Gravity and Strings

4.3 The first-order (Palatini) formalism 125
action, we do not have to introduce any term containing the connection. Thus, the equation
for the connection would not change and the equation for the metric, would become the
Einstein equation with non-vanishing energy–momentum tensor (again, if the connection
were the Levi-Cività connection).
To find the relation between the connection and the metric, we have to solve the second
equation. It is convenient to use a new connection ˜Ɣ, definedby
˜Ɣµν ρ = Ɣµν ρ + 1
d − 1 Tµσ σ δν ρ . (4.65)
Observe that the new connection ˜Ɣ does not completely determine the old one, Ɣ. Infact,
if we shift Ɣ by an arbitrary vector fµ according to
Ɣµν ρ → Ɣµν ρ + fµδν ρ , (4.66)
the connection ˜Ɣ is not modified. Thus, the expression for Ɣ in terms of ˜Ɣ is
Ɣµν ρ = ˜Ɣµν ρ + fµδν ρ , (4.67)
where fµ cannot be determined from ˜Ɣ. The new connection allows us to rewrite the second
equation in the form
∂σ g αβ + ˜Ɣδσ α g δβ + g αδ ˜Ɣσδ β − g αβ ˜Ɣσδ δ = 0. (4.68)
On contracting in the above equation the indices σ with α and σ with β, taking the difference,
and using the property
˜Ɣµρ ρ = ˜Ɣρµ ρ , (4.69)
we arrive at the Maxwell-like equation for the antisymmetric part of g
By contracting now Eq. (4.68) with gαβ/ √ |g|,weobtain
∂αg [αβ] = 0. (4.70)
∂σ ln |g|= ˜Ɣσα α , (4.71)
and, on plugging this back into Eq. (4.68), we obtain an equation for the inverse metric,
∂σ g αβ + ˜Ɣσδ β g αδ + ˜Ɣδσ α g δβ = 0. (4.72)
We now multiply by the inverse “metrics” gγα and gβϕ to obtain, at last,
∂σ gγϕ − ˜Ɣσγ β gβϕ − ˜Ɣϕσ α gγα = 0. (4.73)
Although we have started with the connection Ɣ, the above equation allows us only to
solve for the connection ˜Ɣ in terms of the metric.
It is easy to particularize this general setup for the case that interests us: a symmetric metric
g [µν] = 0 and a torsion-free connection Ɣ[µν] ρ = 0. In this case, Rµν(Ɣ) is automatically