The Principle of Equivalence

The General Theory of Relativity is formulated in terms of mathematics well beyond the scope of our survey course in astronomy (primarily in fields of mathematics that go by the names of tensor analysis and Riemannian geometry). Nevertheless, many of the basic ideas can be understood without extensive mathematics.

### General Relativity: the Principle of Equivalence

One of the most important of these is the Principle of Equivalence, which can be used to derive important results without having to solve the full equations of General Relativity.

There are several ways to formulate the Principle of Equivalence, but one of the simplest is Einstein's original insight: he suddenly realized, while sitting in his office in Bern, Switzerland, in 1907, that if he were to fall freely in a gravitational field (think of a sky diver before she opens her parachute, or an unfortunate elevator if its cable breaks), he would be unable to feel his own weight. Einstein later recounted that this realization was the "happiest moment in his life", for he understood that this idea was the key to how to extend the Special Theory of Relativity to include the effect of gravitation. We are used to seeing astrononauts in free fall as their spacecraft circles the Earth these days, but we should appreciate that in 1907 this was a rather remarkable insight.

### Importance of the Equivalence Principle

An equivalent formulation of the Principle of Equivalence is that at any local (that is, sufficiently small) region in spacetime it is possible to formulate the equations governing physical laws such that the effect of gravitation can be neglected. This in turn means that the Special Theory of Relativity is valid for that particular situation, and this in turn allows a number of things to be deduced because the solution of the equations for the Special Theory of Relativity is beyond the scope of our course, but is not particularly difficult for those trained in the required mathematics.

### Consequences of the Principle of Equivalence

For example, by considering the Principle of Equivalence applied to light travelling across a freely falling elevator, it is possible to conclude that light will follow a curved path in a gravitational field. See this discussion to understand how. Likewise, by considering light travelling upwards in an elevator in free fall, it is possible to conclude that light will be redshifted in a gravitational field.

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