Gravity and Strings

4.1 The Einstein–Hilbert action 117
that the “vector” ω µ (∂g) does not transform as such under GCTs. The solution to these
problems consists in adding a general-covariant boundary term to the original EH action.
We are going to see next how to find the equations of motion and the right boundary term.
4.1.1 Equations of motion
Let us vary the original Einstein–Hilbert action with respect to the metric. For simplicity we
temporarily set χ = 1. Bearing in mind that R(g) = g µν Rµν(Ɣ(g)) and Rµν(Ɣ(g)) depends
on g only through the Levi-Cività connection Ɣ(g) so we can use the Palatini identity
Eq. (3.285),
δRµν =∇µδƔρν ρ −∇ρδƔµν ρ , (4.12)
and using the identities
we immediately find
δSEH =
δg µν =−g να g µβ δgαβ, δg = gg αβ δgαβ, (4.13)
d d x |g| −G µν δgµν + g µν ∇µδƔρν ρ −∇ρδƔµν ρ . (4.14)
Since our covariant derivative is metric-compatible we can absorb the metric in the last
term and combine the two terms into a single total derivative,
δSEH =− d d x |g| G µν
δgµν + d d x |g|∇ρv ρ , (4.15)
M
where
v ρ = g ρµ δƔµν ν − g µν δƔµν ρ . (4.16)
We now have to use the equation that expresses the variation of the Levi-Cività connection
with respect to a variation of the metric in order to find the variation of the action as a
function of the variation of the metric. That expression was given in Eq. (3.282) and with
it we find
v ρ = g ρµ g σν
∇µδgσν −∇σδgµν . (4.17)
Using now Stokes’ theorem Eq. (1.141), we reexpress the integral of the total derivative
terms as an integral over the boundary,
d d x |g|∇ρv ρ = (−1) d−1
d d−1 ρv ρ = (−1) d−1
d d−1 nρv ρ , (4.18)
M
∂M
where d d−1 ρ is defined in Chapter 1,
M
∂M
d d−1 ≡ n 2 d d−1 ρn ρ , (4.19)
and n µ is the unit vector normal to the boundary hypersurface ∂M (n2 =+1 for spacelike
hypersurfaces with timelike normal unit vector and n2 =−1for timelike hypersurfaces
with spacelike normal unit vector). Finally, we expand the integrand
nρv ρ = n µ g σν µ σν
∇µδgσν −∇σδgµν = n h
∇µδgσν −∇σδgµν , (4.20)