Abstract
Introduction
Relativistic 4-forces
The constant velocity profile solution
Future directions

Abstract Gravity is a metric theory, electromagnetism is not.  By using Riemannian quaternions which can have dynamic basis vectors, it becomes possible to merge metric theory with the linear Maxwell equations.  A Lagrangian is proposed which is very similar to the Gupta-Bleuer Lagrangian for quantizing the Maxwell equations.  A significant difference is that the unified Lagrangian has a conserved current.  One scalar potential solution to the equations of motion is found, the inverse square of an interval between two events.  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.  By normalizing the gravitational force law, the constants required by dimensional analysis include hbar, which suggests a connection to quantum mechanics (not explored here).  The normalized gravitational force law is made classical through a series of assumptions.  The solution to the gravitational force second-order differential equation using the inverse square interval potential is found.  The constants are eliminated, and the result is a dynamic 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.  (PACS:12.10.-g).

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 in the form of a pseudo-current.  The pseudo-force that appears in the unified field equations is a manifestation of the Aharanov-Bohm effect in a force law.  

When the force field is normalized to the magnitude of the potential, by dimensional analysis this introduces the constant\hbar.  The resulting force equation for a scalar inverse square potential does not have a closed solution.  The connection back to a classical proposal involves a number of assumptions.  The driving assumption is that Newton's gravitational field should be apparent in the proposal, but in a 4-dimensional form.  The resulting classical gravitational 4-force law does have an exact solution.  By eliminating the constants, a metric equation is the result.  The unified metric has the same coefficients for a Taylor series expansion for a weak field as the Schwarzschild metric of general relativity up to post-Newtonian accuracy.  There is a testable difference between the Schwarzschild metric and this proposal for the next term in the weak field expansion.  

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 (K. C. Freeman, "On the disks of spiral and SO galaxies," Astrophys. J., 160:811-830, 1970).  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 (S. M. Kent, "Dark matter in spiral galaxies I. Galaxies with optical rotation curves", Astron. J., 91(6):1301-1327, 1986. S. M Kent, "Dark matter in spiral galaxies. II. Galaxies with h1 rotation curves," Astron. J., 93(4):816-832, 1987. T. S. van Albada, J. N. Bahcall, and R. Sanscisi, "Distribution of dark matter in the spiral galaxy NGC3198," Astrophys. J. Let., 349:L1, 1990).  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 (A. Toomre, "On the gravitational stability of a disk of stars," Astrophys. J., 139:1217, 1964).  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.

Despite its formulation using quaternions, this unification proposal is strikingly similar to earlier work.  Gupta wanted to quantize the Maxwell equations using a form that was manifestly relativistic in its treatment of time and space.  He worked with this Lagrangian for the vacuum:

The equations of motion for this Lagrangian are:

Although useful, Gupta concluded that there was no new physics beyond the Maxwell equations.  One way to see that is that is by taking the derivative of the Lagrangian with respect to the gauge variables dA_mu/dx_mu.  The result is not a constant current, so there are no symmetries to investigate.  

The coefficients for the scalar are different for the unified field Lagrangian:

Take the derivative of the Lagrangian with respect to the gauge variables:

According to Noether's theorem, a conserved symmetry in the Lagrangian indicates a conserved current.  There is new physics here.  The gauge field is a dynamic variable constrained by the Lagrangian.  Since a gauge is in a sense a form of measure, how things are measured becomes a dynamic variable.  This is a central goal of general relativity.  

A gauge has yet to be chosen, since that involves equating a gauge to a particular scalar value.  For simplicity, choose the Lorenz gauge, where dphi/dt + Div A = 0.  In the Lorenz gauge, the equations of motion are those cited above.  One scalar potential solution to that equation is:

Any interval between two events (tau^2) will make a contribution to the potential field.  The shorter the interval, the greater the contribution.  What are these events? For a test particle and a source, the vast majority are events that signal that the source still exists in spacetime.  Every event that happens to the source contributes to the 1/tau^2 scalar potential.  Most events of a source are like identity functions, mapping a particle onto itself later in spacetime.  In other words, these events are the stuff of inertia, because the more particles participating in identity functions, the greater the inertia.  

Notice that the product, GMA = GM/tau^2, is a scalar analog to the classical gravitational 3-vector field.  It is composed of the ratio of two Lorentz invariant quantities, so will be Lorentz invariant.  

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 constant, 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.  The Aharanov-Bohm effect is very real, so there should be a force that depends on the electric and magnetic fields which does the work.  A pseudo-force is called for, since the effect depends on the total magnetic flux, a pseudo-vector.  Like the pseudo-current added to the Maxwell equations discussed in the first paper, for completeness there should be a pseudo-force equation like the Lorentz 4-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.

The unified force is not relevant for quantum mechanical calculations for a fairly simple reason:it does not contain Planck's constant.  In a subsequent calculation for a metric, it was found that the unified force needed to be normalized to the magnitude of the interval:

The units of the constant k must change to in a way that compensates for the normalization.  Focus on the gravitational force equation.  The potential Box*A* has units of mass.  The normalized potential, Box*A*/|A| has units of 1/length.  The constants are related in the following way:

Substitute this back into the force equation:

Because Planck's constant appears explicitly in this formula, it may play a role in quantum gravity calculations.  

Take the scalar potential which solves the equations of motion, plug that into the unified force equation, and presume no external forces, to create the following second-order differential equation:

There is no closed form solution to these four differential equations.  

Newton's gravitational 3-vector field is good numerically.  Unfortunately, Newton's law of gravitation is not consistent with special relativity (C. W. Misner, K. S. Thorne, and J. A. Wheeler, "Gravitation," Chapter 7, W. H. Freeman, 1970).  One way to derive the field equations of general relativity involves making Newton's law of gravity consistent with the finite speed of light (R. H. Kraichnan, "Special-relativistic derivation of generally covariant gravitation theory," Phys. Rev., 55:1118-1122, 1955).  Up to this point, no constraints have been imposed to make this force relevant only to the classical domain.  The current challenge is to determine what constraints are required to yield the Newton's gravitational 3-vector field in the classical limit, while still respecting the four dimensional nature of spacetime.  

The Newtonian gravitational field is the ratio of the gravitational mass divided by the distance squared.  There is a distance squared factor in the unified force, but it uses an interval between events.  Consistency in the classical region can be obtained if all the intervals between the test particle and the source are spacelike and simultaneous.  In that case:

With the assumption that all intervals are effectively spacelike and simultaneous, the interval square will have the same value as the distance squared.  Mathematically, the behavior will be different, since the interval depends on spacetime, but the distance only on space.  For those that object to only including simultaneous, spacelike events in the tabulation of the potential, it is a compromise require to reach a classical law.  

The equivalence principle of general relativity is required to cancel the mass charge q with the inertial mass m.  

Still, there is something glaringly absent in the discussion so far:where is the source mass? If one looks at the remaining terms, mass would depend linearly on time and space, which is not reasonable.  The potential depends only on simultaneous spacelike events between the test and the source.  Presumably, a source with twice as high an energy density would also produce twice as many events between it and the test particle.  It will be assumed that the events between the test particle and source will vary little between local adjacent points in spacetime.  This is another way of saying that spacetime is quite flat.  The mathematical consequences of that need to be clarified.  Make the following change of variables:

where a and b are constant distances, and\alpha and\beta are small numbers.  Under these conditions for two nearby points in spacetime:

Take derivatives of the potential:

To first order in alpha and beta, the derivative is not a linear function of time or space.  There may well be situations where this is not the case, but for a classical system, this small variation in local spacetime approximation is required.  

What are these constant distances, alpha a and beta b? Time and space have been treated equally thus far, and there is no reason to change that posture.  If these equations are related to gravity, then a relevant distance would be GM/c^2.  This distance hypothesis leads to the following classical gravitational force law:

Solve this second-order differential equation for the spacetime position:

where Ei is the exponential integral, Ei(t)=the integral from negative infinity to t of e^t/t dt.  The exponential integral plays other roles in quantum mechanics, so its presence is interesting.  

Eight constants need to be eliminated:(c_1, C_1) and (c_2, C_2).  Take the derivative of the spacetime position with respect to tau.  This eliminates four constants, (c_2, C_2).  The result is a 4-velocity:

In flat spacetime, beta _mu beta^mu=1, providing four more constraints.  Spacetime is flat if e^(GM/c^2|tau |) goes to 1 because M goes to 0 or tau goes to infinity.  

Solve for c_1^2 and C_1.C_1:

Substitute back into the flat spacetime constraint.  Rearrange into a metric:

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

As expected, this become the Minkowski metric for flat spacetime if M goes to 0 or tau goes to infinity.

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.  More modern efforts have shown that the reason for the connection is due to the conformal mapping of z goes to z^2 (T. Needham, "Newton and the transmutation of force," Amer. Math. Mon., 100:119-137, 1993).  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 in M/|tau|:

Contrast this with the Schwarzschild solution in isotropic coordinates expanded to second order in M/R:

If the magnitude of the spacelike interval is a close approximation to the radius divided by the speed of light, the unified metric will pass the same weak field tests of general relativity as the Schwarzschild metric to post-Newtonian accuracy, which does not use the second order spatial term(C. M. Will, "Theory and experiment in gravitational physics: Revised edition," Cambridge University Press, 1993).

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.  

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 (A. G. Bergmann, V. Petrosian, and R. Lynds, "Gravitational lens images of arcs in clusters," Astrophys. J., 350:23, 1990. S. A. Grossman and R. Narayan, "Gravitationally lensed images in abell 370," Astrophys. J., 344-637-644, 1989. J. A. Tyson, F. Valdes, and R. A. Wenk, "Detection of systematic gravitational lens galaxy image alignments: Mapping dark matter in galaxy clusters," Astrophys. J. Let., 349:L1, 1990).  It will remain to be seen if this proposal is sufficient to work on that scale.

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 (MTW, p. 815).  The flatness problem indicates how unstable the classical big bang theory is, requiring exceptional fine tuning to avoid collapse.  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.