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Falling through to Pellucidar

by John G. Cramer

Alternate View Column AV-32
Keywords: shadow matter, gravitation, hollow Earth, Pellucidar
Published in the April-1989 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 9/19/88 and is copyrighted ©1988, John G. Cramer. All rights reserved.
No part may be reproduced in any form without the explicit permission of the author.


    In 1914, two years after the great success of Tarzan of the Apes, Edgar Rice Burroughs initiated a brand new series with a magazine serial called At the Earth's Core. The hero, one David Innes, arrived in the vast hollow interior of the Earth in a mole machine that had burrowed down through the planet's outer shell. The premise of the series, taken from the ideas of John Symmes, is that the Earth is as hollow as a tennis ball, with a habitable world inside called "Pellucidar". For illumination, Pellucidar has a miniature pseudo-Sun at its center, complete with its own 24-hour satellite. Gravity sticks the inhabitants of Pellucidar to the inside walls of the Earth, just as it sticks us to the Earth's outer surface.

    There are a number of geophysical and astrophysical problems with Burroughs' s setting. Perhaps it is pointless to complain, 75 years too late, about the "hardness" of such science fantasies, for the strong suit of Burroughs' writings always lay in their imaginative sweep, never their scientific accuracy. In the case of Pellucidar, however, Burroughs committed a blunder that sets the teeth of every physicist on edge: he assumed that gravity would pull toward the inner surface of a hollow sphere.

    Consider the following question: Inside a massive hollow sphere, just how much gravitational force would there be at the center, near an inner wall, etc.? The answer is: zero. It's easy to convince yourself that the force at the center would be zero, because, if you think of the sphere as two hemispheres, the gravitational pull from one hemisphere is just cancelled by the equal and opposite pull of the other hemisphere. What perhaps isn't so obvious is that away from the center the same cancellation holds. It occurs because at any point in the hollow region the amount of mass in a section of wall falling within a given angular cone increases with distance as r2 while the gravitational pull from that mass decreases with distance as 1/r2 . The two effects exactly cancel, with the result that there is no gravitational force anywhere inside a hollow sphere. This result, which is a consequence of Isaac Newton's inverse square law of gravitation, has been well known since Newton's day.

    It has some fascinating consequences. For example, suppose we use David Innes' mole machine to bore a Manhattan-to-Sydney subway tunnel straight through the center of the Earth. What would be the pull of gravity inside such a tunnel? For example, take a point ten miles below the surface. Here there will still be a strong but diminished gravitational pull toward the center. You can think of the Earth's mass as being divided into a ten mile thick hollow spherical shell (which exerts no pull), plus a sphere exactly filling the cavity, which exerts a diminished pull because it has less mass than the whole Earth. If you work out this force in detail you will discover that, if the Earth had uniform density all the way through, the force depends directly on the distance r from the center of the Earth. The pull of gravity behaves precisely like the pull of a spring: F=kr where k is a force constant. In physics this is called a Hooke's law force after Sir James Hooke, a contemporary of Newton who demonstrated that springs have this property.

    If a subway car is put into the Manhattan-to-Sydney subway tunnel, the car behaves like the bob of a pendulum. It bobs back and fourth between the two extreme points of its travel at a rate which is independent of the mass of the subway car and depends only on the mass and diameter of the Earth. Assuming that the Earth has a uniform mass density, that there is no friction or air resistance in the tunnel, and that the subway car is moving solely under the influence of gravity with no other forces, the time required for a one-way Manhattan-to-Sydney trip in the subway car is an amazing 43 minutes. (Flying non-stop from JFK Airport to Sydney by 747 at 500 mph would require about 25 hours.)

    Sydney and New York City lie almost along a diameter through the center of the Earth. What about other locations, say Seattle to Moscow, Chicago to Houston, or Tokyo to London. It turns out that under the above assumptions the result is always the same. The transit time from anywhere to anywhere is 43 minutes. Such a subway system would, with no net expenditure of fuel or energy, would provide 43 minute transport between any two point on the Earth's surface.

    Actually, the real situation isn't quite this simple because the Earth is not a sphere of uniform density. If one uses a more realistic model that puts more of the planet's mass near its center because of the greater compressional forces and the higher density there, the Manhattan-to-Sydney transit time is reduced to 38 minutes, while trips that only cut through a small chord of a circle through the Earth's center would require somewhat more than 43 minutes.

    For other bodies of the solar system, planets, satellites, and even the Sun itself, a one way fall-through time along a diameter remains within a factor of two of the same 43 minutes. This is because that time is inversely proportional to the square root of the density (mass per unit volume) of the body involved, and is otherwise independent of both the mass and radius of the body. A fall-through subway on the Moon, which has an average density that is 61% of the Earth's , would have a transit time of 55 minutes. The Sun, with an average density that is only 26% of Earth's, would have a fall-through time of 85 minutes. And so on.

    Because of the obvious technical impossibility of actually putting a subway tunnel through the center of the Earth (not to mention the Sun), doesn't this discussion take almost as many liberties with the geophysics of our planet and the strengths of available construction materials as did the science fantasies of Burroughs himself? The answer to this question would certainly be yes , except that contemporary physics suggests that certain hypothetical particles, the so-called "dark matter candidates", may be riding on just such a "subway", crossing the Earth's diameter every 38 minutes or so as they orbit back and forth past the center of the Earth, trapped in Earth's gravity well but unable to interact non-gravitationally with the matter that produced that well.

    Broadly speaking, these hypothetical invisible particles fall into three classes: axions, WIMPs, and shadow matter. In two previous Alternate View columns dealing with aspects of the dark matter problem I've written about the hypothetical axion remnants of the Big Bang ["The Dark Side of the Force of Gravity", Analog, Feb-85] and about the similarly hypothetical weakly interacting massive particles which may be cooling the interior of the Sun and suppressing neutrino emission ["Neutrinos and WIMPs", Analog, May-86]. I therefore won't discuss these further, except to say that the experimental search for both goes on at many laboratories, but as yet there is no evidence for the existence of either.

    Let me then turn to shadow matter, which is more hypothetical than the other two classes of invisible particles, but which is also of more potential interest to readers and writers of science fiction. Shadow matter is predicted by some variants of superstring theories. In the past few years an army of theoretical physicists led by Ed Whitten of Princeton's Institute of Advanced Studies has attached increasing significance to the fact that, when the mathematical points in space itself are considered to be extra-dimensional strings, ultra-microscopic loops that close back on themselves in six or more extra dimensions, an array of forces and particles is generated that bear a striking resemblance to those of our universe. The extra dimensions invoked by these theories would have thus far escaped our notice because, unlike ordinary space-time, they are exceedingly small [See my AV column "The Other 40 Dimensions", Analog, Apr-85]. They are rolled up on themselves (or "compactified") snail-like, into loops so small as to make an atomic nucleus seem inconceivably vast in comparison.

    One of the variants of superstring theory goes by the name E6 × E8 (pronounced E-six-cross-E-eight). A consequence of this theory is that it describes two sets of particles and forces: the normal forces (strong, electromagnetic, weak) and particles (photons, electrons, neutrinos, quarks, ...) and a set of shadow-forces and shadow-particles that share only gravity in common with the normal world. Thus, our universe could, without our knowledge, be superimposed on another "shadow" universe which has its own light and matter and even stars and planets and animal life which do not interact with ours except through their common gravitational attraction.

    If one could indeed convert a quantity of normal matter, say a subway-car-size vehicle full of passengers, into shadow matter (and if the vicinity of the Earth was empty of shadow matter otherwise) we would have our through-the-Earth subway without need of David Innes' mole machine. The vehicle would fall through the Earth's gravity well, emerging at a diametrically opposite point on the planet 38 minutes later. There it might, as it entered the subway station, be converted back to normal matter.

    Getting into space would be much easier and cheaper with this technology also. No elaborate launch facilities would be needed. A space vehicle could be converted to shadow matter inside a building, and it would then plunge directly toward the center of the Earth. There it could use its rocket fuel far more efficiently because fuel burned deep in a gravity well yields much more net thrust than the same fuel burned in gravity-free space. Moreover, the Earth itself could be made to serve as a fuel source, for the ultra-compressed matter of the Earth's inner core, if it suddenly found itself in the vacuum of empty space, would explode with the volcanic force of a miniature Mt. Saint Helens.

    There is not space here to develop the implications of the existence of shadow matter and the possibility of its conversion to normal matter as fully as they have been developed in my first SF novel, Twistor, which is being published in hardcover by the William Morrow & Co. (March, 1989). Twistor deals with the (fictional) discovery that certain peculiar rotating electromagnetic fields (called "twistor fields" in the novel) change shadow matter into normal matter and vice versa. The notion, while fictional, has a certain plausibility because the shadow and normal particles predicted by E6 × E8 should have identical masses and properties, so that the exchange of one kind of matter for the other might be done at little or no energy cost.

    Is there really such a thing as shadow matter? I have no way of knowing at the moment, and neither does anyone else. Something gives galactic clusters many times more mass than can be explained with present-day physics. The extra mass or "dark matter" may come from shadow matter, or from WIMPs, or from axions, or from something else that no one has yet even thought of. Let me simply say that, from my point of view, one of the most remote and esoteric areas of theoretical physics has predicted something that is the science fiction writer's dream, a doorway to parallel "shadow" worlds, a method of nearly free worldwide transportation, and a cheap route into space.

    It's actually more fun than Pellucidar!

John G. Cramer's 2016 nonfiction book (Amazon gives it 5 stars) describing his transactional interpretation of quantum mechanics, The Quantum Handshake - Entanglement, Nonlocality, and Transactions, (Springer, January-2016) is available online as a hardcover or eBook at: or

SF Novels by John Cramer: Printed editions of John's hard SF novels Twistor and Einstein's Bridge are available from Amazon at and His new novel, Fermi's Question may be coming soon.

Alternate View Columns Online: Electronic reprints of 212 or more "The Alternate View" columns by John G. Cramer published in Analog between 1984 and the present are currently available online at: .

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This page was created by John G. Cramer on 7/12/96.