I want to suggest an idea for consideration, I ask for certain things to be ignored and yet I will accept all the physical laws. First let us consider the earth is a perfect sphere and on its surface are two cities A and B. We can bore, without problems, a tunnel between A and B which will be a straight tunnel and, mathematically, a chord AB on the circle of the circumference of the earth. C is the centre point of the chord. A train will travel through this tunnel and it will ride on magnetic cushions and have minimal friction problems. To accommodate this small friction and the effects of air pressure a calculated impetus is applied and the train begins its journey. As it travels from A to C it will get nearer the centre of the earth O and will in effect be going ‘downhill’’. It will therefore be accelerated by gravity over this part of the journey. As it passes point C to continue on the journey to point B it will have sufficient momentum to go all the way with gravity now having a braking effect as the train is now travelling ’uphill’. The train rolls gently to a halt at the station in the hear of the city B.
Would this work??