

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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