Gravity and Strings

76 A perturbative introduction to general relativity
The 1/r term is not present27 in the Newtonian case28 and, as we did before, we expand W
around its Newtonian value,
ω 2 W ∼ dr −
c
l2
∂
ω 2
ω
2 x l2
+ 2RS dr + −
r 2 ∂x c c r r 2
x=0
∼ WNewtonian + RSω
1
dr
c r 2 − ρ2 ∼ WNewtonian + RSω
c arccosh
r
, (3.164)
ρ
where ρ = cl/ω is clearly the minimal value of r in the path of the massless particle. Following
[644], the variation of W when the particle starts from r = R≫ρ, goes through
r = ρ, and again reaches r = R is
and, according to Eq. (3.156),
W ∼ WNewtonian + 2RSω
c
ϕ ∼ ∂
∂l WNewtonian + 2R 1
√
ρ 1 − ρ/R
R
arccosh , (3.165)
ρ
R→∞
−→ π + 2R
, (3.166)
ρ
and we find that the deviation from the Newtonian value ϕ = π (which means simply
no bending of the light ray) is δϕ = 2R/ρ, ingood agreement with observation. This is an
encouraging result, which indicates that we have found a reasonable relativistic theory of
gravitation worth studying in more detail.
At this point, we remember that we still have to answer the second question posed on
page 67. The answer will prompt us to seek and introduce into our theory corrections that
will make the prediction for the precession of the perihelion of Mercury agree completely
with observations.
3.2.4 The consistency problem
The answer to the second question formulated on page 67 is that, in general, the matter
energy–momentum tensor derived from the free-matter Lagrangian is no longer conserved.
As explained in Chapter 2, the divergence of the energy–momentum tensor is proportional
to the equations of motion derived from the same Lagrangian, but the coupling to gravity
changes these equations. This can be seen in the modified massive-particle action of the
above example but the real scalar field which we studied in Chapter 2 will, however, make
a better example.
27 There are also 1/r 2 corrections, but we take only the most important one.
28 The Newtonian case corresponds to a free massive particle (i.e. vanishing gravitational potential energy)
moving at the speed of light with 2mE ′ = (ω/c) 2 .