Gravity and Strings

3.3 General relativity 99
which can be interpreted as the components of the momentum vector in the direction of the
Killing vector.
This general framework can be applied to any metric in any coordinate system. We can
use it to recover the conserved quantities of a free particle moving in Minkowski spacetime.
First of all, observe that we can always use coordinates adapted to a given Killing vector
k µ : there is a coordinate z such that k µ ∂µ = ∂z and ∂zgµν = 0. Then, there is always a
coordinate system in which the action does not depend on the variable Z(ξ) and hence
the momentum associated with it is conserved as usual. Thus, we are simply encoding
known facts in coordinate-independent form. Second, we can check that the above general
expression gives the usual linear- and angular-momentum components when we use the
Killing vectors of the Minkowski metric:
k (µ) ρ = η µρ , k ([µν])ρ = 2η ρ[µ x ν] , (3.267)
where (µ) and ([µν]) are labels for the d translational and d(d − 1)/2rotational isometries.
To finish this digression, let us mention that the Polyakov-type actions (3.257) and
(3.258) are one-dimensional examples of what is called a non-linear σ -model. 47 The nonlinearity
is associated with the dependence of the metric on the coordinates, which are the
dynamical degrees of freedom.
The principle of equivalence. Accepting that, according to the PGR, the action (3.255)
gives the dynamics of a massive particle in the background given by the metric gµν, we
are led to the discovery of the principle of equivalence of gravitation and inertia (PEGI)
formulated by Einstein in [350, 351]: consider a near-Minkowskian metric gµν = ηµν +
χhµν with χhµν ≪ 1. It is easy to see that, up to second-order terms, the action is precisely
the one given by Eq. (3.116). In particular, we studied the low-velocity (non-relativistic)
limit in order to show that the field hµν describes a gravitational special-relativistic field
and how in the non-relativistic limit that action can be interpreted as the non-relativistic
action of a particle with potential energy Mc 2 χh00/2 proportional to its inertial mass. This
potential energy can be interpreted as a gravitational potential energy, identifying in this
way inertial and gravitational masses and χh00 with 2φ/c 2 ,where φ is the Newtonian
gravitational potential.
Thus, a GCT that, applied to an inertial frame, generates a non-trivial h00 can be seen
as generating a gravitational field. We are identifying the so-called inertial forces with a
gravitational field and we are saying that we cannot distinguish between them. Furthermore,
all the effects of the gravitational field can be eliminated by going to an inertial frame. This
is the essence of the PEGI which we will refine later. One can distinguish among weak (or
Galilean), medium-strong (or Einstein’s), and strong forms of the PEGI [242].
The weak form applies to the dynamics of one particle (precisely our case): one cannot
distinguish whether we are describing its motion in a non-inertial frame or whether there is
a gravitational field present. This implies that the inertial and gravitational masses of any
particle are always proportional, with a universal proportionality constant that, in carefully
chosen units, can be made 1. We have seen that, in the action Eq. (3.116), the inertial and
47 Two useful references on σ -models are [210, 576].