Event Horizon of the Monopole-Quadrupole solution: geometric and thermodynamic properties

We investigate the general geometric properties of the surface of infinite red-shift corresponding to the event horizon of the static and axisymmetric solution of the Einstein vacuum equations that only possesses mass $M$ and quadrupole moment $Q$. The deformation of the Schwarzschild surface $r=2M$ produced by the quadrupole moment is shown, and the range of values of this multipole moment is specified, which preserves a regular, closed, continuous and differentiable surface. Some thermodynamic consequences and speculations ensuing from our results are discussed.

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