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.