Gravity and Strings

2.4 The special-relativistic energy–momentum tensor 33
and the canonical energy–momentum tensor resulting from the use of the general formula
in this case is symmetric and conserved using the above equation of motion:
Tµν(ϕ) =−∂µϕ∂νϕ + 1
2ηµν(∂ϕ) 2 ,
∂ µ Tmatter µν(ϕ) =−∂νϕ∂2ϕ = 0.
on-shell
(2.41)
If we add a total derivative ∂ρ(ϕ∂ ρϕ) to the above Lagrangian, the equations of motion
do not change, as can be seen by using the Euler–Lagrange equations for higher-derivative
theories (2.7). According to Eq. (2.18), the energy–momentum tensor acquires the extra
term
+∂ρ ρµν , ρµν = 2η ν[µ ϕ∂ ρ] ϕ, (2.42)
which is also symmetric but contains second derivatives of the field.
Although the canonical energy–momentum tensor arises as the Noether current associated
with invariance under constant translations, we are going to see that it is a much richer
object and contains information on the response of a theory to spacetime transformations.
Observe that the canonical energy–momentum tensor is not symmetric in general. In
fact, it is symmetric only for scalar fields. However, it can be symmetrized, as we are going
to explain when we study the conservation of angular momentum.
Foreach vector current, we can define the charge Q(ν),
Q(ν) = d
Vt
d−1 xj 0 (ν) =
d
Vt
d−1 xTcan 0 ν. (2.43)
The d conserved charges associated with the energy–momentum tensor are the d components
of a contravariant Lorentz vector, which is nothing but the momentum vector and thus
we have derived the local conservation laws of energy and momentum. It is customary to
write P ν = Q(ν).
2.4.1 Conservation of angular momentum
Let us now consider the infinitesimal Lorentz transformations. The fields appearing in
SRFTs transform covariantly or contravariantly in definite representations of the Lorentz
group. Let us take, for instance, a field ϕα transforming contravariantly in the representation
r of the Lorentz group. The index α goes from 1 to dr, the dimension of the representation
r. If, in the representation r, the generators of the Lorentz group are the dr × dr matrices
α
Ɣr Mµν β, then an infinitesimal Lorentz transformation of the field ϕ can be written in
the form
˜δϕ α = 1
2σ µν αβϕ Ɣr Mµν
β = 1
2σ µν ˜δ(µν)ϕ α . (2.44)
Observe that we can write
where Ɣv is the vector representation given in Eq. (A.60).
˜δ(ρσ )x µ
=
µνx
Ɣv Mρσ
ν , (2.45)