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## Einstein’s General Relativity – part 1 – the Principle of Equivalence

Next year, 2015, marks the centennial of Einstein’s theory of gravity, what we now call the General theory of Relativity (or just “General Relativity” – “GR”). It is widely recognised as one of the greatest achievements in science, and when Arthur Eddington validated one of its predictions in 1919 Einstein was catapulted to the status of an international star. It is often said that, whereas Einstein’s 1905 special theory of relativity (or “special relativity”) would have been thought of by someone else had Einstein not come up with it, general relativity was so far ahead of its time that we may still be waiting for it if it were not for Einstein’s unparalleled genius.

A portrait of Albert Einstein from around the period that he started developing his theory of gravity, General Relativity.

As it turns out, the development of Einstein’s new theory of gravity was not an easy one. Over the course of several blogs I will trace this tortuous path, which took the best part of ten years, mainly because he had to learn the mathematics of curved space and Tensor calculus to be able to express his ideas in equations. Today I will discuss the beginnings of GR, and in particular what we now call Einstein’s “principle of equivalence”, which he thought of in 1907.

## Einstein’s 1905 Special theory of Relativity

I have already blogged about Einstein’s ground-breaking Special theory of Relativity here. Just to recap, based on two assumptions

1. There is no experiment one can do to distinguish between one inertial (non-accelerating) frame of reference and another
2. The speed of light is constant in all inertial (non-accelerating) frames of reference

Einstein was able to show that these two postulates require that strange things happen to space and time when one travels an appreciable fraction of the speed of light. Lengths get shorter, and time passes more slowly. One of the other consequences of this theory is that Einstein predicted that no information can travel faster than the speed of light.

Einstein soon realised, after he had developed his theory, that Newton’s theory of gravity was in violation of special relativity because it violates both of the postulates on which special relativity is based. In Newton’s theory of gravity, the gravitational force between two objects acts instantaneously. So, according to Newton, if the Sun were to disappear, we would instantly notice its absence (the Earth would move in a straight line rather than continue in its orbit).

Secondly, you could have two inertial (non-accelerating) frames of reference in two different gravitational fields (e.g. one on the surface of the Earth and the other on the surface of the Moon), and a simple experiment like the swinging of a pendulum would yield a different result. This is because the force of gravity (which, along with the length of the pendulum’s string, determines its period of motion) would be different in the two places.

## Einstein’s “happiest thought”

In 1907 Einstein was still working in obscurity in the Patent Office in Bern. Although his special theory of relativity had been published two years before, it was yet to have received much attention. It wasn’t until 1908 that he would get his first academic appointment. In his largely boring patent clark job, Einstein had allowed his mind to wander just as he had done leading up to his miraculous year of 1905. This time, it was in pondering how he could fit Newton’s theory of gravity into his own special relativity. One day he had what he would later refer to as the “happiest thought of my life”. In a lecture on the origins of general relativity which he gave at Glasgow University in June 1933 (“The Origins of the General Theory of Relativity”), he expressed this 1907 thought as

If a person falls freely he will not feel his own weight

Very few of us have experienced free-fall, but most of us have been in a lift (elevator). Right at the start, when the lift starts moving, we temporarily feel heavier and our stomach may feel as if it is sinking. When we slow down at the top of the lift’s travel we temporarily experience the opposite, we feel lighter and our stomach may feel as if it is about to hit our diaphragm!

What Einstein realised is that, if a person were in a lift and the cable were to snap so that the lift fell freely towards the Earth, that person would feel weightless whilst the lift was falling. Their feet would come away from the floor of the lift, and if they took e.g. coins out of their pocket, those coins would not fall towards the floor of the lift but instead would appear to “float” next to the person.

Einstein realised in 1907 that being in a lift (elevator) which is falling freely would feel the same as being in empty space – you would feel weightless.

Einstein next illustrated his absolute genius – he went from this idea, which is fairly specific, to the much more general principle of equivalence – which states that:

there is no experiment you can do to distinguish between the effects of a uniform gravitional field and that of uniform acceleration

Einstein’s “happiest thought” led to his principle of equivalence, which simply states that being in a uniform gravitational field feels the same as accelerating in empty space. They cannot be distinguished from each other. The consequences of this idea are profound and far reaching.

## The first mention of what would become “General Relativity”

Einstein was under pressure from his German editor to write up a review of his principle of special relativity, and so in late 1907 he wrote an article entitled “Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen”
(On the Relativity Principle and the Conclusions Drawn from It) which appeared on the 4th of December 1907 in the journal Jahrbuch der Radioaktivität. In a section of this review article he included some ideas as to what would happen if he were to generalise his special theory of relativity to include the effects of gravity. He noted a few consequences (without going into the details as he had yet to work them out) – gravity would alter the speed of light and hence cause clocks to run more slowly (i.e. gravity would slow down time). He even postulated that generalising special relativity to include gravity may explain the drift in the perihelion of Mercury’s orbit, something which had been confusing astronomers for several decades.

## Gravity bends light

One of the more celebrated predictions of Einstein’s general theory of relativity is that gravity should bend light. As I mentioned above, in 1919 this was shown to be the case by England’s foremost theoretical astrophysicist of the day, Arthur Eddington. I will go into the details of what he measured in another blog in this series on general relativity, but to finish this part one I will explain how gravity bends light in Einstein’s model.

To understand how this happens, we have to go back to the principle of equivalence. Remember, this states that whatever is true inside a lift which is accelerating in empty space is also going to be true for a lift which is stationary in a uniform gravitational field.

Imagine that a beam of light enters the lift horizontally on the left hand side of the lift. Because the lift is accelerating, rather than follow a straight path across the lift, it will appear to follow a curve (actually a parabola), and it will exit at a lower point on the right hand side than where it entered (this is exactly the same kind of path as a ball would follow if it is projected horizontally from a platform e.g. 200m above the Earth’s surface).

Through the principle of equivalence, if a beam of light crossing an accelerating lift will follow a curve, so will a beam of light crossing a stationary lift which is in a gravitational field. So, gravity should bend light!

Light traversing a lift which is accelerating will appear to bend (in fact it will follow a parabolic path). Because of the principle of equivalence, light should be similarly affected by gravity.

As Einstein developed the mathematics of his general theory he was able to work out precisely how much a given gravitational field should bend light, and his predicted amount was found to be true for the Sun in a celebrated experiment in 1919 by Arthur Eddington.

In part two of this blog I will discuss some of the mathematical obstacles Einstein faced in bringing his general theory of relativity to fruition.

### 6 Responses

1. You would start blogging this just after I have delivered my book on 1915 and the general theory. Now I can’t pinch your stuff for the book! š

• Pinch it for the 2nd edition š

2. on 18/12/2014 at 15:59 | Reply cliffwilmington

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3. […] I have already explained in this blog here, Einstein’s principle of equivalence tells us that whatever is true for acceleration is true […]

4. […] knew that Einstein was working on a new theory of gravity. As I blogged about here, he had his insight into the equivalence between acceleration and gravity in 1907, and ever since […]