Gravity and Strings

3.1 Scalar SRFTs of gravity 55
where a, b, and e are integration constants. If we want to recover Eq. (3.40) in the weakfield
limit, we have to take a = b = e = 1. Then, we have succeeded and we have found the
action
S = 1
c
d d x
1
2Cc 2
(∂φ) 2
1 + [(d − 2)/c2 ]φ +
1 +
d − 2
φ K − 1 +
c2 d
d − 2 d−2
φ V ,
c2 (3.45)
that gives rise to the equation of motion Eq. (3.37) with T ,the trace of the total energy–
momentum tensor corresponding to the above action, given by
T =
d − 2
2Cc 2
(∂φ) 2
1 + [(d − 2)/c 2 ]φ
+ (d − 2) 1 +
d − 2
φ K − d 1 +
c2 d
d − 2 d−2
φ V.
c2 (3.46)
This result was presented in [405] and [306], but the theory obtained is the one proposed
by Nordström back in 1913 in [730, 731] in terms of different variables: on introducing
the action Eq. (3.45) takes the form
S = 1
d
c
d
x
≡ c 2
1 +
1
d − 2 2
φ , (3.47)
c2 2
(d − 2) 2 Cc 2 (∂)2 + [/c 2 ] 2 K − [/c 2 ] 2d
d−2 V
. (3.48)
In the case in which V = 0, taking into account Eq. (3.5), the equation of motion can be
written in the standard form
∂ 2 =
(d − 3)4πG(d)
N
c 2
T (0)
matter, (3.49)
where T (0)
matter is the trace of the matter energy–momentum tensor obtained from the uncoupled
Lmatter. InNordström’s theory, this is the equation valid in all cases (V = 0).
In this form it is very difficult to see that the theory has the property we wanted (that the
source for the gravitational scalar field is the trace of the total energy–momentum tensor).
There is yet another way of rewriting this theory, which was found by Einstein and
Fokker [365]. This was one of Einstein’s first attempts at building a relativistic theory
of gravity in which the gravitational field is represented by a metric, as suggested by
Grossmann.
3.1.7 The geometrical Einstein–Fokker theory
The Einstein–Fokker theory is based on a conformally flat metric,
gµν ≡ [/c 2 ] 4
d−2 ηµν. (3.50)