Gravity and Strings

100 A perturbative introduction to general relativity
gravitational masses of the particle are identical. We can certainly say that the PGR implies
the weak form of the PEGI.
The medium-strong form extends the rank of applicability from the dynamics of one particle
to all non-gravitational laws of physics. The introduction of the curved metric gµν into
the actions of all known interactions guarantees that it is also a consequence of the PGR.
The strong form applies to all laws of physics, including gravity itself. There is nothing
we can say about this form of the PEGI for the moment, although we already mentioned
in the previous section that GR satisfies it, but let us mention that it is not a direct
consequence of general covariance, for we can write SRFTs in Minkowski spacetime in
general-covariant form.
So far we have considered only gµνs that can be generated by GCTs from ηµν. Our
experience tells us that there are non-trivial gravitational fields in what we would previously
have called inertial frames. These gravitational fields must be described by the metric, too.
To incorporate them into the theory, we are forced to allow for all kinds of metrics gµν
that cannot be transformed into ηµν by a GCT. However, for any arbitrary spacetime metric
at a given point, there will always be coordinate systems defining local inertial frames in
which gµν is equal to ηµν at that given spacetime point P and in which the first derivatives
of gµν vanish at that given point 48 and so all the components of the Levi-Cività connection
Ɣµν ρ (g) also vanish at P. One such system is provided by the Riemann normal coordinates
at the point P(see, for instance, [707]), which have the following properties:
gµν(P) = ηµν, ∂ρgµν(P) = 0,
∂ρ∂σ gµν(P) = 2
3 Rµ(ρσ)ν(P), Rµνρσ (P) = 2∂µ∂[ρgσ ]ν(P).
(3.268)
In this coordinate system, although the first derivatives vanish, the second derivatives
do not. In fact, in general, there is no coordinate system in which both first and second
derivatives at P vanish, because, otherwise, the Riemann tensor would vanish also at P,
which is possible only if it vanishes at P in any coordinate system. This reflects the fact that,
although the gravitational field is encoded in the metric tensor, it is actually characterized
by the Riemann curvature tensor. The two tensors play a role similar in this respect to those
of the vector potential and the field strength in Maxwell electrodynamics. Then, if there is
a non-trivial gravitational field at P, the curvature tensor will not vanish at that point and
the same will be true in any coordinates, including Riemann normal coordinates. Thus, to
what extent is it true that all gravitational effects can be eliminated in the neighborhood
of a point as the PEGI states? The point is that observable gravitational effects depend on
the product of Riemann tensor components and spacetime coordinate intervals that can be
made arbitrarily small and the upshot of this discussion is that the equivalence between
gravitation and inertia will work only locally and for observable effects. The PEGI is only
48 Any real non-singular metric can be diagonalized at a given point using the appropriate coordinate system,
the non-vanishing components being +1s and −1s. The number of −1s minus the number of +1s cannot be
changed by a further coordinate transformation and is an intrinsic property of the metric, an invariant called
the signature. Continuity of the metric implies that the signature is the same at all points of spacetime. We
consider only metrics of signature d − 2, the signature of ηµν in our conventions.