### Einstein's vision II: A unified force equation with constant velocity profile solutions

#### Abstract

A unified relativistic force equation is proposed, created by the product of charge, unified force field and 4-velocity, c d m beta/d tau = kq Box * A* beta*. The Lorentz 4-force is generated by this expression. In addition, there is a Heaviside-dual pseudo-force, perhaps related to the Aharanov-Bohm effect. The gravitational force is the product of the mass, the gravitational field and the relativistic velocity. To explore the gravitational aspects of the unified force equation, a fourth hypothesis postulates a gravitational field, Scalar(Box * A*) = - G M/c^3 | tau |^2, which is analogous to Newton's field using the magnitude of the interval instead of the radius to make it Lorentz invariant. The solution to the gravitational force equation can be rearranged into a dynamic second-order approximation of the Schwarzschild metric of general relativity. The unified metric has a singularity for a lightlike interval. A constant-velocity solution exists for the gravitational force equation for a system with an exponentially-decaying mass distribution. The dark matter hypothesis is not needed to explain the constant-velocity profiles seen for some galaxies. This solution may also have implications for classical big bang theory.

#### Introduction

Gravity was first described as a force by Isaac Newton. In general relativity, Albert Einstein argued that gravity was not a force at all. Rather, gravity was Riemannian geometry, curvature of spacetime caused by the presence of a mass-energy density. Electromagnetism was first described as a force, modeled on gravity. That remains a valid choice today. However, electromagnetism cannot be depicted in purely geometric terms. A conceptual gap exists between purely geometrical and force laws.

The general equivalence principle, introduced in the first paper of this series, places geometry and force potentials on equal footing. Riemannian quaternions, (a_0 i_0,a_1 i_1/3,a_2 i_2/3,a_3 i_3/3), has pairs of (possibly) dynamic terms for the 4-potential A and the 4-basis I. Gauss' law written with Riemannian quaternion potentials and operators leads to this expression: If the divergence of the electric field E was zero, then Gauss' law would be due entirely to the divergence of the basis vectors. The reverse case could also hold. Any law of electrodynamics written with Riemannian quaternions is a combination of changes in potentials and/or basis vectors.

The tensor equation for the relativistic electrodynamic 4-force will serve as a model for a unified force equation. The unified force equation contains the Lorentz 4-force. In the first paper of this series, a classical representation of the quantum Aharanov-Bohm effect appeared in the field equations. The pseudo-force that appears in the unified field equations is a manifestation of the Aharanov-Bohm effect in a force law.

A link for a specific gravitational field between a gravitational force law and a dynamic metric has been found. A Lorentz invariant gravitational field that is a close numerical approximation of Newton's gravitational field. The equation of motion was solved. The solution is of a particular form that can be rearranged into a dynamic metric. Under weak, static conditions, the unified metric is a good approximation of the Schwarzschild metric of general relativity.

Is there an application for the new gravitational force law? Newton's law of gravity is inadequate to describe the motion of spiral galaxies (which are static and weak enough to make Newton's law an appropriate approximation for general relativity). Spiral galaxies are often composed of a spherical bulge and a thin disk whose mass falls off exponentially.[freeman1970] The maximum velocity reaches the velocity expected based on the using the mass to light conversion ratio to calculate the total mass. What has been surprising is that the velocity of the disk far from the center as seen by viewing neutral hydrogen gas spectral lines continues to have that velocity.[kent1986][kent1987][albada1985] There are two problems. First, the velocity profile should show a Keplarian decline unless there is a large amount of dark matter that cannot be seen, but whose presence is inferred to create the flat velocity profile. Second, the disk is not stable to axisymmetric disturbances.[toomre1964] Newtonian theory predicts that galactic disks like the one we live in should have collapsed long ago.

A relativistic force involves a change in momentum with respect to spacetime. For Newton's law, only a change in velocity with respect to time is considered. One could look at the change in mass distribution with respect to space. Using the same Lorentz-invariant gravitational field, a constant-velocity solution is found where the mass falls off exponentially. No dark matter is needed to explain the velocity and mass profile seen in spiral galaxies.

#### Relativistic 4-forces

Define the relativistic 4-force as the change in momentum with respect to the interval. A unified field hypothesis is proposed, modeled on the relativistic electromagnetic 4-force, which involves the product of a charge, a unified field and a 4-velocity: Expand the terms that involve the electromagnetic field on the right-hand side, grouping them by their transformation properties under a spatial inversion:  The first term is the relativistic Lorentz 4-force. The reason this unified force hypothesis is being investigated is due to the presence of this well-known force.

The second term contains a pseudo-scalar and pseudo-3-vector. There is no corresponding classical force. This presents several options. The left-hand side of the equation may be incomplete, perhaps a pseudo-force involving the Aharanov-Bohm current. Alternatively, the operators could be constructed to remove the pseudo-force terms. This would not be consistent with the simplicity mandate followed in this paper.

Analysis of the gravitational force equation turns out to be more direct. Both sides of the force equation are composed of true scalars and 3-vectors: This may be why gravity is always an attractive force: unlike the complete set of terms for the electromagnetic force, all the terms involving gravity force the same way under time or spatial inversion.

What gravitational field should be used in the force equation to generate equations of motion? Newton's gravitational 3-vector field is good numerically. Unfortunately, Newton's law of gravitation is not consistent with special relativity. [misner1970, Chapter 7] One way to derive the field equations of general relativity involves making Newton's law of gravity consistent with the finite speed of light. [kraichnan1955] Test a gravitational field that exploits a close 4-dimensional approximation for a spacelike separation R: the magnitude of the interval between the worldlines of the test and gravitational masses. A hypothesis for the gravitational field is proposed: This field is invariant under a Lorentz transformation since it is the ratio of two invariant scalars along with some constants. For a weak field, the radius over the speed of light and the absolute value of the interval will numerically be the same, but their mathematical behavior will be different.

Assume no relativistic effects concerning the mass. Assume the equivalence principle so that the inertial mass equals the gravitational mass. Plug the gravitational field into the force law, canceling the masses to generate a quaternion equation of motion: The solution for the relativistic velocity is an exponential: Given a real gravitational force, the interval tau  evaluates to a real number. One could explore a solution for an imaginary field, but that will not be investigated in this paper.

General relativity is discussed in terms of curvature, not forces. A metric is a function that involves differential elements of time, space and the interval. Notice that the relativistic velocity that solved the gravitational force equation also has these elements. Look for an algebraic link. Solve for the constants, which evaluate to a 4-velocity in spacetime. Form an invariant scalar under a Lorentz transformation of this constant, and therefore conserved, 4-velocity by taking the scalar of the square. Multiply through by the interval squared to create a function with the form of a metric. To ensure that the metric equals the Minkowski metric in flat spacetime, set the differences of the constants equal to one: If the gravitational field is zero, this generates the Minkowski metric of flat spacetime. Conversely, if the gravitational field is non-zero, spacetime is curved

No formal connection between this proposal and curvature has been established. Instead a mercurial path between a proposed gravitational force equation and a metric function was sketched. There is a historical precedence for the line of logic followed. Sir Isaac Newton in the Principia showed an important link between forces linear in position and inverse square force laws.[newton1934] More modern efforts have shown that the reason for the connection is due to the conformal mapping of z\rightarrow z^{2}.[needham1993] This method was adapted to a quaternion force law linear in the relativistic velocity to generate a metric

For a weak field, write the Taylor series expansion in terms of the total mass over the interval to second-order: The expansion has the same form as the Schwarzschild metric in isotropic coordinates expanded in powers of mass over the radius. If the magnitude of the interval is a close approximation to the radius divided by the speed of light, it will pass the same weak field tests of general relativity.[will1993]

The two metrics are numerically very similar for weak fields, but mathematically distinct. For example, the Schwarzschild metric is static, but the unified metric contains a dependence on time so is dynamic. The Schwarzschild metric has a singularity at R=0. The unified gravitational force metric becomes undefined for lightlike intervals. This might pose less of a conceptual problem, since light has no rest mass.

#### The constant velocity profile solution

In the previous section, the system had a constant point-source mass with a velocity profile that decayed with distance. Here the opposite situation is examined, where the velocity profile is a constant, but the mass distribution decays with distance. Expand the definition of the relativistic force using the chain rule: The first term of the force is the one that leads to an approximation of the Schwarzschild metric, and by extension, Newton's law of gravity. For a region of spacetime where the velocity is constant, this term is zero. In that region, gravity's effect is on the distribution of mass over spacetime. This new gravitational term is not due to the unified field proposal per se. It is more in keeping with the principles underlying relativity, looking for changes in all components, in this case mass distribution with respect to spacetime.

Start with the gravitational force in a region of spacetime with no velocity change: Make the same assumptions as before: the gravitational mass is equal to the inertial mass and the gravitational field employs the interval between the worldlines of the test and gravitational masses. This generates an equation for the distribution of mass: Solve for the mass flow: As in the previous example for a classical weak field, assume the magnitude of the interval is an excellent approximation to the radius divided by the speed of light. The velocity is a constant, so it is the mass distribution that shows an exponential decay with respect to the interval, which is numerically no different from the radius over the speed of light. This is a stable solution. If the mass keeps dropping of exponentially, the velocity profile will remain constant

Look at the problem in reverse. The distribution of matter has an exponential decay with distance from the center. It must solve a differential equation with the velocity constant over that region of spacetime like the one proposed.

The exponential decay of the mass of a disk galaxy is only one solution to this expanded gravitational force equation. The behavior of larger systems, such as gravitational lensing caused by clusters, cannot be explained by the Newtonian term.[bergmann1990][grossman1989][tyson1990] It will remain to be seen if this proposal is sufficient to work on that scale.

#### Future directions

For a spiral galaxy with an exponential mass distribution, dark matter is no longer needed to explain the flat velocity profile observed or the long term stability of the disk. Mass distributed over large distances of space has an effect on the mass distribution itself. This raises an interesting question: is there also an effect of mass distributed over large amounts of time? If the answer is yes, then this might solve two analogous riddles involving large time scales, flat velocity profiles and the stability of solutions. Classical big bang cosmology theory spans the largest time frame possible and faces two such issues. The horizon problem involves the extremely consistent velocity profile across parts of the Universe that are not casually linked.[misner1970, p. 815] The flatness problem indicates how unstable the classical big bang theory is, requiring exceptional fine tuning to avoid collapse.[dicke1979] Considerable effort will be required to substantiate this tenuous hypothesis. Any insight into the origin of the unified engine driving the Universe of gravity and light is worthwhile.

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