Gravity and Strings

118 Action principles for gravity
where hµν = gµν − n2nµnν is the induced metric on the hypersurface ∂M (see Section 1.8).
Thus, we arrive at
δSEH = − d d x |g| G µν δgµν + (−1) d−1
M
− (−1) d−1
∂M
d
∂M
d−1 n µ h σν ∇µδgσν
d d−1 n µ h σν ∇σ δgµν. (4.21)
This is the final form of the variation of the action we were after. Now, we would like to be
able to obtain the Einstein equation by requiring the action to be stationary (so δSEH = 0)
under arbitrary variations of the metric vanishing on the boundary:
= 0. (4.22)
∂M
δgµν
If δgµν is constant on the boundary, then its covariant derivative projected onto the boundary
directions with h µν must vanish:
h σν ∇σ δgµν = 0, (4.23)
and the second of the two boundary terms vanishes. However, the first does not vanish
unless we impose boundary conditions for the covariant derivative of the variation of the
metric. In order to obtain the Einstein equation we must cancel out that boundary term with
the variation of another boundary term added to the Einstein–Hilbert action. This boundary
term is nothing but the integral over the boundary of the trace of the extrinsic curvature of
the boundary given in Eq. (1.149). Observe that
δK = δh µ ν∇µn ν + h µ νδƔµρ ν n ρ . (4.24)
The first term vanishes on the boundary due to our boundary condition (4.22). Using
Eq. (3.282) for δƔ, wefind
δK| ∂M = 1
2 nρ h µσ ∇ρδgµσ . (4.25)
In conclusion, the action that one should use is the following [436, 932]:
SEH[g] = 1
χ 2
d
M
dx √ d 2
|g| R + (−1)
χ 2
d
∂M
d−1 K. (4.26)
Under otherwise arbitrary variations of the metric satisfying Eq. (4.22), we have shown
that the variation of the Einstein–Hilbert action with boundary term (4.26), is just
δSEH =− 1
χ 2
d
M
d x |g| G µν δgµν, (4.27)
and then the vacuum Einstein equation follows, as we wanted.