Gravity and Strings

98 A perturbative introduction to general relativity
where we have introduced the induced metric on the worldline γ
γ(ξ)≡ gµν(X) ˙X µ ˙X ν . (3.262)
We use for it the same symbol as for the auxiliary metric of the Polyakov-type action
because the equation of motion for γ says that γ is the induced worldline metric.
We can easily recognize in Eq. (3.261) the geodesic equation written in terms of an
arbitrary parameter. Curves obeying that equation are called geodesics and are the curves
of minimal (occasionally maximal) proper length between two given points. When ξ = s,
the proper time, then γ = 1 and the third term in Eq. (3.261) vanishes and the standard
form of the geodesic equation is recovered:
¨X λ + Ɣρσ λ ˙X ρ ˙X σ = 0. (3.263)
If the metric is ηµν, itisclear that we recover all the special-relativistic results. Furthermore,
if the metric is related to ηµν through a GCT, it is clear that we will be describing the
same motion (straight lines in spacetime) in some system of curvilinear coordinates. Thus,
even though it is difficult to see, the dynamics of the particle will have the same d(d + 1)/2
conserved quantities associated with the invariances of the Minkowski metric in Cartesian
coordinates. Now that we are dealing with general curvilinear coordinates, it is good to
have a better characterization of the invariances of a metric and how they are associated
with conserved quantities in the dynamics of a particle.
Let us consider the effect of infinitesimal transformations of the form
δX µ = ɛ µ (X),
δgµν = ɛλ∂λgµν. (3.264)
It is worth stressing that these transformations are not GCTs in spacetime (the metric does
not transform in the required way). We know that the action (3.255) is invariant under
arbitrary GCTs. However, under the above transformations
1
δSpp =−Mc dξ ˙X
2 gµν ˙X µ ˙X ν
ρ ˙X σ Lɛgρσ, (3.265)
and is invariant only if ɛ µ = ɛk µ ,where ɛ is an infinitesimal constant parameter and k µ is
aKilling vector satisfying the Killing equation (1.107).
These transformations can be exponentiated, giving a one-dimensional group (for one
Killing vector) that leaves the action invariant. There is a conserved quantity associated
with it via the Noether theorem for global symmetries, 46
P(k) =−
gµν
Mc
kρ
˙X µ ˙X ν
˙X ρ , (3.266)
46 The components of k µ are fixed functions of the spacetime coordinates and the parameters of the group
have to be constant over the worldline; they cannot be arbitrary functions of ξ. Thus, this is a group of
global transformations. These transformations can be gauged by the standard method of introducing a gauge
vector and a covariant derivative, as will be seen in due course.