50 General Relativity
This is most conveniently done using tensor notation, briefly presented in
Section 3.2, which has the advantage of permitting laws of nature to be written in
the same form in all invariant frames.
In Section 3.3 we derive Einstein’s law of gravitation starting from anaction prin-
ciple, the Einstein-Hilbert action. In Section 3.4 we use simple qualitative arguments
to rederive Einstein’s law in the weak field limit which is equivalent to Newton’s law
of gravitation.
3.1 The Principle of Equivalence
Consider the lift in Figure 3.1 moving vertically in a tall tower (it is easy to imagine
an lift to be at rest with respect to an outside observer fixed to the tower, whereas
the more ‘modern’ example of a spacecraft is not at rest when we observe it to be
geostationary). A passenger in the lift testing the law of gravitation would find that
objects dropped to the floor acquire the usual gravitational acceleration푔when the lift
stands still, or moves with constant speed. However, when the outside observer notes
Figure 3.1 The Einstein lift mounted in a nonEuclidean tower. An observer is seen in the
foreground.