Let’s assume to drill a hole at Earth surface at the point P1. The hole passes throughout the centre of the Earth and finishes at the opposite side at the point P2. Let’s assume then that 1) Earth does not rotate and 2) to make vacuum inside this hole.
If I leave an object falling at the point P1, it is said that it will gain velocity falling down to the centre of the Earth. At the centre of the Earth it will have the maximum kinetic energy and the minimum potential energy. After passing the centre of the Earth, the object will continue its run pointing to point2, but progressively loosing its kinetic energy, that will become 0 at P2. Then, it will fall again towards the centre of the Earth again and again.
In this scenario only the component directed towards the centre of the Earth ($z$-direction) is considered for gravity. What about the components along the perpendicular $xy$-plane? What I mean: when the object is at, let’s say, 1500 km below the Earth surface, there will be a relevant gravity all around in the $xy$ plane. These components will delete each other and don’t affect the object trajectory, but do they play some other role or is it like they are totally absent? Do they somehow affect the energy of the object (and its velocity)?
PS: let’s assume the Earth if perfectly homogenous and spherical
PPS: I see that similar questions have been already answered, but my specific question is: do the components in the xy-plane play some (even minimal) role or not?
