Gravity and Strings

4.1 The Einstein–Hilbert action 115
gauge theory of the four-dimensional Poincaré group, which we will obtain by Wigner–
Inönücontraction from the AdS4 case.
Finally, we will briefly review teleparallel formulations and generalizations of GR.
4.1 The Einstein–Hilbert action
In d dimensions, the Einstein–Hilbert action [535] is
SEH[g] =
c 3
16πG (d)
N
M
d d x √ |g| R(g), (4.1)
where R(g) is the Ricci scalar of the metric gµν, G (d)
N is the d-dimensional Newton constant
and M is the d-dimensional manifold we are integrating over. Since we have obtained this
action by imposing consistent coupling of the special-relativistic field theory, we know that
it is canonically normalized and we also know which expression for the force between two
particles it leads to (see Eq. (3.140)). We have introduced here the speed of light in order to
find the dimensions of G (d)
N in “unnatural units,”: M−1 L d−1 T −2 .Recall that the metric gµν
is dimensionless in our conventions. Recall also that the factor of 16π is associated with
rationalized units only in d = 4.
Observe that what will appear in the path integral
Z = Dg e +iSEH/
is the dimensionless combination
where
SEH
2π
=
ℓ d−2
Planck
ℓ d−2
Planck
2π
(4.2)
d d x ···, (4.3)
G(d)
N
=
c3 , (4.4)
is the d-dimensional Planck length. 1 In the absence of any other dimensional quantity this
is the only combination of the constants , c, and G (d)
N with dimensions of length. However,
if there is an object of mass M, there are two more combinations with dimensions of length:
the Compton wavelength associated with the object,
1 Sometimes the reduced Planck length
We have also been using the constant χ defined by χ 2 = 16πG (d)
N /c3 .
−
λCompton =
, (4.6)
Mc
− ℓPlanck = ℓPlanck/(2π). (4.5)