  We start with the Hamiltonian formulation of the first order action of pure
gravity with a full $\mathfrak{sl}(2,\mathbb C)$ internal gauge symmetry. We
make a partial gauge-fixing which reduces $\mathfrak{sl}(2,\mathbb C)$ to its
sub-algebra $\mathfrak{su}(1,1)$. This case corresponds to a splitting of the
space-time ${\cal M}=\Sigma \times \mathbb R$ where $\Sigma$ inherits an
arbitrary Lorentzian metric of signature $(-,+,+)$. Then, we find a
parametrization of the phase space in terms of an $\mathfrak{su}(1,1)$
commutative connection and its associated conjugate electric field. Following
the techniques of Loop Quantum Gravity, we start the quantization of the theory
and we consider the kinematical Hilbert space on a given fixed graph $\Gamma$
whose edges are colored with unitary representations of $\mathfrak{su}(1,1)$.
We compute the spectrum of area operators acting of the kinematical Hilbert
space: we show that space-like areas have discrete spectra, in agreement with
usual $\mathfrak{su}(2)$ Loop Quantum Gravity, whereas time-like areas have
continuous spectra. We conclude on the possibility to make use of this
formulation of gravity to construct a holographic description of black holes in
the framework of Loop Quantum Gravity.
