San José State University

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Thayer Watkins
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 A Model of Negative Mass

Physicists have from time to time toyed with the concept of negative mass, but nothing of significance has come of such speculations. However there does exist a physical model of negative mass. That is the phenomena of bubbles in a fluid.

The defining characteristic for an object to have negative mass is that when a force is applied to it, it accelerates in the opposite direction from the force. For bubbles in a fluid in a gravitational field the apparent effect of the gravity is to cause the bubbles to rise at an accelerating rate. Likewise when a mixture of fluid and bubbles is spun in a centrifuge the bubbles migrate to the center of rotation as if they were subject to a centripetal force; i.e., as if they had negative mass.

## The Apparent Mass of a Bubble

According to Archimedes' Principle a submerged body experiences an upward force equal to the weight of the fluid it displaces. Let ρ0 and ρ1 be the mass densities of the bubble gas and the fluid, respectively. Let g be the acceleration due to gravity and V be the volume of a bubble.

The weight of the fluid which the bubble displaces is Vρ1g and that of the bubble itself is Vρ0g so the force on the bubble is

#### −Vρ1g+Vρ0g = −V(ρ1−ρ0)g

The apparent mass M of the bubble is then

#### M = −V(ρ1−ρ0)

The volume of the bubble depends upon its pressure and that depends upon its depth of submergence. Let z be the vertical coordinate with z=0 being the top of the fluid and z=−D being the bottom. The pressure-depth dependence is then given by

#### p(z) = p0 − ρ1gz

where p0 is the pressure at the top of the fluid.

For an ideal gas the mass density of the bubble gas is given by

#### ρ0(z) = p(z)/(RT)

where T is the absolute temperature and R is the gas constant. To keep things simple T is presumed to be the same at all depths.

The mass density of the fluid would also vary with pressure and temperature but the variation is so small it can be neglected.

The real mass of the bubble mr is constant and is equal to Vρ0. Thus

#### V(z) = mr/ρ0(z)

and the apparent mass of the bubble is

## An Accounting of the Energy Changes Resulting From the Rise of a Bubble

The energy accounting can be carried out from two different perspectives. The first takes into account the fluid and the material of the bubble. The second treates the bubble as being an object of negative mass and ignores the fluid. The two perspectives are equivalent.

When a bubble rises from the bottom of the fluid to the top what is really happening is a volume of fluid from the top is flowing to the bottom taking the place of the bubble. This fluid mass experiences a loss of potential energy equal to V(−D)ρ1gD and the bubble mass experiences a gain in potential energy of mrgD. The net change in the potential energy of the system is then

#### mrgD − V(−D)ρ1gD = [1 − ρ1/ρ0(−D)]mrg

The system also experiences a gain in kinetic energy in form of the mass of the bubble traveling upwards and the mass of the fluid flowing downward to repace the bubble volume. The change preserves energy. The mass of the bubble gains both kinetic energy and potential energy. The downward flowing fluid loses potential energy and gains kinetic energy but not as much as its loss in potential energy. The deficit covers the energy gain of the mass of the bubble.

From the perspective of the bubble being an object of negative mass and the fluid being ignored the bubble loses potential energy as it rises to the top. It thus would have to gain kinetic energy in order for energy to be preserved. Suppose the bubble has, as it approaches the top of the fluid, a velocity v. This means its kinetic energy would have to computed from the formula K=½|M|v²

The volume of the bubble increases as it rises to the top. Therefore it displaces more fluid and thus its mass becomes more negative. This means that the bubble algebraically experiences a loss of mass.

The loss in apparent mass of the bubble as it rises to the top is

#### ΔM = −ρ1mrg[1/ρ0(0) − 1/ρ0(−D)]

Thus it loses potential energy and mass in its rise to the top of the fluid. If the bubble is captured and returned to the bottom of the fluid it would gain in potential energy and in mass.

## Significance of the Model

On a subatomic level there is the phenomenon of the mass deficit of nuclei. The mass of a nucleus is less than the masses of the nucleons which make it up. For example, when a proton and neutron form a deuteron the energy equivalence of the deuteron mass is 2.22 MeV less than the energy equivalences of the proton and neutron. Thus the proton and neutron experience a loss of potential energy as they come together. Presumably the loss of potential energy would be matched by a gain in kinetic energy for the two nucleons. However, the loss of one form of energy, potential energy, is accompanied by a loss of another form of energy, mass-energy. It appears that the loss of potential energy is exactly matched by the loss in mass-energy. This is amazing and perplexing. The fluid bubble model presented above is of interest because it also has losses or gains in potential energy matched with corresponding losses or gains in apparent mass. Model also indicates that in the case of negative mass the kinetic energy is based upon the absolute value of mass rather than its algebraic value.

The kinetic energy is just one term in the expansion of the mass-energy of a particle given by

#### E = m0c²/[1−β²]½

where m0 is the rest mass of the particle and β is v/c, the ratio of velocity to the speed of light. If the kinetic energy of a particle of negative mass is ½|m0|v² then the mass energy formula should be

## Mass and Charge

When a particle of charge q is injected into a magnet field of intensity B with a velocity of v it experiences a Lorentz force F given by

#### F = qv×B

where the redness of a symbol indicates that it represents a vector and × represents the vector product. The magnitude of v×B is equal to the product of the magnitudes of velocity and the magnetic field intensity times the sine of the angle between them.

The centrifugal force on a particle of mass m traveling with a velocity v is equal to mv²/r where r is the radius of curvature of the trajectory of the particle. Thus a particle injected into a magnetic field with a velocity perpendicular to the direction of the magnetic field will travel in a circular orbit of a radius given by

#### mv²/r = qvB or, explicitly r = mv/(qB)

where the direction of the curvature is, for positive mass, determined by the sign of the charge. If the mass is negative then the dirction of the curvature is the same as for a particle with positive mass but with the opposite charge. In general the transformation of m to −m is equivalent to the transformation of q to −q.

Consider a particle and its antiparticle, say an electron and a positron. From the usual accounting one would expect that the merging of the two particles would produce a neutral particle with a mass equal to the sum of their masses. Instead what results is gamma rays with zero rest mass and zero charge. This is consistent with the mass of the two particles being of opposite sign. The masses being of opposite sign would give the appearance of the charges being of opposite sign. The annihilation of the masses would also annihilate the charges because while a particle can have mass and zero charge there is no instance of a particle with charge but zero mass.

If a particle and antiparticle have masses of the opposite sign then in their annihilation mass is conserved as well as energy.

The motion of a charged particle of negative mass subject to the forces due to other charged particles is the same as the motion of a particle of positive mass but the opposite charge.

## Inertial Mass and Gravitational Mass

One of the great mysteries of the world is that the inertial mass of an object and its gravitational mass are equal. This has an important implication. The law for the gravitational attraction of a body of mass M for a body of mass m is given by

#### F = −GmM/r² = ma

where r is the separation distance of the bodies and a is the acceleration of the body of mass m. Division of both sides of the equation by m gives the acceleration a of the particle as

#### a = F/m = −GM/r²

This says that the acceleration of a body subject to gravitational forces is independent of its mass. This is a remarkable principle. If the inertial mass and gravitational mass are always equal then the motion of a particle of negative mass subject to the gravitational attraction of other particles would be the same as that of a particle of positive mass. Since the bubble model indicates that the motion of a negative mass object is in the opposite direction from that of a positive mass object in a gravitational field it must be that the gravitational mgrav is the absolute value of the inertial mass. Thus the law of gravitation should be

## Neutral Particles

For a charged particle negative mass would mean the charge is the opposite sign of the particle with positive mass. For a neutral particle negative mass would make no difference in its overall charge. However the neutron while being electrically neutral does have a charge distribution. This charge distribution gives it a magnetic moment which has been measured as −1.91μN, where μN is the unit called the nuclear magnetron. A neutron with negative mass would appear to have the opposite charge distibution from the neutron and hence have a magnetic moment of +1.91μN. This is the actual case for the antiparticle of the neutron, the antineutron which was identified in 1956. When a neutron and antineutron merge they annihilate each other as would be expected of particles with masses of the opposite sign.

A definitive test of the notion of charge being evidence of positive or negative mass is in the interaction (merger) of particle and the antiparticle of another type of particle. Suppose a neutron interacts with a positron, the antiparticle of the electron. The positron being of negative mass should produce a particle of lesser mass when it merges with a neutron. And of course that is just what happens since the product of the interaction of a neutron with a positron is a proton, a particle of lesser mass.

(To be continued.)