Gravity and Strings

2.3 Conserved charges 31
which is also invariant up to a total derivative
˜δS = d d x∂µs µ (˜δ), (2.29)
and repeating the same steps as those we followed to find the Noether currents, we find a
correction to the Noether current Eq. (2.17):
j µ
N1 (˜δ) =−s µ (˜δ) + ˜δL µ + ∂ρ ρµ ν ˜δx ν ,
ρµ ν = 2L [ρ η µ] ν. (2.30)
If we consider only constant spacetime translations and L µ is a vector density, then ˜δL µ =
s µ (˜δ) and we simply find a correction to the canonical energy–momentum tensor with
the form of a superpotential.
We end this section with an important remark: no change in the superpotential can be
related to the addition of a total derivative to the Lagrangian.
2.3 Conserved charges
Given a conserved current (density) j µ ,bytaking the integral of its time component j0 over
a piece Vt of a constant-time hypersurface we can define a quantity (charge) Q(Vt),
Q(Vt) = d d−1 x j 0 . (2.31)
Vt
If we take the total time derivative of Q(Vt), since the volume of Vt does not depend on
time (the subindex t indicates only that it is in a given constant-t hypersurface, but it is the
same spatial volume for all t) the total time derivative “goes through the integral symbol”
and becomes a partial time derivative of j0 (c = 1):
d
dt Q(Vt) =
Vt
d d−1 x ∂0j 0 . (2.32)
The continuity equation for the current and Stokes’ theorem imply that
d
dt Q(Vt)
= d d−1 x ∂ij i
= d d−2 i j i , (2.33)
Vt
which is interpreted as the flux of charge across the boundary of the volume of Vt. Observe
that the last integral is performed over j i rather than over j i .
This is a local charge-conservation law: the charge contained in the volume of Vt is only
lost (or gained) by the interchange of charge with the exterior; it does not disappear into
nothing and it is not created from nothing. This is what we mean by conserved charge.
If we take the boundary of the volume to spatial infinity, and we assume that the currents
go to zero at infinity (there are no sources at infinity for the charges), then the flux integral
over the boundary vanishes and we see that the total charge contained in space at a given
time is conserved in absolute terms. It is usually denoted by Q (all reference to timedependence
has been eliminated).
∂Vt