Gravitational tunnel

In a shaft through the center of the earth, a mass oscillates from surface to surface.

A gravitation tunnel is a physics-didactic thought experiment in which a train moves through the earth in a tunnel without friction and without drive . Gravitation accelerates the train to the lowest point of the route and then brakes it again. What is asked is the travel time for a straight tunnel through the center point or for a secant , or also for the course of a curved tunnel for a minimal travel time. Usually the earth is assumed to be homogeneous and not rotating.

The problem was already described by the British physicist Robert Hooke in a letter to Isaac Newton in the 17th century . It is sometimes seen as a concept of a transport system instead of a physical arithmetic exercise, but it is predominantly located in the field of science fiction, its feasibility is more than questionable.

Force field

The potential of a two-dimensional isotropic harmonic oscillator

Newton's sphere and shell theorem are used to solve the problem . The former says that a spherically symmetrical mass distribution in the outer space causes the same field as if the mass were concentrated in the center, the latter that a spherical shell does not contribute to the field in its interior. For a position of the train at a distance from the center, only the mass within is considered . This scales with . The law of distance of gravitation contains in the denominator, the local acceleration due to gravity is therefore proportional to . The proportionality factor is the ratio of the values ​​on the surface. ${\displaystyle \textstyle {r}}$ ${\displaystyle \textstyle {r}}$ ${\displaystyle \textstyle {r^{3}}}$ ${\displaystyle \textstyle {r^{2}}}$ ${\displaystyle \textstyle {r}}$ ${\displaystyle \textstyle {g/R}}$

A linear law of distance goes hand in hand with a quadratic potential and harmonic oscillations .

solutions

Vibrations on straight lines

In the formula for the angular frequency of the spring oscillation ,

${\displaystyle \omega ={\sqrt {\frac {D}{m}}}\,\approx 0,0126\,\,{\frac {1}{\textrm {s}}}}$

one identifies with and obtains for the period of oscillation ${\displaystyle \textstyle {D/m}}$ ${\displaystyle \textstyle {g/R}}$

${\displaystyle T={\frac {2\pi }{\omega }}=2\pi {\sqrt {\frac {R}{g}}}\approx 5000\,\,{\textrm {s.}}}$

The travel time is half an oscillation period, i.e. about 42 minutes.

Since it is a harmonic oscillation, the period is independent of the oscillation amplitude. So the travel time through planets of the same density is independent of the diameter and is scaled with ${\displaystyle \textstyle {\rho }}$ ${\displaystyle \textstyle {(\omega }}$ ${\displaystyle \textstyle {\sqrt {g}}}$ ${\displaystyle \textstyle {{\sqrt {\rho }}).}}$

In addition, this period does not only apply to oscillations along the diameter of the earth, but along any secants, which can easily be seen when one expresses in the potential : For each tunnel in -direction is constant and only shifts the zero point of the potential, but does not change its shape . ${\displaystyle \textstyle {r^{2}}}$ ${\displaystyle \textstyle {x^{2}+y^{2}}}$ ${\displaystyle \textstyle {x}}$ ${\displaystyle \textstyle {y}}$ ${\displaystyle \textstyle {y^{2}}}$

Vibrations in - and - direction are independent of each other and result in a circular path on the earth's surface with a 90 ° phase shift and the same amplitude . ${\displaystyle \textstyle {x}}$ ${\displaystyle \textstyle {y}}$ ${\displaystyle \textstyle {R}}$

${\displaystyle v=R\omega \approx 7900\,{\textrm {m/s}}}$

is not only the speed on this circular orbit, the first cosmic speed , but also the maximum speed during free fall across the earth.  ^{P. 96}

Fastest lanes

Faster connections are possible for a short distance. The task of finding the brachistochrone , i.e. the trajectory for the fastest connection between two given points on the surface, is more demanding. The solution is a hypocycloid , i.e. the path of a point on the circumference of a wheel that rolls on the inside of the earth's surface. The circumference of the wheel must be equal to the arc length between the two points. The movement starts and ends vertically in free fall.

For 4800 km (about London – Mecca) Amanda Maxham gives 27.4 minutes of travel time for the hypocycloid. However, this hypocycloid reaches a depth of over 1500 km, while the secant sinks to 447 km. For comparison: The Kola well with a depth of around twelve kilometers has been the deepest well in the world since 1979.

Small hypocycloids run approximately on a non-curved surface, and that these represent brachistochrones for the case of a homogeneous field was already found by Johann I Bernoulli in 1696.

Inhomogeneous mass distribution

Course of the gravitational acceleration according to the PREM (blue) compared to the approximation of constant density (dark green)

The real mass distribution of the earth describes the PREM . Since the earth's core has a much higher density than the mantle, the approximation of constant density is unrealistic, see figure. The approximation of constant gravitational acceleration (not shown) is obviously better. This results in a travel time of

${\displaystyle T_{g=\mathrm {const} }=2{\sqrt {\frac {2R}{g}}}\approx 38{,}0\,\,\mathrm {Minuten.} }$

Alexander Klotz states that the travel time is 38.2 minutes based on the PREM, calculated numerically. But he notes that the original problem with constant density is more instructive.

reception

In the 19th century a concept for a gravitational train was seriously proposed to the Paris Academy of Sciences , but it was ignored there. Especially in the second half of the 20th century, the concept was taken up again by some scientists and mathematically formalized. In ignorance of the older literature, Paul Cooper again suggested gravitational trains as a possible future transport system. The article was picked up by Time magazine and presented to a wider audience.  ^{P. 99f} Physics textbooks use the Gravitrain as an exercise. Furthermore, the concept of the gravitation tunnel is used in works of science fiction , physically correct in Stephen Baxter's book Ultima , with gross physical errors in the film Total Recall .

literature

• Paul W. Cooper: Through the earth in forty minutes , American Journal of Physics 34, 1966, pp. 68-70.
• Alexander Klotz: Gravity tunnel in a non-uniform earth , American Journal of Physics 83, 2015, p. 231, Arxiv
• Alexander Klotz: A Guided Tour of Planetary Interiors , Department of Physics, McGill University Montreal, May 25, 2015, [1]

Web links

• Maria Lubs: How long does it take to fall across the earth?

Individual evidence

1. ↑ ^{a b} Kevin R. Grazier, Stephen Cass: Hollyweird Science: From Quantum Quirks to the Multiverse , 2015, ISBN 3319150723 , from p. 125
2. ↑ " The idea seems so exciting that people tend to take it as a serious engineering project, rather than a Calculus / Physics exercise :-) ," including the smileys from Alexandre Eremenko , on his page Gravity Train Solution below.
3. ↑ http://www.math.purdue.edu/~eremenko/train.html
4. ↑ ^{a b} Martin Gardner: Mathematical Puzzle Tales , 2000, ISBN 088385533X .
5. ↑ ^{a b} Amanda Maxham: Brachistochrone inside the Earth: The Gravity Train, UNLV Department of Physics and Astronomy, September 26, 2008, http://www.physics.unlv.edu/~maxham/gravitytrain.pdf
6. ↑ ^{a b} Alexander Klotz: Gravity tunnel in a non-uniform earth . American Journal of Physics 83, 2015.
7. ↑ Paul W. Cooper, Through the earth in forty minutes , American Journal of Physics 34, 1966, pp. 68-70
8. ↑ To Everywhere in 42 Minutes .
9. ↑ Gerthsen Physics . Springer-Verlag, July 2, 2013, ISBN 978-3-662-07462-6 , p. 1023.