Gravity train

For real Gravity trains, see Gravity railroad.

A gravity train is a theoretical means of transportation intended to go between two points on the surface of a sphere, following a straight tunnel that goes directly from one point to the other 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 enough to cancel this deceleration exactly (so that the train would reach its destination with speed equal to zero).^[1]

Objections

In reality, there are two reasons gravity trains do not exist. First, a lengthy transit distance would pierce the Earth's mantle and traverse a region where rock is more fluid than solid. No materials are known that would withstand the tremendous heat and pressure in the inner core. Temperature is estimated as 5,700 K (5,430 °C; 9,800 °F), and pressure as high as about 330 to 360 gigapascals (3,300,000 to 3,600,000 atm). Secondly, frictional losses would be significant. Rolling friction losses could be reduced by using a magnetically levitated train. However, unless all air is evacuated from the tunnel, frictional losses due to air resistance would render the gravity train unusable. Evacuating the atmosphere to make it a vactrain would eliminate this drag but would require additional power. Such objections would not apply for solid planets and moons that do not have an atmosphere. During this entire trip, the train (and all passengers) would be practically weightless.

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 Paris 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]

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:^[3]

The duration of a trip depends only on the density of the planet and the gravitational constant, but not on the diameter of the planet.
The maximum speed is reached at the middle point of the trajectory.

For gravity trains between points which are not the antipodes of each other, the following hold:

The shortest time tunnel through a homogeneous earth is a hypocycloid; in the special case of two antipodal points, the hypocycloid degenerates to a straight line.
All straight-line gravity trains on a given planet take exactly the same amount of time to complete a journey (that is, no matter where on the surface the two endpoints of its trajectory are located).

On the planet Earth specifically, a gravity train has the following parameters:

The travel time equals 2530.30 seconds (nearly 42.2 minutes), assuming Earth were a perfect sphere of uniform density.
By taking into account the realistic density distribution inside the Earth, as known from the Preliminary Reference Earth Model, the expected fall-through time is reduced from 42 to 38 minutes.^[4]
For a train that goes directly through the center of the Earth, the maximum speed is about 7,900 meters per second (28440 km/h) (Mach 23.2).

To put some numbers in perspective, the deepest current bore hole is the Kola Superdeep Borehole with a true depth of 12,262 meters. While to cover a distance between London and Paris (350 km) via a hypocycloidical path would need the creation of a 55,704-metre-deep hole. This depth isn't only 4.5 times as deep; it will also already need a tunnel that passes inside the Earth's mantle.

Mathematical derivation

Using the approximations that the Earth is perfectly spherical and of uniform density $\rho$ , and the fact that within a uniform hollow sphere there is no gravity, the gravity $a$ experienced by a body within the Earth is proportional to the ratio of the distance from the center $r$ to the Earth's radius $R$ . This is because underground at distance $r$ from the center is like being on the surface of a planet of radius $r$ , within a hollow sphere which contributes nothing.

a={\frac {GM}{r^{2}}}={\frac {G\rho V}{r^{2}}}={\frac {G\rho {\frac {4}{3}}\pi \,r^{3}}{r^{2}}}=G\rho {\frac {4}{3}}\pi \,r

On the surface, $r=R$ so $g=G\rho {\frac {4}{3}}\pi \,R$ giving the gravity at $r$

a={\frac {r}{R}}\,g

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 the surface, so at time $t$ (treating acceleration and velocity as positive downwards):

r_{t}=R-\int _{0}^{t}v_{t}dt=R-\int _{0}^{t}\int _{0}^{t}a_{t}\,dt\,dt

Differentiating twice:

{\frac {d^{2}r}{dt^{2}}}=-a_{t}=-{\frac {r}{R}}\,g=-\omega ^{2}\,r

where $\omega ={\sqrt {\frac {g}{R}}}$ . This class of problems, where there is a restoring force proportional to the displacement away from zero, has general solutions of the form $r=k\cos(\omega t+\phi )$ (plus a constant), and describes simple harmonic motion such as in a spring or pendulum.

In this case $r_{t}=R\cos {\sqrt {\frac {g}{R}}}\,t$ so that $r_{0}=R$ , 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 $\cos {\sqrt {\frac {g}{R}}}\,t$ to sweep out ${\pi }$ radians. Using simple approximations of $g=10m/s^{2},R=6500km$ that time is

$T={\frac {\pi }{\omega }}={\frac {\pi }{\sqrt {\frac {g}{R}}}}\approx {\frac {3.1415926}{\sqrt {\frac {10}{6500000}}}}\approx 2532s$

Straight path between two arbitrary points

Path of gravity train

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 $r$ along this path, the force of gravity depends on distance $x$ to the center of the Earth as above. Using the shorthand $b=R\sin \theta$ for length OC:

g_{r}={\frac {x}{R}}\,g={\frac {\sqrt {r^{2}+b^{2}}}{R}}\,g

The resulting acceleration on the body, because is it on a frictionless inclined surface, is $g_{r}\cos \phi$ :

Diagram of forces on a gravity train on non-diametrical straight line path

a_{r}=g_{r}\cos \phi ={\frac {\sqrt {r^{2}+b^{2}}}{R}}\,g\cos \phi

But $\cos \phi$ is $r/x={\frac {r}{\sqrt {r^{2}+b^{2}}}}$ , so substituting:

a_{r}={\frac {\sqrt {r^{2}+b^{2}}}{R}}\,g\,{\frac {r}{\sqrt {r^{2}+b^{2}}}}={\frac {r}{R}}\,g

which is exactly the same for this new $r$ , distance along AOB away from O, as for the $r$ in the diametric case along ACD. So the remaining analysis is the same, accommodating the initial condition that the maximal $r$ is $r\cos \theta$ = AO the complete equation of motion is

r_{t}=R\cos \theta \cos {\sqrt {\frac {g}{R}}}\,t

The time constant $\omega ={\sqrt {\frac {g}{R}}}$ 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 $\cos \theta$ .

Independence from radius of planet

The time constant $\omega$ depends only on ${\frac {g}{R}}$ so if we expand that we get

{\frac {g}{R}}={\frac {GM/R^{2}}{R}}={\frac {GM}{R^{3}}}={\frac {G\rho \,V}{R^{3}}}={\frac {G\rho \,{\frac {4}{3}}\pi \,R^{3}}{R^{3}}}=G\rho \,{\frac {4}{3}}\pi

which depends only on the gravitational constant and $\rho$ 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

The 1914 book Tik-Tok of Oz has a tube, that passed from Oz, through the center of the earth, emerging in the country of the Great Jinjin, Tittiti-Hoochoo. In the 2012 movie Total Recall, a gravity train is used to commute between Western Europe and Australia. In the game Super Mario Galaxy, there are a few holes that Mario can jump through to illustrate the gravity train effect.

References

↑ Newton, Isaac. Philosophiæ Naturalis Principia Mathematica,.
↑ "To Everywhere in 42 Minutes".
↑ Robin Davis: A Physicist's Pipe Dream
↑ http://scitation.aip.org/content/aapt/journal/ajp/83/3/10.1119/1.4898780

Description of the concept Gravity train and mathematical solution (Alexandre Eremenko web page at Purdue University).
Translations of the previous page into Romanian and Polish.
The Amazing 42-Minute Gravity Sled, at the Hitch-Hiker's Guide to the Galaxy website.

External links

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