Gravity and Strings

1.3 Metric spaces 13
1.3.2 Riemann spacetime Vd
It is defined by the conditions Q = T = 0 which determine the connection to be the Levi-
Civitàconnection Ɣ(g) whose components in a coordinate basis are given by the Christoffel
symbols. In a Riemann spacetime one can construct infinitesimal parallelograms and autoparallel
curves are also geodesics (as in flat spacetime). There are also additional interesting
properties. To start with, we can write the transformation of tensors under infinitesimal
GCTs (Lie derivatives) in terms of covariant derivatives alone (all torsion terms vanish). In
particular, for the metric and r-forms we can write
δξ gµν =−2∇(µξν),
δξ Bµ1···µr =−ξ λ ∇λBµ1···µr − r(∇[µ1|ξ λ )Bλ|µ2···µr ].
(1.59)
Furthermore, we have the usual identity
Ɣρµ ρ
= ∂µ ln |g| , (1.60)
which allows us to write the Laplacian of a scalar function f in this way:
∇ 2 f = 1
µ
√ ∂µ |g| ∂ f , (1.61)
|g|
and the divergence of a completely antisymmetric tensor (k-form) in this way: 7
∇µ1 F µ1µ2···µk 1
µ1µ2···µk
= √|g| ∂µ1 |g| F . (1.62)
The Bianchi identities take the form
R(αβ)γ δ = 0, R[αβγ ] δ = 0, ∇[α Rβγ]ρ σ = 0, Rαβ(γ δ) = 0. (1.63)
The first and fourth identities imply together
Rαβγ δ = Rγδαβ, (1.64)
which in turn implies that the Ricci and Einstein tensors are symmetric. The contracted
Bianchi identity says now that the Einstein tensor is divergence-free:
∇µG µν = 0, (1.65)
which is a crucial identity in the development of general relativity.
The number of independent components of the curvature in d dimensions after taking
into account all these Bianchi identities is (1/12) d 2 (d 2 − 1).
The four-dimensional curvature tensor can be split into different pieces which transform
irreducibly under the Lorentz group: a scalar piece D(0, 0), which is nothing but the Ricci
7 Observe that this implies that the second term on the r.h.s. of Eq. (1.56) times √ |g| is a total derivative.