Gravity and Strings

3.2 Gravity as a self-consistent massless spin-2 SRFT 81
term as a redefinition of the canonical energy–momentum tensor. This redefinition is, however,
important. On the one hand, the equations of motion are going to be corrected and
therefore the addition of terms vanishing on-shell to first order is going to become meaningful
at higher orders and should be considered with care. On the other hand, if we obtain
an action that is invariant under the above gauge symmetry, the equations of motion are
going to satisfy a gauge identity that is, at the same time, the condition for the invariance
of the action. By varying directly the Lagrangian Eq. (3.178) under δ (1)
ɛ ,wefind that it will
be invariant up to O(χ 2 ) if the gravitational energy–momentum tensor that appears in the
equations of motion (3.171) satisfies, to first order in χ,
∂µt (0)µ (0)µ
σ (h) = ∂µ t can σ (h) + Dµ ρ(h)h ρ
σ . (3.185)
On taking explicitly the derivative on the r.h.s., we obtain the gauge identities 34 associated
with invariance under δ (1)
ɛ :
∂µt (0)µ σ (h) = γνρσ D νρ (h), γνρσ = 1
∂νhρσ + ∂ρhνσ − ∂σ hνρ , (3.186)
2
Thus, if we look for invariance under the gauge transformations δ (1)
ɛ , the gravitational
energy–momentum tensor that we will put on the r.h.s. of Eq. (3.171) has to be of the
form
t (0) µσ (h) = t (0)
can µσ (h) + Dµ ρ(h)h ρ σ + ∂ρ ρ µσ , (3.187)
but wecan no longer add on-shell-vanishing terms proportional to D µ ρ(h) because then
the above gauge identities would not be satisfied. Here we see how the requirement of
gauge symmetry constrains the possible energy–momentum tensors. Comparing this situation
with our construction of the scalar theory of gravity in which the energy–momentum
tensor could be asymmetric and did not have to satisfy any kind of conditions, we are much
better off.
Still, the redefined canonical energy–momentum tensor
t (0)
can µσ (h) + Dµ ρ(h)h ρ σ
is not symmetric as we had hoped and we have to find additional terms ∂ρ ρ µσ that cancel
out exactly the antisymmetric part of our energy–momentum tensors. There is only one
systematic procedure for doing this and only for the canonical one: the Belinfante method
explained in Chapter 2 which, unfortunately, requires the addition of on-shell-vanishing
terms. Let us, nevertheless, see where we are taken by this method. It is straightforward
(but long and tedious) to find
ρµ σ =−2∂ [ρ h µ] βh β σ − 2∂βh [ρ σ h µ]β + ∂ [ρ hh µ] σ + ησ [ρ ∂βhh µ]β . (3.188)
The antisymmetric part of the modified canonical tensor t (0)
can µσ + ∂ρ ρ µσ is −Dρ[µh ρ σ ],
34 We are using the zeroth-order gauge identity ∂µD µν (h) = 0, which is, obviously, valid.