The Lorentz Force as Pure Geodesic Motion in Four-Dimensional Spacetime

Transcription

1 The Lorentz Force as Pure Geodesic Motion in Four-Diensional Spacetie Jay R. Yablon 9 Northuberland Drive Schenectady, New York yablon@alu.it.edu Deceber 3, 5, updated Deceber 3, 5 and January, 3, 5 &, 6 bstract: We develop a linear etric eleent ds in ordinary four-diensional spacetie which, when held stationary under worldline variations, leads to the gravitational equations of geodesic otion extended to include the Lorentz force law for both abelian and non-abelian gauge fields. We see that in the presence of an electroagnetic vector potential, all that is needed to obtain this result is to follow the well-known gauge theory prescription of replacing the kinetic oentu p with a canonical oentu π =p +e in the ass / oentu relationship =pp, and then to apply variational calculus to obtain the otion of charged particles in this potential. We also show how by this sae prescription, Maxwell s classical source-free field equations becoe ebedded within canonical extensions of the Rieann tensor and the gravitational field equation. PCS: 4..Fy; 3.5.De; 4..Cv;.5.-q Contents. Introduction.... How Gauge Syetry is used to Introduce Electroagnetic Interactions into Physical Equations Rooted in the Spacetie Metric The Physics of Standing on the Ground and not Passing Through asis and derivation Non-abelian gauge fields Einstein s Equation and Maxwell s Equations Conclusion... 5 References... 5

2 . Introduction The Equivalence Principal first introduced in 97 [] and subsequently refined in 9 [], is the gedanken, i.e. thought experient which lbert Einstein used to successfully navigate fro 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 fro a reference frae assigned to be at rest with velocity v = in a hoogeneous gravitational field which iparts a downward acceleration g to all freely-falling objects, are equivalent to the physical laws observed fro a reference frae likewise assigned to be at rest which is given a teporallyconstant upward acceleration a by applying an external force F = a designed such that a = F / g, where is the total rest ass of all of the aterial bodies to which this force is applied. The basic configuration of this equivalence is illustrated in Figure below. Figure : The (pproxiate) Equivalence Principle: (a) a first observer standing on the earth s surface while releasing a sall object and (b) a second observer standing inside a uniforlyaccelerated housing and likewise releasing a sall object. Up to tidal forces, each will observe and easure and experience equivalent physics. s illustrated in Figure, 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 g 9.8 /s and at which the atospheric pressure p.3 kg/c, with (b) a second observer who is standing inside an equally-pressurized elevator or rocket ship (housing) to which an external Newtonian force F = a is applied such that, taking into account the entire ass of the syste including the housing and the observer and the air and all objects inside, the observer is uniforly-accelerated such that F / = a g. For both observers the +z axis defines the overhead direction, as illustrated.

3 If each observer in Figure is assigned to be at rest with velocity v = dx / dt =, and if each observer releases a sall object in the anner of Galileo s Pisa experient as illustrated, then that object will exhibit a downward free-fall acceleration with respect to the observer which in the forer case is g and in the latter case is a along the z axis, such that the observed, easured agnitudes of these accelerations a = g are the sae. Of course, a hoogeneous 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 fro hoogeneity is easured by the tidal force. Thus, the Equivalence Principal strictly speaking is only an approxiate equivalence denoted by the ɶ in Figure. However, because the area spanned by the Figure observer is iniscule copared to the entire planetary surface area, a precise equivalence is observed for all practical purposes. Underlying this Einsteinian equivalence, is the Galilean equivalence between gravitational interaction ass and the inertial ass of aterial bodies, whereby the gravitational pull on the object dropped in Figure (a) in proportion to its ass is precisely counterbalanced by its ass-proportionate inertial resistance to that pull. Thus, when one conspires to apply a force F / = a g in Figure (b) thereby increasing the force in direct proportion to the ass, one is artificially replicating this Galilean equivalence. So up to the tidal forces, each observer in Figure will observe the exact sae 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 phenoenon as well. Iportantly, this includes that the fact that each observer will feel and easure the sae force applied to his or her feet. Specifically, for illustration, if each observer was to weigh, say, 5 kg at the earth s surface, and if each observer were to stand on a scale during this equivalence experient to take a reading of their weight, then by conspiring to apply a force F / g 9.8 /s in Figure (b), it will be ensured that each observer obtains an identical 5 kg weight easureent fro their scale, as illustrated. Consequently, so long as observer (b) is not easuring 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 / = 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 are fully characterized by the General Theory of Relativity; indeed, this Figure gedanken played a crucial role in Einstein s incorporating these spatial accelerations into the General Theory. However, this weight felt and easured by each observer is not encopassed by the General Theory of Relativity on its own, any ore than the relative spatial accelerations of the dropped objects are encopassed 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 (a) and (b) to be able to stand on the surface and feel a force against their feet, rather than freely fall through the surface as if it was not there, is not a result of gravitation, but rather, results fro what is at botto 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. dditionally, the dropped objects in Figure will

4 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 siply reain on the ground, depending upon the object s elasticity. To account for this net electrostatic repulsion that keeps observers and objects alike fro passing through the floor, one ust consider electroagnetis together with gravitation. That is, the total consideration of all of the physics of the Equivalence Principle in Figure requires that we consider gravitation and electroagnetis together in a unified anner. In essence, developing such a unification of classical gravitation and classical electroagnetis is the fundaental purpose of this paper, and as with the developent of the General fro the Special Theory of Relativity, the Equivalence Principle will provide the priary gedanken for achieving this unification. Specifically, the observer in Figure (b) is only accelerated in relation to the dropped object (or vice versa) because his or her olecules are not able to pass through those of the floor of the elevator or the rocket as the force F is being applied, while the observer in Figure (a) only sees an acceleration through space for the dropped object (or vice versa) because his or her olecules are not able to pass through those of the surface of the earth. Nor can the olecules of the dropped object pass through these surfaces. In either case, it is electroagnetis which stops the geodesic free fall worldlines of pure gravitation. Indeed, consider the converse: if the electrostatic repulsion (and ore generally electroagnetis) 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. So, the dropped object would no longer be accelerating, but would be in free fall right alongside of the observer such that each would be travelling alongside one another with a velocity v = as easured relative to the observer. It is electrostatic repulsion with the botto surface that allows the observers to reain in one reference frae designated to be at rest, while the dropped objects subsist in a second reference frae that is spatially accelerating relative to the first reference frae. s such, electroagnetis is an essential ingredient in a coplete 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 the a easurable weight and causes the objects to eventually strike the surfaces and end their free fall as well. Succinctly: it is electroagnetis which is the barrier to gravitational free fall in the Einstein Equivalence Principle and therefore cause objects which truly are in gravitational free fall to appear as it they are accelerating relative to observers who are not in free fall but are standing on a surface and held in position by gravitation and dee theselves to be at rest. Fro a different view, while selecting a v = rest frae is a totally arbitrary atter of choice, there is a easurable physical difference between assigning v = to the observers and assigning v = to the dropped object in Figure. No atter what assignent is chosen for the rest frae, there will be a visually discernable relative acceleration through space as between the observers and the dropped objects. ut the observers will feel an external force that is easurable with a weighing device, while the dropped objects will not feel any external force until such tie as they strike the surface and have their free fall abruptly ended. This is a real, qualitative, and quantitatively-easurable, physical difference. Indeed, this is one aspect of the suppleentary reark ade by Einstein at the end of [] carefully defining the zero of velocity, in response to a letter fro Max Planck suggesting a clarification to the concept uniforly accelerated. So the accelerations of the dropped objects relative to the observers assigned to v = in Figure is a free-fall acceleration along gravitational geodesic lines of least action. However, if we instead 3

5 were to assign v = to these falling objects and thus place the into what we choose as the rest frae, the accelerations of these observers relative to these objects, although spatially of equal agnitude but opposite direction, would be forced accelerations not along gravitational geodesics, which accelerations can only be brought about by the application of a Newtonian force through the electrostatic repulsive interactions that prevent two objects fro oving through one another and instead ensure a exclusionary collision if one tries to bring two objects into the sae space at the sae tie. Let us now discuss all of this ore quantitatively.. How Gauge Syetry is used to Introduce Electroagnetic Interactions into Physical Equations Rooted in the Spacetie Metric The odern geoetric understanding of gravitation begins with a etric interval: ds gdx dx = (.) where g is the covariant etric tensor and its contravariant inverse is given by g with g = δ δ being the 4x4 Kronecker identity atrix. Here, we shall use diag ( η ) = ( +,,, ) as the etric tensor of the tangent flat Minkowski space. The invariant ds when integrated along a worldline between any two spacetie events and yields s = ds, which is the proper tie along the worldline when the separation between and is tielike with ds > as it is for all aterial bodies, and the proper length when the separation is spacelike with ds <. Of course, ds = for lightlike worldlines. For a assive particle with tielike worldlines, it is coon practice to divide the etric through by ds and then define the four-velocity by u dx / ds, to obtain: dx dx = g = g u u = u u. (.) ds ds It is also coon to thereafter postulate a ass and ultiply both sides of the above through by while defining a four-oentu p u, to obtain: dx dx = g = g ( u )( u ) = g p p = p p ds ds In a local Minkowski frae for which g η thus practice following Dirac [5] to eploy the gaa atrices {, } { γ, } p p γ. (.3) η = p p, it is also coon γ γ = η to write (.3) as = which then separates into two identical linear equations = γ p p/ or γ p =. ut of course is now really ties a 4x4 identity atrix I (4) while the 4x4 4

6 atrix p/ = γ p is decidedly non-diagonal and not any ultiple of an identity. Thus, to write a proper equation, we are required to introduce a four-coponent spinor u( p ) (not to be confused with the four-velocity u ) that is a function only of the four-oentu p and not of spacetie, and write this deconstruction as the eigenvalue equation ( γ ) 5 = p u with γ p operating on the eigenvector u. If we further ultiply through by a Fourier kernel plane wave solutions and define a Dirac wavefunction ip x ip x i e p e =, this eigenvalue equation becoes ip x ( ) ( )( ) ( )( ) ip x ip x p p e u i e u ( i ) e ip x to study ψ e u, then given that = γ ψ = γ = γ = γ ψ, (.4) which is Dirac s equation for a free electron. To introduce electrodynaic interactions which bear the net responsibility for preventing the observers in Figure illustrating the Equivalence Principle fro free-falling through the surfaces that support the, we subject this wavefunction to the local unitary gauge (really phase) i transforation ψ ψ = Uψ = e Λ ψ using a real local phase Λ ( x ) and a transforation factor U i = e Λ which is unitary given that U = U * U =, and we insist that (.4) reain invariant under iλ such transforations. Fro U = e = cos Λ + i sin Λ = a + ib we see that Λ is siply the angle in this phase space, and that U in ψ = Uψ erely has the effect of rotating the orientation of ψ through a coplex two-diensional phase space without changing the agnitude of ψ. For nonabelian gauge theory used to describe, e.g., weak and strong interactions, these angles are prooted i i i i i to Λ = T Λ where T are the Heritian T = T generators of whatever group is being considered and U i = e Λ reains unitary because U = U U =. Gauge theory was of course pioneered by Herann Weyl over 98 to 99 in [6], [7], [8] in order to place electrodynaics on a siilar geoetric 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 requireent for using covariant derivatives to aintain syetry. Recognizing that ( i Λ iλ e ) e ( i ) ( γ ) ψ ( γ ) ψ γ ψ ψ = ψ = + Λ ψ would change the for of (.4) to = i = i Λ and so ruin the invariance by adding the extra ter γ ψ Λ, we aintain the local gauge syetry of (.4) by replacing the ordinary four-gradient in (.4) with a gauge-covariant derivative D + ie where is the four-vector potential of electrodynaics and e is the electric charge strength. With this, Dirac s equation for an interacting electron becoes: ( ) ( ( ) ) ( ) ( ) = iγ D ψ = iγ + ie ψ = iγ eγ ψ = iγ + γ V ψ. (.5) with γ V eγ defining the electroagnetic perturbation. ecause

7 ( ) ( )( ) ( ) iλ iλ iλ iλ Dψ = D e ψ = + ie e ψ = e ψ + i e + Λ e ψ, (.6) = ψ + ψ = + ψ = ψ [ ] iλ iλ iλ iλ e e ie e ie e D and in view of the parallel transforations defined by = + Λ and D D = + ie, (.5) will now transfor as: iλ iλ ( i D ) i D i e D e ( i D ) = γ ψ = γ ψ ψ = γ ψ ψ = γ ψ. (.7) Dirac s equation thus reains invariant under local gauge transforations and at the sae tie as the very consequence of deanding this syetry now accounts for electrons interacting with an electroagnetic potential. This is how we start with a purely gravitational construct, naely the etric ds = g dx dx of (.), and introduce electroagnetic interaction siply by using syetry principles. This is iportant, because we will use these sae syetry principles to introduce the electrostatic repulsion that is responsible for the observers in Figure being able to stand on the ground beneath without passing through. The Klein-Gordon equation eerges in siilar fashion. Here, we start with (.3) in the for of p p = and ultiply fro the right by a scalar wavefunction φ to obtain ( ) ip x = p p φ. If we again consider plane wave solutions of the for φ e using the Fourier kernel, then again given that second derivative, we ay write: ip x ip x i e p e =, thus ip x ip x ip x ( p p ) φ ( p p ) e ( ) e ( ) e = p p e for the ip x = = = = φ, (.8) The net result, ( ) = + φ is the Klein-Gordon equation for a free scalar field φ. To introduce electroagnetic interactions, we deand that this equation be invariant when φ is i subjected to the local unitary gauge transforation φ φ = Uφ = e Λ φ again using a local phase Λ ( x ) which aintains the agnitude of φ but changes its orientation in the coplex phase space. s seen prior to (.5), when we take the derivatives ( i Λ iλ e ) e ( i ) φ = φ = + Λ φ we will obtain additional ters that ruin the syetry. s with (.5), we solve this by prooting to the gauge-covariant derivative D + ie, so that (.8) with an overall sign flip now becoes: ( ) ( ) φ ( )( ) φ ( ie ie e ) φ ( V ) = D D + = + ie + ie + = = + +, (.9) φ 6

8 with the new ters arising fro the gauge syetry used to define an electroagnetic perturbation V ie( + ) e. It will be noticed that the final ter e φ when it appears in the Klein-Gordon Lagrangian 4 ( D )( D ) 4 4 λφ ter and L= φ φ φ 4 λφ with a higher-order, sits at the root of spontaneous syetry breaking. In the course of this developent, two heuristic rules algoriths if one prefers eerge. First, given that ip x ip x i e p e = when one takes the four-gradient of a Fourier kernel as was done to reach (.4) and (.8), one will often interchange i p when oving between configuration space and oentu space. Second, given that local gauge syetry necessitates replacing ordinary derivatives with gauge covariant derivatives D + ie which inherently introduce both an electric charge and an electroagnetic potential, the cobination with the first heuristic rule i p leads to the second heuristic rule interchanging the gauge covariant derivative with a canonical oentu π p e via id = i e p e = π (.) as between configuration and oentu space. With the foregoing review of how electric charges and electroagnetic potentials are introduced by gauge syetry into the Dirac and Klein-Gordon equations steing purely fro the gravitational etric ds = g dx dx of (.), we now turn to the equations of otion that pertain to Figure. Especially, we shall now study the physics of how the observers in Figure are able to stand upon their respective surfaces without passing through, which is what enables the to observe a relative acceleration for the objects which those observers release into free fall. 3. The Physics of Standing on the Ground and not Passing Through To consider equations of otion, once again the starting point is the etric ds = g dx dx of (.) which via s = ds easures the proper tie along a worldline fro event to event. In atheatics generally, if one has a function f ( x ) and wishes to find its inia (or axia), one siply ascertains those places at which its derivative df / dx =. ased on the view that particles and systes will seek the lowest states of energy and follow the paths of least resistance and least distance and least tie, variational physics provides the tools for atheatically deducing generalized least action in various guises. s it turns out, nature has often obliged the view that physical systes will pursue paths of least action, least resistance, least energy, and least proper tie or length, by validating via epirical observation, what is atheatically deduced by variational physics. s regards classical, subliinal, aterial particles oving through spacetie, one deterines the equation of otion by taking and iniizing the variation of the proper tie, = δ s = δ ds. 7

9 In 9 of his landark 96 paper [4], lbert Einstein first calculated the variation = δ ds of the linear etric eleent ds = g dx dx of (.) between any two spacetie events and at which the worldlines of different observers eet so that their clocks and easuring rods and scales can be coordinated at the outset and then copared at the conclusion. 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 otion β d x β dx dx β u u ds = Γ = Γ (3.) ds ds β for a particle in a gravitational field, wherein the g β Γ ( g g g = ) of Christoffel capture all required inforation about the gravitational etric tensor g. In fact, for Figure (a), it is equation (3.) which tells us with specificity, the precise geodesic path that will be followed by the dropped object, until it hits the earth s surface and has its geodesic otion stopped because of the net electrostatic repulsion between its electrons and the earth s electrons. It is the epirical confiration of the otions predicted by (3.) and its integral ds = g dx dx of (.), including perihelion precession, gravitational light deflection, and gravitational redshifts, which provide epirical validation that General Relativity does indeed correctly describe the natural world. Once the freely falling object hits the surface below, (3.) no longer applies alone, but ust be suppleented with soething else. dditionally, neither does (3.) describe the otion of the observers in Figure. These observers are not in free fall and so they do not follow the path of (3.). In fact, given that it is electrostatic repulsion which in all cases stops and bars any free fall along the path (3.), the ipedient to the otion of (3.) is given by the Lorentz force law: β β β d x β dx β β = = = F = a = qf qf u F J, (3.) ds ds because this is the generalized physical law which contains the electrostatic Coulob interaction when taken in the rest frae. Here, F β is the ixed electroagnetic field strength tensor and q is the electric charge oving in this field. y putting the ass next to the acceleration rather β than leaving it in an q / ratio ultiplying F u, we ake clear that this truly is a Newtonian β β force four-vector F = a (distinguish fro F β by the one versus two indexes) that becoes non-zero whenever there are electric charges q with otion specified by u situated in an electroagnetic field F β. nd by using the electric current J qu =, we highlight how the Lorentz force describes electric currents situated and oving in electroagnetic fields. ecause (3.) is a classical force, and because it can be applied to fields F β eanating fro large nubers of charges and applied to currents J containing large nubers of other charges, we can use this to represent the net electroagnetic forces which cause the observers in 8

10 Figure to feel a easurable weight (illustrated as 5 kg) beneath their feet and which cause the objects in Figure to cease their otions (3.) when those objects eventually strike the surface. ut before we proceed, let us clarify this ore deeply: Unless the observer has charged up with a Van de Graaff generator prior to stepping into the experient of Figure or there is a nearby lightning stor or the like, we ay assue that the observer and the housing and the dropped object and all of the other aterial participants in Figure, in the net, are electrically neutral. This is because they are all aterial objects constructed fro atos and olecules containing an equal nuber of electron and protons. However, because the electron shells envelop the nuclei which contain the protons, the surfaces of the aterial bodies in Figure 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 becoes significant enough to stop the observer fro getting any closer to the ground than he or she already is. How close is close? Certainly, the distance aintained between the surface electrons of the observer and those of the ground will be greater than the ohr radius 8 a = ħ / ec = 5.9, because if it were saller, the observer and the ground would be part of the sae olecular syste, and not two separate olecular systes. nd certainly, these electrons do coe closer to one another than a sall fraction of a illieter, 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. So we ay define two objects to be touching physically when their surface electrons get close enough to one another that the surface electrons of each start repelling one another and thereby ake 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 becoe close enough to actuate this touching, the localized environent 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 neutrality is global, not local. So with all of this in ind, we start with the Lorentz force law (3.) and ascribe F β to represent the field strength associated the surface electrons of the ground that are involved in this touching, and ay ascribe J 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 ay assign F β to the observer s and J to the ground s electrons. Now let us return to Figure, and especially Figure (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.). To account for both the gravitation and the electrodynaics, it becoes necessary to suppleent (3.) with (3.), to write the total otion, with both the gravitation and electrodynaics accounted for, in the single equation: β β d x du β dx dx q β dx β q β = = Γ g F u u g F u Γ + + =, (3.3) ds ds ds ds ds 9

11 β β using g F = F to show the field strength with all raised contravariant indexes. The above is well-known, settled physics, see, e.g., the online []. It is iportant at this oent to point out that in (3.3), all we have done is anually suppleent the geodesic equation (3.) for gravitational otion derived via least action variation fro the etric (.) with the Lorentz force law of (3.). This is very unsatisfying, because while (3.) is obtained through the variation = δ ds fro the etric ds = gdx dx of (.) which is entirely geoetric, (3.3) is not the result of any such least action variation. We siply take the Lorentz force and tack it on, because we are fortunate enough to know about the Lorentz force fro a wide range fro electrodynaic studies and observations. Indeed, notably absent fro 9 of Einstein s General Relativity paper [4] which developed the geodesic equation (3.), / d x ds = e / F dx / ds. s was a siilar geodesic developent of the Lorentz force law ( ) ( ) will also be reviewed later in Sections of this paper, subsequent papers by Kaluza [] and Klein [] did succeed in explaining the Lorentz force as a type of geodesic otion and even gave a geoetric explanation for the electric charge itself, but only at the cost of adding a fifth diension to spacetie and curling that diension into a cylinder. To date, a century later, there still does not appear to have been any fully-successful attept to obtain the Lorentz force fro a geodesic variation confined exclusively to the four diensions of ordinary spacetie. In section to follow, we shall solve this long-standing proble. s to how this will be done, we note for the oent that by appending the Lorentz force to the geodesic otion in (3.3), we are appending an electroagnetic expression to a gravitational equation rooted in the spacetie etric (.). nd as reviewed in section, the proper way to introduce electrodynaics into gravitational geoetry, is to ipose gauge syetry. This eans, as suarized in (.), that ordinary derivatives becoe covariant derivatives via i id = i e and ordinary oenta becoe canonical oenta via p π = p e. Following this path, we shall show how ordinary geodesics (3.) becoe gauge-covariant geodesics (3.3) owing to nothing ore than the copletely natural application of gauge syetry. ut for the oent, we shall siply proceed fro that patchwork of (3.3) with the understanding that we shall later derive (3.3) fro least action principles. Thus, let us now use (3.3) to quantitatively explore the counterbalancing of gravitational and electrostatic forces which cause the observers in Figure to reain standing with easurable weight upon the surfaces of Figure, and not be in the gravitational free fall described by (3.). s earlier noted, and related to the Planck-Einstein dialogue at the end of [] as earlier discussed, the observers of Figure ay choose to define their own fraes of reference as rest fraes for which v=, even though they are feeling and can easure a force / weight between theselves and the surfaces upon which they stand. Let us now focus especially on the Figure (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 hi or herself to define v=, then the velocity four- u =,,, in this rest frae. If this observer also vectors in view of (.) will now becoe ( ) choses to define hi or herself as reaining at rest over tie so that u ( τ ) = (,,, ) = constant over the observer s own easureents of proper tie, then the observer s own acceleration

12 β β d x / ds = du / ds = will, by self-declaration, becoe zero. Thus, the left hand side of (3.3) will becoe zero. Further, as discussed several paragraphs back, the observer can choose to have F β represent the surface electric fields of the ground upon which he or she is standing and to have J = qu represent the currents of his or her own surface electrons. s a result, all of the velocity vectors in (3.3) can be set to u = (,,, ). With all of this, (3.3) reduces to: β q q q u u g F β u β u u g F β β β = Γ + = Γ + u = Γ + g F. (3.4) β Next, given that g β Γ ( g g g = ) and given the syetry of the β β etric tensor, we ay deterine that g ( g g ) Γ =. ut the gravitational field in Figure (a) is not tie-dependent, because for soeone standing on the earth s surface, that field always has the acceleration value g 9.8 /s which does not change with tie. β Therefore, g =, and so g β Γ g =. Likewise because of the tie-independence, β β k g =, so that Γ = g k g where k =,,3 runs over the three space diensions only. Using the foregoing, we ay siplify (3.4) to: q = q. (3.5) ( g 3 g g ) k g β g k + g F β g β g β g β 3 g F β = Progressing, we also know that in the linear field approxiation which certainly applies to the observers in Figure (a), the etric tensor g = η + ρh with ρ = 6π G. (Soeties this is written as g = η + κh with κ = 6π G but we use ρ to avoid confusion with the = 8 G = that appears in the Einstein equation κ κ π ρ T = R g R.) Consequently, because η = because the Minkowski tensor is constant, we ay rewrite (3.5) as: k q 3 q ( + 3 ). (3.6) β β β β β β = g kρh + g F = ρ g h + g h g h + g F However, ρ h = Φ is the Newtonian potential of a gravitational field, and for a ass M such as the earth this potential is ρ h = Φ = GM / r at a radial distance r. In Figure (a), we ay indeed take this ass 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 (a), we ay use ρ h = GM z, and eliinate the first two ters g h + g h, thus siplifying (3.6) to: β β / q GM q GM q = g ( ρh ) + g F = g + g F = g + g F z z z β 3 β β 3 β β 3 β 3. (3.7)

13 The above contains four independent equations running over the free index β =,,,3. ut for a spherically-syetric gravitational field which is very closely approxiated by the earth if we neglect the effects of the earth s rotation, only the diagonal coponents of g will be nonzero. Consequently, = and β = 3 are the only indexes in (3.7) that will yield a result other than =. s a result of the spherical syetry, (3.7) reduces to the single equation: GM q GM q GM k Qq = + = + = +. (3.8) z z z z e g g F g g Ez g g 3 bove, we have ade use of the electric field coponent F = E z as well as the Coulob electric field Ez = keq / z at a vertical distance z above a charge Q, where ke = / 4 πε = c / 4π is Coulob s constant and c = ε is the relationship between perittivity ε and pereability through which Maxwell proved that electroagnetic waves propagate at the speed of light c. 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 fro the center of the earth in the Newtonian field. When it coes to utual repulsion between the charges on the feet of the observer and those on the ground, i.e., as regards the observer touching the ground, those interactions are taking place over a uch shorter distance of fractions of a illieter, as earlier discussed. Of course, although the observer in Figure (a) does experience a gravitational field, that 33 diag g diag = +,,,, so g g. field is weak and so approxiates ( ) ( ) ( ) With this approxiation, (3.8) now becoes: e z = + = a z = η GM k Qq F. (3.9) z z bove, aside fro this reduction of (3.8), we have kept in ind that the zero on the left originated at (3.4) when the observer chose his or her reference frae to be the rest frae and reain so over β β tie, and so we set d x / ds = du / ds =. So the overall expression in (3.9) which is equal to β β zero, is an acceleration descending fro d x / ds = du / ds =, and following all of the reductions fro (3.4) through (3.9), it is in particular the z-axis acceleration az = dvz / dt = d z / dt =. This acceleration is equal to zero, not in any absolute sense, but only in a relative sense, because the Figure (a) observer chose to have his or her reference frae be the rest frae and reain so over tie. Fro soe other reference frae, such as that of the dropped object in gravitational free fall that we shall oentarily exaine, this acceleration would not be zero. We can gain further insight into all of this if we ultiply (3.9) through by the ass to look at this in ters of the z-axis force F. Dong so, we now obtain: z M z Qq z = G + k = e a = z Fz. (3.)

14 Here, we arrive at a atheatical 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 frae on a relative basis to be the rest frae over tie. The overall force is zero by choice of reference frae, but there are in fact two offsetting ters which net out to the zero force and the zero acceleration in chosen rest frae of reference, and they are in the for of Newton s law and Coulob s law. To discuss these, it helps to siply ove the Newton s law to the left to obtain: M G = k z e Qq. (3.) z Now let s parse out what this all eans. On the left is Newton s law for the gravitational force and on the right is Coulob s law for the electrostatic force. In (3.) we have oitted any reference to a force, because fro (3.) we see that the net force is zero. Prior to (3.3) we noted that we ight ascribe F β to represent the field strength associated the surface electrons of the ground, and ay ascribe J to represent the current associated with the surface electrons of the observer. In (3.) and (3.) those have respectively descended to Q and q. So when the observer is touching the ground, a net surface charge of Q fro the ground is repelling a net surface charge of Q fro the observer because these charges are close enough, at soe ean or net sall distance z which is sub-illieter but larger than the ohr length. nd so the overall force or repulsion is given in the for of Coulob s law. ut how strong is this? Now we turn to the left side of (3.) which tells us that to net totality of all these electrostatic forces is precisely equal to the expression GM / z where G is Newton s constant, M is the ass of the earth, is the ass of the observer, and z is the distance to the center of the earth. nd it is because these two are precisely equal, that the observer is able to stand on the surface of the earth without passing through and ascribe to hi or herself a rest frae with no net acceleration and no net force notwithstanding the counterbalancing of the Coulob force (which really is a Newtonian force) against the gravitational asseblage GM / z which has force diension d / t and is regarded as a force in Newtonian physics but is really just a result of the geodesic gravitational otion of (3.). Put differently, k Qq / z is the Coulob force between the observer and the ground, and of this force. e GM / z tells us the actual nuerical, easurable agnitude This is what balances electrostatics against gravitation and keeps the Figure (a) observer situated on the surface of the earth rather than falling through the planetary surface. What is experienced in a person s easureent of weight on a scale is the Coulob force blocking the gravitational free-fall otion. This is what also blocks a person on the upper floors of a tall building fro 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 terinates against the Coulob forces at the earth s surface. The relationships (3.) and (3.) are is therefore central to our actual ever-present physical experience of the world. 3

15 ll of the foregoing was calculated in the rest frae 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 choose the rest frae in Figure (a) to travel with the dropped object rather than with the observer. Here, with the results of (3.) and (3.) already in hand, we need nothing fancier than Newtonian physics and soe careful deduction. First, spatially, since the dropped object is accelerating downwards relative to the observer by GM / z which is the Newtonian potential in (3.9), we can likewise say that the observer is accelerating upwards by + GM / z relative to the dropped object in free fall. That is, the otions are equal in agnitude and opposite in direction. Second, no aount of relative otion or acceleration will cause the falling object to see the observer either passing through the earths surface or separating fro the earth s surface. The dropped object will still see the gravitational and Coulob forces counterbalancing one another according to (3.) and (3.). Therefore, using (3.) to represent that the observer is still touching the floor, the dropped object in Figure (a) will see the observer accelerating upwards along the z axis according to: Fz M q Q = az = + G = + k z z e, (3.) and so will also see a non-zero net force acting on the observer given by: M F a G k z Qq z = = + = + z z e. (3.3) So relative to the dropped object, the observer is being pushed upwards by a Coulob force Fz = az = keqq / z, because the observer is touching the floor due to his or her outer electrons with net charge q at a very sall distance z being close enough to the floor s outer electrons with net charge Q so as to generate a repulsive force which forces the observer to accelerate upwards. nd as to the agnitude of this Coulob force, as discussed after (3.), it is siply equal to the gravitational asseblage of ters GM / z given by Newton s law. ut this description of what the dropped object is seeing, is precisely what is illustrated in Figure (b), aside fro tidal forces. In su, Figure (a) represents an observer standing in a gravitational field and dropping an object when the rest frae is taken to be that of the observer and the governing equations are (3.9) through (3.), while Figure (b) represents an observer standing in a gravitational field when the rest frae is taken to be that of the freely-falling object and the governing equations are (3.) and (3.3). new aterial above old aterial below 4

16 4. asis and derivation s the basis for obtaining the Lorentz force fro a geodesic variation in four diensions, we begin with the equation = p p that describes the relativistic relationship between any ass and its kinetic energy-oentu p u ( dx / ds) = =. We then proote this kinetic oentu to a canonical oentu π via the prescription p π = p + e taught by the local gauge (really, phase) theory of Herann Weyl developed over 98 to 99 in [6], [7], [8], and so obtain = p p = π π. It will be appreciated that this prescription is the oentu space equivalent of = / x D = + ie which is the gauge-covariant derivative specified in a configuration space for which the etric tensor of the tangent flat diag η = +,,,. Consequently, deconstructing into a linear Minkowski space is ( ) ( ) equation using the Dirac atrices { γ, γ } η to obtain Dirac s equation ( iγ D ) ψ = in flat spacetie, one can eploy = π π = for an electron wavefunction ψ in an electroagnetic potential, which equation Dirac first derived in [5] for a free electron in a for equivalent to ( iγ ) ψ =, i.e., without yet using D = + ie. So to obtain the Lorentz force fro a geodesic variation in spacetie, we backtrack fro = π π to a linear etric eleent: ( ) ( ) ( ( / ) ) ( / ) ( ) ds = g dx dx ds = g d χ d χ = g dx + ds e dx + ds e / / = g dx dx + e dx ds + e g ds, (4.) which uses a canonical gauge prescription for the spacetie coordinates theselves, naely: ( / ) dx d χ = dx + ds e. (4.) This is just another variation of p π = p + e and D = + ie. Indeed, it is easily ( ) seen that if one ultiplies ( ( / ) ) ( / ) / ds, the result is identical to the canonical ds = dx + ds e dx + ds e in (4.) through by π π =. Now, all we need do is apply a variation = δ ds to the linear eleent (4.) and the Lorentz force naturally eerges as a geodesic equation of otion right alongside of the gravitational equation of otion. Proceeding with this derivation which largely parallels that in the online [9], we first use (4.) to construct the nuber 5

17 dx dx e dx e = g + + g ds ds ds, (4.3) which we then use to write the variation as:. (4.4) dx dx e dx e = δ ds = δ ds g + + g ds ds ds pplying δ to the integrand and using (4.3) to clear the denoinator, this yields: dx dx e dx e δ ds dsδ g g. (4.5) = = + + ds ds ds Dropping the ½ and using the product rule, while assuing that there is no variation in the charge- δ e/ = over the path fro to, we now distribute δ using the to-ass ratio i.e., that ( ) product rule to obtain: dx dx dδ x dx dx dδ x e dx e dδ x δ g + g + g + δ + = ds ds ds ds ds ds ds ds ds.(4.6) + ( e / ) ( δ g + g δ + g δ ) One can use the chain rule in the sall variation δ liit to show that δ g g x = δ and x δ = δ. So the botto line equals δ x ( e / ) ( g g g + + ). Likewise, we ay recondense ( ) g = g + g + g via the product rule. Therefore, the entire integral on the botto line contains a total derivative given by: e e s δ x x g x x ( g ) ds = δ ( ) =. (4.7) This equals zero, because the two worldlines intersect at the boundary events and but have a slight variational difference between and otherwise, so that x δ ( ) = δ x ( ) = while δ x elsewhere. Consequently, the botto line of (4.6) zeros out, leaving us with: dx dx d x dx dx d x e dx e d x = ds δ g + g δ + g δ + δ + δ ds ds ds ds ds ds ds ds. (4.8) Fro here, again using δ g = g δ x and δ = δx, and also re-indexing and using the syetry of g to cobine the second and third ters above, we obtain: 6

18 dx dx dδ x dx e dx e dδ x = ds δ x g + g + δ x + ds ds ds ds ds ds Next, we integrate by parts. ( )( ) ( ) ( ) δ / ( / )( δ ) ( / ) δ. (4.9) First, we use the product rule to replace ( ) ( )( ( )) g dδ x / ds dx / ds = d / ds δ x g dx / ds δ x d / ds g dx / ds and likewise d x ds = d ds x d ds x. ut the ters containing the total derivatives will vanish for the sae reasons that the ters in (4.7) vanished as a result of the boundary conditions δ x = δ x =. s a result, (4.9) now becoes: ( ) ( ) dx dx d dx e dx e d = ds δ x g x g x x δ + δ δ. (4.) ds ds ds ds ds ds pplying the d / ds derivative contained in the second ter above then yields: dg dx dx d x dx e dx e d = ds δ x g δ x g δ x + δ x x δ,(4.) ds ds ds ds ds ds ds for the first tie revealing the acceleration d x / ds in the second ter above. Next, we use the chain rules dg / ds g ( dx = / ds) and d / ds ( dx / ds) rewrite the third and fifth ters above, thus obtaining: = to = dx dx d x dx dx δ x g δ x g δ x g ds ds ds ds ds ds. (4.) e dx e dx + δ x δ x ds ds In the botto line above, we ay renae indexes in the last ter, to find that we x dx / ds x dx δ / ds = δ x F dx / ds using the ay rewrite ( ) δ ( ) ( ) electroagnetic field strength tensor F =, which has now appeared as a result of the variation. So the above now siplifies to: = ds δ x g x g x g + F ds ds ds ds ds ds dx dx d x dx dx e dx δ δ x δ.(4.3) Now we renae indexes so that the δ x ters all contain the index, that is, so all of these ters are δ x. We then factor this out and interchange the first and second ters, obtaining: 7

19 d x dx dx dx dx e dx = dsδ x g + g g F +. (4.4) ds ds ds ds ds ds For aterial worldlines, ds. Likewise, while x ( ) x ( ) δ = δ = at the boundaries, between these boundaries where the variation occurs, δ x. Thus, ultiplying through by ½, for (4.4) to be true the integrand ust be zero, and so we have: d x dx dx dx g dx e dx = g + g + F. (4.5) ds ds ds ds ds ds Now we ove the acceleration ter to the left, split the ter with g = g + g into two halves, renae soe indexes while using the syetry of g, and finally ultiply through by g β and then raise indexes. This all yields: β d x β dx dx e β dx g ( g g g ) + F ds =. (4.6) ds ds ds β ut of course, we recognize that the Christoffel sybols g β Γ ( g g g = ) s a consequence, the above reduces to:. β d x β dx dx e β dx F = Γ +. (4.7) ds ds ds ds In the presence of gravitational and electroagnetic fields, this contains both the equations of gravitational otion and the Lorentz force law, obtained via the geodesic variation of the canonical invariant etric length eleent (4.). In the absence of gravitation, i.e., for g = η over the β spacetie region being considered thus Γ =, this reduces to the Lorentz force law. s a result, we have proved that by using Weyl s canonical prescription in for of dx d χ = dx + ds e / fro (4.) to define the linear etric eleent by ( ) ds = g d d χ χ as shown in (4.), the Lorentz force law of electrodynaics ay indeed be obtained fro a geodesic variation confined exclusively to the four diensions of ordinary spacetie geoetry. 5. Non-abelian gauge fields It will be observed that the field strength tensor eerging in (4.7) is an abelian field strength F = [ ]. It is natural to inquire whether one can use this sae approach to also obtain 8

20 the equation for the otion for charged particles in non-abelian fields F = + ie, = D. [ ] [ ] [ ] It turns out that this is fairly siple to do. First, we again re-index (4.) by interchanging in the botto line. Likewise, we re-index the top line so that all of the coordinate variations have the sae index, naely, are of the for δ x, which we then factor out as earlier. We also ove the acceleration ter to the first position, and we split the final ter on the top line in half once again and re-index to show the cycling of the Christoffel indexes. We also restore the factor of ½ that we earlier dropped after (4.5), and restructure into what is clearly the for of the final result (4.7). So what we now have is: d x dx dx e dx = δ x ds g ( g g g ) + + [ ]. (5.) ds ds ds ds iλ( x ) Fro here, we perfor a local gauge (phase) transforation e = on the gauge fields, and insist that this variation reain invariant under such transforation. Consequently, we ust proote the derivative that acts on the gauge fields to the gauge-covariant D = + ie in the usual way. s a result (5.) now becoes: = δ x ds g + g g g + D ds ds ds d x dx dx e dx ( ) [ ] ds. (5.) The integrand ust still be zero because δ x except at the boundaries. Therefore, this yields the exact sae result as was found in (4.7) with no change whatsoever in for. The only difference is that the field strength is now the non-abelian F = D[ ]. 6. Einstein s Equation and Maxwell s Equations ecause the etric length ds = g d d χ χ of (4.) under a variation = δ ds siultaneously provides a geodesic description of otion in a gravitational field and in an electroagnetic field, and because the prescription dx d χ = dx + ds ( e / ) is no ore than a variant of Weyl s gauge prescriptions p π = p + e in oentu space and D = + ie in configuration space and leads directly as well to Dirac s equation ( iγ D ) ψ = for an interacting ferion, this ay fairly be regarded as a classical etriclevel unification of electrodynaics with gravitation, using four spacetie diensions only. ut the equations of otion in a field are only half the atter. We also need to know the equations for the fields theselves in relation to their sources. Thus we now ask, can the field equation κt = R g R which specifies the gravitational field, be shown to relate in soe direct fashion to Maxwell s field equations for electric and (the absence of) agnetic sources? 9