We have to check that the appropriate limit, General Relativity leads to Newton's theory. The limit we shall use will be that of small velocities and that time derivatives are much smaller than spatial derivatives.
There are two things we must do:
where is small. Since we require that , the inverse metric is given by
To work out the geodesic equations we need to work out what the components of the Christoffel symbols are:
Substituting for etc. in terms of we obtain
The geodesic equations are
But for a slowly moving particle so
Also , so we can neglect terms like . The geodesic equation reduces to
so the ``space'' equation (three- acceleration) is
Since we get
Now
where we have neglected time derivatives over space derivatives. The spatial geodesic equation then becomes
But Newtonian theory has
where is the gravitational potential. So we make the identification
This is equivalent to having spacetime with the line element
This is what we deduced using the Equivalence Principle.
Let's now look at the field equations [ with ]:
Taking the trace we get
This allows us to write the field equations as
Let us assume that the matter takes the form of a perfect fluid, so the stress- energy tensor takes the form:
Taking the trace gives
so the field equations become
The Newtonian limit is . This gives
Look at the 00 component of these equations:
to first order in . Now
to first order in . The (0,0) component of this equation is
and since spatial derivatives dominate over time derivatives, we get
So the field equations are
This is just
Comparing this with Poisson's equation:
we see that we get the same result if the constant is
We can now use this result to write down the full Einstein field equations: