In Special Relativity , in a coordinate system adapted for an inertial frame, namely Minkowski coordinates, the equation for a test particle is:
If we use a non- inertial frame of reference, then this is
equivalent to using a more general coordinate system
. In this case, the
equation becomes
where 
is the metric
connection of 
, which is still a flat metric but
not the Minkowski metric 
. The additional
terms involving which
appear, are inertial forces.
The principle of equivalence requires that gravitational forces, a
well as inertial forces, should be given by an appropriate
. In this case we can no longer
take the spacetime to be flat. The simplest generalization is to
keep as the metric
connection, but now take it to be the metric connection of a
non- flat metric. If we are to interpret the
as force terms, then it
follows that we should regard the as
potentials.
The field equations of Newtonian gravitation
consist of second- order partial differential equations in the
potential 
. In an analogous manner, we would expect that General
Relativity also to
involve second order partial differential equations in the potentials
. The remaining task which will allow us to build
a relativistic theory of gravitation is to construct this set of partial
differential equations. We will do this shortly but first we must
define a quantity that quantifies spacetime curvature.