  Timelike Liouville theory admits the sphere $\mathbb{S}^{2}$ as a real saddle
point, about which quantum fluctuations can occur. An issue occurs when
computing the expectation values of specific types of quantities, like the
distance between points. The problem being that the gauge redundancy of the
path integral over metrics is not completely fixed even after fixing to
conformal gauge by imposing $g_{\mu\nu} = e^{2\b\phi}\tilde{g}_{\mu\nu}$, where
$\phi$ is the Liouville field and $\tilde{g}_{\mu\nu}$ is a reference metric.
The physical metric $g_{\mu\nu}$, and therefore the path integral over metrics
still possesses a gauge redundancy due to invariance under $SL_{2}(\mathbb{C})$
coordinate transformations of the reference coordinates. This zero mode of the
action must be dealt with before a perturbative analysis can be made.
  This paper shows that after fixing to conformal gauge, the remaining zero
mode of the linearized Liouville action due to $SL_{2}(\mathbb{C})$ coordinate
transformations can be dealt with by using standard Fadeev-Popov methods.
Employing the gauge condition that the "dipole" of the reference coordinate
system is a fixed vector, and then integrating over all values of this dipole
vector. The "dipole" vector referring to how coordinate area is concentrated
about the sphere; assuming the sphere is embedded in $\mathbb{R}^{3}$ and
centered at the origin, and the coordinate area is thought of as a charge
density on the sphere. The vector points along the ray from the origin of
$\mathbb{R}^{3}$ to the direction of greatest coordinate area.
  A Green's function is obtained and used to compute the expectation value of
the geodesic length between two points on the $\mathbb{S}^{2}$ to second order
in the Timelike Liouville coupling $\b$. This quantity doesn't suffer from any
power law or logarithmic divergences as a na\"{i}ve power counting argument
might suggest.
