Gravity train

A gravity train is a theoretical means of transportation for purposes of commuting between two points on the surface of a sphere, by following a straight tunnel connecting the two points through the interior of the sphere.

In a large body such as a planet, this train could be left to accelerate using just the force of gravity, since during the first half of the trip (from the point of departure until the middle), the downward pull towards the center of gravity would pull it towards the destination.

During the second half of the trip, the acceleration would be in the opposite direction relative to the trajectory, but, ignoring the effects of friction, the momentum acquired during the first half of the trajectory would equal this deceleration, and as a result, the train's speed would reach zero at approximately the moment the train reached its destination.

[1][better source needed] In the 17th century, British scientist Robert Hooke presented the idea of an object accelerating inside a planet in a letter to Isaac Newton.

A gravity train project was seriously presented to the French Academy of Sciences in the 19th century.

The same idea was proposed, without calculation, by Lewis Carroll in 1893 in Sylvie and Bruno Concluded.

The idea was rediscovered in the 1960s when physicist Paul Cooper published a paper in the American Journal of Physics suggesting that gravity trains be considered for a future transportation project.

[2] Under the assumption of a spherical planet with uniform density, and ignoring relativistic effects as well as friction, a gravity train has the following properties:[3] For gravity trains between points which are not the antipodes of each other, the following hold: On the planet Earth specifically, since a gravity train's movement is the projection of a very-low-orbit satellite's movement onto a line, it has the following parameters: To put some numbers in perspective, the deepest current bore hole is the Kola Superdeep Borehole with a true depth of 12,262 meters; covering the distance between London and Paris (350 km) via a hypocycloidical path would require the creation of a hole 111,408 metres deep.

Not only is such a depth nine times as great, but it would also necessitate a tunnel that passes through the Earth's mantle.

Using the approximations that the Earth is perfectly spherical and of uniform density

, and the fact that within a uniform hollow sphere there is no gravity, the gravitational acceleration

experienced by a body within the Earth is proportional to the ratio of the distance from the center

from the center is like being on the surface of a planet of radius

is In the case of a straight line through the center of the Earth, the acceleration of the body is equal to that of gravity: it is falling freely straight down.

We start falling at the surface, so at time

(treating acceleration and velocity as positive downwards): Differentiating twice: where

This class of problems, where there is a restoring force proportional to the displacement away from zero, has general solutions of the form

, and describes simple harmonic motion such as in a spring or pendulum.

, we begin at the surface at time zero, and oscillate back and forth forever.

The travel time to the antipodes is half of one cycle of this oscillator, that is the time for the argument to

that time is For the more general case of the straight line path between any two points on the surface of a sphere we calculate the acceleration of the body as it moves frictionlessly along its straight path.

The body travels along AOB, O being the midpoint of the path, and the closest point to the center of the Earth on this path.

(directed from point X towards the center of the Earth, along

We have: The resulting acceleration on the body, because it is on a frictionless inclined surface, is

So the remaining analysis is the same, accommodating the initial condition that the maximal

the complete equation of motion is The time constant

is the same as in the diametric case so the journey time is still 42 minutes; it's just that all the distances and speeds are scaled by the constant

so if we expand that we get which depends only on the gravitational constant and

The size of the planet is immaterial; the journey time is the same if the density is the same.

In the 2012 movie Total Recall, a gravity train called "The Fall" goes through the center of the Earth to commute between Western Europe and Australia.