Gravity and Strings

2
Noether’s theorems
In the next chapter, we are going to introduce general relativity as the result of the construction
of a self-consistent special-relativistic field theory (SRFT) of gravity. In this construction,
gauge symmetry and the energy–momentum tensor will play a key role. In this
chapter we want to review Noether’s theorems, the relation between global symmetries and
conserved charges, and the relation between local symmetries and gauge identities. We will
define the canonical energy–momentum tensor as the conserved Noether current associated
with the invariance under constant translations and we will review several ways of improving
it that are associated with invariance under other spacetime transformations (Lorentz
rotations and rescalings). Finally, we will relate these improved energy–momentum tensors
to the energy–momentum tensor used in general relativity.
2.1 Equations of motion
Let us consider an action S[ϕ] for a generic field ϕ, which may have (spacetime or internal)
indices that we do not exhibit for the sake of simplicity. Allowing for Lagrangians
containing higher derivatives of ϕ,wewrite the action as follows:
S[ϕ] = d d x L(ϕ, ∂ϕ, ∂ 2 ϕ,...). (2.1)
In most cases, L is a scalar density under the relevant spacetime transformations (Poincaré
transformations in SRFTs and general coordinate transformations in general-covariant theories).
It is also possible to use a Lagrangian that is a scalar density up to a total derivative, 1
and thus we will make absolutely no assumptions about the transformation properties of
the Lagrangian L.
Under arbitrary infinitesimal variations of the field variable δϕ
δS = d d
x δL = d d
∂L ∂L
x δϕ +
∂ϕ ∂∂µϕ δ∂µϕ + ∂L
∂∂µ∂νϕ δ∂µ∂νϕ
+ ··· . (2.2)
1 For instance, in general relativity one may want to eliminate the piece of the Lagrangian with second derivatives,
which is a total derivative, but then the rest is not a scalar density.
26