### 11.1 Field equations

We already showed that under the conformal transformation the action (10.10) is transformed to the Einstein frame action:
Recall that in the Einstein frame this gives rise to a constant coupling between non-relativistic matter and the field . We use the unit , but we restore the gravitational constant when it is required.

Let us consider a spherically symmetric static metric in the Einstein frame:

where and are functions of the distance from the center of symmetry. For the action (11.1) the energy-momentum tensors for the scalar field and the matter are given, respectively, by

For the metric (11.2) the and components for the energy-momentum tensor of the field are

where a prime represents a derivative with respect to . The energy-momentum tensor of matter in the Einstein frame is given by , which is related to in the Jordan frame via . Hence it follows that and .

Variation of the action (11.1) with respect to gives

where is the field Lagrangian density. Since the derivative of in terms of is given by Eq. (2.41), i.e., , we obtain the equation of the field  [59442]:
where a tilde represents a derivative with respect to . From the Einstein equations it follows that
Using the continuity equation in the Jordan frame, we obtain
In the absence of the coupling this reduces to the Tolman–Oppenheimer–Volkoff equation, .

If the field potential is responsible for dark energy, we can neglect both and relative to in the local region whose density is much larger than the cosmological density (). In this case Eq. (11.8) is integrated to give

Substituting Eqs. (11.8) and (11.9) into Eq. (11.7), it follows that

The gravitational potential around the surface of a compact object can be estimated as , where is the mean density of the star and is its radius. Provided that , Eq. (11.13) reduces to Eq. (5.15) in the Minkowski background (note that the pressure is also much smaller than the density for non-relativistic matter).