Gravity and Strings

70 A perturbative introduction to general relativity
It is convenient to use the variable ¯hµν and the De Donder gauge 23 ∂ µ ¯hµν = 0. Then a
solution can be immediately obtained for d ≥ 4:
¯hµν =−ηµ0ην0
and the non-vanishing components of hµν are 24
χ Mc 1
, (3.123)
(d − 3)ω(d−2) |xd−1|
d−3
h00 ≡ 2
χc2 φ, hii
2
χ
=
φ, φ =−
(d − 3)χc2 2Mc3 1
. (3.124)
2(d − 2)ω(d−2) |xd−1|
d−3
The notation we have chosen suggests, correctly, that φ can be identified with the Newtonian
potential as in the scalar SRFT of gravity (Eq. (3.7)). Also, as in the case of the
scalar SRFT of gravity, we have to see how it affects the motion of test particles in order to
confirm it.
The gravitational field of a massless point-particle. The action and energy–momentum tensor
for a free massless particle moving in Minkowski spacetime are given, respectively, by
Eqs. (3.32) and (3.34). After coupling to the gravitational field hµν, the modified action is
S[X µ (ξ), γ (ξ)] =− p
dξ
2
√ γγ −1 ηµν + χhµν(X) ˙X µ ˙X ν . (3.125)
This time the gravitational field cannot be absorbed into a redefinition of the worldline
metric γ (unless hµν ∝ ηµν) and a massless particle interacts with the gravitational field.
Let us first find the gravitational field produced by a massless particle by solving the
equation D µν (h) = (χ/c)T µν
pp , where the energy–momentum tensor has to be calculated
for a solution of the equations of motion of the free massless particle ˙P µ = P µ Pµ = 0. It
is convenient to use light-cone coordinates u,v,and xd−2 defined by
u = 1 √ 2 (t − z), v = 1
√ 2 (t + z), (xd−2) = (x 1 ,...,x d−2 ), (3.126)
where z ≡ x d−1 ,inwhich the Minkowski metric takes the form
⎛
0
= ⎝ 1
1
0
⎞
⎠. (3.127)
ηµν
−I(d−2)×(d−2)
Asolution describing the particle moving at the speed of light along the z axis toward +∞
is given by
U = Xd−2 = 0, V = ξ, γ = 1. (3.128)
23 In this case we cannot impose a traceless gauge because the particle’s energy–momentum tensor itself is not
traceless.
24 To compare this with Thirring’s results [888] it has to be taken into account that Thirring’s energy–
momentum tensor is twice ours and that its coupling constant f = χ/2.