Lectures on String Theory

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and also
g kl (x + δx) = g kl
(x i + ξ i − 1 )
2 Γi j 1 j 2
ξ j 1
ξ j 2
+ · · · = g kl (x) + ∂ k g ij ξ k + · · ·
Thus, at the linearized level we find that
)
g ij(ξ) ′ =
(δ i k − Γ k inξ
)(δ n j l − Γ l jmξ
)(g m kl (φ) + ∂ p g kl ξ p + · · ·
)
= g ij (x) +
(∂ n g ij − Γ k ing kj − Γ k jng ki ξ n + · · ·
} {{ }
D ng ij
We see that
g ′ ij(ξ) = g ij (x) + D k g ij (x)ξ k + · · ·
(D.1)
It is important to realize that the coordinates ξ k depend on x because they define
the tangent vector to the manifold at the point x. Under reparametrizations of
x → x ′ the object ξ k (x) transforms as a vector! The expansion (D.1) is covariant
under general coordinate transformations x → x ′ because it includes the tensorial
quantities only.
In the case of the minimal connection (connection compatible with the metric)
we have D k g ij = 0. This, the expansion of the metric start from the quadratic order
in ξ i . Extending this calculation to higher orders in ξ one finds
g ′ ij(ξ) = g ij (x) − 1 3 R ik 1 jk 2
ξ k 1
ξ k 2
− 1 3! D k 1
R ik2 jk 3
ξ k 1
ξ k 2
ξ k 3
+ · · ·
Also we recall the transformation property of the Christoffel connection
(
Γ k′ ∂xk′
p ′ q ′(ξ) =
∂x k
∂x p
Γ k pq
∂x p′
∂x q
∂x ′q′ + ∂2 x k
∂x p′ ∂x q′ )
,
where x i ≡ φ i and x ′i ≡ φ i + π i . Expanding the r.h.s. of the last equation in ξ i it is
easy to see that expansion does not contain the constant piece because the constant
terms in the bracket cancel against each other, in other words, in the Riemann normal
coordinates we have
Γ k′
p ′ q ′(ξ) = O(ξ)
In the Riemann normal coordinate system die to the vanishing of Γ k pq at ξ = 0 the
geodesic equation takes a form of the free motion ¨λ = 0. This is coordinate system
corresponds to the rest frame of a freely falling observer.