Gravity and Strings

54 A perturbative introduction to general relativity
We are looking for a theory with the equation of motion
∂ 2 φ = CT, (3.37)
where T is the trace of the total energy–momentum tensor, which should include contributions
from the scalar gravitational field, matter fields, and interaction terms. The energy–
momentum tensor of φ in the free theory is quadratic in ∂φ.Toobtain it on the r.h.s. of the
equation of motion, we must add to the Lagrangian a term of the form φ(∂φ) 2 .However,
this term will also contribute to the new energy–momentum tensor, and, to produce it on
the r.h.s. of the new equation of motion, we need a term φ 2 (∂φ) 2 in the Lagrangian, and so
on. Thus, we need to introduce an infinite number of corrections to the scalar Lagrangian.
As for the interaction terms, they contain the trace of the matter energy–momentum
tensor, and thus we need to make some assumption about the form of the matter Lagrangian
in order to make some progress: we will take it to be of the form
Lmatter = K − V, (3.38)
where K is quadratic in the first partial derivatives of the matter fields and V is just a
function of the fields. This implies that
Tmatter = (d − 2)K − dV, (3.39)
and the action Eq. (3.3), which we can consider the lowest order in an expansion in small
φ,takes the form
S = 1
d
c
d
1
x
2Cc2 (∂φ)2
d − 2
+ 1 + φ K − 1 +
c2 dφ
c2
V . (3.40)
It is reasonable to expect that the full action, with all the φ corrections, takes the form
S = 1
d
c
d
1
x
2Cc2 f (φ)(∂φ)2
+ g(φ)K − h(φ)V , (3.41)
where f, g, and h are functions of φ to be found by imposing the condition that the equation
of motion of φ can be written in the form Eq. (3.37), where T is the trace of the total
energy–momentum tensor of the above Lagrangian, which is easily found to be
T = (d − 2) 1
2Cc 2 f (φ)(∂φ)2 + (d − 2)g(φ)K − dh(φ)V. (3.42)
The φ equation of motion coming from Eq. (3.41) is
∂ 2 φ =− 1
2 ( f ′ /f )(∂φ) 2 + Cg ′ /( fK) − Ch ′ /( fV), (3.43)
and, on comparing this with Eqs. (3.37) and (3.42), one finds
f =
1
a + [(d − 2)/c2 d
, g = f/b, h = ( f/e) d−2 , (3.44)
]φ