therefore
An extension might be:
where is an unknown object that relates to gravitational field.

Promotes to G.C.:

To determine , we write down the properties of :

is a tensor.

is made out of at most second derivatives of the metric tensor, e.g. , or .

.

On non-relativistic limit, .

These requirements uniquely determines the form of .

therefore, has the form .
Take the non-relativistic limit, that is, and the field is weak (), and so is the affine connection and the Riemann tensor. Also, .

Conclusion (Einstein's field equation):

Equivalent form:

In vacuum, , therefore , this is how gravitational waves are calculated, which does not imply Riemann tensor is . In fact, even when is , can still oscillate through spacetime.

(then a discussion about the cosmological constant).