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In a first attempt to describe the effect on neutrino travel inside the Earthcaused by general relativity in the case of a dense Earth, we have neglectedthe Earth's rotation, the Earth's ellipticity and also the surface terrainvariation, nevertheless we have focused our attention on the densitydescription of the Earth interior provided by geophysic's models such as PREM.Assuming a non rotating Earth, the general relativistic effect on neutrinotravelling from CERN to GRAN-SASSO happened to produce a delay of $\deltat=4.1863 \, picosecond$.

The road toward a general relativistic metric inside the Earth and its effect on neutrino travel from CERN to GRAN-SASSO Laboratory