In another solar system there exists a planet approximately the size of Earth. This planet is a solid sphere of uniform density.
The inhabitants send messages to each other by placing the message in a capsule and placing the capsule in a message tube. These message tubes are straight lines through the planet from one point on the surface to another.
A capsule moves through a tube only under the influence of gravity (no friction force between the capsule and the tube and no forces due to the planet's rotation).
What is the ratio between the time it takes to send a message from the North Pole to the South Pole and the time it takes to send a message from the South Pole to the Equator?
Note: The gravitational effects of the message tubes are negligible compared to the gravitational effects of the planet.
Note: Assume a mechanism at each end of a message tube to catch an incoming capsule (easy to do since the velocity will be zero).