12.1 Lovelock scalar invariants

One may wonder whether it is possible to remove these spin-2 ghosts. To answer this point, one should first introduce the Lovelock scalars [399Jump To The Next Citation Point]. These scalars are particular combinations/contractions of the Riemann tensor which have a fundamental property: if present in the Lagrangian, they only introduce second-order derivative contributions to the equations of motion. Let us give an example of this property [399]. Soon after Einstein proposed General Relativity [226] and Hilbert found the Lagrangian to describe it [301], Kretschmann [372] pointed out that general covariance alone cannot explain the form of the Lagrangian for gravity. In the action he introduced, instead of the Ricci scalar, the scalar which now has been named after him, the Kretschmann scalar:
∫ 4 √ --- αβγδ S = d x − gR αβγδR . (12.1 )

At first glance this action looks well motivated. The Riemann tensor R αβγδ is a fundamental tensor for gravitation, and the scalar quantity P1 ≡ R αβγδR αβγδ can be constructed by just squaring it. Furthermore, it is a theory for which Bianchi identities hold, as the equations of motion have both sides covariantly conserved. However, in the equations of motion, there are terms proportional to ∇ μ∇ νR μαβν together with its symmetric partner (α ↔ β). This forces us to give in general at a particular slice of spacetime, together with the metric elements gμν, their first, second, and third derivatives. Hence the theory has many more degrees of freedom with respect to GR.

In addition to the Kretschmann scalar there is another scalar αβ P2 ≡ RαβR which is quadratic in the Riemann tensor R αβ. One can avoid the appearance of terms proportional to ∇ μ∇ νR μ(αβ)ν for the scalar quantity,

𝒢 ≡ R2 − 4R R αβ + R R αβγδ, (12.2 ) αβ αβγδ
which is called the Gauss–Bonnet (GB) term [572Jump To The Next Citation Point67Jump To The Next Citation Point]. If one uses this invariant in the action of D dimensions, as
∫ D √ --- S = d x − g𝒢, (12.3 )
then the equations of motion coming from this Lagrangian include only the terms up to second derivatives of the metric. The difference between this scalar and the Einstein–Hilbert term is that this tensor is not linear in the second derivatives of the metric itself. It seems then an interesting theory to study in detail. Nonetheless, it is a topological property of four-dimensional manifolds that √ --- − g𝒢 can be expressed in terms of a total derivative [150], as
√ --- α − g𝒢 = ∂α𝒟 , (12.4 )
where
α √ --- αβγδ μν ρ [ σ σ λ ] 𝒟 = − g𝜖 𝜖ρσ Γ μβ R νγδ∕2 + Γ λγΓ νσ∕3 . (12.5 )
Then the contribution to the equations of motion disappears for any boundaryless manifold in four dimensions.

In order to see the contribution of the GB term to the equations of motion one way is to couple it with a scalar field ϕ, i.e., f (ϕ)𝒢, where f (ϕ) is a function of ϕ. More explicitly the action of such theories is in general given by

∫ [ ] 4 √ --- 1 1 2 S = d x − g 2F (ϕ)R − 2ω (ϕ )(∇ϕ ) − V (ϕ) − f(ϕ )𝒢 , (12.6 )
where F(ϕ ), ω (ϕ ), and V(ϕ ) are functions of ϕ. The GB coupling of this form appears in the low energy effective action of string-theory [275Jump To The Next Citation Point273Jump To The Next Citation Point], due to the presence of dilaton-graviton mixing terms.

There is another class of general GB theories with a self-coupling of the form [458Jump To The Next Citation Point],

∫ √ --- [1 ] S = d4x − g -R + f(𝒢 ) , (12.7 ) 2
where f(𝒢) is a function in terms of the GB term (here we used the unit κ2 = 1). The equations of motion, besides the standard GR contribution, will get contributions proportional to ∇μ ∇νf,𝒢 [188Jump To The Next Citation Point189]. This theory possesses more degrees of freedom than GR, but the extra information appears only in a scalar quantity f,𝒢 and its derivative. Hence it has less degrees of freedom compared to Kretschmann gravity, and in particular these extra degrees of freedom are not tensor-like. This property comes from the fact that the GB term is a Lovelock scalar. Theories with the more general Lagrangian density R∕2 + f (R,P1, P2) have been studied by many people in connection to the dark energy problem [142110Jump To The Next Citation Point521420Jump To The Next Citation Point58564166543180]. These theories are plagued by the appearance of spurious spin-2 ghosts, unless the Gauss–Bonnet (GB) combination is chosen as in the action (12.7View Equation[465Jump To The Next Citation Point153Jump To The Next Citation Point447] (see also [110181109]).

Let us go back to discuss the Lovelock scalars. How many are they? The answer is infinite (each of them consists of linear combinations of equal powers of the Riemann tensor). However, because of topological reasons, the only non-zero Lovelock scalars in four dimensions are the Ricci scalar R and the GB term 𝒢. Therefore, for the same reasons as for the GB term, a general function of f (R) will only introduce terms in the equations of motion of the form ∇ μ∇ νF, where F ≡ ∂f ∕∂R. Once more, the new extra degrees of freedom introduced into the theory comes from a scalar quantity, F.

In summary, the Lovelock scalars in the Lagrangian prevent the equations of motion from getting extra tensor degrees of freedom. A more detailed analysis of perturbations on maximally symmetric spacetimes shows that, if non-Lovelock scalars are used in the action, then new extra tensor-like degrees of freedom begin to propagate [5726730246515330399]. Effectively these theories, such as Kretschmann gravity, introduce two gravitons, which have kinetic operators with opposite sign. Hence one of the two gravitons is a ghost. In order to get rid of this ghost we need to use the Lovelock scalars. Therefore, in four dimensions, one can in principle study the following action

∫ √ --- S = d4x − gf(R, 𝒢). (12.8 )
This theory will not introduce spin-2 ghosts. Even so, the scalar modes need to be considered more in detail: they may still become ghosts. Let us discuss more in detail what a ghost is and why we need to avoid it in a sensible theory of gravity.
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