pursuit of a new theory was joined by one of the best mathematicians of his day – who threatened to solve the final puzzles before Einstein.

 From space and time to spacetime

Einstein introduced relativity theory in 1905, one of four groundbreaking papers he published that year, soon after completing his PhD. These papers were completed in the spare time Einstein could find while he was working at the Swiss patent office, and he kept his work stashed in a drawer he called his ‘Department of Theoretical Physics.’ Einstein’s Department had more revolutionary ideas than most actual departments.

At the age of 16, Einstein wondered what would happen if he could ride on a light beam. What would a light beam look like if he were moving at the same speed next to it?

This thought experiment revealed a paradox regarding two ideas.

The first idea is called the relativity principle, and goes back to Galileo. It states all observers moving at uniform speed should observe the same laws of physics. Galileo considered a ship, but a better modern example is an airplane. Onboard a flight with no turbulence, everything in the cabin behaves in the same way it does when the plane is at rest on the tarmac. If the airplane lacked windows, there would be no way to tell the difference between these two situations.

The second idea is the constancy of the speed of light. The theory of electromagnetism describes light as an electromagnetic wave. These waves can’t stand still, instead always moving (in a vacuum) with a specific speed, the speed of light. The fact this speed is always the same is puzzling. Einstein’s contemporaries thought of light waves as moving through a substance called ‘the ether.’ How fast light apparently moves should change, depending on how an observer moves through the ether. Young Einstein, riding on a light beam, should say light is, in fact, not moving at all. But that would conflict with Galileo’s idea.

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Years after his youthful thought experiment, Einstein discovered, in 1905, the relativity principle could be combined with the constant velocity of light. All that was needed was to change the concepts of space and time. Einstein rejected the absolute character of time. Consider, for example, Alice moving on a train, and Bob standing by the side of the tracks. Suppose there are two lights set up on the train at equal distances from Alice. If the light flashes reach Alice at the same time, she will say that they flashed simultaneously. But Bob will disagree, because from his point of view, Alice is moving towards the light coming from the front of the train and racing away from the light at the other end. Since Alice and Bob agree on the speed of light, they disagree about whether the lights flashed at the same time.

It took Einstein to recognize there is, in fact, nothing wrong with this counterintuitive conclusion. This insight leads to a deeper understanding of space and time. Alice and Bob may disagree about the spatial distance and time elapsed between two events, such as the lights flashing, depending on their perspective. The objective facts, ones that Alice, Bob and other observers will agree to, are claims about the space-time distance between events.

A problem of gravity

This deeper understanding led to new problems. One of the most striking involves the most successful physical theory in Einstein’s day, Newton’s theory of gravity. This theory describes gravity as a force of attraction pulling bodies together. The force between two bodies depends on how far apart they are at a given moment, and it acts instantaneously. But this conflicts with Einstein’s new conception of space and time. The distance between two bodies, and the time when the attractive force pulls them together, are no longer objective.

Einstein’s contemporaries tried the most straightforward repair of Newton’s theory, which would make it compatible with the new spacetime ideas. But starting in 1907, Einstein pursued a different approach. He thought the problem had much deeper roots.

He drew inspiration from philosophical criticisms of Newton’s theory by Ernst Mach. Einstein followed Mach in thinking there was a deep flaw with any theory that privileged some descriptions of the motion of objects. We usually describe motion with respect to some background object, such as the table over which the billiard balls roll. On Mach’s view, motion is always a relation among bodies. Mach interpreted Newton as picking out motions with respect to “absolute space” as privileged. But “absolute space” is not another body, and not something we can observe.

Einstein followed Mach in thinking Newton had made a deep mistake. But he was worried his own earlier theory had made a similar mistake, since the relativity principle only applied to observers in uniform motion. (It is, after all, easy to tell the difference between a turbulent airplane ride and sitting at rest on the runway.) Einstein took to calling his own earlier theory special relativity because of this. He thought the key to solving the problem of gravity was a general theory, according to which all states of motion are relative.

In 1907, Einstein had ‘the happiest thought of his life,’ which he also explained later with a simple thought experiment. Suppose you are in a sealed elevator without windows. As with the discussion of Galileo’s principle above, there are two situations that appear the same from inside the elevator.

First, the elevator could be in a building on Earth, so the Earth’s gravity will have its usual effects. But, second, if the elevator were on an accelerating spaceship, the effects would be the same (if it were accelerating by the right amount). Einstein concluded from this thought experiment there must be a close connection between acceleration and gravity. Generalizing relativity to cover accelerated motion would also, Einstein thought, help to solve the problem of gravity. Einstein later called this idea the principle of equivalence, and it guided his search for a new theory.

Another thought experiment suggested by his friend, Paul Ehrenfest, led Einstein to connect gravity with ‘curved’ geometry.

Ehrenfest asked what would happen if one tried to determine the geometry of a rotating disk, according to Einstein’s special theory of relativity. One of the counterintuitive consequences of Einstein’s theory is length contraction: An object moving with respect to an observer will appear shorter than it does at rest.

Ehrenfest considered an observer at the centre of a rotating disk, and realized due to length contraction, rulers laid around the circumference of the disk would be shorter, whereas those along the radius would not. (The latter are not in motion with respect to the observer.) More rulers would fit around the circumference, meaning the usual Euclidean result (C = 2pr) would not hold.

This suggested a connection with new ideas in geometry regarding non-Euclidean geometries. Einstein would exploit this more general conception of geometry in his theory, which describes gravity in terms of curved spacetime.

 Race for the prize

By 1913, Einstein had developed a theory in which gravity was represented in terms of curved spacetime, called the Entwurf theory (based on the German title of Einstein’s paper). But he continued to worry about whether the theory ‘generalized’ the relativity principle in the right way. (At this point, he was concerned about whether the theory should be ‘generally covariant,’ which describes the mathematical form of the theory.)

When Einstein presented his theory in Göttingen in June 1915, the leading centre for mathematical physics of his day, he did not hide his ambivalence about the Entwurf theory or its problems.

Einstein probably expected the Göttingen experts would help him to make progress on his theory. But his presentation inspired David Hilbert, one of Europe’s best mathematicians, to focus intently on the problem of gravitation. Hilbert saw Einstein’s work as fitting into his own project of combining gravity and electromagnetism. Einstein was joined by a companion for the final stretch of his rough and winding road.

Einstein’s pace of work on general relativity quickened, spurred by concerns Hilbert would solve the problems, and find a fully satisfactory theory, before he did. The two were in constant correspondence. In November 1915, Einstein gave a series of four Thursday lectures, culminating in the presentation of the final theory on Nov. 26.

But did Hilbert finish first?

Hilbert presented a paper the week before Einstein’s. Hilbert’s paper, preserved in the archives, does not have Einstein’s equations, and Hilbert never claimed priority over Einstein for the theory.

Conclusion

Einstein achieved worldwide fame in 1919 when an expedition led by the British physicist Arthur Eddington observed light bending around the sun, as Einstein had predicted. General relativity has remained a central part of physics since its discovery, and physicists have continued to explore the complexities of the theory and its further implications.

Our understanding of Einstein’s path to discovery has been enriched by the work of many historians and philosophers of science. Several philosophers at Western have studied different aspects of Einstein’s work, including Robert DiSalle, Bill Harper, Carl Hoefer, Wayne Myrvold and myself.

Christopher Smeenk is the new Director of the Rotman Institute of Philosophy, having stepped into the role in September.

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MORE READING: For those interested in more about Einstein’s scientific work, the best introduction to his work is the online book Einstein for Everyone by John Norton. Also recommended are the Cambridge Companion to Einstein, edited by Michel Janssen and Christoph Lehner, for a more advanced discussion and Robert DiSalle’s book Understanding Space-Time considers Einstein’s relativity theory in relation to philosophical analysis of spacetime concepts.