In this notes, we are particularly interested in some special model of general relativity. More precisely, here we consider the scalar fields since, for example, a real scalar field more or less provides one of the simplest sources of stress-energy in GR.

**1. Derivation of stress-energy tensor**.

To derive the stress-energy tensor associated to a real scalar field, we make use of the Einstein-Hilbert action. Suppose that the full action of the theory is given by the Einstein–Hilbert term plus a term describing any matter fields appearing in the theory, then we have

The action principle then tells us that the variation of this action with respect to the inverse metric is zero, yielding . Calculating this equation gives

where the right hand side is nothing but the stress-energy tensor .

In modern cosmology, one can introduce on the spacetime a real scalar field with potential as a smooth function of . A particular Einstein field theory is specified by the choice of an action principle with

To find its associated stress-energy tensor, we first look at . Since the term does not depend on the metric, we get

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