Riemannian curvature tensor 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. 


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*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?
 A: The Einstein-Hilbert action, from which the Einstein field equations are derived, is given by,
$$S = \frac{1}{16\pi G_N} \int d^4x \, \sqrt{-g} \, \mathcal{R}$$
where $\mathcal{R}$ is the Ricci scalar, dependent on the metric $g_{\mu\nu}$ and its derivatives. The self-interaction and non-linearity of general relativity arises from this factor. To see this, let us work in natural units wherein $\hbar = c = 1$, and define the Planck mass $M_{pl}$,
$$8\pi G_N = M_{pl}^{-2}$$
The relevant coupling in the quantum theory is $1/M_{pl}$, so let us consider perturbations of the form,
$$g_{\mu\nu} = \eta_{\mu\nu} + \frac{1}{M_{pl}}h_{\mu\nu}$$
around flat spacetime with metric $\eta_{\mu\nu}$. We can then see that the action may be written schematically, neglecting messy index contractions as,
$$S= \int d^4x \, (\partial h)^2 + \frac{1}{M_{pl}} h (\partial h)^2 + \frac{1}{M_{pl}^2} h^2 (\partial h)^2 + \dots$$
Hence, we see that an infinite series of interaction terms are encoded in $S$, which include self-interactions of the field with itself. Whilst there are many issues to quantizing gravity, this is certainly one of them. In renormalization group language, the perturbation theory is an expansion in $E^2 / M^2_{pl}$, for some energy scale $E$, and so gravity is strong at high energies, and includes what are known as irrelevant couplings.
