Extended Gravity Models 107where훹represents some matter fields. Many proposals have been made to general-
ize this action. Suppose that the universe contains a scalar field휙not present in the
Einstein equation. There are then different options how to introduce it, even the met-
ric frame may need to be reformulated. Since the Einstein equation describes how
matter affects the geometry and the curvature of space-time, the scalar field in the
energy-momentum tensor will also affect the geometry.
The Robertson–Walker metric in Equation (2.32) determines the geometry of
space-time by the components푔휇휈in Equation (2.33), the affine connections훤휎휈휇 in
Equation (3.13), and the Ricci scalar푅in (3.18) in what is called theEinstein frame.
If the matter field휙is introduced into the action in such a way that no terms휙푅nor
퐹(휙)푅mixing matter and geometry appear, we are in the Einstein frame. The opposite
case is theJordan frame. The two frames are only mathematically different, because
one can transform expressions in one frame into one in the other by aconformal
transformationsuch as푔̂휇휈=휁^2 (푥)푔휇휈,where휁^2 (푥)is the conformal factor. There is no
physical difference, only a choice of mathematical convenience.
Let us now rewrite the Einstein–Hilbert action in the most general form in the
Jordan frame as
푆total=
∫
d푛푥√
−푔
[
1
2 휅
푓(푅,휙)+phi(푔휇휈,휙,휕휙)+푀(푔휇휈,훹)]
. (5.85)
Here we have replaced the curvature scalar푅by a general function푓(푅,휙),and
enlarged the dimensionality of space-time from 4 to푛. The Lagrangian density is then
usually of the form
phi=−푀2
2휔(휙)(휕휙)^2 −푉(휙), (5.86)
where we have denoted(휕휙)^2 ≡(∇훼휙)(∇훼휙). For a canonical scalar field the function휔
equals unity.
Two simple versions of extended gravity are those which lack a scalar field휙but
are nonlinear in푅,andscalar-tensortheories. Some nonlinear functions푓(푅)studied
are of the form(푅+훼푅^2 )and(푅−훼∕푅). The dynamical equations are then written in
the Einstein frame.
Scalar-tensor theories are of the form푓(푅,휙)=퐹(휙)푅in the Jordan frame, of which
the classical (1961)Brans–Dicketheory with퐹(휙)=휙is the simplest. Some other
examples of scalar-tensor theories are퐹(휙)=exp(−휙)and퐹(휙)=훼휙^2.
If the theory has no scalar field휙one proceeds as for the Einstein–Hilbert action,
varying the action푆with respect to the metric푔휇휈and setting the variation equal to
zero. For scalar-tensor theories the equations of motion follow from setting the varia-
tion of푆with respect to휙equal to zero, and next differentiating the component of the
energy–momentum tensor푇휇휈which depends on휙. After that, one again proceeds to
vary the action푆with respect to the metric푔휇휈and setting the variation equal to zero.
Note that since the Ricci scalar푅contains second derivatives of the metric, one
ends up with fourth order terms of the metric which are difficult to analyze. A more
tractable form of extended gravity can be obtained by using thePalatini variation.In
this formulation the affine connection훤휎휈휇in Equation (3.13) is treated as an indepen-
dent variable from the metric푔휇휈. Varying the action separately with respect to푔휇휈