2 Field Equations in the Metric Formalism
We start with the 4-dimensional action in f (R) gravity:
where
,
is the determinant of the metric
, and
is a matter
Lagrangian
that depends on
and matter fields
. The Ricci scalar
is defined by
, where the
Ricci tensor
is
In the case of the torsion-less metric formalism, the connections
are the usual metric connections
defined in terms of the metric tensor
, as
This follows from the metricity relation,
.