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
[594
, 42
]:
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).