On pp. 41–42, Einstein arrived at an even more encouraging result. According to the early version
of the equivalence principle, rotation should be equivalent to some gravitational field. This is
illustrated in the figure below. Standing on a rotating merry-go-round your velocity constantly
changes direction, which means that you are experiencing a centripetal acceleration. This feels as if there is a force—known as the centrifugal force—that is trying to throw you off the merry-goround.
According to the equivalence principle, that situation should be fully equivalent to one in
which you are standing on a merry-go-round at rest in a peculiar centrifugal gravitational field.
One can easily calculate the metric field describing space and time from the point of view of an
observer in uniform rotation. The early version of the equivalence principle requires that this metric
field can also be interpreted as a gravitational field. This means that it should be a solution of
the field equations of the theory. To check this, Einstein used the same approximation procedure
he had used to find the field of the sun for the perihelion calculations. In first-order approximation, it is easily verified that the metric field for the rotating observer is indeed a solution of the field equations of the Einstein-Grossmann theory. Moreover, this first-order metric field has the same form as the first-order metric field for the case of the rotating shell found on pp. 36–37, which fits nicely with the idea that this metric field can be interpreted as the field produced by the distant
stars rotating with respect to the observer. Einstein then substituted this first-order field into the
field equations in a second-order approximation and checked whether the metric field of the rotating
observer is also a solution to the equations at this further level of approximation. He concluded
that it is. Next to the final result of his calculation on p. 41, he wrote: “stimmt” (“is
correct”). The relevant passage is reproduced at the top of the next page.