Compact stars: To cross or go around? That is the question

The travel times of light signals between two antipodal points on the surface of a compact star are calculated for two different trajectories: a straight line that passes through the center of the star and a semi-circular trajectory that connects the antipodal points along the surface of the star. Interestingly, it is explicitly proved that, for highly dense stars, the longer trajectory (the one that goes along the surface of the star) may be characterized by the {\it shorter} travel time as measured by asymptotic observers. In particular, for constant density stars we determine {\it analytically} the critical value of the dimensionless density-area parameter $\Lambda\equiv 4\pi R^2\rho$ that marks the boundary between situations in which a direct crossing of the star through its center has the shorter travel time and situations in which the semi-circular trajectory along the surface of the star is characterized by the shorter travel time as measured by asymptotic observers [here $\{R,\rho\}$ are respectively the radius of the star and its density].