_{1}

^{*}

The necessary derivation of negative mass in dispersion dynamics suggests cosmic applications. The method analyzes functional relationships between particle angular frequency, wave vector, rest mass and electromagnetic or nuclear potential, f(
ω, k, m
_{0}, V) = 0. A summary of consequential predictions of the dynamics leads to a calculation of ways in which negative mass might influence such phenomena as the rotational velocities that are observed in spiral galaxies. The velocities are found to be not Newtonian in the simple two body approximations for our solar system; but nearly constant with increasing orbital radii. It has moreover been suggested that the motion is due to halo structures of dark matter or dark energy. However, the motion is simply described by many-body gravitation that is transmitted along elastic spiral arms. In this context, we calculate possible effects of negative mass, but without observational confirmation.

Dispersion dynamics [_{o} of a free body, f(E, p, m_{o}) = 0. In wave mechanics, this translates to f(ω, k, m_{o}, V) = 0, by substitution with angular frequency in Planck’s law; with wave vector in the de Broglie hypothesis; and with potential V ≠ 0 for a bound particle. An immediate consequence is that the product of the group velocity and phase velocity in a free particle is equal to the square of the speed of light c^{2}. Further derivation will show (described below) that antiparticles have negative mass. This concept has a history of speculation [

Prior expectation supposed that central attraction in a galaxy would result in Newtonian planetary velocities where the inverse square gravitational attraction balances centrifugal forces due to orbital motions: then a planet of mass m orbiting a sun M at radius ~r reacts against the Gravitational attraction GMm/r^{2}, where G is the gravitational constant. The planet in an approximately stable orbit would have a velocity v = G M / r i.e. inversely proportional to the square root of the orbital radius, for simplicity supposed circular. In fact, the velocities that are observed are almost independent of r [

The most fundamental property of modern physics is wave-particle duality. It is best expressed by the stable wave packet: self-evidently stable as the travelling wave group for a free particle or photon:

ϕ = A ⋅ exp ( X 2 2 σ 2 + X ) , with X = i ( k ¯ x − ω ¯ t ) (1)

(^{2}/2σ^{2}), depends on the square of X which is a function of four variables. Two are already considered, so we are left with the variables x and t that describe the profile. Since the other variables are all stable, this profile is also stable. From the start of quantum mechanics, and following Dirac’s opinion [

From these wave mechanics are also derived Planck’s law, E = ħω, the de Broglie hypothesis p = ħk, and several conservation rules. Solve f(ω, k, m_{0}, V) = 0, first for the free particle with rest mas m_{o} in a potential V = 0.

Operation of the relativistic Klein-Gordon equation, ( □ 2 − m 0 2 ) ϕ ( x ) = 0 , on Equation (1) yields, as output, an algebraic equation in second order:

ℏ 2 ω 2 = ℏ 2 k 2 c 2 + m 0 2 c 4 (2)

ħ being the reduced Planck constant and c the speed of light. This is the same equation as is obtained from Einstein’s relativistic formula, E 2 = p 2 c 2 + m 0 2 c 4 , by substituting for energy using Planck’s law and for momentum using the de Broglie hypothesis. The equation can be simplified with appropriate units c = 1 = ħ. Differentiation then gives a new result in relativity, for the product of group velocity dω/dk, (see e.g. [

d ω d k ⋅ ω k = v g ⋅ v p = 1 (=c^{2} in generalized units) (3)

The result is plotted in the positive quadrant of _{0} = 1. The group velocity is well behaved: it tends to zero at low k and to c at large k exactly as in the special theory of relativity^{1}. The phase velocity is faster than the speed of light c and is singular when k → 0: within this rest frame, time is Newtonian within the coherence σ. This has significance in the reduction of the wave packet during a quantum transition [

The Uncertainty Principle can be derived from Equation (1) by Fourier transforms [

states involved in atomic spectral line emissions or absorption. Without quantization, as in the Bohr atom, wave functions would destructively self-interfere. Notice that in the calculation of lines such as the Lyman α for the hydrogen atom, m_{o} cancels.

Two important facts are: the expected mass energy or density, integrated in time over the packet in Equation (1), is equal to ℏ ω ¯ ; while the expected momentum, integrated over space, is equal to ℏ k ¯ [

This description is a physical, non-axiomatic^{2} [

Dirac’s calculation for the speed of the electron [

Two extreme regimes are commonly identified: relativistic when m 0 c ≪ p , and non-relativistic when m 0 c ≫ p .

Relativistically, at high k ≫ m 0 , both the group velocity and phase velocity tend to the speed of light: v_{g}, v_{p} → c, as in the massless photon travelling in free space. Then d ω / d k = ω / k = ν ′ λ , the product of oscillational frequency ν' with wavelength. Conductance depends on the group velocity.

At low k ≪ m 0 , non-relativistic values approximate:

E = m 0 ( 1 + p 2 c 2 / ( m 0 2 ) ) 1 / 2 ≈ m 0 + p 2 / 2 m 0 (4)

In classical mechanics, the mass energy is ignored as a constant in mechanical or chemical changes, as it is in Schrödinger’s equation―which is likewise non-relativistic. Moving on from the free particle, when the potential V ≠ 0 is included, the Schrödinger eigenvalue ε corresponds to the result of the virial theorem, so that the expectation value for 〈 | V | 〉 ≈ − 2 〈 | p 2 / 2 m 0 | 〉 (

ε ≈ 〈 | ℏ 2 k 2 / 2 m 0 | 〉 (5)

Using the simplified units previously described. The kinetic energy for the free particle is positive; while the eigenvalue in a potential V ≠ 0 is about the same absolute value, but negative. Whether free or bound, the group velocity (and conductance etc.) is given by:

C = E − m 0 2 p = ω − m 0 2 k (6)

or ε/2p in Schrödinger’s system. This is the velocity that is proportional to the Lorentz force of magnetism for a charged particle moving in free space.

In constrained crystal fields, the force depends on the dispersive curvature d^{2}ω/dk^{2} of electron energy bands near the Fermi surface [_{H}. For this reason, R_{H} is positive in Al, though in the majority of metals it is negative, as in Cu.

The analysis of the function f(ω, k, m_{0}, V) = 0 summarized above, modifies the conventional notion of rest mass m_{0}. Equation (2) has solutions when energy is positive or negative. In the Schrödinger equation, the energy is negative since the expectation value for the potential 〈 | − V | 〉 is negative, but, in value, double the expectation for kinetic energy 〈 | p 2 / 2 m 0 | 〉 , which is positive. This is typically the case for bound states in atoms. However, for the free particle, when V = 0, there remain positive and negative solutions. In his relativistic equations, Dirac ascribed his negative eigenvalues to antiparticles, and we follow him in so ascribing the negative solution for energy in Equation (2). Two consequences follow: firstly, when k = 0, in the rest frame, the result for relativistic mass E = m’c^{2} implies that for the antiparticle, both m’ and m_{0} are negative. Secondly the supposed alternative that E < 0 while m > 0 leads to anomalies when |k| = |m_{0}| [_{g} combined with zero net energy and zero v_{p}. Neither of these consequences is observed, and the condition is generally inconsistent with physical theory. Fundamental conclusions of Dispersion Dynamics are that the antiparticle has negative mass, and also that they have negative kinetic energy. This solution for the equation is a simple convention that supplies consistency and is clearer than Dirac’s concept of hole states which is not generally favored [

However, the consistency reopens some old doubts. With a new representation for the wave function of antiparticles, we also find a non-paradoxical explanation for the Feynman-Stückelburg principle: “An antiparticle travelling forward in time does not exist”. Though stated to explain transition and scattering probabilities, the principle seems to contradict cloud chamber and bubble chamber images of particle creation. By contrast, Dispersion Dynamics has, “An antiparticle travelling forward in time has negative momentum.” This causes the same arrow reversal in Feynman diagrams as before. However, it implies also consistency with the physically constant electronic charge to mass, ratio e/m, i.e. having the same sign for both charged particle and antiparticle.

The best known part of the universe is our own solar system. Newton gave us the law of gravitation for planetry motion, to which perturbations due to general relativity are added. Galactic orbits are more complicated. Their rotational velocities do not follow, in simple form, Newton’s law of gravitation [

We get an idea of the force by comparing with electrostatics, which also follows the inverse square rule, i.e. proportional to r^{−2}. Both forces operate between two bodies; but in electrostatics we are used to applying many-body interactions. Planetry forces are comparatively simple: the sun on the planets, the moon on the earth etc. The orbit of the moon around the earth is perturbed by the distant sun; producing a wobble on the earth’s orbit. In galaxies by contrast, matter that orbits a galactic center is influenced also by other masses, some nearer; others further away. The observational fact that they mostly travel with similar velocities, implies multiple forces that are more than perturbations. Consider the gravitational forces between three masses: a solar center M, and two orbiting masses, m_{1} and m_{2}, traveling at similar velocities on the same plane. At a moment when the three masses lie sequentially on a straight radial line, the centripetal forces are G M m 1 / r 1 2 ; G M m 2 / r 2 2 ; G M m 3 / r 3 2 ; and − G M m 1 m 2 / ( r 1 − r 2 ) 2 . The last has a centrifugal reaction on m_{2} (since r_{3} > r_{2}) that adds to inertia opposing the central force. Where there are a large number n of masses, moving with similar velocities far from the galactic center, i.e. with m 1 / ( r 1 − r n ) 2 > M / r 1 2 etc., a many-body resultant force might be written, similar to electrostatics:

F = G M m 1 / e g r 2

where e_{g} is a gravitational permittivity. This would not be a material property but a consequence of environment. We could use gravitational permittivity in the context of gravitational lensing; but meanwhile we shall see how this many-body effect is due to galactic radius while causing and stabilizing it.

Before proceeding to investigate what effect negative mass in antimatter might have on rotational velocities, we need to understand familiar positive mass in regular matter. We will then carry forward what we know about dispersion dynamics to speculation about gravity in antimatter.

• For exploratory purposes, we suppose that dark matter is simply normal matter at typical temperatures with low visibility whether due to halo or other configurations, though possibly including by extension, uncharged particles, whether light or heavy.

• Start with a massive spherical core.

• Continue with a planar disc to represent smoothly distributed matter in a bright galaxy.

• Let many-body gravitational bonding cause constant rotational velocities.

• Calculate the structure of the galaxy for general stability by considering the gravitational forces balanced against inertia under observed rotational constraints.

• Adapt the calculations for spiral arms supposing these are gravitationally coupled to adjacent matter

• Consider negative mass.

Various measurements demonstrate approximately constant galactic rotational velocities [

F = Σ G M ( r ) ≈ supposedly G ρ ( r ) (7)

is extended in the many-body case to a force that is, in a simple approximation, proportional to a generalized galactic density ρ ( r ) that is treated as a continuum, and presumed to follow some power of radius, r^{a}. We need to calculate what value of a will provide a constant galactic rotational velocity. The rotational velocities will then depend accordingly, v ( r ) ≈ G ρ ( r ) / r . Begin by examining the forces acting on an elemental ring in the galactic disc.

We treat the galaxy as approximately planar to which is added a spherical hub (

differentials instead of integrated infinitesimals. Then, along a given axis, the gravitational force F(r) due to a ring of mean radius r_{1}, can be calculated from the formula:

F ( r ) = 2 G ρ ( r 1 ) ∑ θ = 0 π cos ( a tan ( r 1 sin ( θ ) / ( r − r 1 cos ( θ ) ) ) ) ( r − r 1 cos ( θ ) ) 2 + r 1 2 sin 2 ( θ ) Δ θ ⋅ Δ r (8)

where ρ ( r ) is the mass density in the ring. As is well known in electrostatics, the potential inside a conducting sphere is zero; the elemental ring is different. At first view, the result shown in

ΣF(r)∙ρ(r) → 0. The density ρ(r) can therefore be simulated by varying the index on the power law proportional to r^{−}^{1.15}. Notice that this rule applies only to the neighboring matter; not to the central hub. When we add the central hub, the power law will change. Supposing stability, we can now return to calculate rotational velocities. The calculated value of r^{a} gave a first estimate of the mean mass density distribution in the plane beyond the central hub. Adjustments must now be made to correct for two suppositions: the galaxies that are still evolving are not in equilibrium; and the central hub cannot be neglected, so is included in the following calculation.

The elemental rings illustrated in

radius, where the galactic center approximates, by supposition, to spherical symmetry, the zero potential is applied in ^{a}. Notice that the elemental volume, Δ r ⋅ Δ z ⋅ r Δ ϑ , depends on the first power of r, so that the summation represented in Equation (8) implies the mass density distribution ρ ≈ 1 / r if the elemental volume is treated as constant; or alternatively, the force converts to rotational velocity when ρ is substituted realistically, i.e. when the variable function is formed to be constant as in

To simulate the gravitation on a spiral arm, the density is converted to a net resultant force on a particular body due to matter distributed about radii that are greater or less than its own.

With the results so far given, next consider effects on rotational velocities due to spatial variations in matter and antimatter. Electromagnetic forces in charged particles are greater than gravitational forces, which typically therefore, are more difficult to measure. In uncharged systems, or in galaxies where negative and positive charges are neutralized, we suppose as before, that particles and antiparticles are gravitationally repulsive due to their opposite masses. Then the two types of matter are held apart by gravitational forces and massively explosive annihilations between colliding galactic structures are avoided. Meanwhile, contrary to the standard view that antimatter is rare and asymmetrical, suppose it is thinly distributed in space, but separated from matter by gravitational repulsion. To illustrate,

By comparison,

Many-body gravitation provides a simple explanation for observed plateaux in velocity/radius ratios in spiral galaxies. In contrast wave-particle duality, expressed in the stable wave packet, demonstrates the necessity for negative mass in particle physics. Invisible halos and other structures that might surround or imbue spiral galaxies, when analyzed by suppositions dependent on negative mass, are not conducive to the plateaux that have previously been regarded as anomalous. Many-body gravitation is a simpler explanation, for the plateaux that are observed in rotational velocities, than dark matter or dark energy [

Bourdillon, A.J. (2018) Many-Body-Gravitational Solutions for Galactic Rotational Velocities, Including Possible Negative Mass. Journal of Modern Physics, 9, 1304-1316. https://doi.org/10.4236/jmp.2018.96079