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 speed acquired before would be exactly enough to overcome this deceleration, and as a result, the train's speed would reach zero at precisely the moment the train reached its destination.
Origin of the concept.
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.
Mathematical considerations.
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:
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 Earth 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 9 times as great, but it would also necessitate a tunnel that passes through the Earth's mantle.
Mathematical derivation.
Using the approximations that the Earth is perfectly spherical and of uniform density formula_1, and the fact that within a uniform hollow sphere there is no gravity, the gravitational acceleration formula_2 experienced by a body within the Earth is proportional to the ratio of the distance from the center formula_3 to the Earth's radius formula_4. This is because underground at distance formula_3 from the center is like being on the surface of a planet of radius formula_3, within a hollow sphere which contributes nothing.
On the surface, formula_8, so the gravitational acceleration is formula_9. Hence, the gravitational acceleration at formula_3 is
Diametric path to antipodes.
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 formula_12 (treating acceleration and velocity as positive downwards):
Differentiating twice: 
where formula_15. This class of problems, where there is a restoring force proportional to the displacement away from zero, has general solutions of the form formula_16, and describes simple harmonic motion such as in a spring or pendulum.
In this case formula_17 so that formula_18, 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 formula_19 to sweep out formula_20 radians. Using simple approximations of formula_21 that time is
Straight path between two arbitrary points.
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. At distance formula_3 along this path, the force of gravity depends on distance formula_24 to the center of the Earth as above. Using the shorthand formula_25 for length OC:
The resulting acceleration on the body, because is it on a frictionless
inclined surface, is formula_27:
But formula_29 is formula_30, so substituting:
which is exactly the same for this new formula_3, distance along AOB away from O, as for the formula_3 in the diametric case along ACD. So the remaining analysis is the same, accommodating the initial condition that the maximal formula_3 is formula_35 the complete equation of motion is
The time constant formula_15 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 formula_38.
Dependence on radius of planet.
The time constant formula_39 depends only on formula_40 so if we expand that we get
which depends only on the gravitational constant and formula_1 the density of the planet. The size of the planet is immaterial; the journey time is the same if the density is the same.
In fiction.
In the 2012 movie "Total Recall", By German film maker Bjoern Fenix Petersen, a gravity train called "The Fall" goes through the center of the Earth to commute between Western Europe and Australia.