Gravity and Strings

where dρ 2 + ρ 2 d 2 (2)
7.1 Schwarzschild’s solution 199
≡ d x 2
3
and ρ =|x3|.
16. Yet another system of coordinates: let us consider some arbitrary coordinate system
{y α } and let us take four scalar functions labeled by µ = 0, 1, 2, 3ofthe coordinates
y α and H µ (y), which we require to be harmonic
∇ 2 H µ = 1
αβ
√ ∂α |g| g ∂β H
|g| µ
= 0. (7.27)
Now we can define new coordinates x µ ≡ H µ (y), which are called harmonic coordinates.Inthe
system of harmonic coordinates, the above equation takes the form of
a condition on the metric:
∂α
|g| g αµ
= 0. (7.28)
If we expand the metric in a perturbation series around flat spacetime,
gµν = ηµν + χh (0) µν + χ 2 h (1) µν + ···,
g µν = η µν − χh (0)µν + χ 2 h (0)µρ h (0) ρ ν − h (1)µν ,
g = 1 + χh (0) + χ 2 h (1) + 1
(0) 2 (0)µν (0)
h − h h 2
µν ,
1
|g| =1 + 2χh(0) + 1
4χ 2 2h (1) + h (0) 2 − 2h (0)µν h (0)
µν ,
µν µν
|g| g = η − χ ¯h (0)µν + χ 2 −h (1) µνh (0)µρ h (0) ρ ν − h (0) h (0)µν
+ 1
(1) (0) 2 (0)αβ (0) µν
2h + h − 2h h 4
αβ η ,
(7.29)
where, as usual, h ≡ hρ ρ and ¯h (0) µν ≡ h (0) µν − 1
2ηµνh (0) .Onsubstituting these into
the above equation, we find that the linear perturbation h (0) µν of the metric in
harmonic coordinates is in the harmonic gauge, Eq. (3.57), but the next order is
not.
To set the Schwarzschild solution in a harmonic coordinate system it turns out that
we just have to shift the Schwarzschild radial coordinate r ≡ rh − ω/2 toobtain
ds 2 =
rh + ω/2
(dct)
rh − ω/2
2 −
rh − ω/2
dr
rh + ω/2
2 h + (rh − ω/2) 2 d 2 (2) , (7.30)
and reexpress the metric in terms of coordinates x3 (having nothing to do with the
isotropic coordinates introduced before) such that rh =|x3| using rhdrh =x3 · d x3