Gravity and Strings

3.2 Gravity as a self-consistent massless spin-2 SRFT 87
There are two types of terms: terms of the form ∂k∂k and of the form k∂µ∂νk, which
give finite contributions, and terms of the form k∂ 2 k, which give singular contributions
(∂ 2 k ∼ δ (3) (x3))but only at the origin x3 = 0 and have to be absorbed into a renormalization
of the source. In the Rosenfeld case, it is just a renormalization of the mass, but in the
second case the mass is not renormalized and, instead, the source’s energy–momentum
tensor has singular terms Tsource ij ∼ δijδ (3) (x3), which do not fit within the concept of a
point-particle. Since we are mainly interested in obtaining corrections to the gravitational
field of massive, finite-sized bodies of spherical symmetry (the Sun, for instance), we opt
for hiding this problem in the closet with the other skeletons for the moment and simply
ignore these terms.
By taking the derivatives on the r.h.s. of the above expressions, we find
1 (0)
t Ros 00
c (h(0) ) = 7 R
2
2 S
χ 2
1
,
|x3| 4
1 (0)
t Ros ij
c (h(0) ) =− R2 S
χ 2
x i x j
1 (0)
t GR 00
c (h(0) ) = 3 R
2
2 S
χ 2
1
,
|x3| 2
1 (0)
t GR ij
c (h(0) ) = 7 R2 S
χ 2
1
−
|x3| 6 2 δij
x i x j
1
−
|x3| 6 2 δij
1
|x3| 4
1
|x3| 4
,
.
(3.210)
To solve these equations, we could try to eliminate all the off-diagonal terms in the
energy–momentum tensor by a gauge transformation, as did Thirring in [888]. However, as
observed in [725], the gauge transformation that one has to use is ɛµ ∼ ∂µ ln r, which does
not go to zero at infinity and, furthermore, takes us out of the De Donder gauge in which
we want to solve the equation. This clearly invalidates Thirring’s results.
However, we can solve directly the first of Eqs. (3.207) in the De Donder gauge: observe
that the r.h.s. of this equation,
∂ 2 ¯h (1) µν = 1
c t (0) µν(h (0) ), (3.211)
is divergence-free. For the Rosenfeld energy–momentum tensor we obtain [725]
¯h (1) 00 =− 7 R
4
2 S
χ 2
1
, ¯h
|x3| 2
(1) ij =− 1 R
4
2 S
χ 2
xi x j
, (3.212)
|x3| 4
and, by combining this correction and the linear term into gµν = ηµν + χh (0) µν + χ 2h (1) µν,
we obtain the spherically symmetric metric
ds 2 Ros =
1 − RS
r − R2 S
r 2
c 2 dt 2
− 1 + RS
r + R2 S
r 2
dr 2
− 1 + RS 3 R
+
r 4
2 S
r 2
r 2 d 2 (2) ,
(3.213)
where we have defined r =|x3| and used dr = xi dxi /|x3|, dxidxi = dr2 + r 2d2 , etc.