In Einstein's field equations, it includes only energy momentum tensor of the matter alone. However, it doesn't include the energy momentum tensor of the field. In Professor Hamber lectures on General theory of Relativity, he said that the energy momentum tensor of the fields are present in the Ricci curvature tensor and the Ricci scalar. In the Ricci tensor, there are products of Christoffel symbols.
Does this product imply that there is a self interaction (i.e) interaction of field with itself?
Also why energy momentum tensors are expressed as square of the derivatives of the metric tensor?
What about its first derivative and cube or fourth power of derivatives of metric?