Jay R. Yablon

On the Geometric Foundations of Classical Electrodynamics

Jay R. Yablon 910 Northumberland Drive

Schenectady, New York 12309-2814 yablon@alum.mit.edu

January 26, 2016

Abstract: To be added PACS: 04.20.Fy; 03.50.De; 04.20.Cv; 11.15.-q Contents PART I: A REVIEW OF EINSTEIN’S EQUIVALENCE PRINCIPLE, GAUGE THEORY, AND GRAVITATIONAL GEODESIC MOTION ......................................................................1

1. Introduction ............................................................................................................................1

2. How Gauge Symmetry is used to Introduce Electromagnetic Interactions into Physical Equations Rooted in the Spacetime Metric ..................................................................................4

3. The Physics of Standing on the Ground in a Gravitational Field, and not Passing Through .....8

4. “Nowhere You Can Be that Isn’t Where You’re Meant to Be”: A Review of the Least Action Derivation of Gravitational Geodesic Motion ............................................................................ 15

PART II: THE LORENTZ FORCE AS PURE GEODESIC MOTION IN FOUR-DIMENSIONAL SPACETIME ................................................................................................ 19

5. Why Associating the Canonical Momentum in Gauge Theory with Mass Times Velocity Leads to the Wrong Equation of Motion .................................................................................... 19

6. Derivation of the Lorentz Force as Geodesic Motion, in Four Spacetime Dimensions Only .. 21

7. How Electrodynamic Interactions Dilate and Contract Time, just as do Motion and Gravitation in Special and General Relativity ............................................................................ 26

8. Non-abelian gauge fields ...................................................................................................... 33

9. Relation to Five-Dimensional Kaluza-Klein Theory ............................................................. 34

PART III: UNIFICATION OF GRAVITATIONAL AND ELECTROMAGNETIC FIELDS .... 34

10. Einstein’s Equation and Maxwell’s Equations .................................................................... 34

11. Conclusion ......................................................................................................................... 40

References ................................................................................................................................ 40

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PART I: A REVIEW OF EINSTEIN’S EQUIVALENCE PRINCIP LE, GAUGE THEORY, AND GRAVITATIONAL GEODESIC MOTION 1. Introduction The Equivalence Principal first introduced in 1907 [1] and subsequently refined in 1911 [2], is the gedanken, i.e. thought experiment by which Albert Einstein successfully navigated from the Special [3] to the General [4] Theory of Relativity, and which he later called the “happiest thought” of his life. This principle holds that the physical laws observed from a reference frame assigned to be at rest with velocity 0v = , in a homogeneous gravitational field which imparts a downward acceleration g− to all freely-falling objects, are equivalent to the physical laws observed from a reference frame likewise assigned to be at rest which is given a temporally-constant upward acceleration a by applying an external force F ma= designed such that

/a F m g= ≡ , where m is the total rest mass of all of the material bodies to which this force is applied. The basic configuration of this equivalence is illustrated in Figure 1 below.

Figure 1: The (Approximate) Equivalence Principle: (a) a first observer standing on the earth’s

surface while releasing a small object and (b) a second observer standing inside a uniformly-accelerated housing and likewise releasing a small object. Up to tidal forces, each will observe

and measure and experience equivalent physics. As illustrated in Figure 1, the Equivalence Principal is often discussed by contrasting (a) a first observer who is standing on the surface of the earth at which the gravitational field produces a free-fall acceleration of 29.81 m/sg ≅ and at which the atmospheric pressure 21.03 kg/cmp ≅ , with (b) a second observer who is standing inside an equally-pressurized elevator or rocket ship

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(housing) to which an external Newtonian force F ma= is applied such that, taking into account the entire mass m of the system including the housing and the observer and the air and all objects inside, the observer is uniformly-accelerated such that /F m a g= ≡ . For both observers the +z axis defines the overhead direction, as illustrated.

If each observer in Figure 1 is assigned to be “at rest” with velocity / 0d dt= =v x , and if each observer releases a small object in the manner of Galileo’s Pisa experiment as illustrated, then that object will exhibit a downward free-fall acceleration with respect to the observer which in the former case is –g and in the latter case is a− along the z axis, such that the observed, measured magnitudes of these accelerations a g= are the same. Of course, a “homogeneous”

gravitational field is a physical fiction except locally, because gravitational fields of consequence are generated by large, substantially-spherical heavenly bodies so that the field lines point toward the center of those bodies and thus are not strictly parallel to one another, which deviation from homogeneity is measured by the tidal force. Simply put: the earth is not flat. Thus, the Equivalence Principal strictly speaking is only an approximate equivalence denoted by the ≡ɶ in Figure 1. However, because the area spanned by the Figure 1 observer is miniscule compared to the entire planetary surface area, a precise equivalence is observed for all practical purposes. Underlying this Einsteinian equivalence, is the Galilean equivalence between the gravitational interaction mass and the inertial mass of material bodies, whereby the gravitational pull on the object dropped in Figure 1(a) which is in proportion to its mass, is precisely counterbalanced by its mass-proportionate inertial resistance to that pull. Thus, when one conspires to apply a force

/F m a g= ≡ in Figure 1(b) thereby increasing the force in direct proportion to the total mass of the housing / observer system, one is artificially replicating this Galilean equivalence. So up to the tidal forces, each observer in Figure 1 will observe the exact same downward acceleration in space for the dropped object, and will likewise observe identical physics for the behavior in space of light and, given identical air pressurization, for all other physical phenomenon as well. Importantly, this includes the fact that each observer will feel and measure the same force applied to his or her feet. Specifically, for illustration, if each observer was to weigh, say, 52 kg at the earth’s surface, and if each observer were to stand on a scale during this equivalence experiment to take a reading of their weight, then by conspiring to apply a force

2/ 9.81 m/sF m g≡ ≅ in Figure 1(b), we will have ensured that each observer obtains an identical 52 kg weight measurement from their scale, as illustrated. Consequently, so long as observer (b) is not measuring tidal forces, it will not be possible for observer (b) to distinguish whether he or she is being upwardly accelerated with a Newtonian force /F m a g= ≡ , or is standing upon the earth’s surface in the planetary gravitational field g− .

Now, the accelerations in space observed for the dropped objects relative to the observers in Figure 1 are fully characterized by the General Theory of Relativity; indeed, this Figure 1 gedanken played a crucial role in Einstein’s incorporating these spatial accelerations into the General Theory. However, this weight felt and measured by each observer is not encompassed by the General Theory of Relativity on its own, any more than the relative spatial accelerations of the dropped objects are encompassed by the Special Theory of Relativity which on its own only deals with relative velocities. This is because the ability of the observers in both of Figures 1(a) and (b) to be able to stand on the surface and feel a force against their feet, rather than freely fall through

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the surface as if it was not there, is not a result of gravitation. Rather, it results from what is at bottom the net electrostatic repulsion between electrons carried by the observer and electrons carried by the supporting surface, which electrons are illustratively designated by the e− and which repulsion is designated by the վ in Figure 1.

Additionally, the dropped objects in Figure 1 will not free fall forever. They also carry

electrons which via an electrostatic repulsion with the ground will eventually stop the object’s free fall and cause the object to rebound or simply remain on the ground, depending upon the object’s elasticity. To account for this net electrostatic repulsion that keeps observers and objects alike from passing through the floor, one must consider electromagnetism together with gravitation. That is, the total consideration of all of the physics of the Equivalence Principle in Figure 1 requires that we consider gravitation and electromagnetism together in a unified manner. In essence, developing such a unification of classical gravitation and classical electromagnetism is the fundamental purpose of this paper. As with the development of the General from the Special Theory of Relativity, the Equivalence Principle, with a focus on how observers are barred by electrostatic repulsion from passing through the surfaces upon which they stand, will provide the primary gedanken for achieving this unification.

Specifically, the observer in Figure 1(b) is only accelerated in relation to the dropped object (or vice versa) because his or her molecules are not able to pass through those of the floor of the elevator or the rocket as the force F is being applied. Likewise, the observer in Figure 1(a) only sees an acceleration through space for the dropped object (or vice versa) because his or her molecules are not able to pass through those of the surface of the earth. Nor can the molecules of the dropped object pass through these surfaces. In all cases, it is electromagnetism which stops the geodesic free fall worldlines of pure gravitation. Indeed, consider the converse: if the electrostatic repulsion (and more generally electromagnetism) did not exist, then each observer would continue in free fall through the floor of the housing or the surface of the earth, as would the dropped object. Thus, as measured relative to the observer, the dropped object would no longer be accelerating, but would be in free fall right alongside of the observer and the observer would regard the dropped object as being at rest in the observer’s frame of reference. It is electrostatic repulsion with the bottom surface that allows the observers to remain in one reference frame designated to be at rest, while the dropped objects subsist in a second reference frame that is spatially accelerating relative to the first reference frame. As such, electromagnetism is an essential ingredient in a complete characterization of the Einstein Equivalence Principle, because in reality, it is the net electrostatic repulsion between electrons which supports the observers on the surfaces and gives them a measurable weight and causes the objects to eventually strike the surfaces and end their free fall as well. Succinctly: it is electromagnetism which is the barrier to gravitational free fall, and which causes objects in gravitational free fall to appear as it they are accelerating relative to observers who are not in free fall. Observers not in free fall are able to stand on a surface without passing through and so witness the relative free-fall acceleration of the dropped object, because of electromagnetism.

From a different view, while selecting a 0=v rest frame is a totally arbitrary matter of choice, there is a measurable physical difference between assigning 0=v to the observers and assigning 0=v to the dropped object in Figure 1. No matter what assignment is chosen for the rest frame, there will be a visually discernable relative acceleration through space as between the

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observers and the dropped objects. But the observers will feel an external force that is measurable with a weighing device, while the dropped objects will not feel any external force until such time as they strike the surface and have their free fall abruptly ended. This is a real, qualitative, and quantitatively-measurable, physical difference. Indeed, this is one aspect of the supplementary remark made by Einstein at the end of [1] carefully defining the zero of velocity, in response to a letter from Max Planck suggesting a clarification to the concept ‘uniformly accelerated’. So the accelerations of the dropped objects relative to the observers assigned to 0v = in Figure 1 is a free-fall acceleration along gravitational geodesic lines of least action. However, if we instead were to assign 0v = to these falling objects and thus place them into what we choose as the rest frame, the accelerations of these observers relative to these objects, although spatially of equal magnitude but opposite direction, would be forced accelerations not along gravitational geodesics. Such accelerations can only be brought about by the application of a Newtonian force through the electrostatic repulsive interactions that prevent two objects from moving through one another and instead ensure a collision if one tries to bring two objects into the same space at the same time. Let us now review all of this more quantitatively. 2. How Gauge Symmetry is used to Introduce Electromagnetic Interactions into Physical Equations Rooted in the Spacetime Metric

The modern geometric understanding of gravitation begins with a metric interval:

2ds g dx dxµ νµν= (2.1)

where gµν is the covariant metric tensor and its contravariant inverse is given by g gµσ µ

σν νδ=

with µνδ being the 4x4 Kronecker identity matrix. Here, we shall use ( ) ( )diag 1, 1, 1, 1µνη = + − − −

as the metric tensor of the tangent flat Minkowski space. The invariant ds when integrated along

a worldline between any two spacetime events A and B yields B

As ds= ∫ , which is the proper time

along the worldline when the separation between A and B is timelike with 2 0ds > as it is for all material bodies, and it is the proper length when the separation is spacelike with 2 0ds < . Of course, 2 0ds = for lightlike worldlines.

For a massive particle with timelike worldlines, it is common practice to divide the metric through by 2ds and then define the four-velocity by /u dx dsµ µ≡ , to obtain:

1dx dx

g g u u u uds ds

µ νµ ν σ

µν µν σ= = = . (2.2)

It is also common to thereafter postulate a mass m and multiply both sides of the above through by 2m while defining a four-momentum p muµ µ≡ , to obtain:

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( )( )2 dx dxm g m m g mu mu g p p p p

ds ds

µ νµ ν µ ν σ

µν µν µν σ

= = = =

. (2.3)

In a local Minkowski frame for which gµν µνη→ thus 2m p pµ ν

µνη= , it is also common

practice following Dirac [5] to employ the gamma matrices { }12 ,µ ν µνγ γ η= to write (2.3) as

{ }2 12 ,m p pµ ν

µ νγ γ= which then separates into two identical linear equations m p pσσγ= ≡ / or

0p mσσγ − = . But of course m is now really m times a 4x4 identity matrix (4)I while the 4x4

matrix p pσσγ=/ is decidedly non-diagonal and not any multiple of an identity. Thus, to write a

proper equation, we are required to introduce a four-component spinor eigenvector ( )u pσ (not to

be confused with the four-velocity uµ ) that is a function only of the four-momentum pσ and not

of spacetime, and write this deconstruction as the eigenvalue equation ( )0 p m uσσγ= − with

p mσσγ − operating on the eigenvector u. If we further multiply through by a Fourier kernel

ip xeσ

σ− to study plane wave solutions using a Dirac wavefunction ip xe uσ

σψ −= , then given that ip x ip xi e p e

σ σσ σ

σ σ− −∂ = , this eigenvalue equation becomes

( ) ( )( ) ( )( ) ( )0 ip x ip xp m p m e u i m e u i mσ σ

σ σσ σ σ σσ σ σ σγ ψ γ γ γ ψ− −= − = − = ∂ − = ∂ − . (2.4)

This is Dirac’s equation for a free electron. To introduce electrodynamic interactions which bear the net responsibility for preventing the observers in Figure 1 illustrating the Equivalence Principle from free-falling through the surfaces that support them, we subject this wavefunction to the local unitary gauge (really phase) transformation iU eψ ψ ψ ψΛ′→ = = using a real local phase ( )xµΛ and a transformation factor

iU eΛ= which is unitary given that 2

* 1U U U= = , and we insist that (2.4) remain invariant under

such transformations. From cos siniU e i a ibΛ= = Λ + Λ = + we see that Λ is simply the angle in this phase space, and that U in Uψ ψ′ = merely has the effect of rotating the orientation of ψ through a complex two-dimensional phase space without changing the magnitude of ψ . For non-abelian gauge theory used to describe, e.g., weak and strong interactions, these angles are promoted to i iTΛ = Λ where iT are the Hermitian †i iT T= generators of whatever group is being considered

and iU eΛ= remains unitary because 2 † 1U U U= = . Gauge theory was of course pioneered by

Hermann Weyl over 1918 to 1929 in [6], [7], [8] in order to place electrodynamics on a similar geometric footing as gravitation, which is a point that will be developed at length in sections … of the present paper. The particular parallel between gauge theory and gravitational theory that we shall review at present, is the requirement for using covariant derivatives to maintain symmetry.

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Recognizing that ( ) ( )i ie e iσ σ σ σψ ψ ψΛ Λ′∂ = ∂ = ∂ + ∂ Λ would change the form of (2.4) to

( ) ( )0 i m i mσ σ σσ σ σγ ψ γ ψ γ ψ′= ∂ − = ∂ − − ∂ Λ and so ruin the invariance by adding the extra term

σσγ ψ− ∂ Λ , we maintain the local gauge symmetry of (2.4) by replacing the ordinary four-gradient

σ∂ in (2.4) with a gauge-covariant derivative D ieAσ σ σ≡ ∂ + , where Aσ is the four-vector

potential of electrodynamics and e is the electric charge strength. With this, Dirac’s equation for an interacting electron becomes:

( ) ( )( ) ( ) ( )00 i D m i ieA m i m e A i m Vσ σ σ σ σσ σ σ σ σ σγ ψ γ ψ γ γ ψ γ γ ψ= − = ∂ + − = ∂ − − = ∂ − + . (2.5)

with 0V e Aσ

σγ γ≡ − defining the electromagnetic perturbation. Because

( ) ( ) ( ) ( )[ ]

i i i i

i i i i

D D e ieA e e i eA e

e e ieA e ieA e D

σ σ σ σ σ σ σ

σ σ σ σ σ

ψ ψ ψ ψ ψ

ψ ψ ψ ψ

Λ Λ Λ Λ

Λ Λ Λ Λ

′ = = ∂ + = ∂ + + ∂ Λ

′ ′ ′= ∂ + = ∂ + =, (2.6)

and in view of the parallel transformations defined by A A Aσ σ σ σ′→ ≡ + ∂ Λ and

D D ieAσ σ σ σ′ ′→ ≡ ∂ + , (2.5) will now transform as:

( ) ( )0 i ii D m i D m i e D me i D mσ σ σ σσ σ σ σγ ψ γ ψ ψ γ ψ ψ γ ψΛ Λ′ ′ ′ ′ ′= − = − = − = − . (2.7)

Dirac’s equation thus remains invariant under local gauge transformations and at the same time – as the very consequence of demanding this symmetry – now accounts for electrons interacting with an electromagnetic potential. This is how we start with a purely gravitational construct, namely the metric 2ds g dx dxµ ν

µν= of (2.1), and introduce electromagnetic interactions merely by

using symmetry principles. This is important, because later in this paper we will use these same symmetry principles to introduce the electrostatic repulsion that is responsible for the observers in Figure 1 being able to stand on the ground beneath without passing through. The Klein-Gordon equation emerges in similar fashion. Here, we start with (2.3) in the form of 2 0p p mσ

σ − = and multiply from the right by a scalar wavefunction φ to obtain

( )20 p p mσσ φ= − . If we again consider plane wave solutions of the form ip xe

σσφ −= using the

Fourier kernel, then again given that ip x ip xi e p eσ σ

σ σσ σ

− −∂ = , thus ip x ip xe p p eσ σ

σ σσ σσ σ

− −−∂ ∂ = for the

second derivative, we may write:

( ) ( ) ( ) ( )2 2 2 20 ip x ip xp p m p p m e m e mσ σ

σ σσ σ σ σσ σ σ σφ φ− −= − = − = −∂ ∂ − = −∂ ∂ − , (2.8)

The net result, ( )20 mσσ φ= ∂ ∂ + is the Klein-Gordon equation for a free scalar field φ .

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To introduce electromagnetic interactions, we demand that this equation be invariant when φ is subjected to the local unitary gauge transformation iU eφ φ φ φΛ′→ = = again using a local

phase ( )xµΛ which maintains the magnitude of φ but changes its orientation in the complex phase

space. As seen prior to (2.5), when we take the derivatives ( ) ( )i ie e iσ σ σ σφ φ φΛ Λ′∂ = ∂ = ∂ + ∂ Λ

we will obtain additional terms that ruin the symmetry. As with (2.5), we solve this by promoting

σ∂ to the gauge-covariant D ieAσ σ σ≡ ∂ + , so that (2.8) with an overall sign flip now becomes:

( ) ( )( )( )( ) ( )

2 2

2 2 2

0 D D m ieA ieA m

m ie A ieA e A A m V

σ σ σσ σ σ

σ σ σ σ σσ σ σ σ σ

φ φ

φ φ

= + = ∂ + ∂ + +

= ∂ ∂ + + ∂ + ∂ − = ∂ ∂ + +, (2.9)

with the new terms arising from the gauge symmetry used to define an electromagnetic

perturbation ( ) 2V ie A A e A Aσ σ σσ σ σ≡ ∂ + ∂ − . It will be noticed that the final term 2e A Aσ

σ φ when

it appears in the Klein-Gordon Lagrangian ( )( ) 2 2 41 1 12 2 4D Dµ

µφ φ µ φ λφ− −L = with a higher-order 41

4 λφ term and m µ→ , sits at the root of spontaneous symmetry breaking.

In the course of this development, two widely-used heuristic rules – algorithms if one

prefers – emerge, known as the “minimal prescription.” First, given that ip x ip xi e p eσ σ

σ σσ σ

− −∂ =

when one takes the four-gradient of a Fourier kernel as was done to reach (2.4) and (2.8), one will often interchange i pσ σ∂ ↔ when moving between configuration space and momentum space.

Second, given that local gauge symmetry necessitates replacing ordinary derivatives with gauge covariant derivatives D ieAσ σ σ≡ ∂ + which inherently introduce both an electric charge and an

electromagnetic potential, the combination with the first heuristic rule i pσ σ∂ ↔ leads to the

second heuristic rule interchanging the gauge covariant derivative with a canonical momentum p eAσ σ σπ ≡ − via

iD i eA p eAσ σ σ σ σ σπ= ∂ − ↔ − = , (2.10)

as between configuration and momentum space. With the foregoing review of how electric charges and electromagnetic potentials are introduced by gauge symmetry into the Dirac and Klein-Gordon equations stemming purely from the gravitational metric 2ds g dx dxµ ν

µν= of (2.1), we now turn to the equations of motion that pertain

to Figure 1. Especially, we shall now study the physics of how the observers in Figure 1 are able to stand upon their respective surfaces without passing through, which is what enables them to observe a relative downward acceleration for objects which those observers release into free fall.

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3. The Physics of Standing on the Ground in a Gravitational Field, and not Passing Through To consider equations of motion, once again the starting point is the metric

2ds g dx dxµ νµν= of (2.1) which via

B

As ds= ∫ measures the proper time along a worldline from

event A to event B. In mathematics generally, if one has a function ( )f x and wishes to find its minima (or maxima), one simply ascertains those places at which its derivative / 0df dx= . Based on the view that particles and systems will seek the lowest states of energy and follow the paths of least resistance and least distance and least time, variational physics provides the tools for mathematically deducing generalized “least action” minima in various guises. As it turns out, nature has often obliged the view that physical systems will pursue paths of least action, least resistance, least energy, and least proper time or length, by validating via empirical observation, what is mathematically deduced by variational physics. As regards classical, subliminal, material particles moving through spacetime, one determines the equation of motion by taking and

minimizing the variation of the proper time, 0B

As dsδ δ= = ∫ , and one finds that the equation of

motion so-derived accords with what is empirically observed.

In §9 of his landmark 1916 paper [4], Albert Einstein first calculated the variation 0B

Adsδ= ∫

of the linear metric element 2ds g dx dxµ νµν= of (2.1) between any two spacetime events A and B

at which the worldlines of different observers meet so that their clocks and measuring rods and scales can be coordinated at the outset A and then compared at the conclusion B. In so doing, as we shall soon review in detail here ([9] contains a very good online review of this), he deduced the geodesic equation of motion

2

2

d x dx dx

ds ds dsu uβ β µ ν

µ µ

β µ ν

ν ν= −Γ = −Γ (3.1)

for a particle in a gravitational field, wherein the ( )12 g g ggβ

µν α µβα

µ να ν αµν−Γ = − ∂∂ − ∂ of

Christoffel capture all required information about the gravitational metric tensor gµν . In fact, for

Figure 1(a), it is equation (3.1) which tells us the precise geodesic path that will be followed by the dropped object, until it hits the earth’s surface and has its geodesic motion stopped because of the net electrostatic repulsion between its electrons and the earth’s electrons. It is the empirical confirmation of the motions predicted by (3.1) and its integral 2ds g dx dxµ ν

µν= of (2.1), including

perihelion precession, gravitational light deflection, and gravitational redshifts, which provide empirical validation that General Relativity does indeed correctly describe the natural world, or at least does so more accurately than Newtonian gravitation.

Once the freely falling objects hits the surfaces below the observers in Figure 1, (3.1) no longer applies alone, but must be supplemented with something else. Additionally, neither does (3.1) describe the motion of the observers in Figure 1. These observers are not in free fall and so they do not follow the path of (3.1). In fact, given that it is electrostatic repulsion which in all

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cases stops and bars any free fall along the path (3.1), the impediment to the motion of (3.1) is given by the Lorentz force law:

2

2

d x dxF ma m qF qF u F J

ds ds

β σβ β β β σ β σ

σ σ σ= === = , (3.2)

because this is the generalized physical law which contains the electrostatic Coulomb interaction when taken in the rest frame. Here, F β

σ is the mixed electromagnetic field strength tensor and q

is the electric charge moving in this field. By putting the mass m next to the acceleration rather than leaving it in an /q m ratio multiplying F uβ σ

σ , we make clear that this truly is a Newtonian

force four-vector F maβ β= (distinguish from F βσ by the one versus two indexes). This force

becomes non-zero whenever there is an electromagnetic field 0F βσ ≠ and there are electric

charges 0q ≠ situated in this field. By also using the electric current J quσ σ= , we highlight how the Lorentz force describes electric currents situated and moving in electromagnetic fields. As we shall review momentarily, in electrostatics the above is an equation for a repulsive force between a charge q+ and an electrical field strength ( )0

01/ 4 /kF Qπε= =E x provided by a second charge

Q+ . For an attractive force one simply reverses the sign.

Because (3.2) is a classical force, and because it can be applied to fields F βσ sourced by

large numbers of charges and applied to currents Jσ containing large numbers of other charges, we can use (3.2) to represent the net electromagnetic forces which cause the observers in Figure 1 to feel a measurable weight (illustrated as 52 kg) beneath their feet and which cause the objects in Figure 1 to cease their motions (3.1) when those objects eventually strike the surface. But before we proceed, let us clarify more deeply how we may do this:

Unless the observer has charged up with a Van de Graaff generator prior to stepping into

the experiment of Figure 1 or there is a nearby lightning storm or the like, we may assume that the observer and the housing and the dropped object and all of the other material participants in Figure 1, in the net, are electrically neutral. This is because they are all material objects constructed from atoms and molecules containing an equal number of electron and protons. However, because the electron shells envelop the nuclei which contain the protons, the surfaces of the material bodies in Figure 1 will expose electrons, not protons. So when we say that the observer is “touching” the ground, what we are really saying is that the surface electrons of the observer are close enough to the surface electrons of the ground so that the repulsion between these two sets of electrons becomes significant enough to stop the observer from getting any closer to the ground than he or she already is. How close is “close”? Certainly, the distance maintained between the surface electrons of the observer and those of the ground will be greater than the Bohr radius

80 / 5.292 10 mmea m cα −= = ×ℏ , because if it were smaller, the observer and the ground would be

part of the same molecular system, and not two separate molecular systems. And certainly, these electrons do come closer to one another than a small fraction of a millimeter, because otherwise one would be able to visually discern a separation between the observer and the floor and so they would no longer be “touching.”

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Consequently, we may define two objects to be “touching” – physically – when they get close enough to one another that the surface electrons of each start repelling one another and thereby make it no longer possible for those objects to get any closer. That is, “touching” is defined by the activation of electrostatic repulsion between the surface electrons of the two “touching” objects. To those electrons which do come close enough to actuate this “touching,” the localized environment at the very short distances involved is not electrically neutral, even though as a whole, the observer and the ground are in fact electrically neutral. So this is another way of saying that net electrical neutrality is global, not local.

With all of this in mind, we start with the Lorentz force law (3.2), and may ascribe F β

σ to

represent the field strength associated the surface electrons of the ground that are involved in this “touching,” and may ascribe Jσ to represent the current associated with the surface electrons of the observer (or the dropped object when it strikes the ground) which are involved in the touching. Or, vice-versa, we may assign F β

σ to the observer’s and Jσ to the ground’s electrons. For

purposes of the development from here, we shall make the former assignments. With F β

σ representing the net fields of the surface electrons of the ground and Jσ

representing the net currents of the surface electrons of the observers, let us return to Figure 1. We now focus especially on Figure 1(a) which involves gravitational fields and, because the observer is standing on the ground, which also involves electrostatic repulsions and thus the Lorentz force law (3.2). To account for both the gravitation and the electrodynamics, it becomes necessary to supplement (3.1) with (3.2), and so write the total motion, with both the gravitation and electrodynamics accounted for, in the single equation:

2

2

d x du dx dx q dx qg F g F u

ds ds ds ds m ds mu uβ β µ ν

µν µν

β β µ ν σβα βα σ

σα σα= − +Γ= −Γ+ = , (3.3)

using g F Fβα β

σα σ= to show the field strength with all raised contravariant indexes. The above

(3.3) is well-known, settled physics, see, e.g., the online [10]. It is important at this moment to point out that in (3.3), all we have done is manually supplement the geodesic equation (3.1) for gravitational motion – derived via least action variation from the metric (2.1) – with the Lorentz force law of (3.2). This is very unsatisfying, because

while (3.1) is obtained through the variation 0B

Adsδ= ∫ from the metric 2ds g dx dxµ ν

µν= of (2.1)

which is entirely geometric, (3.3) is not the result of any such least action variation. We simply take the Lorentz force and tack it on, because we are fortunate enough to know about the Lorentz force from a wide range from electrodynamic studies and observations. Indeed, notably absent from §9 of Einstein’s General Relativity paper [4] which first developed the geodesic equation

(3.1), was a similar development of the Lorentz force law ( ) ( )2 2 // /d x ds e m F dx dsαµ µ α= . As will

also be reviewed later in Sections … of this paper, subsequent papers by Kaluza [11] and Klein [12] did succeed in explaining the Lorentz force as a type of geodesic motion and even gave a geometric explanation for the electric charge itself, but only at the cost of adding a fifth dimension to spacetime and curling that dimension into a cylinder. To date, a century later, there still does

Jay R. Yablon

11

not appear to have been any fully-successful attempt to obtain the Lorentz force from a geodesic variation confined exclusively to the four dimensions of ordinary spacetime.

In section … to follow, we shall solve this long-standing problem. As to how this will be done, we note for the moment that by appending the Lorentz force to the geodesic motion in (3.3), we are appending an electromagnetic expression to a gravitational equation rooted in the spacetime metric (2.1). And as reviewed in section 2, the proper way to introduce electrodynamics into gravitational geometry, is to impose gauge symmetry. This means, as summarized in (2.10), that ordinary derivatives become gauge-covariant derivatives via i iD i eAσ σ σ σ∂ ⇒ = ∂ − and ordinary

momenta become canonical momenta via p p eAσ σ σ σπ⇒ = − . Following this path, we shall

show how ordinary geodesics (3.1) become gauge-covariant geodesics (3.3) owing to nothing more than the completely natural application of gauge symmetry. But for the moment, we shall simply proceed from that patchwork of (3.3) with the understanding that we shall later derive (3.3) from least action principles. Thus, let us now use (3.3) as given, to quantitatively explore the counterbalancing of gravitational and electrostatic forces which cause the observers in Figure 1 to remain standing with measurable weight upon the surfaces of Figure 1, and to not be in the gravitational free fall described by (3.1) absent the Lorentz force. As earlier noted, and related to the Planck-Einstein dialogue at the end of [1] which was earlier discussed, the observers of Figure 1 may choose to define their own frames of reference as rest frames for which v=0, even though they are feeling and can measure a force / weight between themselves and the surfaces upon which they stand. Let us now focus especially on the Figure 1(a) observer standing on the surface of the earth in a gravitational field and dropping an object into a brief free fall. If this observer choses him or herself to define v=0, then the velocity four-vectors in view of (2.2) will become ( )1,0,0,0uσ = in this rest frame. If this observer also choses

to define him or herself as remaining at rest over time so that ( )( ) 1,0,0,0uσ τ = = constant over the

observer’s own measurements of proper time, then the observer’s own acceleration 2 2/ / 0d x ds du dsβ β= = will become zero, by self-declaration. Thus, the left hand side of (3.3)

will become zero. Further, as discussed several paragraphs back, the observer can choose to have F βα represent the surface electromagnetic fields of the ground upon which he or she is standing at relative rest, and to have J quσ σ= represent the currents of his or her own surface electrons.

As a result, all of the velocity vectors in (3.3) can be set to ( )1,0,0,0uσ = . With all of this, (3.3)

reduces to:

00 000

00

000

q q qg F u g F u g F

m mu

mu u uβα σ βαβ µ ν β β

µνβα

σα α α−Γ −Γ −Γ= + = + = + . (3.4)

Next, given that ( )12 g g ggβ

µν α µβα

µ να ν αµν−Γ = − ∂∂ − ∂ and given the symmetry of the

metric tensor, we may determine that ( )100 002 0 02gg gβαβ

α α−Γ = ∂ − ∂ . But the gravitational

field in Figure 1(a) is not time-dependent, because for someone standing on the earth’s surface, that field always has the acceleration value 29.81 m/sg ≅ which does not change with time.

Therefore, 0 0 0g α∂ = , and so 100 002 g gα

βαβ−Γ = ∂ . Likewise because of the time-independence,

Jay R. Yablon

12

0 00 0g∂ = , so that 100 002

kkg gββ−Γ = ∂ where 1,2,3k = runs over the three space dimensions

only, excluding the time dimension. Using all of the foregoing, we may simplify (3.4) to:

( )1 2 31 100 1 00 2 00 3 020 0020 k

kg gq q

g g F g g g g Fm m

g gβ βα β β β βαα α∂= ∂= + +∂+ +∂ . (3.5)

Progressing, we also know that in the linear field approximation which certainly applies to

the observers in Figure 1(a), the metric tensor g hµν µν µνη ρ= + with 16 Gρ π= . (Sometimes

this is written as g hµν µν µνη κ= + with 16 Gκ π= , but we use ρ to avoid confusion with the 21

28 Gκ π ρ= = that appears in the Einstein equation 12T R g Rµν µν µνκ− = − .) Because the

Minkowski tensor µνη is constant, and given that g hµν µν µνη ρ= + in this linear approximation,

the gradient g hα µν α µνρ∂ = ∂ . Consequently, we may rewrite (3.5) in terms of 00h as:

( )1 100 1 00

1 2 32 00 32 20 0000 k

k q qg g F g g g g F

m mh h h hβ βα β β β βα

α αρ ρ= + = + +∂+∂ ∂ ∂ . (3.6)

However, 1 002 hρ = Φ is the Newtonian potential of a gravitational field, and for a mass M

such as the earth this potential is 1002 /h GM rρ = Φ = − at a radial distance r. In Figure 1(a), we

may indeed take this mass M to be that of the earth, and the distance r to be the radius of the earth. So for the observers situated exclusively along the z axis in relation to the earth’s center, such as the observer of Figure 1(a), we may use 1

002 /h GM zρ = − , and eliminate the first two terms

1 00 01

2 02g gh hβ β+∂ ∂ , thus simplifying (3.6) to:

( )3 3 3 30 0

13 00 0 02 230

q q GM q GM qg g F g g F g g F g g F

m m z z m z mhβ βα β βα β βα β βα

α α α αρ ∂ = + = Φ+ = − + = +∂ ∂ ∂ .(3.7)

The above contains four independent equations running over the free index 0,1,2,3β = . But for a spherically-symmetric gravitational field which is very closely approximated by the earth if we neglect the effects of the earth’s rotation, only the diagonal components of gµν will be non-

zero. Consequently, 0α = and 3β = are the only indexes in (3.7) that will yield a result other than 0 0= . As a result of this spherical symmetry, (3.7) reduces to the single equation:

33 30 33 3300 00 002 2 2 2

10 e

z

k QqGM q GM q GMg g F g g E g g

z m z m z m z= + = + = +

′. (3.8)

Above, we have used the electric field component 30

zF E= , and then used the Coulomb electric

field 2/z eE k Q z′= at a vertical distance z′ above a charge Q , where 02

01/ 4 / 4ek c πµπε= = is

Coulomb’s constant and 0 0c ε µ= is the relationship between permittivity 0ε and permeability

0µ through which Maxwell proved that electromagnetic waves propagate at the speed of light c.

Jay R. Yablon

13

Note that we use z′ rather than z for the electrostatic distance, because z is already being used to represent the distance of the observer from the center of the earth in the Newtonian field. When it comes to mutual repulsion between the charges on the feet of the observer and the charges on the ground, i.e., when it comes to the observer “touching” the ground, those interactions are taking place over a much shorter distance which are fractions of a millimeter yet larger than the Bohr radius, as earlier discussed. Of course, although the observer in Figure 1(a) does experience a gravitational field, that

field is weak and so approximates ( ) ( ) ( )diag diag 1, 1, 1, 1g µνµν η≅ = + − − − , so 33

00 1g g≅ − ≅ − .

With this approximation, (3.8) now becomes:

2 2

10 e z

z

k Qq FGMa

z m z m= − + = =

′. (3.9)

Above, aside from this reduction of (3.8), we have kept in mind that the zero on the left originated at (3.4) when the observer chose his or her reference frame to be the rest frame and remain so over time, and so we set 2 2/ / 0d x ds du dsβ β= = . Consequently, the overall expression in (3.9) which is equal to zero, is an acceleration descending from 2 2/ / 0d x ds du dsβ β= = . Following all of the reductions from (3.4) through (3.9), this zero has become the z-axis acceleration

2 2/ / 0z za dv dt d z dt= = = . This is why we have also appended /z za F m= to the right side of

(3.9). This acceleration is equal to zero, not in any absolute sense, but only in a relative sense, because the Figure 1(a) observer chose to have his or her reference frame be the rest frame and remain so over time. From some other reference frame, such as that of the dropped object in gravitational free fall that we shall momentarily examine, this acceleration would not be zero. We can gain further insight if we multiply (3.9) through by the mass m to look at this in terms of the z-axis force zF . Doing so, we now obtain:

2 20 e z z

Mm QqG k ma F

z z= − + = =

′. (3.10)

Here, we arrive at a mathematical description of the offsetting forces which enable the observer to stand on the ground in a gravitational field, not pass through the ground, and declare his or her reference frame on a relative basis to be the rest frame over time. The overall force is zero by choice of reference frame, but there are in fact two offsetting terms which net out to the zero force and the zero acceleration in chosen rest frame of reference, and they are in the form of Newton’s law and Coulomb’s law. To discuss these, it helps to simply move the Newton’s law to the left to obtain:

2 2e

Mm QqG k

z z=

′. (3.11)

Now let’s parse out what this all means.

Jay R. Yablon

14

On the left is Newton’s law for the gravitational “force” and on the right is Coulomb’s law for the electrostatic force. In (3.11) we have omitted any reference to a force, because from (3.10) we see that the net force in the chosen rest frame is zero. Prior to (3.3) we noted that we may ascribe F β

σ to represent the field strength associated with the surface electrons of the ground, and

may ascribe Jσ to represent the current associated with the surface electrons of the observer. In (3.10) and (3.11) those have respectively descended to Q and q. So when the observer is “touching” the ground, a net surface charge of Q from the ground is repelling a net surface charge of q from the observer because these charges are close enough to actuate this repulsion, at some mean small distance z′ which is a fraction of a millimeter but larger than the Bohr length. And so the overall force of this repulsion is given in the form of Coulomb’s law.

But how strong is this repulsion, numerically? For this we turn to the left side of (3.11)

which tells us that the net totality of all these electrostatic forces is precisely equal to the expression 2/GMm z where G is Newton’s constant, M is the mass of the earth, m is the mass of the observer,

and z is the distance to the center of the earth. This expression 2/GMm z is precisely measurable. And it is because these two expressions 2/ek Qq z′ and 2/GMm z are equal, that the observer is

able to stand on the surface of the earth without passing through and is able ascribe to him or herself a rest frame with no net acceleration and no net force notwithstanding the counterbalancing of the Coulomb force (which really is a Newtonian force) against the gravitational assemblage

2/GMm z which has force dimension 2/md t and is regarded as a force in Newtonian physics but is really just a result of the geodesic gravitational motion of (3.1). Put differently, 2/ek Qq z′ is the

net Coulomb electrostatic repulsive force between the observer and the ground, and 2/GMm z is not a force per se, but rather tells us the actual numerical, measurable magnitude of this net Coulomb repulsive force which is a force per se.

This is what balances electrostatics against gravitation and keeps the Figure 1(a) observer

situated on the surface of the earth rather than falling through the planetary surface. What is experienced in a person’s measurement of weight on a scale is the Coulomb force blocking the gravitational free-fall motion. This is what also blocks a person on the upper floors of a tall building from falling through the floors to the ground, and it also keeps an airplane passenger inside the airplane rather than passing through the fuselage toward a fatal free-fall to earth that would occur when the free fall terminates against the Coulomb forces at the earth’s surface. The relationships (3.10) and (3.11) are is therefore central to our actual ever-present physical experience of the world.

All of the foregoing was calculated in the rest frame of the observer, but to fully understand

the relativistic explanation of these forces and accelerations and the Equivalence Principle, we should also see how this is all described if we now choose the rest frame in Figure 1(a) to travel with the dropped object rather than with the observer. Here, with the results of (3.10) and (3.11) already in hand, we need nothing fancier than Newtonian physics and some careful deduction.

First, spatially, since the dropped object is accelerating downwards relative to the observer

by 2/GM z− which is the Newtonian potential in (3.9), we can likewise say that the observer is accelerating upwards by 2/GM z+ relative to the dropped object in free fall. That is, the motions

Jay R. Yablon

15

are equal in magnitude and opposite in direction. Second, no amount of relative motion or acceleration will cause the falling object to “see” the observer either passing through the earths’ surface or separating from the earth’s surface. The dropped object will still “see” the gravitational and Coulomb “forces” counterbalancing one another according to (3.10) and (3.11). Therefore, using (3.11) to represent that the observer is still “touching” the floor, the dropped object in Figure 1(a) will “see” the observer accelerating upwards along the z axis according to:

2 20z

z e

F M q Qa G k

m z m z= = + = + ≠

′, (3.12)

and so will also “see” a non-zero net force acting on the observer given by:

2 20z z e

Mm QqF ma G k

z z= = + = + ≠

′. (3.13)

So relative to the dropped object, the observer is being pushed upwards by a Coulomb force

2/z z eF ma k Qq z′= = for which the measurable numerical magnitude is 2/GMm z , because the

observer is “touching” the floor due to his or her surface electrons with net charge q at a very small distance z′ being close enough to the floor’s surface electrons with net charge Q so as to generate a repulsive force which forces the observer to accelerate upwards. And again, as discussed after (3.11), the magnitude of this Coulomb force is simply equal to the gravitational assemblage of terms 2/GMm z given by Newton’s law. Aside from tidal forces, this description of what the dropped object is “seeing,” is precisely what is illustrated in Figure 1(b): a force F ma= pushing the observer upwards with an acceleration a g= . This is simply the other half of Einstein’s Equivalence Principle. In sum, Figure 1(a) represents an observer standing in a gravitational field and dropping an object when the rest frame is taken to be that of the observer and the governing equations are (3.9) through (3.11), while Figure 1(b) represents an observer standing in a gravitational field when the rest frame is taken to be that of the freely-falling object and the governing equations are (3.12) and (3.13). That is the Equivalence Principle, mathematically explicated in complete detail. 4. “Nowhere You Can Be that Isn’t Where You’re Meant to Be”*: A Review of the Least Action Derivation of Gravitational Geodesic Motion We have just shown how the combined gravitational and electrodynamic equation of motion (3.3) assumes a central role in the physics of an observer standing on the ground in a gravitational field, and not passing through that ground, and leads to the counterbalancing of electrostatic and Newtonian gravitational forces represented in the rest frame of an observer by equations (3.10) and (3.11). This is essential to a complete mathematical explication of the Equivalence Principle, because without these electrostatic repulsions, the observer could never be held in a position to observe gravitational free-fall accelerations but would instead free fall right alongside of the freely-falling objects being observed. * The Beatles, All You Need is Love

Jay R. Yablon

16

But as discussed following (3.3), it is highly unsatisfying to have to append the Lorentz

force law (3.2) to the geodesic equation of motion (3.1). For, while the geodesic motion (3.1) is

obtained from the metric tensor (2.1) by 0B

Adsδ= ∫ which minimizes the proper time, i.e., moves a

particle on a worldline from event A to event B in the least possible proper time, there does not appear to date to be any known derivation, in the four dimensions of spacetime alone, whereby the Lorentz Force law is also seen as a principle of generalized “least action,” in this case, least proper time. As noted, and as will be studied in detail in section …, Kaluza [11] and Klein [12] did obtain such a derivation in five dimensions. But to date, there does not appear to have been any such derivation in four dimensions alone. We now turn out attention to solving this long-standing problem, and will use the review of gauge theory in section 2 as the basis for doing so. As a foundation for revealing the Lorentz force law one in which a charged particle in an electromagnetic field follows a path through spacetime which minimizes its proper time, let us begin with a review of how the gravitational geodesic equation (3.1) is obtained from the spacetime metric (2.1). In essence, we shall be reviewing the calculation which Einstein first presented in §9 of his 1916 paper [4] on General Relativity, but with an eye toward laying the foundation for deriving the Lorentz force in the same way and simplifying that derivation. The online reference [9] provides a very good summary of this derivation as well. To derive the equation of gravitational geodesic motion (3.1) from the spacetime metric (2.1), we first turn (2.1) into (2.2) as shown, and then take the square root of (2.2) to write:

1dx dx

gds ds

µ ν

µν= . (4.1)

We then use this “1” to write the variational minimization as:

0B B

A A

dx dxds ds g

ds ds

µ ν

µνδ δ= =∫ ∫ . (4.2)

Applying δ to the integrand and using the “1” of (4.1) to clear the denominator yields:

1 10

2 2

B B B

A A A

dx dxg

ds ds dx dxds ds ds g

ds dsdx dxg

ds ds

µ ν

µν µ ν

µνµ ν

µν

δδ δ

= = =

∫ ∫ ∫ . (4.3)

The variation then distributes via the product rule according to:

10

2

B B

A A

dx dx d x dx dx d xds ds g g g

ds ds ds ds ds ds

µ ν µ ν µ ν

µν µν µνδ δδ δ

= = + +

∫ ∫ . (4.4)

Jay R. Yablon

17

Now, one can use the chain rule in the small variation δ → ∂ limit to show that g x gα

µν α µνδ δ ∂= . Indeed, the generic calculation that yields this result for any field φ (taking

δ ≅ ∂ ), is:

x

xx

xx x

x

αα α α

α α ααφ δφφ δ δφ φδ δ∂= ≅ =∂

∂∂ =

∂∂

∂. (4.5)

Additionally, we may use the symmetry of gµν to combine the second and third term inside the

parenthesis in (4.4). Thus, (4.4) becomes:

10 2

2

B B

A A

dx dx d x dxds ds g

ds ds ds sx g

d

µ ν µ να

α µν µνδδ δ

= =

∂ +

∫ ∫ . (4.6)

The next step is to integrate by parts. From the product rule, we may obtain: d dx d dx d dx d x dx d dx

x g x g x g g x gds ds ds ds ds ds ds ds ds ds

ν ν ν µ ν νµ µ µ µ

µν µν µν µν µνδδ δ δ δ = + = +

.(4.7)

It will be recognized that the first term after the second equality in (4.7) is the same as the final term in (4.6) up to the factor of 2. So we use (4.7) in (4.6) to write:

10 2 2

2

B B

A A

dx dx d dx d dxds ds x g x g

ds ds ds ds ds dx g

s

µ ν ν νµ µ

µνα

µα µν νδ δ δδ

= = + −

∂∫ ∫ . (4.8)

The middle term in the above, which is a total integral, is equal to zero because of the boundary conditions on the variation. Specifically, this middle term is:

0B

B B

A AA

d dx dx dxds x g d x g g x

ds ds ds ds

ν ν νµ µ µ

µν µν µνδ δ δ = = =

∫ ∫ . (4.9)

This definite integral is zero because the two worldlines intersect at the boundary events A and B but have a slight variational difference between A and B otherwise, so that ( ) ( ) 0x A x Bσ σδ δ= =

while 0xσδ ≠ elsewhere. Therefore we may zero out the middle term and rewrite (4.8) as:

10 2

2

B B

A A

dx dx d dxds ds x g

ds ds dsx g

dsα

µ

α

νµ

νµν

ν

µδδ δ

= = −

∂∫ ∫ . (4.10)

Next, in the final term above, we distribute the /d ds to each of gµν and /dx dsν via the

product rule, so that this becomes:

Jay R. Yablon

18

2

2

10 2 2

2

B B

A A

dgdx dx dx d xds ds x x gx g

ds ds ds ds ds

µ ν ν νµνµ

να

α µµ

ν µδδ δ δ = = − −

∂∫ ∫ . (4.11)

For the first time, we see an acceleration 2 2/d x dsν . It is then straightforward to apply the chain

rule to deduce that ( )/ /dg ds g dx dsαµν α µν= ∂ , which is a special case of the generic relation for any

field φ given by: d dx dx

ds x ds ds

α α

ααφ φ φ∂= =∂

∂. (4.12)

As a result, (4.11) now becomes:

2

2

10 2 2

2

B B

A A

dx dx dx dx d xds ds x g x g

ds ds ds dsx g

ds

µ ν α ν νµ µ

αα

µν µα µ ννδδ δ δ = = − ∂ −

∂∫ ∫ . (4.13)

At this point we have a coordinate variation in front of all terms, but the indexes are not the same. So we need to re-index to be able to factor out the same coordinate variation from all terms. So we rename the summed indexes µ α↔ in the second and third terms and factor out

the resulting xαδ from all three terms. And we also use the symmetry of gµν to split the middle

term into two, then cycle all indexes, then factor out all the terms containing derivatives of gµν .

The result of all this re-indexing, also moving the outside coefficient of ½ into the integrand, is:

( )2

2

10

2

B B

A Ax g

dx dx d xds ds g g g

ds ds ds

µ ν ν

µ να ν αα

µ αα µ ννδδ = = − ∂ − ∂ −

∂∫ ∫ . (4.14)

Now we are ready for the final steps. Because the worldlines under consideration are for material particles, the proper time 0ds≠ . Likewise, while ( ) ( ) 0x A x Bσ σδ δ= = at the

boundaries, between these boundaries where the variation occurs, 0xσδ ≠ . Therefore, for the overall expression (4.14) to be equal to zero, the expression inside the large parenthesis must be zero. Consequently:

( )2

2

10

2

dx dx d xg g g

dg

s ds ds

µ ν ν

µ να ν αµν µα αν= −∂ ∂ − ∂ − . (4.15)

From here, we multiply through by gβα , apply ( )12 g g ggβ

µν α µβα

µ να ν αµν−Γ = − ∂∂ − ∂ for the

Christoffels, flip the sign and put the acceleration term first to obtain:

2 2

2 20

d x dx dx d x

ds ds s dsu u

d

ββ β µ ν

µν µ

β

ν

µ ν

= + = +Γ Γ . (4.16)

Jay R. Yablon

19

This is the geodesic equation (3.1) for the motion of a particle in free fall in a gravitational field. Given its derivation as the least-proper-time variation of the spacetime metric 2ds g dx dxµ ν

µν= of

(2.1), it is not uncommon to regard 2ds g dx dxµ νµν= as the first integral of this equation of motion.

PART II: THE LORENTZ FORCE AS PURE GEODESIC MOTION IN FOUR-DIMENSIONAL SPACETIME 5. Why Associating the Canonical Momentum in Gauge Theory with Mass Times Velocity Leads to the Wrong Equation of Motion The gravitational geodesic equation (3.1) a.k.a. (4.16) is extremely attractive theoretically, because it is directly rooted in the spacetime metric 2ds g dx dxµ ν

µν= and so has a natural

interpretation in terms of objects travelling through geodesics in the spacetime geometry because those are the simplest paths to follow, minimizing the proper time to get from event A to event B. It is also very attractive as shown in section 2 that the same metric is at the root of the Dirac and Klein Gordon equations, and that to introduce electromagnetism, all one needs to do is apply gauge symmetry. And, it is empirically attractive that all of the foregoing have been uniformly and consistently validated by the natural world as providing empirically-correct (really, to-date-contradicted) descriptions of nature.

But then, as shown in section 3, these objects travelling through paths of least proper time in gravitational free fall hit an obstacle: They fall to the point where their surface electrons grow close enough to the surface electrons of some other body such as the surface of the earth, and the electrostatic repulsion of this event causes an abrupt end to their travels along gravitational geodesics. Now, the Lorentz Force enters the picture also, in a way most generally described by (3.3). So the question arises: once this occurs, will these objects still be following paths of least

proper time? Or, is there no variation of the same form 0B

Adsδ= ∫ which can be used to obtain

equation (3.3) with the Lorentz force without going to a fifth dimension as was done by Kaluza [11] and Klein [12]?

To tackle this question, we return to ( )( )( )20 ieA ieA mσ σσ σ φ= ∂ + ∂ + + , which is the

Klein-Gordon equation (2.9). We start here, because it is clear how these potentials have entered this metric-rooted equation by requiring local gauge symmetry, so we avoid applying the heuristic rule / algorithms (2.10) and stay directly in contact with the physics requirement for local gauge

symmetry. If we return to using plane wave solutions of the form ip xeσ

σφ −= , then again given that ip x ip xi e p e

σ σσ σ

σ σ− −∂ = hence ip x ip xe p p e

σ σσ σσ σ

σ σ− −−∂ ∂ = , it is readily shown that this equation in

momentum space is:

( )( )( ) ( )2 20 p eA p eA m mσ σ σσ σ σφ π π φ= − − − = − . (5.1)

Jay R. Yablon

20

Here, we can strip off the wavefunction and still maintain a proper equation (unlike for the Dirac equation, see prior to (2.4)), and after moving 2m to the left, write:

( )( )2m p eA p eAσ σ σσ σ σπ π= = − − . (5.2)

Now, a key question arises: How do the terms in (5.2) relate to the expression

/mu mdx dsσ σ= for the particle mass times the four-velocity? Prior to applying gauge theory, this was one and the same as the kinetic momentum, /p mu mdx dsσ σ σ= = . But what about now?

One possibility is that /p mu mdx dsσ σ σ= = is still the correct relationship. The other possibility

is that /p eA mu mdx dsσ σ σ σ σπ = − = = , in other words, that the canonical momentum is now equal to the mass times the four velocity. In the latter case, this would mean that the original kinetic momentum is now modified to be /p mu eA mdx ds eAσ σ σ σ σ= + = + , and is no longer mass times velocity. It is now mass times velocity, plus charge times potential. It is commonly believed that the correct answer is that p mu eAσ σ σ= + , because this

superficially applies a heuristic algorithm that mu pσ σ= gets promoted to muσ σπ= which seems

to be in accord with the algorithms of (2.10) that promote ieAσ σ σ∂ → ∂ + and p p eAσ σ σ→ − .

But we need to be careful about relying just on algorithms and should not try to answer this in the abstract. We should ascertain what equation of motion is associated with each of these two choices, and then choose the answer that gives us the motion that is actually empirically-observed. Specifically, what changes as between 2m p pσ

σ= in (2.3) and 2m σσπ π= in (5.2) is that in the

latter relationship, there are now electric charges and electromagnetic potentials. And because of this, we should expect the equation of motion resulting from minimizing the variation via

0B

Adsδ= ∫ will be (3.3) including the Lorentz force law, and not simply (3.1) for the gravitational

motion absent the Lorentz force law. So let’s calculate. Starting with (5.2), let’s test the later possibility that /mu mdx dsσ σ σπ = = . If that is the case, then:

( ) ( )2 2 2 dx dxm p eA p eA m u u m

ds ds

σσ σ σ σ σ

σ σ σ σπ π= = − − = = . (5.2)

Dividing out the 2m and showing the metric tensor, we then obtain:

1dx dx

gds ds

µ ν

µν= , (5.3)

which is (2.2) all over again. If we then take the square root as in (4.1) and use this as the “1” in

the variation 0B

Adsδ= ∫ to specify the path of least proper time as in (4.2) and then carry out the

calculation from (4.2) through (4.16), we will obtain the exact same equation (4.16) for the

Jay R. Yablon

21

gravitational geodesic motion, with the Lorentz force nowhere to be seen. So although we have used gauge theory to naturally introduce electric charges and electromagnetic potentials into the

interacting Klein-Gordon equation ( )( )( )20 ieA ieA mσ σσ σ φ= ∂ + ∂ + + , this has done nothing to

change the equation motion. All we have – even with electrodynamics now in the picture – is gravitational motion. This simply cannot be correct. With electrodynamics in play, there must be a Lorentz force law. This leaves the second alternative: that /p mu mdx dsσ σ σ= = still remains the correct relationship between physical momentum and mass times velocity, even after gauge symmetry has been applied. The heuristic algorithm does not apply in the naive form whereby mu pσ σ= gets

promoted to muσ σπ= . Rather, just as gauge theory heuristically promotes ieAσ σ σ∂ → ∂ + and

p p eAσ σ σ→ − , it must also promote (3.1) to (3.3), that is, it must promote:

� �

2 2

2 2

d x dx dx d x dx dx q dxg F

ds ds ds ds ds ds m ds

D D ieA

p p eA

β µ ν β µ ν σβαβ

σα

σ σ σ σ σ

σ σ σ σ σ

βµν µν

π π

= → = +

= ∂ → = ∂ += → =

−Γ −Γ

−

������� ������� �������

, (5.4)

How do we prove this to be the case? By showing that this alternative of maintaining

/p mu mdx dsσ σ σ= = as the mass / velocity / kinetic momentum relationship leads directly to the Lorentz force law in the form of (5.4), which is the motion empirically observed when electric charges are present in electromagnetic fields. We validate general relativity by the classical motions it predicts being in accord with motions that are observed, and we must do the same here. In fact, as we shall now show, if we do maintain the usual relationship /p mu mdx dsσ σ σ= = even

after gauge symmetry as brought the terms eAσ into the our physical equations, the equation of motion will change to the empirically-confirmed (5.4) above, which contains both the gravitational motions together with the Lorentz force in precisely the manner that is schematically illustrated above. This, and not mu p muσ σ σ σπ= → = , we shall see, is the correct heuristic parallel. 6. Derivation of the Lorentz Force as Geodesic Motion, in Four Spacetime Dimensions Only

Let us return to ( )( )2m p eA p eAσ σ σσ σ σπ π= = − − in (5.2), but now, set the kinetic

momentum to /p mu mdx dsσ σ σ= = , just as we do prior to requiring local gauge symmetry. Thus:

( )( ) ( ) ( )2 dx dxm p eA p eA m eA m eA mu eA mu eA

ds ds

σσ σ σ σ σ σσ

σ σ σ σ σ σπ π = = − − = − − = − −

.(6.1)

We then divide out 2m to obtain:

Jay R. Yablon

22

2 2

2 2

1

2 2

dx e dx e e eA A u A u A

ds m ds m m m

dxe e dx e dx eu u A u A A A A A

m m ds ds m ds m

σσ σ σσ

σ σ σ

σ σσ σ σ σσ

σ σ σ σ σ

= − − = − −

= − + = − +

. (6.2)

For the square root, making gµν explicit in the first term, contrast (4.1), we easily find:

2

21 2

dx dx e dx eg A A A

ds ds m ds m

µ ν σσ

µν σ σ= − + . (6.3)

Then, contrast (4.2), if we use this as the “1” in the variation:

2

20 2

B B

A A

dx dx e dx eds ds g A A A

ds ds m ds m

µ ν σσ

µν σ σδ δ= = − +∫ ∫ , (6.4)

and carry out the rest of the calculation that was done from (4.2) through (4.16), the result, as we shall now show, is that we obtain (3.3) for the gravitational geodesic motion together with the Lorentz force motion. Let us proceed. First, we apply the variation δ to the integrand and use the “1” of (6.3) to clear the denominator, contrast (4.3), which yields,

2

2

10 2

2

B B

A A

dx dx e dx eds ds g A A A

ds ds m ds m

µ ν σσ

µν σ σδ δ = = − +

∫ ∫ . (6.5)

We then use the product rule to distribute the variation, contrast (4.4), as such:

2 2

2 2

10

22 2 2

B B

A A

dx dx d x dx dx d xg g g

ds ds ds ds ds dsds dse dx e d x e e

A A A A A Am ds m ds m m

µ ν µ ν µ ν

µν µν µν

σ σσ σ

σ σ σ σ

δ δδδ

δδ δ δ

+ +

= =

− − + +

∫ ∫ . (6.6)

Note, we have assumed that there is no variation in the charge-to-mass ratio – i.e., that ( )/ 0e mδ =

– over the path from A to B.

The top line in (6.6) is exactly what we had in (4.4). So we can save all the steps taken from (4.4) though (4.14) and use (4.14) to rewrite the top line as such:

Jay R. Yablon

23

( )2

2

2

2

21

02

2 2 2

B B

A A

dx dx d xg g g

ds ds dsds ds

e dx e d x eA A A A

m d

x g

s m ds m

µ ν ν

µ να ν αµ αν

σ σσ

σ σ

αα µν

σ

δδ

δδ δ

− ∂ − ∂ −

= = − − +

∂

∫ ∫ . (6.7)

Up to a raised index, this already puts the Christoffels ( )12 g g ggβ

µν α µβα

µ να ν αµν−Γ = − ∂∂ − ∂ and

the acceleration 2 2/d x dsν in place, so we may now turn our focus to the new terms involving the gauge fields Aσ and the electric charge e.

From the generic result (4.5) with Aσφ → we deduce xA Aα

ασ σδ δ= ∂ . Using this in the

two places above where Aσδ appears, we may write:

( )2

2

2

2

21

02

2 2 2

B B

A A

dx dx d xg g g

ds ds dsds ds

x g

xe dx e d x e

A A A Am ds m ds

xm

αα µν

α

µ ν ν

µ να ν αµ αν

σ

α

σσ

σ σα

α σ

δ

δ δδ

δ

− ∂ − ∂ −

∂

∂ ∂

= = − − +

∫ ∫ . (6.8)

We may also use the generic relation (4.12) with Aσφ → to obtain / /dA ds A dx dsα

σ α σ= ∂ . Then,

for the second term on the bottom line, to set up an integration-by-parts, we may use this along with the product rule to form:

( ) dAd d x dx d xA x x A x A A

ds ds ds ds ds

σ α σσ σ σσ

σ σ α σ σδ δδ δ δ= + = ∂ + . (6.9)

Similarly, for the final term on the second line of (6.8), the product rule easily informs us that

( )12A A A Aα

σσ σα

σ∂ = ∂ . Using these two results in (6.8) leads to:

( )

( ) ( )

2

2

2

2

21

02

2 2

B B

A A

dx dx d xg g g

ds ds dsds ds

e dx e d dx eA A x x A A A

m ds m ds ds m

x g

x x

αα µν

α αα

µ ν ν

µ να ν αµ αν

σ α

ασ σ σ

σ σ α σ σ

δδ

δ δδ δ

− ∂ − ∂ −

= = − − − ∂ +

∂

∂ ∂

∫ ∫ .(6.10)

The two terms containing total integrals above are equal to zero because of the boundary

conditions on the definite integral in the variation. Specifically, for the term with ( ) /d A x dsσσδ

above we have:

( ) 0B B

AA

dds A x A x

dsσ σ

σ σδ δ= =∫ . (6.11)

And for the very last term in (6.10) we have:

Jay R. Yablon

24

( ) ( ) ( ) 0B

B B

A AA

x x xx d

dsA A ds A A ds A A

xα α α

α α ασ σ σ

σ σ σδ δ δ= = =∂∂∂∫ ∫ . (6.12)

As in (4.9), these total integrals are zero because of the variational boundary conditions

( ) ( ) 0x A x Bσ σδ δ= = . So setting these terms set to zero, (6.10) is now:

( )2

22

10

22 2

B B

A A

dx dx d xg g g

ds ds dsds ds

e dx e dxA x A

m d

g

xs m s

x

d

µ ν ν

µ να ν αµ ανα

α µν

αα

σ ασ

σ α σ

δδ

δ

δ

− ∂ − ∂ −

= = − + ∂

∂

∂∫ ∫ . (6.13)

Above, all of the coordinate variations are indexed as xαδ with the exception of the final term. So in this final term we rename the summed indexes α σ↔ . Then with a little restructuring which includes factoring out xαδ from all terms throughout, we obtain:

( )

( )

2

22

10

22

B B

A A

dx dx d xg g g

ds ds dsds dse dx

A As

x

m d

gµ ν ν

µ να ν αµ αα ν

σ

σ α

ν

σ

µα

α

δ δ

− ∂ − ∂ −

= =

+ ∂ −

∂

∂

∫ ∫ . (6.14)

It will now be seen, very importantly, that:

F A Aσα σ α σα∂ − ∂= (6.15)

is the covariant (lower-indexed) electromagnetic field strength tensor. As might have been expected, the variation has turned the gauge potential first introduced via D ieAσ σ σ σ∂ → = ∂ + to

ensure gauge symmetry, into the field strength that appears in the Lorentz force law. So, using (6.15) in (6.14) and moving the lead coefficient of ½ inside, leads to:

( )2

2

10

2

B B

A A

dx dx d x e dxds ds g g g F

ds ds dsg

m dsxα

µ ν ν σ

µ να ν αµ ν αµ σν ααδ δ = = − ∂ − ∂ − +

∂∫ ∫ . (6.16)

Now were are back at (4.14), but with an extra field strength term. As before, the proper

time 0ds≠ for material worldlines, and between these boundaries where the variation occurs 0xσδ ≠ . So the large parenthetical expression must be zero, enabling us to extract:

( )2

2

1

2

dx dx d x e dxg g g F

ds ds d m dsg

sα

µ ν

µ

ν σ

µ να ν αν µ αν σα− ∂ − ∂ − +∂ . (6.17)

Jay R. Yablon

25

Moving the acceleration to the left, flipping the sign via F Fσα ασ= − , multiplying through

by gβα , applying ( )12 g g ggβ

µν α µβα

µ να ν αµν−Γ = − ∂∂ − ∂ , putting the field strength into

contravariant form and reducing, we finally obtain:

2

2

d x dx dx e dxg F

ds ds ds m ds

β µ ν σβα

ασβ

µν−Γ= − . (6.18)

This is precisely the same as the combined gravitational and Lorentz motion equation (3.3) with two minor exceptions: the sign of the last term is reversed, and what appears in (6.18) is the electron charge e with 2 / 4e cα π= ℏ that approaches 1/137.036 at low probe energies, which originally entered when we introduced the gauge-covariant derivative D ieAµ µ µ µ∂ → = ∂ + to

maintain the gauge symmetry of the Klein-Gordon equation (2.9) for an interacting field. So the particle worldline motion being described in (6.17) is the classical motion for a single negatively-charged electron in an electromagnetic field F βα . So if we now generalize e q− → to a collection of charge quanta with a positive charge sign convention, then (6.18) precisely reproduces the Lorentz force law appearing in (3.3).

With (6.18), we have shown how the Lorentz force law emerges naturally as the least-proper-time motion from the gauge-symmetric Klein-Gordon equation (2.9) through the same

variation 0B

Adsδ= ∫ that yields the gravitational geodesic motion (3.1) a.k.a. (4.16). But, recalling

section 5, this only happens if we keep the mass times velocity muσ equal to the kinetic momentum

in gauge theory equal to via p muσ σ= , rather than setting this mass times velocity muσ to the

canonical momentum via muσ σπ = . As we showed in section 5, choosing /mu mdx dsσ σ σπ = = yields only the gravitational motion without the Lorentz force, which is not correct for charges travelling in electromagnetic fields. And as we have shown here at (6.18), setting

/p mu mdx dsσ σ σ= = leads to the correct, observed gravitational and the electrodynamic motion. Because the empirically-validated motion for electric charges in electromagnetic fields is given by (3.3) which is (6.18) for a single electron, the empirical data tells us that /p mu mdx dsσ σ σ= = is the correct association even after imposing gauge symmetry, and we confirm the schematic (5.4) of how gauge theory and nothing more leads to the Lorentz force law using the variation

0B

Adsδ= ∫ . This solves the long-standing question of how to obtain the Lorentz force law from a

least-action variation in the four dimensions of spacetime only, without resort to the fifth dimension of Kaluza and Klein.

Now, beyond simply having another way to derive the Lorentz force, let us explore what

all of this means for how we understand the physical structure and dynamics of the natural world.

Jay R. Yablon

26

7. How Electrodynamic Interactions Dilate and Contract Time, just as do Motion and Gravitation in Special and General Relativity On the surface, there are good reasons why it might be thought that the mass times velocity muσ once local gauge symmetry and its covariant derivative D ieAµ µ µ µ∂ → = ∂ + has been

applied, should be associated with the canonical momentum in the form muσ σπ = , rather than with the kinetic momentum in the form p muσ σ= . And yet, the former leads to the wrong equation of motion (4.16), see (5.3) and (4.1), with only gravitational motion, while the latter leads to the correct equation of motion (6.18) and more generally (3.3), including gravitational and Lorentz motions. So let us work through what this different choice as association p muσ σ= teaches us about the natural world.

To start, let us promote the single negative electron charge e− in (6.2) to a generalized collection of charges q+ with a positive sign convention via e q− → + as discussed following (6.18). It should be clear that with this change we can proceed with the exact same variation

0B

Adsδ= ∫ used in section 6 to arrive at (6.18) in the form of the generalized gravitational and

Lorentz force law (3.3). Then, let us multiply (6.2) through by 2ds to spell out the underlying metric structure. The result is:

22 2

22q q q q

ds dx ds A dx ds A dx dx ds A dx ds A A d dm m m m

σ σ σ σ σ σσ σ σ σ σ σχ χ = + + = + + =

, (7.1)

where after the final equality we have defined the “canonical coordinates”:

qd dx ds A

mσ σ σχ ≡ + . (7.2)

To be clear: if we now divide (7.1) by 2ds and use this in the variation 0B

Adsδ= ∫ as in (6.4), we

will end up with (3.3) which we reproduce in pertinent part below:

2

2

d x dx dx q dxg F

ds ds ds m ds

β µ ν σβα

σαβ

µν−Γ += , (7.3)

and which is the result (6.18) with e q− → + . So (7.1) is the spacetime metric for which the geodesic equation of motion is (7.3), and with this positive sign in front of the Coulomb portion of the motion, the electrostatic force is repulsive. Thus, we need to understand what (7.1) teaches us that has not previously been understood, about the physical structure of the natural world. Now, the spacetime metric should not depend on the nature of the test particles moving within spacetime, and yet, (7.1) contains the ratio /q m which would appear to make the metric so-dependent. Specifically, because various systems of particles may have both different electric

Jay R. Yablon

27

charges and different charge to mass ratios, the metric (7.1) would appear to depend on the particular type of test particle whose geodesic was being determined, and so could not be a property of the background spacetime and electromagnetic fields.

Indeed, the fact that various systems of particles have different /q m ratios is a fundamental distinction between gravitation and electrodynamics: In gravitation, the gravitational interaction mass gm is equivalent to the inertial mass, g im m m= = . This is the Galilean

equivalence principle of the legendary Pisa experiment. In electromagnetism, the electrical interaction mass q is definitively not equal to the inertial mass m, and this is the source of the /q m ratio in (7.3) and (7.1) and everywhere else throughout classical electrodynamics. And this is why it has been so conceptually difficult to understand electrodynamic motion as geodesic motion.

Specifically, consider the following simple gedanken: Posit a region of spacetime with

constantgµν µνη= = over the entire region so that 0βµνΓ = in (7.3). Posit an electromagnetic

field 0F βα ≠ in this same region, so that (7.3) is the Lorentz force law alone with no gravitation involved. Finally, posit two distinct systems A and B of electrically-charged mass which have two different charge-to-mass ratios ( ) ( )/ /

A Bq m q m≠ . If these two systems are given the identical

motion /u dx dsσ σ= at a given identical event (specific location x at a specific time t) in F βα , then a brief time later, these two systems will not end up in the same place, because their accelerations 2 2/d x dsβ will be different. For example, if ( ) ( )/ 2 /

B Aq m q m= , and if the field is

a Coulomb field and the motion is at rest ( )1,0,0,0uσ = , the acceleration of the B system will be

twice the acceleration of the A system, 2 2 2 2( ) / 2 ( ) /d x B ds d x A dsβ β= . And this, notwithstanding

that we make no change to the background field F βα to favor one system over the other and that the proper time ds is still an invariant, ( ) ( )ds A ds B= . So when we are accustomed to gravitation where all masses move the same way in the same field and so we can easily understand the motion as being that of a geodesic, and then we turn to electromagnetism where different charge and mass systems can and do move differently in the exact same background field, it becomes conceptually challenging to understand this as geodesic motion. And yet, (7.3) is the result of applying the

variation 0B

Adsδ= ∫ to the metric (7.1), and so it is indeed geodesic motion in the sense of motion

that follows a worldline of minimum proper time. So, how do we understand this? The key is to carefully consider the “electrical coordinates” ( )/d dx ds q m Aσ σ σχ = +

defined in (7.2), while writing (7.1) to explicitly show the metric tensor:

2ds g d dµ νµν χ χ= . (7.4)

When we do so, and upon careful analysis, we realize that the Lorentz force motion results from the dilation or contraction of space and time coordinates no different in character from the space and time dilation seen in special relativity for observer with differing motions, and no different in character from the space and time dilation and contraction seen in general relativity for differing

Jay R. Yablon

28

locations in a gravitational field. The very existence of an electric charge in an electromagnetic field causes a dilation or contraction in space and time. Let us look at this more closely. First, go to Special Relativity and its Lorentz transformations. Consider an observer in a frame of reference defined to be at rest, 0=v , and consider two other observers with relative velocities ′v and ′′v . We may use the Minkowskian relationship

2 (0) (0) ( ) ( ) ( ) ( )ds dx dx dx dx dx dxµ ν µ ν µ νµν µν µνη η η′ ′ ′ ′ ′′ ′′ ′′ ′′= = =v v v v (7.5)

to interrelate the coordinates dxµ , dx µ′ , dx µ′′ as among these three observers’ frames with relative velocities 0, ′v and ′′v . The metric interval ds is invariant as among these three reference frames,

( ) ( ) ( )ds ds ds′ ′′= =0 v v , i.e., the state of motion does not affect ds. Likewise, the metric tensor

µνη is unchanged as among these three reference frames, ( ) ( ) ( )µν µν µνη η η′ ′′= =0 v v . All that

changes are the coordinates dxµ , dx µ′ , dx µ′′ , and these do so in the a well-known manner of a Lorentz transformation which exhibits space and time dilation and mixing. Indeed, taking the square root of the above, we may write:

(0) (0) ( ) ( ) ( ) ( )ds dx dx dx dx dx dxµ ν µ ν µ νµν µν µνη η η′ ′ ′ ′ ′′ ′′ ′′ ′′= = =v v v v . (7.6)

And so, this motion is not a property of the background spacetime field µνη or the metric interval

ds, but merely an attribute of each observer. And yet observers in each reference frame, by virtue of their own relative motion, measure space and time differently from one another and perceive time and space dilations and mixing when they observe and measure time and length and mass in one another’s frames of reference. Second, go to General Relativity, and consider three observers arranged to be at rest at different potentials in a static gravitational field gµν , at the three respective locations 0=x , ′x

and ′′x . Because all three are at rest (which removes any Lorentz transformation dilations), the space coordinate differentials of these observers ( )0,0,0k k kdx dx dx′ ′′= = = with 1,2,3k = .

Therefore the metric:

2 0 0 0 0 0 000 00 00( ) ( ) ( )vds g dx dx g dx dx g dx dx g dx dxµ

µν ′ ′ ′ ′′ ′′ ′′= = = =0 x x . (7.7)

So with 0dx dt= , the measurements of time found from clocks employed by these observers is interrelated by:

00 00 00( ) ( ) ( )ds g dt g dt g dt′ ′ ′′ ′′= = =0 x x , (7.8)

and so is relatively dilated or contracted depending on the value of 00g at each observer’s locale.

Once again, in (7.7), as in (7.5), the metric interval ds is invariant as among these three locations

in the gravitational field, ( ) ( ) ( )ds ds ds′ ′′= =0 x x . Likewise, 00 00 00( ) ( ) ( )g g g′ ′′≠ ≠0 x x , but

Jay R. Yablon

29

yet the underlying field gµν is unchanged: all we are doing in (7.8) is measuring this one

background field gµν at three different places.

So if the Lorentz force arises as yet another variety of space and time dilatation and contraction specified by (7.2), it helps to explore how this occurs in more detail. We start with the metric (7.1), but flip the sign q q→ − so that the charge is negative and the Coulomb force is attractive and can be thus compared on an equal footing to gravitation. So we now write the metric (7.1) for attractive gravitation and electromagnetism as:

2 q qds g d d g dx ds A dx ds A

m mµ ν µ µ ν ν

µν µνχ χ = = − −

(7.9)

We now consider three-part gedanken. First, we consider a neutral mass m arranged to be at rest at a given location x in a static electromagnetic potentialAµ . Because a neutral mass has 0q = ,

(7.9) reverts to the usual 2ds g dx dxµ νµν= . Second, we keep this neutral mass at the exact same

location, or at least at an equipotential position, but add a non-zero (negative because of the sign above) charge to the mass which we designate as q′ . Third, we still keep the neutral mass at equipotential, and add a different non-zero charge negative that we designate as q′′ . Because of the physics, we cannot change the invariant metric proper time element dsnor can we change the background fields gµν and Aµ . Therefore ds ds ds′ ′′= = , g g gµν µν µν′ ′′= = and A A Aσ σ σ′ ′′= = .

By the terms of this gedanken we are not changing the mass so m m m′ ′′= = . All that will change is the charge 0 0 0q q q′ ′′= → ≠ → ≠ with q q′ ′′≠ , and the related coordinates

dx dx dxµ µ µ′ ′′→ → which dilate or contract. So focusing now on the space and time dilations or contractions, it will be seen that when the only change we make is from 0 0q q q′= → = ≠ , and again when the only change we make is from 0 0q q q′′= → = ≠ , the coordinates will transform in these respective cases as:

( )( )

/

/

dx dx dx ds q m A

dx dx dx ds q m A

µ µ µ µ

µ µ µ µ

′ ′→ = −

′′ ′′→ = −, (7.10)

which is restructured in terms of the original neutral coordinates dxµ as:

( ) ( )/ /dx dx ds q m A dx ds q m Aµ µ µ µ µ′ ′ ′′ ′′= + = + . (7.11)

Therefore, we may use (7.11) to write the metric in all three cases as:

2

(0) (0) ( , ) ( , ) ( , ) ( , )v

q q q qds g dx dx g dx ds A dx ds A g dx ds A dx ds A

m m m m

g dx dx g dx q A dx q A g dx q A dx q A

µ ν µ µ ν ν µ µ ν νµν µν µν

µ ν µ µ ν µ µ ν νµν µν µν

′ ′ ′′ ′′ ′ ′ ′′ ′′= = + + = + +

′ ′ ′′ ′′= = =,(7.12)

Jay R. Yablon

30

and taking the square root, we obtain:

(0) (0) ( , ) ( , ) ( , ) ( , )vds g dx dx g dx q A dx q A g dx q A dx q Aµ ν µ µ ν µ µ ν νµν µν µν′ ′ ′′ ′′= = = . (7.13)

This should be contrasted with (7.6) and (7.8) as an expression of how metric interval ds is invariant and the underlying background fields gµν and Aµ are unchanged when we add these

charges, so that the only change is to the charge itself and to the coordinates. At this point, we divide (7.10) through by ds to obtain:

dx dx q qu A u A

ds ds m m

dx dx q qu A u A

ds ds m m

µ µµ µ µ µ

µ µµ µ µ µ

′ ′ ′′ = = − = −

′′ ′′ ′′′′ = = − = −. (7.14)

Because it we necessary in section 6 to set the kinetic momentum to the mass times velocity, /p mu mdx dsσ σ σ= = in order to derive the observed Lorentz force from a variation

0B

Adsδ= ∫ , rather than setting the canonical momentum /mu mdx dsσ σ σπ = = which as shown

in section 5 yields an incorrect motion that lack the Lorentz force, we may also multiply the above through by m to write:

dx dxp mu m m q A mu q A

ds ds

dx dxp mu m m q A mu q A

ds ds

µ µµ µ µ µ µ

µ µµ µ µ µ µ

′′ ′ ′ ′= = = − = −

′′′′ ′′ ′′ ′′= = = − = −. (7.15)

The components of the vectors in the above are ( ), , ,x y zp E p p pµ = , ( ), , ,dx dt dx dy dzµ = and

( ), , ,x y zA A A Aµ φ= . Now let’s look at the 0µ = time components in the above, namely:

0 0 0 0

0 0 0 0

dt dt dtE p mu m m q A m q mu q

ds ds dsdt dt dt

E p mu m m q A m q mu qds ds ds

φ φ

φ φ

′′ ′ ′ ′ ′ ′= = = = − = − = −

′′′′ ′′ ′′ ′′ ′′ ′′= = = = − = − = −. (7.16)

Because by the terms of the gedanken the neutral mass with coordinate elements dxµ is to be at rest, 0 / 1u dt ds= = , so the above further reduce to:

Jay R. Yablon

31

0 0 0

0 0 0

dtE p mu m m q A m q

dsdt

E p mu m m q A m qds

φ

φ

′′ ′ ′ ′ ′= = = = − = −

′′′′ ′′ ′′ ′′ ′′= = = = − = −. (7.17)

Now let’s examine a number of features of (7.17). First, at rest, the potential ( ),0,0,0Aµ φ= , and this will contain the Coulomb potential

/ek Q rφ = . So the energies in this potential will be:

0 0 0

0 0 0

e

e

dt QqE p mu m m q A m q m k

ds rdt Qq

E p mu m m q A m q m kds r

φ

φ

′ ′′ ′ ′ ′ ′= = = = − = − = −

′′ ′′′′ ′′ ′′ ′′ ′′= = = = − = − = −. (7.18)

Therefore,

0

0

e

e

QqU E m q A k

rQq

U E m q A kr

′′ ′ ′= − = − = −

′′′ ′′ ′′= − = − = −, (7.19)

which is precisely the empirically-observed electrostatic potential for an attractive Coulomb interaction. Therefore, we see that these are empirically-correct results. Second, we take the final four terms in (7.18) and divide through by m, then multiply through by / 1ds dt= because the neutral charge is at rest, to obtain:

00 0

00 0

11 1 1

11 1 1

e

e

E p dt ds dt dt q q Qqu A k

m m ds dt ds dt m m m r

E p dt ds dt dt q q Qqu A k

m m ds dt ds dt m m m r

φ

φ

′ ′ ′ ′ ′ ′ ′ ′′= = = = = = − = − = −

′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′′′= = = = = = − = − = −

�����������������

�����������������

. (7.20)

The expression with the lower brackets shows how for an attractive Coulomb force there is a contraction of time which occurs in direct proportion to the reductions of energy (7.18) in the Coulomb potential. Therefore it a repulsive force dilates the time. So, just as relative motion in special relativity brings about a kinetic energy 21

2 mv in the non-relativistic limit as shown below:

Jay R. Yablon

32

2

2

2

2

1

211 1

121

dt dt mE m m m mv

ds dt vE dt dt

vm ds dt v

′ ′′ = = = ≅ +−

′ ′ ′= = = ≅ +

−

(7.21)

which is directly tied to a time dilation 2 212/ 1/ 1 1dt dt v v′ = − ≅ + , so to the placement of charges

in a Coulomb potential is directly tied to a time contraction, with 212/ /eq m k Qq mr vφ′ ′− = − ⇔ +

playing analogous roles in the time dilation. Third, from the bracketed terms in (7.20), we may now obtain the ratio:

11 1

11 1

e

e

q Qqkdt m m r

q Qqdt km m r

φ

φ

′′ ′′− −′′

= =′ ′′ − − (7.22)

for the rate at which time flows for the q′ and q′′systems in the electromagnetic potential. This provides a direct point of comparison to gravitation. Specifically, as used earlier in (3.6) and (3.7), in the linear field approximation g hµν µν µνη ρ= + , and 1

002 /h GM rρ = Φ = − is the Newtonian

potential. Therefore, 00 00 00 001 1 2 1 2 /g h h GM rη ρ ρ= + = + = + Φ = − . So from (7.8), we may

obtain the ratio:

00

00

1( ) 1 2 ( ) 1 ( )

1 ( )( ) 1 2 ( ) 1

MGgdt r

Mdt g Gr

−′ ′′′ ′+ Φ + Φ ′= = ≅ =′ ′′+ Φ′′ ′′+ Φ −

′′

x x xxx x

, (7.23)

where in the final expression we use the series approximation 1 2 1+ Φ ≅ + Φ for 2 1Φ≪ . For / 1q mφ ≪ we may approximate ( )1 / 1/ 1 /q m q mφ φ− ≅ + and so write (7.22) as:

1

1 1 1

11 1 1

e

e

q Qq qkdt m m r m

q Qq qdt km m r m

φ φ

φ φ

′′ ′′ ′− − +′′

= = ≅′ ′ ′′′ − − + (7.24)

We then may highlight the contrast between the gravitational an electrodynamic time dilations:

11 ( )

1 ( ) 1

qdt dt m

qdt dtm

φ

φ

′+′′ ′ ′′+ Φ≅ ⇔ ≅ ′′′ ′′ ′+ Φ +

xx

, (7.25)

Jay R. Yablon

33

Fourth and finally, let us examine the electrodynamic dilation and contraction of space as

a result of the foregoing. Obtaining the space components of (7.15) we have:

d dm m m q m q

ds dsd d

m m m q m qds ds

′′ ′ ′ ′= = = − = −

′′′′ ′′ ′′ ′′= = = − = −

x xp u A u A

x xp u A u A

. (7.26)

But at rest, / 0d ds= =u x and 0=A , so these yield 0′ ′′= = =p p p and 0d d d′ ′′= = =x x x and there is no space dilation of contraction. We see therefore that at rest, electrodynamics only dilates or contracts the time, not the space. Any contractions or dilations of the space coordinates therefore come about simply through the ordinary relative motion of special relativity.

As a result of all of the foregoing, the spacetime metric (7.1), (7.4) does not depend on the nature of the test particles moving within the spacetime. Even though various types of particle system such as the q′ and q′′ systems have both different electric charges and different charge to mass ratios, the metric (7.1), (7.4), (7.9) does not depend on the particular type of test particle whose geodesic is being determined, and remains a property of the background spacetime and electromagnetic fields. So too, do the background fields gµν and Aµ themselves, remain

independent of the nature of the test particles. The Lorentz force comes about by yet another variety of space and time dilatation and contraction which is heretofore unrecognized, with the space dilation arising only from relativistic motion, and as a result of all this, the Lorentz force is simply a heretofore unrecognized form of geodesic motion in four spacetime dimensions only,

because it is arrived at by a least action variation 0B

Adsδ= ∫ of the metric (7.1), (7.4), (7.9) which

minimizes the proper time along particle worldlines. This time dilation produces electrostatic potentials (7.19) which are fully in accord with empirical observation, the electrostatic potential energies contract or dilate the time in a manner fully analogous to the time dilations (7.21) for relative motion and kinetic energy in special relativity, and also fully analogous as in (7.25) to the time dilations occurring in a gravitational potential in general relativity. 8. Non-abelian gauge fields It will be observed that the field strength tensor emerging in (6.15) from the variation

0B

Adsδ= ∫ is an abelian field strength [ ]F Aσασ α=∂ . It is natural to inquire whether one can use this

same approach to also obtain the equation for the motion for charged particles in non-abelian fields

[ ][ ] [ ],F A ie A A D Aασ α σ α σ α σ= ∂ + = . It turns out that this is fairly simple to do. First, we return to

(6.14) which we rewrite via ( )12 g gg gβ

αβ µν α µ µ να νν αµ− ∂ − ∂− Γ = ∂ and [ ]A A Aσ α σ αα σ∂ ∂ − ∂= as:

2

[ ]20B B

A A

dx dx d x e dxds ds gx g A

ds ds ds m dsα β

αβ µν

µ ν ν σ

αν σ αδδ = = − + ∂

− Γ∫ ∫ . (8.1)

Jay R. Yablon

34

From here, we perform a local gauge (phase) transformation ( )i xA A e Aµ

σ σ σΛ′→ = on the

gauge fields, and insist that this variation remain invariant under such transformation. Consequently, we must promote the derivative that acts on the gauge fields to the gauge-covariant

D ieAµ µ µ µ∂ → = ∂ + in the usual way. As a result (8.1) now becomes:

2

[ ]20B B

A A

dx dx d x e dxds ds gx D A

ds dg

s ds m dsα β

αβ µ

µ ν ν σ

ανν σ αδδ = = − +

− Γ∫ ∫ . (8.2)

Therefore, this yields the exact same result as was found in (6.16) with no change whatsoever in form other than that we have already incorporated β

µνΓ . The only difference is that the field

strength is now the non-abelian [ ]F D Aασ α σ= . As twice before, the proper time 0ds≠ for material

worldlines and between the variation boundaries 0xσδ ≠ . Therefore we may extract:

2

2

d x dx dx e dxg F

ds ds ds m ds

β µ ν σβα

ασβ

µν−Γ= − , (8.3)

precisely the same as (6.18), but now with the non-abelian field strength:

,F A A ie A A D A D Aβα β α α β β α β α α β = ∂ − ∂ + = − . (8.4)

9. Relation to Five-Dimensional Kaluza-Klein Theory To be added PART III: UNIFICATION OF GRAVITATIONAL AND ELECTROMAGNETIC FIELDS 10. Einstein’s Equation and Maxwell’s Equations

Because the metric length 2ds g d dµ νµν χ χ= of (7.4) under a variation 0

B

Adsδ= ∫

simultaneously provides a geodesic description of motion in a gravitational field and in an electromagnetic field, and because the prescription ( )/dx d dx ds e m Aµ µ µ µχ→ = + is no more

than a variant of Weyl’s gauge prescriptions p p eAµ µ µ µπ→ = + in momentum space and

D ieAµ µ µ µ∂ → = ∂ + in configuration space and leads directly as well to Dirac’s equation

( ) 0i D mµµγ ψ− = for an interacting fermion, this may fairly be regarded as a classical metric-

level unification of electrodynamics with gravitation, using four spacetime dimensions only. But the equations of motion in a field are only half the matter. We also need to know the equations for the fields themselves in relation to their sources. Thus we now ask, can the field equation

Jay R. Yablon

35

12T R g Rµν µν µνκ− = − which specifies the gravitational field, be shown to relate in some direct

fashion to Maxwell’s field equations for electric and (the absence of) magnetic sources? Because the Lorentz force in (6.18) is obtained by dilating or contracting the differential coordinate elements via ( )/dx dx dx ds q m Aµ µ µ µ′ ′→ = − as in (7.10) without in any way altering

the metric tensor gµν as is done, for example, in Kaluza-Klein theory, one might incorrectly

conclude that the electromagnetic interaction does not affect spacetime curvature as represented by the Riemann tensor Rα

βµν with the field dynamics specified by 12T R g Rµν µν µνκ− = − .

However, one must keep in mind that the Riemann tensor may be defined via

; ;,R V Vαβµν α ν µ β ≡ ∂ ∂ as a measure of the extent to which the gravitationally-covariant

derivatives ; V V Vαµ β µ β µβ α∂ = ∂ − Γ operating on a vector Vβ do not commute. Likewise, the field

strength tensor Fνµ may be defined via ,ieF V D D Vνµ β ν µ β ≡ as a measure of the extent to which

the gauge-covariant derivatives ( )D V ieA Vµ β µ β= ∂ + do not commute when operating on this

same vector Vβ . Indeed, this latter definition results in [ ] [ ] ,F D A A ie A Aνµ ν µ ν µ ν µ = = ∂ + for a

non-abelian gauge theory defined such that , 0A Aν µ ≠ , which simplifies to [ ]F Aνµ ν µ= ∂ for an

abelian theory such as electrodynamics in which , 0A Aν µ = .

Therefore, let us now apply Weyl’s canonical prescription to the gravitationally-covariant

derivatives by employing:

( ); ;V V V D V ieA V Vα αµ β µ β µβ α µ β µ µ β µβ α∂ = ∂ − Γ → = ∂ + − Γ , (10.1)

for vectors Vβ a.k.a. first-rank tensors, and likewise extended for second and higher-rank tensors.

This is the same prescription that in the form ( )/d dx ds q m Aσ σ σχ = + of (7.2) yielded the

Lorentz force law in (6.19). If we then use these derivatives (10.1) to define a gauge-enhanced canonical Riemann tensor α βµνℜ as:

; ;,V D D Vαβµν α ν µ β ℜ ≡ , (10.2)

it can be expected as a consequence of ,ieF V D D Vνµ β ν µ β ≡ that the electrodynamic fields Fνµ

and potentialsAµ will appear in this Riemann tensor. Further, because [ ]F D Aνµ ν µ= encompasses

both abelian and non-abelian field strengths, one would expect that the gravitational field equations using α

βµ βµαℜ = ℜ and σσℜ = ℜ can be related not only to abelian electrodynamics, but also to

non-abelian such as weak and strong interactions. So let us expressly calculate this enhanced canonical α

βµνℜ using (10.2) and see what results.

Jay R. Yablon

36

We first calculate:

( ) ( ) ( )( )( )( ) ( )( )

;; D V ieA ieA V V

ieA V V ieA V V

Dν ν

τ τµν

αµ β ν µ µ β µβ α

α ατ τ β τβ α µ µβ µ αν τ τ

= + ∂ + − Γ

∂ + − Γ ∂ + − Γ

∂

− Γ − Γ (10.3)

as well as the like expression interchanging µ ν↔ , then subtract the latter from the former and reduce using index renaming and the symmetries of the objects in the resulting equations. Many terms cancel, but with the vector Vα still attached as the operand on the right, what remains is:

( ) ( )( )

; ;; ;; ;,V D D V D V D V

ie F V

D Dαβµν α ν µ β µ β ν β

α α α α α

ν µ

τ τν µµβ ν νβ µβ µ ν ν αβτ τ β µδ∂

ℜ ≡ = −

= − Γ + Γ Γ − Γ −∂ + Γ Γ, (10.4)

including a non-abelian field strength:

( )[ ] [ ] [ ], ,a a a a b c b c a a abc b cF T F A ie A A T A ie T T A A T A ef A Aµν µν µ ν µ ν µ ν µ ν µ ν µ ν = = ∂ + = ∂ + = ∂ − ,(10.5)

which becomes abelian in the event , 0A Aµ ν = . When we explicitly display the group structure

constants abcf for the non-abelian Hermitian generators aT via ,abc a b cif T T T = , we see that

[ ]a a abc b cF A ef A Aνµ ν µ ν µ= ∂ − is real and so a aieF ieT Fµν µν= in (10.4) is a complex Hermitian field

owing to the aT . With Vα removed and some index renaming and lowered to covariant form, the

canonical Riemann tensor in (10.4) is then seen to be:

g g g g g ieg F

R ieg F

σ σν µ β

τ τ τ τ ταβµν ατ βµν ατ βµ ατ βν ατ σµ ατ σνν αβ µν

αβµν

β

αβ µν

µ∂ ∂ + Γ Γℜ = ℜ = − Γ + Γ Γ − Γ −

= − (10.6)

As expressed by R ieg Fαβµν αβµν αβ µνℜ = − , the terms containing Christoffels are no

different from the usual in Rαβµν . But the new term ieg Fαβ µν− resulting from the same gauge

prescription D ieAµ µ µ µ∂ → = ∂ + that likewise brought the Lorentz force law into (6.18) changes

several aspects of αβµνℜ in relation to the ordinary Rαβµν . First, while for the last two indexes

αβµν αβνµℜ = −ℜ as with Rαβµν , for the first two indexes αβµν βαµνℜ ≠ −ℜ due to the presence of the

symmetric gαβ next to the antisymmetric Fµν in the term g Fαβ µν . Thus, αβµνℜ is non-symmetric

in ,α β . Second, noting that a aF T Fµν µν= is Hermitian, the term ieg Fαβ µν provides an similar

complex aspect to αβµνℜ , so that overall, this enhanced αβµνℜ is a complex object. Third, as a

consequence of both these matters, the real part of αβµνℜ has the usual symmetries of Rαβµν , while

the new complex part has the mixed symmetry of g Fαβ µν .

Jay R. Yablon

37

It is readily seen from (10.6) after some re-indexing that the canonical Ricci tensor:

ieF R ieFα α α α αµν µνα µν µα σν σα µν µν

σ σα ν µ να µν µℜ = ℜ = − Γ + Γ Γ − Γ + = +∂ ∂ + Γ Γ , (10.7)

concisely R ieFµν µν µνℜ = + , is likewise non-symmetric, with the usual gravitational terms being

real and symmetric, and the new, Hermitian electrodynamic term being antisymmetric in ,µ ν .

Finally, because 0Fσσ = , the canonical Ricci scalar is the usual:

g g g g g Rµν στ α στ α στ α στ α

µν στ σα βσβ β

α τ σα σ βτ αℜ = ℜ = − Γ + Γ Γ − Γ Γ∂ ∂ Γ =+ (10.8)

with no residual terms from electrodynamics, that is, Rℜ = . If we now construct ; ; ;σ αβµν µ αβνσ ν αβσµ∂ ℜ + ∂ ℜ + ∂ ℜ , then because (10.6) informs us that

R ieg Fαβµν αβµν αβ µνℜ = − , all of the Christoffel terms will zero out as a result of the second Bianchi

identity ; ; ; 0R R Rσ αβµν µ αβνσ ν αβσµ∂ + ∂ + ∂ = , simply due to the inherent structure of the Riemannian

geometry itself. All that will remain are terms containing the field strength, so that:

( )[ ]( )

; ; ; ; ; ;

; ; ;, , ,

ieg F F F

ieg A A A A A A

σ αβµν µ αβνσ ν αβσµ αβ σ µν µ νσ ν σµ

αβ σ µ ν µ ν σ ν σ µ

∂ ℜ + ∂ ℜ + ∂ ℜ = − ∂ + ∂ + ∂

= − ∂ + ∂ + ∂

(10.9)

Specifically: We see here that the Hermitian part of ; ; ;σ αβµν µ αβνσ ν αβσµ∂ ℜ + ∂ ℜ + ∂ ℜ contains the

terms ; ; ;F F Fσ µν µ νσ ν σµ∂ + ∂ + ∂ which specify magnetic charges. Because exterior calculus teaches

that the differential forms 0dF ddA= = , this set of terms must be equal to zero for any abelian

gauge theory with , 0A Aµ ν = . And because this set of terms must be zero if , 0A Aµ ν = , this

means that for an abelian interaction the canonical Riemann tensor obeys the analogous Bianchi identity ; ; ; 0σ αβµν µ αβνσ ν αβσµ∂ ℜ + ∂ ℜ + ∂ ℜ = as a consequence of 0dF ddA= = which is Maxwell’s

magnetic charge equation. Next, given the identity (10.9), let us double-contract two pairs of indexes to find the

canonical extension of the contracted Bianchi identity ( )1; 2 0R g Rµν µνµ∂ − = which is used to

ensure local energy conservation in the Einstein equation via ( )1; ; 2 0T R g Rµν µν µνµ µ∂ = ∂ − = , for

these canonically-extended αβµνℜ . Further, let us consider an abelian interaction , 0A Aµ ν = , so

that (10.9) will clearly be zero:

( ); ; ; ; ; ; 0ieg F F Fσ αβµν µ αβνσ ν αβσµ αβ σ µν µ νσ ν σµ∂ ℜ + ∂ ℜ + ∂ ℜ = − ∂ + ∂ + ∂ = . (10.10)

We want to contract indexes carefully for reasons that will momentarily become apparent, so let’s go step by step and pay close attention to the indexes and their symmetries. First, let us

Jay R. Yablon

38

raise the index α . Given that αβσµℜ is antisymmetric in the last two indexes but unlike Rα

βσµ not

in the first two indexes, let us interchange σ µ↔ and reverse the sign in the final term ;ν αβσµ∂ ℜ .

Finally, let’s contract α with σ . As in (10.7), and as for any fourth rank tensor, we obtain the Ricci tensor extension σ

βν βνσℜ ≡ ℜ and the trace tensor generally by contracting the first and last

indexes together. In the term with the field strengths after this we will have a σ βδ so we pass β

through σ into the field strength expression to yield:

( ); ; ; ; ; ; 0ie F F Fσσ βµν µ βν ν βµ β µν µ νβ ν βµ∂ ℜ + ∂ ℜ − ∂ ℜ = − ∂ + ∂ + ∂ = . (10.11)

This now expresses Maxwell’s magnetic charge equation very directly.

Now we move to the next contraction. In the term with F Fνβ βν= − we flip indexes and

sign. Then, we raise β and contract it with µ . Setting ββℜ = ℜ we rename the second term

; ;β σ

β ν σ ν∂ ℜ = ∂ ℜ and in the third term we rename ; ;σ

ν ν σδ∂ = ∂ . Finally, we take what is now ;σ∂

in front of all three extended Riemann terms and factor it all the way to the left, while multiplying everything through by ½. The result is:

( ) ( );1 1 1 1; ; ;2 2 2 2 0ie F F Fσβ σ σ β β βσ βν ν ν βν β ν ν βδ∂ ℜ + ℜ − ℜ = − ∂ − ∂ + ∂ = . (10.12)

The term ;

; ;F F Fβ β ββν β ν ν β∂ − ∂ + ∂ is clearly identical to zero given the antisymmetry of Fβν (and

will likewise be zero even for non-abelian fields). So the first-rank identity we obtain for the

canonical Riemann extension is ( )1 1 1; 2 2 2 0σβ σ σσ βν ν νδ∂ ℜ + ℜ − ℜ = . One may be inclined to further

simplify σβ σβν νℜ → ℜ to arrive at ( )1

; 2 0σ σσ ν νδ∂ ℜ − ℜ = which is precisely the analog of

( )1; 2 0R Rσ σσ ν νδ∂ − = , but that would be a subtle but important mistake. Precisely because σβ

βνℜ

is non-symmetric (neither symmetric nor antisymmetric) in its first two indexes as discussed at (10.6), we cannot contract via the second and third indexes, but only via the first and fourth indexes. That is, βσ σ

νβ νℜ = ℜ is correct, but σβ σβν νℜ = ℜ is not.

This brings us back to in (10.7) which we write as R ieFµν µν µνℜ = + in contravariant form. Applying the four-gradient ;µ∂ operator throughout yields:

; ; ; ;R ie F R ieJµν µν µν µν νµ µ µ µ∂ ℜ = ∂ + ∂ = ∂ + , (10.13)

which enables us to pinpoint the electric source current ;J Fν µν

µ= ∂ . Given that Rℜ = as found

in (10.8), let us next subtract ( )1;2 gµνµ∂ ℜ throughout from the above (note that ; 0gµν σ = always

because of the metricity of gµν ), and then apply the ordinary contracted Bianchi identity

( )1; 2 0R g Rµν µνµ∂ − = , to obtain:

Jay R. Yablon

39

( ) ( )1 1; ;2 2g R g R ieJ ieJµν µν µν µν ν νµ µ∂ ℜ − ℜ = ∂ − + = . (10.14)

So, very importantly, we see that the current ieJν is identified with the non-zero

( )1; 2 0gµν µνµ∂ ℜ − ℜ ≠ and that the actual “zero” is the one in ( )1 1 1

; 2 2 2 0σβ σ σσ βν ν νδ∂ ℜ + ℜ − ℜ =

of (10.12). Indeed, we may subtract (10.12) from (10.14) with renaming µ σ→ and lowering ν to find that:

( )1 1; 2 2ieJ σ σβ

ν σ ν βν= ∂ ℜ − ℜ . (10.15)

This shows us not only that σβ

βνℜ cannot be contracted to σνℜ using the second and third indexes,

but that the electric current itself is a measure of the non-symmetry of these first two αβ indexes

in R ieg Fαβµν αβµν αβ µνℜ = − from (10.6). This is why it was so important to be very careful with

the index contractions leading to (10.12). However, absent electric charge sources, that is, for the source-free 0Jν = , the above will yield σ σβ

ν βνℜ = ℜ and then this contraction is permitted.

With all of this, we now move to Einstein’s equation itself. We know that local conservation of the energy tensor is expressed by ; 0Tσ

σ νκ− ∂ = , and we ordinarily enforce this by

setting ( )1; ; 2 0T R Rσ σ σσ ν σ ν νκ δ− ∂ = ∂ − = because of the Bianchi identity of ( )1

; 2R Rσ σσ ν νδ∂ −

with zero. But (10.12) gives us a different identity to zero (enforced by the antisymmetry of Fβν

which is inviolate even for non-abelian gauge theories) which can serve the same purpose and also has some very direct electrodynamic connections. So we instead use the identity (10.12) to write the conservation equation:

( )1 1 1; ; 2 2 2 0Tσ σβ σ σσ ν σ βν ν νκ δ− ∂ = ∂ ℜ + ℜ − ℜ = (10.16)

which in turn integrates sans cosmological constant into a field equation:

1 1 12 2 2Tσ σβ σ σ

ν βν ν νκ δ− = ℜ + ℜ − ℜ . (10.17)

Were it possible to contract σβ σ

βν νℜ = ℜ , (10.17) would take the exact form 12Tσ σ σ

ν ν νκ δ− = ℜ − ℜ of the Einstein equation. We have shown at (10.15) that this contraction

is only permitted for source-free fields, 0Jν = .

Finally, if we start from (10.16) and use (10.15) to replace 1 1

; ;2 2 ieJσβ σσ βν σ ν ν∂ ℜ = ∂ ℜ − and

then move the current to the same side of the equation so that all sources (gravitational and electric) are together, we may write the local energy conservation law in the form:

Jay R. Yablon

40

( )1; ; 2T ieJ ieJσ σ σσ ν ν σ ν ν νκ δ− ∂ + = ∂ ℜ − ℜ = . (10.18)

Summarizing, the three main field equations we have found, written exclusively in terms

of the canonically-extended αβµνℜ , from (10.15), (10.11) and (10.17) respectively, with some

simple reindexing, are:

( )1; 2

; ;;

1 1 12 2 2

0

ieJ g

T

ν µν µνµ

σβµν µ βν ν βµσ

µ µβ µ µν βν ν νκ δ

= ∂ ℜ − ℜ = ∂ ℜ + ∂ ℜ − ∂ ℜ− = ℜ + ℜ − ℜ

. (10.19)

The first two are Maxwell’s electric and magnetic charge equations, while the final equation is a canonical gauge extension of the Einstein equation for gravitation which by identity will always produce a locally-conserved energy ; 0Tσν

σ∂ = .

11. Conclusion

It has been shown how the Lorentz force law (6.18) can be obtained from a geodesic variation confined exclusively to the four dimensions of ordinary spacetime geometry as a consequence, at bottom, of simply applying Weyl’s gauge prescription D ieAµ µ µ µ∂ → = ∂ + to

dilate or contract the spacetime coordinate elements by ( )/dx d dx ds e m Aµ µ µ µχ→ = + . It has

also been shown how this same prescription embeds Maxwell’s equations a canonically-extended version (10.19) of the gravitational field equation. None of this appears to have been found before.

As a consequence of what has been shown here, it may well be possible to unify gravitation not only with electrodynamics, but – because the F στ obtained in (8.4) and (10.4) encompass a

non-abelian field strength [ ] ,a a abc b cF A ef A Aνµ ν µ ν µ = ∂ − – with the remaining weak and strong

interactions as well, because the canonical gauge prescriptions p p eAµ µ µ µπ→ = + and

D ieAµ µ µ µ∂ → = ∂ + and now ( )/dx d dx ds e m Aµ µ µ µχ→ = + remain at the root of the entire

development. The main questions that would remain following such a unification, would be as to the specific non-abelian gauge groups that operate physically at any given energy ranging up to the Planck mass, and how the symmetry of those groups becomes broken at lower energies down to the phenomenological group (3) (2) (1) (3) (1)C W Y C emSU SU U SU U× × → × and the fermions on

which these groups act. The author has previously published on these questions, and even shown how the three generations of quarks and leptons originate, and why their left-chiral projections engage in CKM mixing, at [13]. References

[1] A. Einstein, On the Relativity Principle and the Conclusions Drawn from It, Jahrbuch der Radioaktivität, 4, 411–462 (1907)

Jay R. Yablon

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[2] A. Einstein, On the Influence of Gravitation on the Propagation of Light, Annalen der Physik (ser. 4), 35, 898–908 (1911) [3] A. Einstein, On the Electrodynamics of Moving Bodies, Annalen der Physik (ser. 4), 17, 891–921 (1905)

[4] A. Einstein, The Foundation of the General Theory of Relativity, Annalen der Physik (ser. 4), 49, 769–822 (1916) [5] P. A. M. Dirac, The Quantum Theory of the Electron, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 117 (778): 610 (1928) [6] H. Weyl, Gravitation and Electricity, Sitzungsber. Preuss. Akad.Wiss., 465-480. (1918). [7] H. Weyl, Space-Time-Matter (1918) [8] H. Weyl, Electron und Gravitation, Zeit. f. Physik, 56, 330 (1929) [9] https://en.wikipedia.org/wiki/Geodesics_in_general_relativity#Deriving_the_geodesic_equation_via_an_action [10] https://en.wikipedia.org/wiki/Geodesics_in_general_relativity#Extension_to_the_case_of_a_charged_particle [11] T. Kaluza, Zum Unitätsproblem in der Physik, Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.): 966–972 (1921) [12] O. Klein, Quantentheorie und fünfdimensionale Relativitätstheorie, Zeitschrift für Physik A 37 (12): 895–906 [13] J. R. Yablon, Grand Unified SU(8) Gauge Theory Based on Baryons which Are Yang-Mills Magnetic Monopoles, Journal of Modern Physics, Vol. 4 No. 4A, 2013, pp. 94-120. doi: 10.4236/jmp.2013.44A011, http://www.scirp.org/Journal/PaperDownload.aspx?paperID=30822 (2013)