Gravity and Strings

94 A perturbative introduction to general relativity
to find, case by case, the corrections to the lowest-order coupled system, which we know
is inconsistent. A weakness of the geometrical point of view is that there is no generalcovariant
energy–momentum tensor of the gravitational field itself. As we have seen, there
is a Lorentz-covariant energy–momentum tensor (or pseudotensor) of the gravitational field
embedded in the Ricci tensor together with the wave operator, but it cannot be promoted
to a general-covariant tensor, as can be understood from the equivalence principle. This
obscures the physical interpretation of vacuum solutions of the geometrical theory, which
are not strictly speaking vacuum solutions since the whole spacetime is filled by a nontrivial
gravitational field that acts as a source for itself. We will come back to this point in
Chapter 6.
Where does this principle of general covariance come from? We started from a theory
with an Abelian gauge symmetry 38 δ (0)
ɛ hµν =−2∂(µɛν).Weargued that this symmetry was
necessary in order to have a consistent theory of free massless spin-2 particles. Then we
coupled this free theory to the conserved energy–momentum tensor of the matter fields,
saw the need to introduce a self-coupling of the spin-2 field, and argued that the form of
the coupling should be dictated by gauge invariance with respect to the corrected trans-
formations δ (1)
ɛ hµν =−2∂(µɛν) − χLɛhµν, which combined the Abelian gauge symmetry
we started from and “localized” translations in such a way that the commutator of two δ (1)
ɛ
infinitesimal transformations gives another δ (1)
ɛ transformation. This is the only possible
extension of the Abelian δ (0)
ɛ transformations [739, 933] and the algebra is the algebra of
infinitesimal GCTs. 39 In fact, we can easily see how the full gauge transformation δ (1)
ɛ hµν
arises from the effect of a GCT on the metric gµν = ηµν + χhµν, just by substituting and
38 (0)
Any two of these gauge transformations commute because δ ξ1 δξ2hµν = δ (0)
ξ1+ξ2 hµν.
39 As shown in [933], it is possible to have a self-coupled spin-2 theory with only “normal spin-2 gauge
symmetry” (δ (0)
ɛ ). For instance, we can add to the Fierz–Pauli Lagrangian a term proportional to some (for
instance, the third) power of the linearized Ricci scalar
∂ 2 h − ∂µ∂νh µν , (3.249)
which is exactly invariant under δ (0)
ɛ .Ofcourse, the resulting higher-derivative theory cannot have the same
interpretation, since the r.h.s. of the equation of motion is not the gravitational energy–momentum tensor.
Also, we can couple the linear theory to matter and obtain an interacting theory that is invariant under δ (0)
ɛ :
we just have to add to the free-matter Lagrangian and the Fierz–Pauli Lagrangian an interaction term of the
form
d d xh µν Jµν(ϕ), (3.250)
where Jµν is any symmetric, identically conserved tensor built out of ϕ and its derivatives. This excludes
the matter energy–momentum tensor, which is conserved only on-shell. Since Jµν is identically conserved,
the modification introduced into the equations of motion for matter by the coupling to gravity is immaterial.
Local Jµνs can be constructed from local four-index tensors with the symmetries Jµρνσ = J[µρ][νσ] =
Jνσµρ, defining
Jµν = ∂ ρ ∂ σ Jµρνσ. (3.251)
These Jµνs are called Pauli terms in [943]. It is also possible to define identically conserved nonlocal Jµνs,
for instance the non-local projection of the energy–momentum tensor Eq. (3.170). In all these cases, we see
that the spin-2 field does not couple to the total energy–momentum tensor and the quantum theories are not
consistent, according to [942].