# General Relativity from Newtonian Gravity

Start with Newtonian gravity field, the validity of which is locally verified for weak fields :

(1)

which is, expressed in terms of the energy-momentum tensor

(2)

Now attempt a first ansatz to formulate this in a Lorentz-invariant manner :

(3)

which further leads to a covariant formulation of the form

(4)

Our task will now be to determine the unknown tensor G. We impose the following conditions on that tensor :

1. G is a Riemann tensor
2. G is composed of the first and second derivatives of the metric tensor
3. The energy-momentum tensor obeys the usual symmetry and conservation laws and ; these properties then by default must also apply to our tensor G
4. The theory must reduce to Newton’s gravity for weak fields

Using the above four points, the Bianchi identities, as well as the general ansatz

(5)

plus a little tensor algebra, one find that the easiest tensor which satisfies all of the above conditions is

(6)