Gravity and Strings

In a complete revolution
3.2 Gravity as a self-consistent massless spin-2 SRFT 75
ϕ =− ∂
W.
∂l
(3.156)
By expanding W around the Newtonian WNewtonian as a power series in the relativistic correction
δ = 2R 2 Sm2c 2 and observing that
∂W
∂δ
=− ∂W
,
∂l 2
(3.157)
we obtain
and
W ∼ W | δ=0 + δ ∂W
∂δ
δ=0
δ=0
= WNewtonian − R2 S m2 c 2
W = WNewtonian − R2 S m2 c 2
l
= WNewtonian + R2 S m2 c 2
= WNewtonian − δ 1 ∂WNewtonian
2l ∂l
∂WNewtonian
, (3.158)
∂l
l
l
∂WNewtonian
∂l (3.159)
ϕNewtonian,
where we have used Eq. (3.156) for WNewtonian.Onsubstituting this into Eq. (3.156) we find
ϕ = ϕNewtonian + R2 S m2 c 2
l 2 ϕNewtonian. (3.160)
Newtonian orbits are closed, so in one revolution ϕNewtonian = 2π and the deviation from
the Newtonian value is, according to this theory
δϕ = 2π R2 Sm2c 2
l2 . (3.161)
This result is 4
of the actual value; that is, it is close (better than the value given by the
3
scalar SRFT of gravity) but not quite right. We will have to find a correction to our theory
in order to obtain the right value.
The second effect that we want to calculate is the deflection of a light ray (or a massless
particle) by the central gravitational field of a massive body, given by Eq. (3.124). To first
order in χ we can simply take the Hamilton–Jacobi equation for a relativistic massive particle,
Eq. (3.144), and set m = 0 [644]. The resulting equation can be solved as in the massive
case with the replacement of E =−∂tS by ω =−∂tS. ForWwe obtain the equation
W = dr µλ−1
ω
2 −
c
l2
. (3.162)
R2 On expanding µ and λ in powers of 1/r,weobtain, for the solution Eq. (3.124),
ω 2
ω
2 1 l2
W ∼ dr + 2RS − . (3.163)
c c r r 2