Gravity and Strings

56 A perturbative introduction to general relativity
Only the conformal factor is dynamical. The equation of motion for the metric (i.e. for
)is
(d − 1)(d − 3) 16πG
R(g) =
d − 2
(d)
N
c2 Tmatter, (3.51)
where R(g) is the Ricci scalar for the metric gµν and Tmatter is calculated from the canonical,
special-relativistic fully covariant energy–momentum tensor Tmatter µν, bycontracting both
indices with g µν .
Alternatively, the Einstein–Fokker theory can be formulated by giving the above equation
for an arbitrary metric, but adding another equation,
Cµν ρσ (g) = 0, (3.52)
where Cµν ρσ is the Weyl tensor. This equation implies that the metric is conformally flat
and can be written, in appropriate coordinates, in the form (3.50).
Using the formulae in Appendix E, we find
This, together with
R(g) =
4(d − 1)
d − 2 [/c2 d+2
−
] d−2 ∂ 2 [/c 2 ]. (3.53)
Tmatter = [/c 2 4
−
] d−2 T (0)
matter, (3.54)
gives Eq. (3.49).
Einstein and Fokker did not give a Lagrangian for gravity coupled to matter, and therefore
they had to postulate how gravity affects the motion of matter. Here, the power of
the Einstein–Fokker formulation of Nordström’s theory becomes manifest: Einstein and
Fokker suggested replacing the flat spacetime metric ηµν by the conformally flat metric
gµν everywhere in the matter Lagrangian. This prescription can be used in most matter
Lagrangians (not involving spinors). For instance, for the massive particle, it leads to
Spp[X µ
(ξ)] =−Mc
=−Mc
∼−Mc
dξ gµν(X) ˙X µ ˙X ν
dξ [(X)/c 2 ] 2
d−2 ηµν ˙X µ ˙X ν (3.55)
dξ [1 + φ(X)/c 2
+ ···] ηµν ˙X µ ˙X ν ,
which is, to lowest order in φ, our old result. In general, the equation of motion simply tells
us that massive particles move along timelike geodesics with respect to the metric gµν. This
is a very powerful statement that goes far beyond Nordström’s original theory.
For the massless particle, we also find that the coupling can again be absorbed into the
worldline auxiliary metric. There is no bending of light in this theory. However, one can argue
[349] that, although there is no global bending, there is local bending of light rays. As
explained in [349], local bending is a kinematical effect associated with accelerating reference
frames and occurs, via Einstein’s equivalence principle of gravitation and inertia (to be