Gravity and Strings

4.3 The first-order (Palatini) formalism 123
When we studied vector and tensor fields living on a general background, we adopted as
sign of their masslessness the existence of gauge transformations leaving their equations of
motion invariant. If we interpret the above equations as the equations of a scalar field living
on a background metric gµν, wemay wonder how we can tell whether the scalar field is
massless. The only kind of local transformations that we can define for a scalar field are
the above Weyl transformations and we can define as a massless field one whose equation
of motion is invariant under them. Therefore we could consider the conformal scalar as a
massless scalar. This means, in particular, that the equation of motion of a massless scalar
in a spacetime satisfying Rµν = gµν is
∇ 2 d(d − 2)
+
4(d − 1)
Kc = 0, (4.56)
and, as usual, the term is not a mass term but, on the contrary, its presence ensures the
masslessness of the scalar field.
4.3 The first-order (Palatini) formalism
This formalism [752] consists in writing an action in which the metric and the connection
(which contains the dependence on the derivatives of the metric) are considered independent
variables. The connection is, therefore, not the Levi-Cività connection. It is assumed
to be torsion-free, i.e. Ɣ[µν] ρ = 0, but no other properties (metric-compatibility, for example)
are assumed. The first-order action contains only derivatives of the connection and it
is linear in them. To obtain the equations of motion, one now has to vary the metric and the
connection independently. The connection equation of motion gives us the standard relation
between the connection and the metric and the metric equation is, after substitution of
the solution to the other equation, nothing but the Einstein equation.
The first-order action turns out to be essentially the Einstein–Hilbert action: 3
S[gµν,Ɣµν ρ
] =
d d x √ |g| g µν Rµν(Ɣ). (4.57)
All the dependence on the metric is concentrated in the factor √ |g| g µν since the Ricci
tensor depends only on the connection and its derivatives as shown in Eq. (1.33).
We stress that, since the connection is here a variable, and it is not the Levi-Cività connection,
one cannot use the standard property
d d x |g|∇µξ µ
= d d
µ
x ∂µ |g| ξ
. (4.58)
The calculations are simpler using as a variable the density
3 We set χ = 1 throughout this section.
g µν = |g| g µν . (4.59)