From acceleration togeometry

In exploring the equivalence of
gravity and acceleration as well
as the role of tidal forces,
Einstein discovered several
analogies with the geometry of
surfaces. An example is the
transition from an inertial
reference frame (in which free
particles coast along straight
paths at constant speeds) to a
rotating reference frame (in
which extra terms
corresponding to fictitious
forces have to be introduced in
order to explain particle motion):
this is analogous to the
transition from a Cartesian
coordinate system (in which the
coordinate lines are straight
lines) to a curved coordinate
system (where coordinate lines
need not be straight).
A deeper analogy relates tidal
forces with a property of
surfaces called curvature. For
gravitational fields, the absence
or presence of tidal forces
determines whether or not the
influence of gravity can be
eliminated by choosing a freely
falling reference frame. Similarly,
the absence or presence of
curvature determines whether
or not a surface is equivalent to
a plane. In the summer of 1912,
inspired by these analogies,
Einstein searched for a
geometric formulation of
gravity.[12]
The elementary objects of
geometry – points, lines,
triangles – are traditionally
defined in three-dimensional
space or on two-dimensional
surfaces. In 1907, the
mathematician Hermann
Minkowski introduced a
geometric formulation of
Einstein's special theory of
relativity in which the geometry
included not only space, but also
time. The basic entity of this new
geometry is four-dimensional
spacetime. The orbits of moving
bodies are lines in spacetime; the
orbits of bodies moving at
constant speed without changing
direction correspond to straight
lines.[13]
For surfaces, the generalization
from the geometry of a plane –
a flat surface – to that of a
general curved surface had been
described in the early 19th
century by Carl Friedrich Gauss.
This description had in turn been
generalized to higher-dimensional
spaces in a mathematical
formalism introduced by
Bernhard Riemann in the 1850s.
With the help of Riemannian
geometry, Einstein formulated a
geometric description of gravity
in which Minkowski's spacetime is
replaced by distorted, curved
spacetime, just as curved
surfaces are a generalization of
ordinary plane surfaces.[14]
After he had realized the validity
of this geometric analogy, it
took Einstein a further three
years to find the missing
cornerstone of his theory: the
equations describing how matter
influences spacetime's
curvature. Having formulated
what are now known as
Einstein's equations (or, more
precisely, his field equations of
gravity), he presented his new
theory of gravity at several
sessions of the Prussian
Academy of Sciences in late
1915.