A cool transit system could connect every capital to every other capital through straight tunnels only was the Earth a perfect nonrotating sphere with homogenous density, and we somehow made the tunnels.
This physics goldie oldie was known to Newton and was featured in an 1883 article “De Paris à Rio-Janeiro en 42 minutes 11 secondes” in some French journal. Forty two minutes it would take such gravity trolley between any two surface points connected by a straight underground tunnel, for as it turns out the time of journey depends only on the Earth’s density and gravitational constant. How much did we say again? Did we just get the question to the ultimate answer?
Note: I tried and visualized it in 3d with a globe and all the 240 world capitals but my computer wouldn’t hear much of it. In the end I found these eight capitals lying approximately on the same great circle.
Circumstances naturally renders such transportation completely unrealistic. On Earth that is, so we imagine sufficiently advanced extraterrestrials who built themselves a planet, a homogeneous solid sphere, and tunneled it settlement to settlement, or presumably had the tunnels already in the planet blueprint. They also utilized magnetic levitation and vacuumed the tubes of air (would their planet likely have an atmosphere) so as to reduce the rolling friction and air resistance to zero.
Under the influence of gravity only. If perforating a planet through and through is not a problem — how much greener can one get?