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Anti-Gravity and Anti-Mass

by John G. Cramer

Alternate View Column AV-14
Keywords: negative mass, antigravity, antimatter, gravitational repulsion space drive
Published in the July-1986 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 12/13/85 and is copyrighted © 1985, John G. Cramer. All rights reserved.
No part may be reproduced in any form without the explicit permission of the author.


    One of the great and persistent technological dreams of science fiction has been the invention which would nullify or reverse the force of gravity. H. G. Wells in The First Men in the Moon did it in 1901 with Cavorite, a substance that shields objects behind it from gravitational lines of force. James Blish in the Cities in Flight series used the Spindizzy, a device that converts rotation and magnetism into gravity fields and forces. And, of course, "floaters", "null-g speeders" and "grav sleds" have abounded as techno-props in science fiction stories for many years.

    And yet the control of gravity is no closer today than it was in Wells' time. If anything, as we have come to understand more about gravity the problem looks more difficult. It is now clear that gravitational attraction arises from the distortion and curvature of space itself, and that truly enormous amounts of mass-energy must be present to produce or change this curvature. Nature seems to have conspired to keep us stuck to the surface of this planet. We are down at the bottom of a deep gravity well, with only large and expensive rockets to pull us out.

    But perhaps there is a loophole in the equations of Newton and Einstein. This AV column concerns the idea of negative mass, one such potential loophole. This idea too has been anticipated in science fiction. In Gene Wolfe's The Citadel of the Autarch, the fourth volume of the "Book of the New Sun" trilogy, the Autarch transports the wounded Severian in a "flier". He explains that it obtains its lift from a boulder-size chunk of antimatter iron which is repelled by gravity rather than attracted. The question which we will consider here is whether antimatter, or more generally negative mass, could provide negative gravity and gravitational repulsion in this way.

    To begin this discussion we'll have to consider some elementary Newtonian physics. I beg the reader's indulgence in the use of a bit of math; it seems the only way to make certain points. The characteristic of matter which we call mass is related to three quite different phenomena: inertia, gravity, and energy. We must therefore distinguish inertial mass from gravitational mass and both from mass-energy. Inertial mass is Nature's way of telling you to stay where you are. It's the tendency of matter to resist acceleration or changes in speed. The equation of Newton's second law, F=ma, is a mathematical representation of this. The more massive (m) an object is, the more force (F) we have to use in pushing on it to make a change (a) in its speed.

    Gravitational mass actually has two aspects: a mass experiences a force in the presence of a gravity field, and it also produces a gravity field. Both of these functions are represented in Newton's law of gravitation: F = Gm1m2/r2. This equation says that an object with mass m1 experiences a force F when it is a distance r away from another object with mass m2. So m2 makes the gravity field and m1 experiences the force produced by it. Here G is Newton's gravitational constant (6.67 × 10-11 N-m2/kg2). Actually, one purpose of G is to fix up things so we can use the same mass values in both inertial and gravitational contexts. And inertial and gravitational mass are always found to be equivalent. That's the basis for the Equivalence Principle, one of the basic assumptions of Einstein's General Relativity theory. The Equivalence Principle been tested very carefully in the Eötvös-Dicke experiment. The interested reader might want to consult the novel Newton and the Quasi-Apple by our own Stanley Schmidt to see what happens when someone tinkers with the Equivalence Principle.

    Finally, there's the mass-energy, as embodied in Einstein's famous equation E=mc2, where E represents energy and c is the velocity of light. In normal MKS units c is a very big number (3 × 108 m/s). The equation says that you get an awful lot of energy when you convert mass into energy. That conversion could be done by lowering a mass m into a black hole on a super-strong and massless rope which turns an electrical generator as the mass was lowered. The amount of energy you could get from the generator by the time the mass m reached the event horizon of the black hole is mc2. A single kilogram (2.2 pounds) of mass could be used in this way to produce 25 billion kilowatt-hours of electrical energy. At current power rates, that's about a billion dollars worth of energy.

    So what happens if we assume that we can have objects with negative mass? Let's start with the effects of a negative inertial mass. If we let the m in Newton's second law have a negative value then the object will behave in a way we can only describe as backwards or perverse. It will be accelerated in a direction opposite the direction of the applied force. If we push it north, it will accelerate south. I know people who behave with that sort of contrariness, but it certainly isn't a kind of behavior that can be observed in ordinary objects.

    Negative mass in Newtonian gravitation has two implications. A negative mass in a gravitational field would experience a force in the opposite direction from the force which a normal mass experiences in the same field. It would also produce a negative gravitational field which would repel normal masses. A negative value of the mass-energy would mean that we could gain energy by creating the object, but it would cost us energy to get rid of it.

    But Newton's theory of gravity can't really be used as a reliable guide to the effects of negative mass, because we know that it is only an approximation to the best gravity theory we have, Einstein's general theory of relativity. Fortunately for this discussion general relativity was used in the late 1950's by the British physicist Sir Hermann Bondi to investigate the effects of negative mass. Bondi pointed out that when general relativity is considered purely as a theory of gravity, mass never actually appears. It first appears when the equations are solved in a way devised by the German physicist K. Schwartzschild. Then mass appears as a constant of integration. Bondi noticed that this mass constant could be made either positive or negative. He was able to show that when m is made negative, both the inertial and the gravitational mass effects are reversed. The results of Bondi's calculations can be summarized in a few words: a positive mass attracts all nearby masses whether positive or negative; an negative mass repels all nearby masses whether positive or negative.

    It is not hard to interpret Bondi's result using Newtonian gravity. Consider first a small negative mass m- in the field of a nearby large positive mass M+. Because m- has negative gravitational mass, the gravitational force on it is reversed and pushes away from M+. But because m- also has negative inertial mass, it responds to this force perversely, so that it is accelerated toward rather than away from M+. The double change in sign (gravitational and inertial) results in no change on observed effect and attraction remains attraction. Now consider a small positive mass m+ in the field of a nearby large negative mass M-. In this situation, the gravitational field of M- is repulsive, as Bondi has calculated, and m+ is pushed away from M-. If we substitute a small negative mass m- for m+, the result is the same because of the reversal of both gravitational and inertial mass, as described above. So M- repels all masses, positive or negative.

    There is a curious corollary of this result, which Bondi pointed out in his paper. Consider a pair of equal and opposite positive and a negative mass placed close to each other. The negative mass is attracted to the positive mass, while the positive mass is repelled by the negative mass. Thus the two masses will experience equal forces and accelerations in the same direction (in violation of Newton's third law) and the system of two particles will accelerate, seemingly without limit. The negative mass will chase the positive mass with constant acceleration.

    What about the mass-energy of a negative mass like m-? Bondi doesn't deal directly with this point, but the answer is implied by his calculations. It was mentioned above that if a positive mass m+ were lowered into a black hole on a strong massless rope that turned a generator, the energy from the generator by the time the mass reached the event horizon of the black hole would be m+c2. We can try the same trick with a negative mass m- and use this to measure it's mass-energy. Since m- is also attracted to the black hole, it would appear that it should have the same mass-energy as m+. But the problem comes in attaching the rope to an object with negative inertial mass. If we want to slowly lower m- into the black hole we have to support it by pushing it down, not pulling it up. Therefore, we have to do work against gravity to slowly lower m- into the black hole, and so its mass-energy is -|m-|c2. In other words, a negative mass will have negative mass-energy. It costs energy to get rid of it. The net mass-energy of equal positive and negative masses will be zero.

    Another question that we can now answer is whether, as Gene Wolfe's Autarch has implied in The Citadel of the Autarch, antimatter has negative mass. It does not. We know this from recent experiments with antiprotons at the LEAR facility at the CERN laboratory in Geneva in which antiprotons are scattered from normal matter protons at low energies. Antiprotons have a negative electrical charge, or at least they appear to. But if they had negative inertial mass of the type Bondi considered, they would really have positive electrical charges but would appear to be negatively charged because they respond perversely to every electrical force applied to them. In analogy with the case of gravity, the anti-proton would be attracted to a nearby proton, but the positively charged proton would be repelled by the positive charge of the antiproton. The two particles would both accelerate in the same direction, with the anti-proton chasing the proton. No such bizarre behavior is observed in the LEAR experiments, which can be taken as experimental evidence that antiprotons have positive mass. The Autarch's flier would never get off the ground (but might, if the antimatter slipped, make a big hole in it).

    The idea that negative mass can be made to chase positive mass (or vice versa), producing uncancelled forces and free acceleration, sounds as if it has the makings of a space drive. However, the problem mentioned above of attaching the rope to m- applies here too. When we try to hitch up the negative mass to the floor of our space ship to make use of this free acceleration, its negative inertial mass produces a force in the opposite direction from that from the positive mass. The forces on the ship are equal and opposite, just as Newton said, and the space ship doesn't go anywhere.

    The conclusion that we can draw from all of this is that Einstein's general theory of relativity does seem to have a loophole which would allow for the possibility of negative gravity from an object with a negative mass. But that kind of negative gravity doesn't appear to be very useful for flying around. If we wanted to use gravitational repulsion to float away from the Earth, we would have to make the Earth's mass negative, not a mass on our "grav sled". Close, folks, but no cigar.


Hermann Bondi, "Negative Mass in General Relativity", Reviews of Modern Physics 29, 423 (1957).

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