"The Alternate View" columns of 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

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 =
Gm _{1}m_{2}/r^{2}**. This equation says that an
object with mass

Finally, there's the mass-energy, as embodied in Einstein's famous equation
**E=mc ^{2}**, where

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**_{+}**c ^{2}**. We can try the same trick with a
negative mass

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

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.

**Reference:**

Hermann Bondi, "Negative Mass in General Relativity",

Reviews of Modern Physics29, 423 (1957).

** SF
Novels by John Cramer: **
my two hard SF novels,

**AV
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*This page was created by John G. Cramer
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