3.5 Other Theories
4.1.1 Action, field equations and transformations
The f (R) generalisations of Einstein’s equations are derived from a Lagrangian den- sity of the form
L =√−gf(R), (286)
where the factor of√−g is included, as usual, in order to have the proper weight. This is clearly about as simple a generalisation of the Einstein-Hilbert density as one could possibly conceive of. The field equations derived from such an action are automatically generally covariant and Lorentz invariant for exactly the same reasons that Einstein’s equations are. Unlike the Einstein-Hilbert term, however, the field equations that one obtains from the least action principle associated with Eq. (286) depend on the varia- tional principle that one adopts. Different possibilities are the ‘metric variation’ where the connection is assumed to be the Levi-Civita one, the ‘Palatini approach’ in which Eq. (286) is varied with respect to the metric and connection independently, and the ‘metric-affine’ approach in which the same process occurs but the matter action is now taken to be a functional of the connection as well as the metric. In this section we will mostly be concerned with the metric variational approach, although we will also outline how the other procedures work below.
Metric variational approach
Let us now derive the field equations in the metric variational approach. Integrating Eq. (286) over a 4-volume, including a matter term and varying with respect to gµν
yields δS = Z dΩ√−gh 1 2f g µνδg µν+ fRδR + χ 2T µνδg µν i , (287)
where fRmeans the functional derivative of f with respect to R, χ is a constant, and Tµν
is the energy-momentum tensor defined by a variation of the matter action with respect to gµν in the usual way. Assuming the connection is the Levi-Civita one we can then
write
fRδR' −[fRRµν+ fR;ρσ(gµνgρσ− gµρgνσ)]δgµν, (288)
where' is used here to mean equal up to surface terms [568]. Looking for a stationary point of the action, by setting the first variation to zero, then gives
fRRµν−
1
2f gµν− fR;µν+ gµνfR= χ
2Tµν. (289)
These are the f (R) field equations with which we will be primarily concerned in this section. It can be seen that for the special case f = R the LHS of Eq. (289) reduces to the Einstein tensor, and the field equations are second-order in derivatives of the metric. For all other cases, except an additional constant, the equations in (289) are fourth-order in derivatives.
Conformal transformation in the metric variational approach
As with scalar-tensor theories, the f (R) theories of gravity derived from the metric variational approach can be conformally transformed into a frame in which the field
equations become those of General Relativity, with a minimally coupled scalar field. This is sometimes referred to as ‘Bicknell’s theorem’ in the case of f (R) theories, particularly when the minimally coupled scalar field is in a quadratic potential [155]. In the general case we consider conformal transformations of the form [101, 853]
¯
gµν = fRgµν, (290)
together with the definition
φ≡ r
3
χln fR, (291)
which allows the field equations from the metric variational principle, Eqs. (289), to be transformed into ¯ Rµν−1 2¯gµνR =¯ χ 2 φ,µφ,ν−1 2g¯µν¯g ρσφ ,ρφ,σ− ¯gµνV +χ 2T¯µν. (292) Here ¯Tµν is a non-conserved energy-momentum tensor, and we have defined
V = V (φ)≡ (RfR− f) χf2
R
. (293)
Theories derived from an action of the form (286) can therefore always be conformally transformed into General Relativity with a massless scalar field (as long as fR6= 0), and
a non-metric coupling to the matter fields.
Legendre transformation in the metric variational approach
As well as conformal transformations, one can also perform Legendre transformations on f (R) theories in the metric variational approach. Such transformations allow the field equations of f (R) to take the form of a scalar-tensor theory (albeit it a slightly strange one). These transformed theories maintain the universal metric coupling of the matter fields, unlike the case of conformal transformations.
The first step here is to notice that the Eq. (286) can be written in the equivalent form
L =√−g [f(χ) + f0(χ)(R− χ)] , (294)
where χ is a new field, and the prime denotes differentiation. Variation with respect to χ then gives
f00(χ)(R− χ) = 0, (295)
so that χ = R for all f00(χ)6= 0. Substitution of this result back into Eq. (294) then
immediately recovers Eq. (286), showing that the two Lagrangian densities are indeed equivalent. What is more, the special case f00(χ) = 0 can be seen to correspond to the
Einstein-Hilbert action.
Now, if we make the definition
φ≡ f0(χ), (296)
and assume that φ(χ) is an invertible function, then we can define a potential Λ(φ)≡ 1
2[χ(φ)φ− f(χ(φ))] . (297)
In terms of this new scalar field we can then write the density (294) as
L =√−g [φR − 2Λ(φ)] , (298)
which is clearly just a scalar-tensor theory, as specified in Eq. (97), with vanishing cou- pling constant, ω = 0. As we have not transformed the metric, the coupling of this field to any matter fields that are present remains unchanged.
The Palatini procedure
Starting again from the density (286) we can also proceed in a entirely different way to the metric variational approach just described. Instead of assuming the connection from which the Ricci scalar is constructed is the Levi-Civita one, we could instead treat the metric and connection as independent fields. For the case of General Relativity a variation with respect to the connection then simply results in the connection being shown to be the Levi-Civita one, so that the difference between the metric variational approach and the Palatini approach is a semantic one. For the case of f (R) theories, however, the Palatini approach leads to an entirely different set of field equations.
Starting with an integral of Eq. (286) over some 4-volume, and extremising with respect to gµν now gives
fRRµν−
1 2gµνf =
χ
2Tµν, (299)
where Tµν is once again the usual energy-momentum tensor. In this expression Rµν
is now defined independently from the metric, and R should be taken to be given by gµνR
µν. The next step is the variation of Eq. (286) with respect to Γµνσ, which results
in
√
−ggµνfR;σ= 0, (300)
where the covariant derivative here should be understood to be taken with respect to Γµ
νσ, which is not the Levi-Civita connection unless fR =constant (as is the case in
GR). It is remarkable that the field equations (299) do not involve any derivatives of the metric, and only first derivatives of the connection. These are a different set of field equations to Eq. (289), and should be considered a different set of theories to the f (R) in which R is a priori taken to be constructed from the Levi-Civita connection.
It can be noted from Eq. (300) that even if the connection is not compatible with the metric gµν, we can still define a new metric, ¯gµν = fRgµν, with which it is compatible.
Rewriting the full field equations under this conformal transformation we see that we recover General Relativity with a minimally coupled scalar field in a potential, but no kinetic term:
L =√−¯g ¯R− 2V (φ), (301)
where φ ≡ fR and V (φ) = (R(φ)φ− f(φ))/2φ2. Here R(φ) and f (φ) are given by
inverting the definition of φ, and ¯R should be understood to be constructed from the metric connection compatible with ¯gµν. Transforming back to the original conformal
frame this theory then can be shown to be equivalent to a scalar-tensor theory with ω =−3/2 and Λ = (RfR− f)/2 [999].
While being an interesting variant on the metric variational incarnation of the f (R) theories, there are a number of very severe problems in proceeding with the Palatini
procedure in this way. Not least of these is the apparent ill-posed nature of the Cauchy problem in the presence of most matter fields, which is discussed in [766]. Without a well-posed initial value problem f (R) gravity in the Palatini formalism lacks predictive power, and hence is not a good candidate for a viable theory of gravity. Furthermore, the Palatini approach to f (R) gravity also appears to introduce problematic strong couplings between gravity and matter fields at low energies [509, 640], and singularities in systems that are usually well described by weak fields [87, 88, 89]. For these reasons we will focus on f (R) theories in the metric variational approach in the sections that follow.
For further details of the Palatini approach to f (R) gravity, and the results that follow from it, the reader is referred to [1167]. For studies of weak field gravity in the Palatini formalism the reader is referred to [92, 429, 881, 999, 30, 1000, 1163, 1001, 1078, 224, 665, 31, 1077], and for cosmology to [1251, 509, 883, 28, 882, 884, 29, 1162, 1164, 724, 34, 727, 258, 793, 1232, 725, 781, 782, 497]. An interesting class of theories that interpolate between the Palatini approach to f (R) theories and the metric variational approach to f (R) theories is investigated in [39, 726].
The metric-affine approach
One further approach to the f (R) theories of gravity is the ‘metric-affine’ formula- tion. Here one again considers the metric and connection to be independent, as in the Palatini procedure, but now allows the matter action to be a function of both metric and connection (rather than metric alone, as is the case in Palatini and metric variational formalisms). The relevant action for the theory then becomes [1168]
S = Z
dΩ√−gf(R) + Sm(gµν, Γµνσ, Ψ), (302)
where R = gµνR
µν, and Rµν is taken to be a function of the connection only, as in the
Palatini procedure. One can therefore think of the action (302) as a generalisation of the Palatini action, which is recovered when the dependence of the matter action, Sm,
on Γµ
νσ vanishes.
As is the case in General Relativity, the invariance of the Ricci scalar under the projective transformation Γµ
νσ → Γµνσ+ λνδµσ can lead to inconsistency of the field
equations, as matter fields do not generically exhibit an invariance of this type. This invariance can be cured by adding to the action an additional Lagrange multiplier term of the form S =RdΩ√−gBµΛν
[νσ], and results in the field equations
fRRµν−
1 2gµνf =
χ
2Tµν, (303)
together with Γµ[µν]= 0, and 1 √ −g h √ −gfRgµσ;σδνρ− √−gfRgµν;ρ i + 2fRgµσΓν[σρ] = χ 2 ∆ρµν− 2 3∆ σ[ν σ δ µ] ρ , (304) where ∆ νρ
µ ≡ −(2/√−g)δSm/δΓµνρ. It can then be shown that ∆ [νρ]
µ = 0 corresponds
to a vanishing of the torsion, and ∆µ(νρ) 6= 0 introduces non-zero non-metricity [1168].
The metric-affine approach has not been studied as intensively as the other approaches to f (R) gravity that we have already discussed, and will not feature heavily in the sections that follow.