DIRD Negative mass Propulsion

Negative Mass Propulsion

Defense Intelligence Reference Document, Defense Futures. DIA-08-1101-023, 3 January 2011 (ICOD: 30 August 2010).

Prepared by the Defense Intelligence Agency. This product is one in a series of advanced technology reports produced in FY 2010 under the Defense Intelligence Agency Advanced Aerospace Weapon System Applications (AAWSA) Program.

Copyright warning: further dissemination of the photographs in the original publication is not authorized.

Summary

It is easy to prove that there are negative masses all around us, albeit hidden behind positive masses. But their use for propulsion by reducing the inertia of matter, for example in the limit of macroscopic bodies with zero rest mass, depends on a technical solution to free them from their imprisonment by positive masses. It appears that there are basically two ways this might be achieved: 1. By the application of strong electromagnetic or gravitational fields or by high particle energies; 2. By searching for places in the universe where nature has already done this separation, and from which the negative masses can be mined.

The first of these two possibilities is for all practical means excluded, because if possible at all, it would depend on electromagnetic or gravitational fields with strengths beyond what is technically attainable, or on extremely large particle energies likewise not attainable.

With regard to the second possibility, it has been observed that non-baryonic cold dark matter tends to accumulate near the center of galaxies, or places in the universe which have a large gravitational potential well. Because of the equivalence principle of general relativity, the attraction towards the center of a gravitational potential well, produced by a positive mass, is for negative masses the same as for positive masses. Large amounts of negative masses might have over billions of years been trapped in these gravitational potential wells.

Now it just happens that the center of the moon is a potential well, not too deep that it cannot be reached by making a tunnel through the moon, not possible for the deeper potential well of the earth, where the temperature and pressure are too high. Making a tunnel through the moon, provided there is a good supply of negative mass, could revolutionize interstellar space flight. A sequence of thermonuclear shape charges would be required to make such a tunnel technically feasible.

1. Introduction

If we extend the law of gravity to negative masses, but hold onto the equivalence of inertial and gravitational masses, we have to distinguish between the following four cases, if a test particle is placed near a gravitational field producing mass (Table 1):

| Case | Gravitational field producing mass | Mass of test particle | Motion of test particle | |---|---|---|---| | 1 | positive | positive | attraction | | 2 | positive | negative | attraction | | 3 | negative | positive | repulsion | | 4 | negative | negative | repulsion |

Under the principle of equivalence if a negative test mass particle would be placed in the gravitational field of earth, it would not fall upwards, as happens in science-fiction antigravity machines. A test particle, regardless of whether it has positive or negative mass, would there always fall down. It would fall upwards only if placed in the field of a large negative mass.

A somewhat different situation arises if both masses, the field producing mass and the mass of the test particle, have the same absolute value but are permitted to have different signs. There we have to distinguish between the four cases shown in Figure 1, which sets a pair of like and unlike masses side by side and marks the resulting force as attraction, repulsion, or self-acceleration.

If both masses are positive, we have the usual Newtonian attraction. For negative masses, the force has the same magnitude but is repulsive. A quite different situation exists if one mass is positive and the other one is negative. With both forming a mass dipole, the system becomes self-accelerating, because one mass is repelled and the other one attracted. With the two masses having opposite sign, the total energy and momentum of the combined system remains zero for all times, leaving intact the conservation laws of energy and momentum. Under its self-acceleration, the mass dipole would eventually reach the velocity of light. It is this property of self-acceleration without expenditure of energy that has intrigued many researchers and raised the prospect of a propulsion system without limits. We remark that even without an appreciable gravitational interaction, a mass dipole with zero, or close to zero inertial mass, could be accelerated to very high velocities with negligible jet power and energy.

No matter how strange the properties associated with negative masses appear to be, there can be little doubt that they can be incorporated into Einstein’s gravitational field theory as long as they do not violate the principle of equivalence. In particular, the well known Schwarzschild solution for a positive mass M can be extended to a negative mass, simply by replacing M with minus M, where gamma is Newton’s constant.

One therefore has to raise the question if nature has not made use of negative masses somewhere. Over and over again we have found that what is possible, within the framework of the fundamental laws of physics, exists. Only one important physical set of laws, Einstein’s special theory of relativity, appears to forbid the existence of negative masses. This is because in a relativistic quantum field theory the particle number is not a conserved quantity, and the existence of negative masses would make all matter unstable against decay into negative masses.

Apart from Einstein’s purely kinematic interpretation of special relativity, being the expression of a Minkowskian space-time structure, there is an older alternative dynamic interpretation by Lorentz and Poincaré. In it space and time are absolute, but it can explain all relativistic effects as well. It assumes the existence of an aether, with all objects in absolute motion through the aether suffering a Lorentz contraction and time dilation. If this aether has a grainy structure, characterized by some smallest length — for example the Planck length, about 10⁻³³ cm — then according to Heisenberg’s uncertainty principle special relativity would ultimately break down at a high energy. If the length is very small, this energy can be so high as to be far beyond the capabilities of any existing particle accelerator or even beyond the high energy of cosmic ray particles, making both interpretations of special relativity experimentally indistinguishable at the energies presently available.

2. The Theory of Bondi

The first attempt to introduce negative masses into general relativity to describe a mass dipole was made by H. Bondi [1]. For a uniformly accelerating mass dipole Bondi uses the axially symmetric metric by Weyl and Levi-Civita [2], whose two potential functions satisfy a coupled pair of second-order equations; inserting that metric into Einstein’s nonlinear gravitational field equations yields four nonlinear partial differential equations for the components of the stress-energy tensor.

(The Weyl–Levi-Civita metric and the four field equations derived from it are omitted for length; the complete text is at the source.)

In solving these equations Bondi assumes that the potential is small, which then also implies that the second function is small to second order, reducing the solution of the problem to the linear Laplace equation of the scalar Newtonian potential in empty space. Making this assumption, Bondi can reproduce the uniform acceleration of the mass dipole, as it is expected from an elementary analysis. It is here that we must disagree with Bondi, because it can be shown that the nonlinearity of the gravitational field equation leads to a very different result. The nonlinearity also sheds light on why it is so difficult to separate negative from positive masses, whereby negative masses are all around us, but imprisoned by positive masses.

The author had the pleasure to meet Prof. Bondi on a common flight from Graz, Austria in 1993 — we both are members of an academy which had a meeting in that year in Graz — and ask him how his solution can be correct since it does not include the field of the positive gravitational field mass of a mass dipole. This problem will be analyzed in the next section, and its solution has far reaching consequences.

3. Hund’s Nonlinear Newtonian Theory of Gravity

As explained by Hund [3], already Newton’s theory of gravity, in conjunction with the postulates of special relativity, leads to a nonlinear theory of gravity. With this model theory of gravity the nonlinearity of the gravitational field can be much better explained than with Einstein’s theory.

Hund begins with the force acting on a mass in a merry-go-round: the mass times the gravitational acceleration, plus the mass times the velocity divided by c, crossed into a second field which plays the role of a Coriolis field.

If the gravitational acceleration is produced by real masses with density rho, one has in Newton’s theory that the divergence of the gravitational field equals minus four pi gamma rho. In the merry-go-round the field has a vertical component from the earth’s gravitational field, but also a radial component from the radial centrifugal acceleration, which is not source-free. With the centrifugal acceleration equal to omega squared times r, where omega is two pi over the period of revolution and r is the radial distance from the center, the divergence of the field is two omega squared.

Comparing these two expressions, one sees that the centrifugal force corresponds to a gravitational repulsion of a homogeneous mass density equal to minus omega squared divided by two pi gamma. For a typical merry-go-round one has a period of 10 seconds, hence omega of about 0.6 per second. For this example the mass density comes out as minus 10⁶ g/cm³, taken as an absolute value about equal to the mass density of a white dwarf.

That mass density is not fictitious but represents physical reality. According to Einstein’s E equals m c squared one obtains a mass density for an electric field. Replacing the permittivity of free space with minus one over gamma, one obtains the energy density of the gravitational field, which is minus g squared over eight pi gamma and is therefore negative. It possesses a correspondingly negative mass density.

Besides the gravitational field, we have on a merry-go-round the Coriolis force field, equal to two c times omega. By setting the Coriolis field equal to the gravitational field and inserting it into the mass density of the field, one recovers exactly the mass density found above. This means the centrifugal force is the gravitational force associated with the mass density of the Coriolis field.

But where is this huge negative mass coming from? The obvious answer is by the very large vacuum energy, making itself felt by going to an accelerated frame of reference.

In Mach’s principle the motion of the distant galaxies as seen in an accelerated frame of reference is responsible for the inertial forces. But this idea is wrong because if by some miracle the distant galaxies were to be set into motion, it would take a long time before their fields propagating with the velocity of light would reach the earth.

Adding the mass of the Coriolis field to the source side of the Newtonian field equation, and then moving it across, one obtains a modified Poisson equation in which the Laplacian of the Newtonian potential equals four pi gamma rho minus one half c squared times the square of the potential gradient. According to that equation the positive mass as the source of the Newtonian potential is reduced by the negative mass of its field. Einstein’s theory leads to almost the same, except that there the negative gravitational mass density is twice as large.

The earth is therefore embedded in a sea of negative mass. Making a substitution of the potential for a new variable transforms the nonlinear equation into a linear one. If the density is a delta function at the origin, one recovers the ordinary Newtonian potential of a point mass.

For a sphere of constant density the solution is the hyperbolic sine of k r divided by k r, where k squared equals four pi gamma rho over c squared. For small radii the potential reduces to two pi over three times gamma rho r squared, with the force per unit mass equal to four pi over three times gamma rho r, as in Newton’s theory. In general the field strength has a form which, in the limit of large r, becomes a constant — which implies the self-shielding of a large mass by the negative mass of its own gravitational field surrounding the mass. The shielding becomes important at the distance R equal to c divided by the square root of four pi gamma rho.

Inserting into that expression the average mass density of the universe, R becomes the radius of the universe. The negative gravitational mass of the universe there shields its positive mass.

4. The Theory of Bondi Revisited

We are now in a position to revisit the theory by Bondi. In his treatment of the positive-negative-mass dipole two body problem, he did not take into account the gravitational field of this configuration. The gravitational potential of a point mass remains the same as in Newton’s theory. This means that the gravitational potential interaction energy for two positive equal masses, which is minus gamma m squared over r, is changed for a mass dipole into plus gamma m squared over r.

In the theory of Bondi this positive field mass must be added to the positive mass, resulting in a mass pole-dipole, which is a mass pole with a superimposed mass dipole. As we will see in the next section this fact has very important consequences [4].

5. The Zitterbewegung Phenomenon as a Manifestation of Negative Masses

There is no fundamental physical principle standing in the way which forbids the existence of negative masses. If this is true, the question is: Where are these negative masses? The recently noticed large bubbles or voids observed in intergalactic space could possibly be explained by the repulsive force of negative masses assumed to occupy the voids, but alternative, less exotic, explanations have been offered as well. However, there is at least one fundamental phenomenon which strongly speaks for the existence of negative masses. It is the spin of the fermions, like the spin of the electron.

Fermions are described by Dirac’s relativistic wave equation. This equation has both positive and negative energy components and because of the mass-energy relation, it therefore must have negative mass components. According to Schrödinger [4], it is these negative mass components which lead to the phenomenon of the spin. Since the overall mass of the electron is positive, the occurrence of negative masses in the Dirac equation must mean that the electron is a mass pole with a superimposed mass dipole [5].

The spin is definitely not an intrinsic rotational motion of a finite size particle, as older models had suggested it to be. The original model by Uhlenbeck and Goudsmit, for example, cannot possibly be correct because it requires superluminal rotation velocities for an electron with the classical radius e squared over m c squared.

If we consider the linear motion of a mass dipole (Figure 2, which draws a positive and a negative mass displaced from one another and moving together), we immediately see that its translation generates angular momentum. Construction of a mass pole with a superimposed mass dipole can simply be done by choosing the positive mass slightly larger than the magnitude of its negative counterpart. For such a pole-dipole particle the center of mass lies outside the line connecting both masses (Figure 3, which shows that center of mass tracing a circle of radius r sub c).

Because it is self-accelerating the circular motion will eventually reach the velocity of light. The connection with Schrödinger’s analysis is reached if one puts in Planck’s constant and the electron mass: the radius becomes h-bar over two m c, whereby the circular motion around the center of mass produces just the angular momentum h-bar over two as in Dirac’s equation.

The velocity-of-light result was first obtained by Breit [6], according to which Dirac’s equation predicts a local electron velocity equal to the velocity of light. Even though its time-averaged velocity is always less than the velocity of light, this means that the electron, represented by the pole-dipole configuration, makes a circular luminal motion onto which a subluminal motion of the center of mass is superimposed. The trajectory of the resulting motion is a screw-line, but it is the motion of the center of mass only which one identifies with the time-averaged electron velocity. The result derived from this simple pole-dipole model is in beautiful agreement with Schrödinger’s analysis of the Dirac equation, in which the luminal rotational motion emerges as a fluctuation of the electron coordinate, called by Schrödinger Zitterbewegung, German for quivering motion.

(The detailed pole-dipole algebra — the definitions of the mass pole and mass dipole in terms of the positive and negative masses, the location of the center of mass, and the evaluation of the angular momentum in the limit where the circular velocity approaches c — is omitted for length; the complete text is at the source.)

Experimentally, the electron is indistinguishable from a point. This would make the separation of the two masses zero. In reality its size must be finite but in principle can be very small. This means that the positive mass, and the magnitude of the negative mass, are likely to be very much larger than the observed electron mass.

It has been conjectured by Hönl and Papapetrou [5] that the electron is a pole-dipole particle where the surplus positive energy comes from the positive gravitational interaction energy of a very large positive mass with a likewise very large negative mass of the same magnitude. According to this hypothesis one would have for the electron rest mass energy an expression in which the gravitational interaction energy of the pair supplies the residue. Combining that with the pole-dipole radius one can compute the bound mass. The result is [5] about 6 by 10⁻¹³ g, larger by a factor 3.6 by 10¹¹ times the mass of the proton.

We therefore see that there are huge amounts of negative masses bound to positive masses in Dirac spinors. It shows that it cannot be a simple matter to free the negative masses from the positive masses. And it explains why the masses of the elementary particles are so much smaller than the Planck mass, about 10⁻⁵ g.

6. Planck Aether Hypothesis

We make here the proposition that the fundamental group is SU2, and that by Planck’s conjecture the fundamental equations of physics contain as free parameters only the Planck length, of order 10⁻³³ cm, the Planck mass and the Planck time, built from Newton’s constant gamma, Planck’s constant h and the velocity of light c.

The assumption that SU2 is the fundamental group means that nature works like a computer with a binary number system. Since SU2 is isomorphic to SO3, the rotation group in three dimensions, this explains why natural space is three-dimensional.

The Planck aether conjecture is the assumption that the vacuum of space is densely filled with an equal number of positive and negative Planck mass particles, with each Planck length volume on the average occupied by one Planck mass, with the Planck mass particles interacting with each other by the Planck force over a Planck length, and with Planck mass particles of equal sign repelling and those of opposite sign attracting each other. The particular choice made for the sign of the Planck force is the only one that keeps the Planck aether stable. While Newton’s action-reaction remains valid for the interaction of equal Planck mass particles, it is violated for the interaction of a positive with a negative Planck mass particle, even though globally the total linear momentum of the Planck mass plasma is conserved, with the recoil absorbed by the Planck aether as a whole.

It is the local violation of Newton’s action-reaction which leads to quantum mechanics at the most fundamental level, as can be seen as follows. Under the Planck force — the Planck mass times c squared, divided by the Planck length — the velocity fluctuation of a Planck mass particle interacting with a Planck mass particle of opposite sign is equal to c, and hence yields a momentum fluctuation equal to the Planck mass times c. But since the position uncertainty is the Planck length, and because the Planck mass times the Planck length times c equals h, one obtains Heisenberg’s uncertainty relation for a Planck-mass particle.

Accordingly, the quantum fluctuations are explained by the interaction with hidden negative masses, with energy borrowed from the sea of hidden negative masses.

According to Newtonian mechanics and Planck’s conjecture, the interaction of a positive with a negative Planck-mass particle leads to a velocity fluctuation equal to c with a displacement of the particle equal to half the Planck length. Therefore, a Planck-mass particle immersed in the Planck aether makes a stochastic quivering motion — a Zitterbewegung — with a velocity fixed by the average number density of positive or negative Planck mass particles, which is one per Planck volume.

Writing the aether velocity as the gradient of a Hamilton action function, the Lagrange density for the Planck aether can be formed. Variation of that Lagrangian with regard to the action function leads to the continuity equation of the Planck aether. Variation with regard to the number density leads to a second equation, and with the Madelung transformation the two together yield the Schrödinger equation.

(The Lagrange density, the two variational equations and the Madelung transformation are omitted for length; the complete text is at the source.)

In the Planck aether hypothesis all particles, save and except the Planck mass particles, are quasi-particles of the Planck aether, like the phonons, rotons, excitons, and so on, of condensed matter physics, and by the wave structure of the Planck aether are Lorentz invariant. In forming quantized vortices, the Planck aether also has vortex waves, simulating Maxwell’s and Einstein’s electromagnetic and gravitational waves. Dirac spinors are made possible by the negative masses of the Planck aether.

Quantum mechanics predicts for each harmonic oscillator the zero-point energy one half h-bar omega, which has to be multiplied with the volume element in frequency space to obtain the zero-point energy spectrum — a spectrum proportional to the cube of the frequency.

Now that spectrum turns out to be just the only spectrum that is Lorentz invariant. But it is also the only one which does not lead to a friction force on a charged particle moving through an electromagnetic spectrum with this frequency dependence. This means that special relativity is a consequence of quantum mechanics, leading to the zero point vacuum energy, and can be interpreted by saying that the zero-point vacuum energy generates the Minkowski space-time.

7. Dynamic Interpretation of Lorentz Invariance

A cut-off at the Planck frequency generates a distinguished reference system in which the zero-point energy spectrum is isotropic and at rest. In this distinguished reference system, the scalar potential from which the forces are to be derived satisfies the inhomogeneous wave equation, with the sources being those of the body itself. For a body in static equilibrium at rest in the distinguished reference system, that reduces to a Poisson equation.

If the body is set into absolute motion with velocity v along the x-axis, the coordinates of the reference system at rest with the moving body are obtained by the Galilei transformation, which introduces mixed space-time derivative terms into the wave equation. After the body has settled into a new equilibrium in which nothing depends on the new time coordinate, comparison with the static case shows that the equation is the same if the body is uniformly contracted by the factor square root of one minus v squared over c squared, because the sources are contracted by the same factor.

Since the zero-point energy is invariant under a Lorentz transformation, the quantum potential changes in the same way as the scalar potential. The body therefore sustains its static equilibrium under that contraction if set into absolute motion, explaining the Lorentz contraction dynamically.

The clock retardation effect can be derived from the contraction effect, and from there the Lorentz transformation. Following Builder [9] this original interpretation of Lorentz invariance by Lorentz and Poincaré has been worked out in every detail by Prokhovnik [10]. To derive the clock retardation effect from the contraction effect one considers a light clock, which is a rod with mirrors attached to its two ends in between which a light signal is sent forth and back.

(The light-clock derivation — the inclination of the rod, the anisotropic to-and-fro light velocities, the resulting time dilation by the Lorentz factor independent of the rod’s inclination, and the synchronization argument showing that the one-way velocity of light cannot be measured — is omitted for length; the complete text is at the source.)

With solid bodies held together by electromagnetic forces, clocks made from solid matter should behave like light clocks. As it was claimed by Poincaré, it should for this reason be possible to obtain the Lorentz transformations solely from the contraction effect with a proper convention about the synchronization of clocks.

From an absolute point of view the propagation of light is isotropic only in the distinguished reference system, but anisotropic in a reference system in absolute motion against the distinguished reference system. This anisotropy remains hidden due to the impossibility to measure the one-way velocity of light. The impossibility is expressed in the Lorentz transformations themselves, containing the scalar c squared rather than the vector c, through which an anisotropic light propagation would have to be expressed.

8. Negative Mass Interpretation of the Aharonov-Bohm Effect

In Maxwell’s equations the electric and magnetic fields can be expressed through a scalar potential and a vector potential. The fields remain unchanged under the gauge transformation of the potentials, where the added function is called the gauge function. Imposing the Lorentz gauge condition on the potentials, the gauge function must satisfy the wave equation.

By making a gauge transformation of the Hamilton operator in the Schrödinger wave equation, the wave function transforms by a phase factor, leaving invariant the probability density.

To give gauge invariance a hydrodynamic interpretation, we compare the electromagnetic force on a charge with the force acting on a test body of mass m placed into the moving Planck aether. This force follows Euler’s equation. Complete analogy between the two is established if one sets the scalar potential equal to minus m over two e times the aether velocity squared, and the vector potential equal to minus m c over e times the aether velocity. The scalar and vector potentials then shift the phase of a Schrödinger wave by a definite amount.

The corresponding expressions for a gravitational field can be directly obtained from the equivalence principle [3]. If the acceleration and the angular velocity of the universe are taken relative to a reference system assumed to be at rest, the inertial forces in this system take the form of a Lorentz force, with an effective electric field built from the acceleration, the angular acceleration and the centrifugal term, and an effective magnetic field equal to minus two c times the angular velocity. These effective fields satisfy the same source-free equations as their electromagnetic counterparts and can be derived from a scalar and a vector potential.

Applied to a rotating reference system, the effective scalar potential is minus one half the square of the cross product of angular velocity and position, and the effective vector potential is minus c times that cross product — apart from the factor m over e, the same as in the hydrodynamic case.

For weak gravitational fields produced by slowly moving matter, Einstein’s linearized gravitational field equations permit a gauge condition replacing the Lorentz gauge. For a stationary gravitational field the vector potential changes the phase of the Schrödinger wave function, leading to a phase shift on a closed path equal to the mass over h-bar c times the line integral of the vector potential.

In the hydrodynamic interpretation suggested by the Planck aether hypothesis, the phase shifts caused by either the magnetic or the gravitational vector potential result from a circular flow of the Planck aether. The principle of equivalence can precisely relate this circular flow to the angular velocity of a rotating platform. Applying the phase-shift formula to the Sagnac effect for photons of frequency nu, by putting m c squared equal to h nu, gives the phase shift four pi omega times the enclosed area divided by c squared — the same as predicted by quantum mechanics.

We now compute the phase shift by a magnetic vector potential. To make a comparison with the gravitational vector potential in the Sagnac effect, we consider the magnetic field produced by an infinitely long cylindrical solenoid of radius R. Inside the solenoid the field is constant, vanishing outside. If the magnetic field inside the solenoid is H, the vector potential is one half H r inside, and one half H R squared over r outside.

The vector potential on a closed path leads to a phase shift proportional to e H over h-bar c times the enclosed flux. As noted by Aharonov and Bohm [11], there is a phase shift outside the solenoid, even though the magnetic field there is zero, because the curl of the vector potential vanishes there.

Expressing the vector potential through the hypothetical circular aether velocity, one finds that inside the coil the velocity profile is the same as in a rotating frame of reference, having outside the coil the form of a potential vortex. If expressed in terms of the aether velocity the phase shift becomes the same as for the vector potential created by a gravitational field, and hence the same as in the Sagnac experiment and the neutron interference experiment.

But for the magnetic vector potential the aether velocity can easily become much larger than in any rotating platform experiment. The velocity reaches a maximum at the coil radius. For electrons this maximum velocity divided by c is 3 by 10⁻⁴ times H times R, where H is measured in gauss. For a field of 10⁴ gauss, this would mean that the maximum velocity approaches c for a radius greater than 0.3 cm. If this were the same aether velocity felt on a rotating platform, it would lead to an enormous centrifugal and Coriolis field inside the coil, obviously not observed.

The Planck aether model can give a simple explanation for this paradox. The Planck aether consists of two superfluid components, one composed of positive Planck masses and the other one of negative Planck masses. The two components can freely flow through each other, making possible two configurations, one where both components are co-rotating and one where they are counter-rotating. The co-rotating configuration is realized on a rotating platform, where it leads to the Sagnac and neutron interference effects. This suggests that in the presence of the magnetic vector potential the two superfluid components are counter-rotating. Outside the coil, where the curl of the vector potential vanishes, the magnetic energy density vanishes, implying that the magnitudes of both velocities are exactly the same. Inside the coil, where the curl is non-zero, there must be a small imbalance in the velocity of the positive over the negative Planck masses to result in a positive energy density.

9. Negative Masses in Cosmology

In the Planck aether theory, the negative gravitational field energy surrounding a mass is due to an excess of negative over positive mass, and the negative mass surrounding a highly collapsed spherical body or black hole is of the same order of magnitude as the positive mass accumulated inside the collapsed body. An assembly of interacting positive and negative masses can, in general, not lead to a thermodynamic equilibrium. If all the positive masses are separated from the negative ones, it is sufficient to require that each mass species reaches thermodynamic equilibrium. It was shown by Vysin [12] that an assembly of negative masses can acquire thermodynamic equilibrium provided the temperature is negative. The kinetic energy of a negative Planck mass is negative, and an assembly of negative Planck masses has, for this reason, a negative temperature. It therefore can reach thermodynamic equilibrium.

We now make the following hypothesis: if an assembly of positive and negative masses, with the total energy equal to zero, is brought together, the temperature and hence entropy of the mixture will go to zero.

This hypothesis is the only one consistent with Nernst’s theorem. It, of course, implies that an assembly of positive and negative masses can perfectly mix because otherwise no equilibrium can be reached. To satisfy this hypothesis, we assume that the negative masses have negative entropy. Only the assumption that an assembly of negative masses has negative entropy permits an analytic continuation of the entropy from positive to negative temperatures. If the entropy for negative temperatures would be counted positive, the derivative of entropy with respect to temperature would be discontinuous at zero.

For the entropy of a mixture of positive and negative masses to become zero requires an exact correlation in the disorder of the positive mass gas with the disorder of the negative mass gas. This is certainly true if the negative mass is equal to the negative mass of the gravitational field of the positive mass, because the Newtonian gravitational field of each particle, all the way down to the smallest dimension, is precisely correlated to the position of the particle. The entropy of the positive mass of matter and the entropy of the negative mass of its gravitational field — correlated to the entropy of the positive mass which is the source of this field — might therefore be called complementary, like a positive and negative photographic image. The expansion from a very small phase space volume would then be possible, because if the positive and negative masses are densely packed within the same volume, not only their energy, but also their entropy would cancel.

The time needed to bring back the universe to its original low entropy state is the Poincaré recurrence time. While under normal conditions this time is huge, it may in a dense mixture of positive and negative masses with a divergent acceleration become quite small. This might explain why the initial entropy of the universe is very small.

The Planck aether hypothesis gives a plausible explanation for the observed vanishing of the sum of all charges, like the electric, color and weak charges. With the phenomenon of charge explained to result from the zero point fluctuations of Planck masses bound in vortex filaments, and with an equal number of positive and negative Planck masses, the sum of all the charges must vanish. That this should also be true for the gravitational charges finds its expression in the compensation of the positive energy in the universe by its negative gravitational energy. This compensation explains why the flatness parameter is one.

Furthermore, with the sum of all gravitational charges equal to zero, the cosmological constant, playing the role of a kind of charge, demands that it also be zero. Finally, with the negative entropy of the negative Planck masses playing the role of a kind of photographic negative for the positive entropy of the positive Planck masses, the total entropy, made up from the sum of the positive and negative Planck masses, would also be equal to zero.

In summary: the sum of all charges is zero, with the cosmological consequence that the flatness parameter minus one, the cosmological constant and the entropy are all zero.

The horizon problem is here resolved by superluminal electromagnetic and gravitational shock waves during the high density phase of the cosmological evolution, not by an inflationary expansion of space.

10. The Cusp/Core Problem in Galactic Halos

We have seen that there appears to be strong evidence for the existence of negative matter in the universe. And we have also given reasons that negative matter might be hidden behind positive matter, forming pole-dipole Dirac spinor configurations neutralizing the negative matter. This still leaves open the question as to whether in certain regions of space there might be a surplus of negative over positive matter, and if negative matter can be mined from such regions.

It has been conjectured by Forward [13] that negative matter might be located in the intergalactic voids, explaining the bubble structure of the metagalaxy. The negative mass in the voids would produce gravitational potential hills repelling all matter, positive and negative, like the positive matter of the galaxies produces gravitational potential wells attracting all matter, positive and negative. But the accumulation of negative matter in the gravitational wells of positive matter reduces and flattens the depth of the wells. This simple fact might explain the unsolved cusp/core problem of galactic halos [14].

But what happens in the center of galaxies must also happen to a lesser degree in the center of the sun, the planets, the earth and the moon. To mine negative matter, if it should exist there, excludes the sun, and also all planets that have a hot molten core. This does not exclude the moon, however, having the deepest potential well near the earth, with only hot rocks in its center accessible by advanced nuclear mining technology, permitting in principle constructing a tunnel through the moon [15].

11. Searching for Negative Matter in the Gravitational Potential Well of the Moon

The radius of the moon is 1.74 by 10⁸ cm and the gravitational acceleration at its surface is about 1.62 by 10² cm/s². At a distance r less than the lunar radius from its center the gravitational acceleration falls linearly with r, and the pressure balance equation integrates to a central pressure equal to one half the density times the surface gravity times the radius.

With an average lunar density of about 3.33 g/cm³, one finds that the maximum pressure is about 5 by 10¹⁰ dyn/cm², or roughly 50,000 atmospheres.

The temperature can be estimated from the ideal gas relation, where Boltzmann’s constant is 1.38 by 10⁻¹⁶ erg per kelvin and the atomic number density of the rocks is about 10²³ per cm³. For that maximum pressure one finds a temperature of about 4 by 10³ K. Both the pressure and the temperature are technically manageable — the pressure with layers of shattered rocks around a tunnel passing through the center of the moon, and the temperature with some cooling. Seismic measurements suggest that the center of the moon is made up of hot rocks.

If appreciable amounts of negative matter have accumulated over billions of years in the center of the moon, it is more likely that this matter is in the form of ultra-light matter, perhaps by an order of magnitude lighter than ordinary matter. There are indications that a Swedish research group has found evidence for the existence of an ultra-dense phase of deuterium, about more than 100,000 times more dense than water [16]. Suppose that in the center of the moon the accumulation of negative matter has led to a form of matter which is 100,000 times lighter than steel, but still has the strength of steel. This would not lead to a negative-positive mass self-chasing mass dipole as envisioned by Forward [13], but to something very important for space flight, because it would dramatically reduce the energy requirements to accelerate a space craft made from such ultra-light material.

The question as to whether there is such an unusual substance in the center of the moon can probably be answered by seismic wave tomography, obtained by nuclear explosions set off on the surface of the moon.

12. Making a Tunnel through the Moon

The cohesive energy of rocks is of the order 10¹⁰ erg/cm³. Therefore, the explosive yield needed to shatter a spherical volume scales with the cube of its radius times that cohesive energy.

The energy released in a kiloton nuclear explosion is about 4 by 10¹⁹ erg. With this energy, the radius of the crushed rocks would be about 10³ cm, or 10 m, and with a 10 kiloton explosion it would be twice as large.

To make a tunnel, a cylindrical, rather than spherical, volume of crushed rocks is desired. For this reason a thermonuclear shape charge or an explosive lens is better suited to shatter the rocks.

In the center of the moon the temperature is several thousand degrees centigrade. With the heat diffusion equation, the diffusion time for a layer of thickness x scales as x squared over the heat diffusion coefficient. For lunar rocks the coefficient is about 4 by 10⁻³ cm²/s. Taking the example of a 20 m layer, one finds a diffusion time of about 10⁹ s, some 30 years, and at the high rock temperatures the heat diffusion time would be uncomfortably long.

The situation is drastically changed for a layer of crushed rocks, because there it is possible to remove heat by a coolant pumped through the porous medium of the crushed rocks. At the high temperatures of several thousand degrees centigrade, a liquid alkali metal — for example lithium, abundantly available on the moon — could be used as a coolant. The velocity the coolant diffuses into the crushed rocks is determined by Darcy’s law. If the pressure gradient is provided by the gravitational force, the coolant velocity is about 1 cm/s.

The time needed for the liquid metal to pass through a 20 m thick layer is then about 2 by 10³ seconds, roughly 1 hour. The specific heat per unit volume of the coolant is about 3 by 10⁷ erg per cm³ per kelvin, and for a temperature of 3 by 10³ K one has about 10¹¹ erg/cm³.

The heat per unit volume which has to be removed from the crushed rocks is of the order of the rock pressure. In the center of the moon, where that pressure is 5 by 10¹⁰ dyn/cm², this energy is 5 by 10¹⁰ erg/cm³. It thus follows that the volume of the liquid coolant must be about one half of the rock volume to be cooled. The same coolant can be used many times over after the heat is removed from it, which could be done on the surface of the moon by radiation or perhaps better by heat exchangers transferring the heat to lunar sand.

Without a thick layer of shattered rocks surrounding the tunnel, the pressure acting on the tunnel wall would be large, in particular in the center of the moon. Because of friction between particles of the shattered rock, large shear stresses can be sustained, changing the pressure distribution in the rock and reducing the pressure gradient and hence the pressure on the tunnel wall. A more detailed calculation [15] gives the pressure distribution in the shattered rock tunnel wall as a power law in the ratio of radius to tunnel radius.

(The pressure integral for the tunnel wall, the radius of shattering as a function of depth, and the evaluation of the total shattering energy by way of Euler’s beta function are omitted for length; the complete text is at the source.)

Inserting a tunnel radius of 2 by 10³ cm, a lunar radius of 1.74 by 10⁸ cm, a cohesive binding energy of 10¹⁰ erg/cm³ and a central-to-surface pressure ratio of 5 by 10⁴, one finds that the total energy required to shatter the rocks and make a tunnel from the center of the moon to its surface is about 2 by 10²⁴ erg, or 5 by 10⁴ kilotons — 50 megatons.

It must be emphasized that this energy must be quite nonuniformly released along the tunnel shaft. Nuclear fusion explosions below a yield of 10 kiloton become uneconomical, with only a fraction of the energy in the fissionable material — needed to make a critical assembly — released. For a 10 kiloton fission explosion the shatter radius is about 20 m. With a tunnel radius of about 10 m, the shattering condition sets the depth at which such a shot becomes worthwhile at about 10 km.

For a depth less than 10 km the nuclear explosion with a yield below 10 kiloton would suffice, a yield which is uneconomical. It is for this reason suggested that one uses altogether thermonuclear explosive devices where the cost per yield is much lower. To penetrate and shatter the rocks more efficiently, jet-generating thermonuclear explosive lenses could be used. The thermonuclear detonation wave ignited at one point is there shaped into a jet-producing conical explosion by placing obstacles in the path of the wave. The ignition can be done by a fission explosive, but conceivably also by a powerful laser beam, with the laser beam projected down the tunnel shaft, triggering the thermonuclear explosive positioned at the lower end.

With the above-given estimate of about 50 megaton needed to dig the tunnel shaft, the number of thermonuclear explosive devices making use of the detonation wave lens technique could for this reason be quite reasonable, and certainly much less than the number of required fission explosives.

After nuclear explosions have crushed the rocks and the heat is removed, the tunnel wall has to be made from some kind of ceramic material, since water with which to make concrete is only sparsely available on the moon. But for the wall to last, its temperature must be kept low. The low heat conductivity of rocks, requiring little cooling, is there of considerable help. For the crushed rocks the heat conduction coefficient should not be very different than for solid rocks. The heat diffusion time for a 20 m layer of rocks is about 30 years. This means that a small, continuous removal of the heat through the injection and circulation of a liquid metal into the crushed rocks should keep down the temperature of the tunnel wall and its environment.

13. Conclusion

The purpose of this study is the question as to whether negative mass might exist, and if negative mass propulsion is possible at all. It is unlikely to be possible in the fashion speculated by Forward [13] — but also see Winterberg [17] — where a negative mass is chasing a positive mass without the expenditure of any energy. Rather, it might perhaps be possible through the existence of an ultra-light form of matter with the tensile strength of ordinary matter on a macroscopic scale where positive matter is bound to negative matter, as happens with Dirac spinor particles on a microscopic scale. This is the speculative existence of macroscopic bodies approaching zero rest mass, of importance for space flight because such matter would greatly reduce its energy requirements.

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