Gravity and Strings

74 A perturbative introduction to general relativity
where W is a function only of r.Onsubstituting into the above equation, we find that W is
given by
W =
dr µλ−1 2 E
−
c
l2
R2 − m2c2 µ. (3.148)
In the absence of the gravitational field λ = µ = 1, and R = r. Ondefining the nonrelativistic
energy E ′ = E − mc2 ,assuming that E ′ ≪mc2 so that
2 E
− m
c
2 c 2 = m 2 c 2
E
mc2
2
− 1 = m 2 c 2
′ E
mc2 2 ′ E
+ 2
mc2
∼ 2mE ′ , (3.149)
and substituting in the integrand, we obtain W for a classical free particle of energy E ′ .In
the presence of a spherically symmetric gravitational field, vanishing at infinity, on making
the same approximation E ′ ≪mc 2 ,expanding
λ ∼ 1 + λ1 λ2
µ1 µ2
+ + ···, µ∼1 + + + ···,
r r 2 r r 2
R 2 ∼ r 2
1 + R1
+ ··· , (3.150)
r
and expanding the expression under the square root to order O(1/r 2 ),wefind
W ∼
dr 2mE ′ − λ1m2c 2
−
r
l2 − [λ1(λ1 − µ1) − λ2]m2c 2
r 2
. (3.151)
For the solution Eq. (3.124)
µ1 =−λ1 = RS ≡ 2MG (4)
N /c2 , R1 = 0, (3.152)
where we have introduced RS, the Schwarzschild or gravitational radius of an object of
mass M, and we obtain from Eq. (3.151)
W ∼
dr 2mE ′ + RSm2c 2
−
r
l2 − 2R 2 Sm2c 2
r 2
. (3.153)
We should first compare this expression with the Newtonian expression26
WNewtonian = dr 2mE ′ + RSm2c 2
−
r
l2
. (3.154)
r 2
The second term is the Newtonian potential energy. We see in Eq. (3.153) that there is
an O(1/r 2 ) relativistic correction to the Newtonian potential. The main consequence will
be that the orbits will not be closed and the perihelions will shift. To evaluate the angular
difference between two consecutive perihelions we reason, following [644], as follows. The
equation for the orbit can be found from
ϕ = βϕ − ∂W
. (3.155)
∂l
26 We assume that M ≫ m so that the reduced mass can be approximated by m.