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92 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
Jordan–Brans–Dicke (JBD) [2] cosmology is the most simple scalar–
tensor theory that under FRW metrics with perfect fluids its nonlinear
field equations comprise two unknown variables which makes their integration
more difficult, and its solutions hard to come by in contrast
with GR which only has a single function –the scale factor– to determine.
Consequently, a fruitful avenue exploited to obtain cosmological
solutions in JBD, first done in [3] and later elsewhere [4], has been to
find the way to combine the two aforementioned variables into a single
one. This or any other method employed to obtain barotropic, fluid solutions
in this, and other similar but more general scalar–tensor theories
[5] has been fully accomplished in non flat spaces only for the vacuum,
incoherent radiation, and stiff matter [6] so this paper shows how to
advance the “single function” method further in order to integrate the
field equations for other perfect fluids.
The Lagrangian for scalar–tensor gravity theories with no self interaction
can be expressed as
where is the spacetime Ricci curvature scalar, is the scalar field,
a dimensionless coupling function while G, Newton’s gravitational
constant together with c, the speed of light are put equal to 1. Finally,
is the Lagrangian for the matter fields. JBD assumes
and the variation of its action with respect to and generates the
following field equations
and
is the energy–momentum tensor for a perfect fluid where is the pressure,
and its energy density. is its four velocity, its trace,
and its covariant derivative is