Specific Examples of Negative Mass

         We use the term "mass" to refer to the conserved quantity, and never use it for the rest mass of a particle, since rest mass is not conserved. Mass has gravitational properties and inertial properties. We see no reason for questioning the equivalence of gravitational and inertial mass. Energy is viewed as comprising mass in any volatile form.

         Vector quantities, which have both direction and magnitude, are denoted by bold face characters. When the symbol for a vector quantity is not in boldface, it denotes the magnitude of the vector quantity. Asterisks are sometimes used to denote multiplication. Weight is considered to be a real force, as is centrifugal force and the Coriolis force. In this, we rely on the relativistic principle that we may view the universe from any coordinate frame we choose.

         We do not, however, disparage purists who prefer to consider geodesics in Minkowsky space, rather than considering gravitational forces as contributing to the acceleration of a mass. Our attempt is to stay as close as possible to everyday experience. SI units are employed. These include the meter, second, kilogram, Joule, Newton, Volt, Ampere, Coulomb, etc.

         A local observer formulation is always employed. Distance between nearby points in space is measured by a local meter stick. The time interval between two nearby events is measured by a local clock. Mass at a point in space is measured by local instruments. Quantities extending over large regions are measured by a sum or integral of incremental values made by local observers, in the limit as the size of the increments approaches zero. This is consistent with the use of a metric tensor to define the properties of the space-time continuum since the incremental distances and times related by a metric tensor are measured by local instruments.

         Negative mass is not antimatter. When matter and antimatter contact each other, they release energy, and hence mass. For example, an electron contacting a positron causes emission of gamma rays, which have positive energy and, therefore, positive relativistic mass. (Gamma rays, like other photons, do not have rest mass, but, since they have energy, they do have relativistic mass.)

         Negative mass is not a shield against gravity. A region of space having negative mass acts as a source of the gravitational field vector which, at any point, is added to all the other gravitational fields at that point.

         A chunk of material having negative mass in a laboratory on Earth would have negative weight. However, if we released it, it would not fall toward the ceiling. Since its mass is negative, it would accelerate in the direction opposite to the direction of the force on it. Since the force on it is upward, it accelerates downward. All masses, whether positive or negative, tend to accelerate toward a positive mass such as Earth. All masses, whether positive or negative, tend to accelerate away from a negative mass.

         If I had a chunk of matter in my hand, which had negative mass, it would feel pretty strange. If I exert a northward force on it, it accelerates toward the south. If I exert an eastward force on it, it accelerates toward the west. However, lets not be put off by strange feelings. After all, a gyroscope also feels strange. If I have a spinning gyroscope with a vertical axis and try to tilt it toward the north, it tilts toward the east or west, depending on the direction of its rotation.

         A chunk of matter having negative mass would be quite difficult to control. It would have to be held in position by a restraint having a positive dF/dx. (Most springs have a negative dF/dx, that is, they push back against any displacement. A magnet near an iron wall experiences a positive dF/dx. A spring with a bent knee, like a chevron, can also have a positive dF/dx.) Also, the angular position of the chunk of matter would have to be controlled by springs with a positive derivative of torque with respect to angular displacement, just the opposite of the balance spring in a clock.

         Furthermore, to prevent vibratory motion of the negative mass, all possible vibratory velocities would have to be energized, rather than being damped as occurs naturally for positive masses. If, at an instant, it was moving north, then the system holding it would have to push north on it to reduce that velocity. If it was moving east, then the system would have to push east on it to reduce that velocity. This would also be true for angular vibrations. If it started to rotate to the right, as seen by an observer, then it would have to be torqued toward the right, as seen by that observer, to decrease the velocity of that rotation. Furthermore, if it could vibrate in a bending mode like a tuning fork, or vibrate in any other deformation mode, that mode would have to be energized to decrease the mode.

         Let us speculate on a particle having a negative mass interacting with an atom of ordinary matter. Could they annihilate, leaving nothing? We see many problems. Mass, momentum and angular momentum would have to be conserved, as seen by all observers. Electrical charge would have to be conserved. What about entropy? The entropy of both the negative mass particle and the ordinary atom would have to be destroyed. Hence, some mechanism would be needed which would increase entropy elsewhere as it annihilated the two particles, to avoid a decrease in entropy.

         If a particle having negative mass entered Earth's atmosphere and had any interaction with atoms of the atmosphere, it would energize those atoms. Hence, the energy of the negative mass particle would have to become more negative to have an energy balance. If the negative mass particle was traveling at less than the speed of light, its speed would increase as its energy went more negative. One can imagine it passing through the earth like a neutrino, increasing its velocity as it went, and emerging from the Earth with more than sufficient velocity to escape. Hence, we do not expect any fortunate prospector to chance upon a nugget of matter having negative mass.

         However, what about force fields? We take the view that the electric field, the magnetic field, the gravitational field, and the Coriolis field all are observable realities each having, at any point in space, as seen by a particular observer, a mass density, a momentum density, a stress tensor and other physical properties. Do any of these fields have a negative mass density?

         Clearly, the electric field does not. A charged capacitor stores positive energy and hence positive mass. Likewise, the magnetic field does not have a negative mass density. An inductor carrying a current stores positive energy and hence positive mass. But what about the gravitational field.... ?

         Our method of calculation is as follows: We set up a thought experiment involving a system comprising masses that can be positioned in two configurations. In an initial configuration of the system, there is no gravitational field in a particular region of space. In a final configuration of the system, there is a gravitational field in that region. We avoid making changes to gravitational fields in any other region. We then do an energy balance, considering the energy we obtained or expended when we moved the masses from the initial configuration to the final configuration.

         Of course, it is taken as understood that energy is a type of mass. A charged battery has more mass than the same battery when it is discharged. Likewise, a charged capacitor has more mass than the same capacitor uncharged. We may consider energy to be a volatile form of mass. By contrast, matter contains mass in a stable form. If, in our thought experiment, we find that positive energy is necessary to create a gravitational field in the particular region we selected, then we will conclude that the gravitational field has a positive mass density. However, if we find that energy is produced when the gravitational field is created, then we will conclude that the gravitational field has a negative mass density.

         We consider a spherical shell of positive charges having a lesser radius, R_L, which we permit to expand to a greater radius, R_G. We assume that the distribution of charges on the spherical shell is uniform, and we denote the total charge as Q.

         Figure G1 shows the configuration when the charges are at the lesser radius, R_L. Figure G2 shows the configuration when the charges are at the greater radius, R_G. In Figure G1, the heavy circle shows the charges at the lesser radius, R_L and the greater radius is shown in phantom. In Figure G2, the heavy circle shows the charges at the greater radius, R_G, and the lesser radius is shown in phantom. In both figures, the vector denoted R is the vector to any point in space, either inside the spheres, or outside.

         We now look for regions of space in which the electric field has been created or eliminated as a result of moving the charges from R_L to R_G. It is well understood that inside a uniformly charged sphere, the electric field due to the charged sphere is zero. Hence, for both the configuration shown in Figure G1 and the configuration shown in Figure G2, the electric field inside the smaller sphere having radius R_L is zero. Therefore, there is no change in the electric field inside of R_L

         It is also well understood that, anywhere outside of a uniformly charged sphere, at a radius R from the center of the sphere, the electric field is given by:

         The quantity ε₀ is the permittivity of free space, which equals 8.85 * 10^-12 in SI units. Hence, for both the configuration shown in Figure G1 and the configuration shown in Figure G2, the electric field outside the larger sphere having radius R_G has the value given by Equation G1. Therefore, there is no change in the electric field outside of R_G.

         There is, however, a change in the electric field in the region for which R is greater than R_L but less than R_G. In both Figure G1 and Figure G2, the three inequalities define the three regions, and the corresponding three equations give the electric field in the corresponding region. We see that expanding the charged sphere from R_L to R_G only affects the electric field in the region between the two spheres, where R_L < R < R_G.

         From this point, we will calculate the pressure on the charged sphere as it is expanded from R_L to R_G, and we will calculate the energy we obtain as we expand the sphere from R_L to R_G. To simplify the problem, we consider a case in which R_L and R_G are very nearly equal. We now denote the radius of the shell as R_S.

         The pressure equals the surface charge multiplied by the electric field just outside the shell divided by 2. (The factor of 2 is because the electric field seen by the charge is averaged through the thickness of the charged spherical shell, and it is zero inside the shell.)

         Just outside the shell, the electric field, E is:

         The surface charge is:

         The pressure on the spherical shell, which is outward, is, therefore:

         From simple mechanical considerations, we find that the energy density of the electric field equals the pressure of the field on the spherical shell and division by c² yields the mass density. (The speed of light is denoted c.) The mass density of the electric field, therefore, is:

         As the spherical shell expanded from R_L to R_G, we consumed electric field and obtained positive mass. Hence, as surprises no one, the electric field has positive mass.

         Figure 3 shows the configuration when the mass particles are disposed at the greater radius, R_G, and Figure G4 shows the configuration when the mass particles are disposed at the lesser radius, R_L. In Figure G3, the heavy circle shows the mass particles at the greater radius, R_G and the lesser radius is shown in phantom. In Figure G4, the heavy circle shows the mass particles at the lesser radius, R_L, and the greater radius is shown in phantom. In both figures, the vector denoted R is the vector to any point in space, either inside the spheres, or outside.

         We now look for regions of space in which the gravitational field has been created or eliminated as a result of moving the mass particles from R_G to R_L. It is well understood that inside a massive spherical shell having a uniform surface density, the gravitational field due to the spherical shell is zero. Hence, for both the configuration shown in Figure G1 and the configuration shown in Figure G2, the gravitational field inside the smaller sphere having radius R_L is zero. Therefore, there is no change in the gravitational field inside of R_L.

         It is also well understood that, anywhere outside of a spherical shell having a uniform surface mass density, and a total mass M, at a radius R from the center of the sphere, the gravitational field is given by:

         (Equation G2 resembles Newton's law of gravity, where G is the gravitational constant. In SI units, G = 6.672*10^-11. Equation G2 may be obtained from the law of gravity by integrating that law over all the mass particles in the spherical shell. It is noted that Equation G2 is an approximation because it ignores the mass density of the gravitational field itself. It is extremely accurate, however, for a system having much less mass or density than a black hole.) We will discuss this approximation later.

         Hence, for both the configuration shown in Figure G3 and the configuration shown in Figure G4, the gravitational field outside the larger sphere having radius R_G has the value given by Equation G2. Therefore, there is no change in the gravitational field outside of R_G.

         There is, however, a change in the gravitational field in the region for which R is greater than R_L but less than R_G. In both Figure G3 and Figure G4, the three inequalities define the three regions, and the corresponding three equations give the gravitational field in the corresponding region. We see that contracting the shell of mass particles from R_G to R_L only affects the gravitational field in the region between the two spheres, where R_L < R < R_G.

         From this point, we will calculate the pressure on the spherical shell as it is contracted from R_G to R_L, and we will calculate the energy we obtain as we contract the spherical shell from R_G to R_L. To simplify the problem, we consider a case in which R_G and R_L are very nearly equal. (Our calculation would exactly apply in the limit in which the thickness becomes vanishingly small compared to the radius of the shell.) We now denote the radius of the shell as R_S.

         The pressure equals the mass per unit area of the shell multiplied by the gravitational field just outside the shell divided by 2. (The factor of 2 is because the gravitational field seen by the mass particles is averaged through the thickness of the spherical shell of mass particles, and it is zero inside the shell.)

Just outside the shell, the gravitational field, g is:

         M is the total mass of the shell and G is the gravitational constant. The mass per unit area on the shell is:

         The pressure on the spherical shell due to the gravitational field, is, therefore:

         The pressure is inward, and the spherical shell moved inwardly from R_G to R_L. Hence, we obtained positive energy as we lowered the mass particles from R_G to R_L.

         Thus, we obtained energy as we created a gravitational field in the region for which R_L < R < R_G. Therefore, conservation of energy requires that the gravitational field have a negative energy density. The magnitude of the energy density equals the magnitude of the pressure on the spherical shell. Hence, the energy density of the gravitational field is:

         Thus, the mass density of the gravitational field is:

         At this point, it is of interest to calculate the divergence of the gravitational field vector due to its own mass density. To do this, we rewrite Newton's law of gravity in differential form as follows, where M_V is the mass density of space, g is the gravitational field vector and div is the divergence operator:

         Combining Equations G3 and G4, we obtain for the divergence of the gravitational field vector due to the mass density of the gravitational field itself the following:

         We note that the divergence of the gravitational field vector due to its own mass density is positive because its mass density is negative. (The Earth has a positive mass density and causes a negative divergence of the gravitational field vector.)

         At this point, we can look at the approximation we made previously in Equation G2, in which we ignored the mass density of the gravitational field itself. First of all, the mass density of the gravitational field itself is small compared to the mass that causes it, for masses which do not approximate black hole conditions. Second, only the mass of the gravitational field within the thickness of the shell of mass particles affects the gravitational field immediately outside the shell. Since our calculation is for the limit in which the thickness of the shell becomes vanishingly small compared to the radius of the shell, the mass of the gravitational field itself does not cause an error in our calculation of its mass density. Thus, our calculation of the mass density of the gravitational field should be valid, even in the vicinity of a black hole.

         Since in the vicinity of a black hole, the gravitational field becomes very great, we anticipate that there is a very high negative mass density surrounding a black hole.

         We do not intend here to disparage Schwarzshield and others who have performed analyses of the geometry of the space-time continuum near black holes. However, those analyses may produce results so esoteric that the negative mass density of the gravitational field is not obvious. In the present analysis, we wish to stay as close as possible to ordinary experience, using credible thought experiments, but to point out that we find the mass density of the gravitational field to be negative.

         Extrapolating the present analysis, we wonder: Do other attractive agents, for example, the fields or particles which mediate the strong nuclear force that binds the protons together in an atomic nucleus also have negative mass? Also, would repulsive agents such as that which causes the outward acceleration of distant galaxies, be better understood as comprising negative mass, rather than "dark energy"?

• A pendulum clock.
  
  
• A clock based on a mechanical simple harmonic oscillator comprising a spring and a massy object attached to the spring.
  
  
• A clock based on a pulse of light reflected back and forth between a pair of mirrors.
  
  

• The pendulum clock would slow down because the mass of the Earth and hence its gravitational field strength would be reduced.
  
  
• The clock based on a spring and attached massy object would speed up because of the reduced mass of the object.
  
  
• The clock based on the reflected light pulse would continue at the same rate.