In order to have a complete theory of gravity, we need to know
How particles behave in curved spacetime.
How matter curves spacetime.
The first question is answered by postulating that free particles [ i.e. no force other than gravity ] follow timelike or null geodesics. We will see later that this is equivalent to Newton's law in the week field limit [ ].
The second requires the analogue of . We first consider the vacuum case [ ] .
The easiest way to do this is to compare the geodesic deviation equation derived in the last section with its Newtonian analogue. In Newtonian theory the acceleration of two neighboring particles with position vectors and are:
so the separation evolves according to:
This gives us
This clearly is analogous to the geodesic deviation equation
provided we relate the quantities and
Both quantities have two free indices, although the Newtonian index runs from 1 to 3 while in the General Relativity case it runs from 0 to 3.
The Newtonian vacuum equation is which implies that