In Einstein gravity ( and
) the field equations (9.2
) and (9.4
) are identical
to the equations (2.7
) and (2.4
), respectively. However, the difference appears for the f (R) models
which include non-linear terms in
. While the kinetic term
is present in Eq. (2.7
),
such a term is absent in Palatini f (R) gravity. This has the important consequence that the
oscillatory mode, which appears in the metric formalism, does not exist in the Palatini formalism.
As we will see later on, Palatini f (R) theory corresponds to Brans–Dicke (BD) theory [100
]
with a parameter
in the presence of a field potential. Such a theory should be
treated separately, compared to BD theory with
in which the field kinetic term is
present.
As we have derived the action (2.21) from (2.18
), the action in Palatini f (R) gravity is equivalent to
Using the relation (9.3), the action (9.5
) can be written as
There is another way for taking the variation of the action, known as the metric-affine
formalism [299, 558
, 557
, 121]. In this formalism the matter action
depends not only on the metric
but also on the connection
. Since the connection is independent of the metric in this approach,
one can define the quantity called hypermomentum [299], as
. The usual
assumption that the connection is symmetric is also dropped, so that the antisymmetric quantity called the
Cartan torsion tensor,
, is defined. The non-vanishing property of
allows the
presence of torsion in this theory. If the condition
holds, it follows that the Cartan
torsion tensor vanishes (
) [558]. Hence the torsion is induced by matter fields with the
anti-symmetric hypermomentum. The f (R) Palatini gravity belongs to f (R) theories in the
metric-affine formalism with
. In the following we do not discuss further f (R) theory
in the metric-affine formalism. Readers who are interested in those theories may refer to the
papers [557, 556].
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