At first glance this action looks well motivated. The Riemann tensor is a fundamental tensor for
gravitation, and the scalar quantity
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
together with its symmetric partner (
). This forces us to give in general at
a particular slice of spacetime, together with the metric elements
, 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 which is quadratic in the
Riemann tensor
. One can avoid the appearance of terms proportional to
for the
scalar quantity,
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 , i.e.,
, where
is a function of
. More explicitly the action of such theories
is in general given by
There is another class of general GB theories with a self-coupling of the form [458],
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 and the GB term
. 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
, where
. Once more,
the new extra degrees of freedom introduced into the theory comes from a scalar quantity,
.
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 [572, 67, 302, 465, 153, 303, 99]. 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
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.http://www.livingreviews.org/lrr-2010-3 | ![]() This work is licensed under a Creative Commons License. Problems/comments to |