The Geometrodynamic Foundation of Electrodynamics

Transcription

1 Jay R. Yablon, May 4, 16 The Geometroynamic Founation of Electroynamics 1. Introuction The equation of motion for a test particle along a geoesic line in cure spacetime as ν specifie by the metric interal c τ = g x x with metric tensor g was first obtaine by lbert Einstein in 9 of his lanmark 1916 paper [1] introucing the General Theory of relatiity. The infinitesimal linear element τ = s / c for the proper time is a scalar inariant that is inepenent of the chosen system of coorinates. Likewise the finite proper time τ = τ measure along the worlline of the test particle between two spacetime eents an has an inariant meaning inepenent of the choice of coorinates. Specifically, the geoesic of motion is stationary, with the ariational equation = δ τ. (1.1) Simply put, the particle goes from eent to eent in the physically minimum (shortest) possible proper time. fter carrying out the well-known calculation of [1] for which there is a ery goo online reiew at [], this equation of motion is foun to be: β β ν x u β x x β ν u u ν = = Γ τ τ τ τ = Γ, (1.) β 1 with the Christoffel connections efine by g βα Γ ( g g g α ν να ν α ) an the relatiistic four-elocity by u x / τ. This geoesic path (1.) is the path along with the proper time is minimize, again, the shortest proper time between two eents. If the test particle to which we ascribe a mass m > also has a non-zero net electrical charge q an the region of spacetime in which it is moing also has a nonzero electromagnetic fiel strength F βα βα β α α β efine by F in relation to the gauge potential fourector α, with F βα containing the electric an magnetic fiel biectors E an, then the equation of motion is no longer (1.), but is supplemente by an aitional term which contains the Lorentz force law, namely: β β ν x u β x x q βα x β ν q βα u = Γ g F u u g F α = Γ α = + +, (1.3) τ τ τ τ m cτ m c The aboe is also well-known, obsere, settle physics, see, e.g., the online reiew [3]. Gien that the graitational geoesic (1.) specifies a path of minimize proper time (1.1), the question arises whether there is a way to obtain (1.3) from the same ariation (1.1), thus reealing the electroynamic motion to also be one of particles moing through spacetime along 1

2 Jay R. Yablon, May 4, 16 paths of minimize proper time. Philosophically, it cannot be argue other than that this woul be a esirable state of affairs. ut physically the ifficulty rests in how to o this without ruining the integrity of the metric an the backgroun fiels in spacetime by making them a function of the charge-to-mass ratio q / m, because this ratio is an must remain a characteristic of the test particle alone. It is not an cannot be a characteristic of the metric τ or the metric tensor g or the gauge fiel α or the fiel strength F βα which efine the fiel-theoretical spacetime backgroun through which the test particle is moing. n, at bottom, this ifficulty springs form the inequialence of the electrical mass a.k.a. charge q an the inertial mass m, ersus the Galilean equialence of the graitational an inertial mass. In (1.3), this is capture by the fact that m oes not appear in the graitational term Γ β u u ν, while the the q / m ratio oes appear in the electroynamic Lorentz force term that we rewrite as ( / ) q m F β u in natural c = 1 units. This may also be seen ery simply if we compare Newton s law with Coulomb s law. In the former case we start with a force F = GMm / r (with the minus sign inicating that graitation is attractie) an in the latter F = keqq / r (for which we choose an attractie interaction), where G is Newton s graitational constant an the analogous ke = 1/ 4 πε = c / 4π is Coulomb s constant. If the graitational fiel is taken to stem from M an the electrical fiel from Q, then the test particle in those fiels has graitational mass m an electrical mass q. ut the Newtonian force F = ma always contains the inertial mass m. So in the former case, because the graitational an inertial mass are equialent, the acceleration β ν a = Fm = GMm / mr = GM / r an these two masses cancel, hence Γ u u without any mass in (1.3). ut in the latter case the acceleration = = e = ( ) β because the electrical an inertial masses are not equialent, hence ( q / m) F u a F / m k Qq / mr q / m k Q / r e containing this same ratio in (1.3), an the motion is istinctly epenent on the electrical an inertial masses q an m of the test particle, een though ifferent charges q with ifferent masses m may all be moing through the exact same backgroun fiels. So, were we to pursue the philosophically attractie goal of unerstaning electroynamic motion to be the result of particles moing through spacetime along paths of minimize proper time, whereby (1.1) applies to electroynamic motion just as it oes to graitational motion, the τ q / m of q / m. n this in turn metric element τ woul inescapably hae to be a function ( ) woul appear to iolate the integrity of the metric element τ as well as the metric tensor g in τ =, because these woul all seem to be epenent upon the attributes q an m of ν c g x x the test particles that are moing through the spacetime backgroun. Were this to be reality an not just seeming appearance, this woul be is physically impermissible. Consequently, espite there being many known eriations of the Lorentz force law, there oes not, to ate, appear to be an acceptable rooting of the Lorentz force law in the ariational equation = δ τ which woul reeal electroynamic motion to also be geoesic motion. n this is precisely because it is not unerstoo how to o this while simultaneously maintaining the integrity of fiel theory such that the metric an the backgroun fiels o not epen upon the attributes of the test particles which

3 Jay R. Yablon, May 4, 16 may moe through these fiels. n this in turn, is because electrical mass is not equialent to inertial mass in contrast to what is the Galilean equialence of graitational an inertial mass. τ which is In this paper we shall show that we can in fact hae a metric element ( q / m) a function of the electrical-to-inertial mass ratio q / m, from which the ariational equation = δ τ oes yiel the combine graitational an electroynamic equation of motion (1.3), yet for which the metric τ an the metric tensor g an the gauge fiel α an the fiel strength F βα are all inepenent of this q / m ratio. This seemingly-paraoxical result of haing the metric be a mathematical function of q / m yet be physically inepenent of q / m reeals that when a first test particle with electrical mass q an inertial mass m is place in a fiel F βα, an a secon test particle with electrical mass q an inertial mass m an a ifferent ratio q / m q / m is situate at equipotential in the same fiel F βα, the obserably-ifferent Lorentz motions for these two ifferent test particles een though they are at equipotential is the consequence of the fact that time oes not flow at the same rate for these two test particles in ery much the same way that time oes not flow at the same rate for two reference frames in special relatiity which are in motion relatie to one another. Specifically, it will be appreciate that the Lorentz motion in (1.3) also contains a set of coorinates x β, so that in the absence of graitation with g = η an Γ =, the first test particle will hae a Lorentz motion gien by: β x q βα x η F α =. (1.4) τ m cτ Orinarily, it is assume that for the secon test particle, the motion is gien by this same equation (1.4), merely with the substitution of q q an m m, that is, by: =. (1.5) τ m cτ β x q βα x η F α The particular assumption is that there is no change to the rate at which time flows as between (1.4) an (1.5), an more generally the assumption is that x in (1.4) is ientical to x in (1.5). Yet, as we shall see, it is impossible to hae both (1.4) an (1.5) emerge through the ariation = δ τ from the same metric element τ, an simultaneously maintain fiel theory integrity, unless the coorinates are ifferent, wherein x in (1.4) is not ientical to what must now be x x x in (1.5). In fact, as we shall establish here, the ery physics of haing electric charges in electromagnetic fiels actually inuces a change in coorinates as between these two test charges with ifferent q / m q / m, ery similar to the coorinate change ia a Lorentz transformations 3

4 Jay R. Yablon, May 4, 16 inuce by relatie motion, whereby the electroynamic motion of the secon test charge is gien not by (1.5), but by:. (1.6) τ m cτ β x q βα x = η F α Here, x β β β an x x are ifferent coorinates, yet they are interrelate by a efinite transformation one to the other. Most importantly, this results in time itself being inuce to flow ifferently as between these two sets of coorinates, making time ilation an contraction a funamental aspect of electroynamics as it alreay is for the special relatiistic theory of motion an the general relatiistic theory of graitation. In fact, what is really happening physically is that the placement of a charge in an electromagnetic fiel is physically inucing a change of coorinates x β ( q / m) x β ( q / m ) in the ery same way that relatie motion between the β β coorinate systems x ( ) an x ( ) of two ifferent reference frames with elocities an ν inuces a Lorentz transformation x β ( ) x β ( ) which with relates both coorinate systems to ν ν one another ia the inariant element c τ = c τ = η x ( ) x ( ) = η x ( ) x ( ) with the same metric tensor η = η. The metric which in fact yiels (1.3) from (1.1) so as to inclue electroynamic motion is: q ν q ν ν c τ = g x + τ x + τ = g Dx Dx, (1.7) mc mc Dx x + q mc τ. n it will be seen that upon multiplying through where we efine ( / ) by m c an iiing through by τ this becomes: ν 4 x x ν m c = g mc + q mc + q τ τ (1.8) for the square energy equialent of the inariant rest mass. y way of preiew, this gauge- Dx x + q / mc τ is a irect outgrowth of the gauge- coariant coorinate interal ( ) coariant eriaties D + ie an canonical momenta π = p e which emerge from gauge theory an in particular from the manate for gauge (really, phase) symmetry. This metric (1.7) is clearly a function of q / m an so has the appearance of epening on the ratio q / m. ut when we now place the secon test charge with the secon ratio q / m q / m in the exact same metric measure by the inariant element τ an moing through the exact same fiels g an, this metric becomes: 4

5 Jay R. Yablon, May 4, 16 q ν q ν ν c τ = c τ = g x + τ x + τ = gdx Dx. (1.9) m c m c So espite τ being a function of the q / m ratio, this τ = τ as a measure proper time element is actually inariant with respect to the q / m ratio because the ifferences between ifferent q / m an q / m are entirely absorbe into the coorinate transformation x x which is wholly analogous to the Lorentz transformation of special relatiity. So τ = τ is a function of charge q an mass m yet is inariant with respect to the same, an there is no paraox in haing τ = τ be a function of, yet be inariant uner, a rescaling of the q / m ratio. Likewise, the fiels g = g an = are inepenent of the charge an the mass of the test particle, because again, eerything emanating from the ifferent ratios q / m an q / m is absorbe into a coorinate transformation x x. In fact, this transformation x x is efine so as to keep τ = τ an g = g an = βα βα an by implication the fiel strength biector F = F all unchange, just as Lorentz transformations are efine so as to maintain a constant spee of light for all inertial reference frames inepenently of their state of motion. This coorinate transformation, which applies to the test particle in question whereby eery ifferent test particle carries its own coorinates, causes time to ilate for electrical attraction an to contact for repulsion, with a imensionless ratio t / τ = x / τ γ em that integrally epens upon the magnitue of the likewise imensionless ratio q / mc of electromagnetic interaction energy q to the test particle s rest energy t / τ γ 1/ 1 / c 5 mc. This supplements the ratio τ = γ = for a clock = = for motion in special relatiity an t / g 1/ g at rest in a graitational fiel in the oerall prouct combination t / τ = γ emγ gγ goerning time ilation when all of motion an graitation an electromagnetic interactions are present. Operationally, the electromagnetic contribution to this time ilation or contraction woul be measure by comparing the rate at which time is kept by otherwise ientical geometroynamic clocks or oscillators which are electrically charge with ifferent q / m ratios, an then place at rest into a backgroun potential = ( ϕ, ) = ( ϕ, ) at equipotential, where ϕ is the proper potential. More generally, this woul be measure by electrically charging otherwise ientical clocks an then placing them into the potential to hae iffering q / mc = qϕ / mc ratios. Empirically, for q ϕ / mc 1, the interaction energies Eem = Fr = + keqq / r sans integration constant for an attractie Coulomb force F = keqq / r are relate to these electromagnetic time ilations in a fashion ientical to how the kinetic energy in special relatiity is obsere to be the 1 1 quantity E = m in mc γ = mc / 1 / c mc + m for nonrelatiistic elocities c. n in fact, these electromagnetic time ilations are miniscule for eeryay electromagnetic interactions, as are special relatiistic time ilations for eeryay motion. Testing of t / τ changes for electroynamics may perhaps be best pursue with experimental approaches similar to those use to test relatiistic time ilations, as will be elaborate further here.

6 Jay R. Yablon, May 4, 16 Importantly, unlike graitational time reshifts or blueshifts which are a consequence of spacetime curatures, these electromagnetic time ilations o not stem irectly from curature, an they only affect curature inirectly through any changes in energy to which they gie rise because graitation still sees all energy. Hermann Weyl s ill-fate attempt from 1918 until 199 in [4], [5], [6] to base electroynamics on real graitational curature roote in the inariance of Λ(, ) Lagrangians an fiel equations uner a non-unitary local transformation ψ ψ e t x = ψ which re-gauges the magnitue of a waefunction, rather than uner the correct transformation iλ( t, x) ψ ψ = Uψ = e ψ which simply reirects the phase, foreclose this possibility, because the latter correct transformation is associate with an imaginary, not real curature that places a factor i = 1 into the geoesic eiation when expresse in terms of the commutatiity, ; ; of spacetime eriaties, as will be reiewe here. The alteration of time flow in electroynamics, is therefore much more akin to the time ilation of special relatiity, than it is to the reshifts an blueshifts of general relatiity, an may transpire entirely in flat spacetime but to the egree that the electroynamic energies associate with any particular time alterations may reach sufficient magnitue to cure the nearly spacetime. lso importantly, the similarity of the ratios ϕ / mc an / q c as the riing number in γ em an γ = 1/ 1 / c respectiely, is more than just an analogy. Just as m < mc is a funamental limit on the motion of material subluminal particles, so too, when we follow the electroynamic time ilations an contractions through to their logical conclusion, an if we likewise require that particle an antiparticle energies are always positie an time always flows forwar in accorance with Feynman-Stueckelberg an that the spee of light must remain the material limit that it is known to be, then it turns out that qϕ < mc is a material limit on the strength of the interaction energy between a test charge q with mass m interacting with the sources of the proper potential ϕ. n, it turns out that when ϕ = keq / r is the Coulomb potential whereby this limit becomes keqq / r < mc, we fin that there is a lower physical limit on how close two interacting charges can get to one another, thereby soling the long-staning problem of how to circument the r = singularity in Coulombs law. In short, in orer to be able to obtain equation (1.3) for graitational an electroynamic motion from the minimize proper time ariation (1.1) in a way that preseres the integrity of the metric an the backgroun fiels inepenently of the q / m ratio for a gien test charge an thereby achiees the philosophically-attractie goal of unerstaning electroynamic motion to be geoesic motion just like graitational motion, we are require to recognize that attractie electroynamic interactions inherently ilate an repulsie interaction inherently contract time itself, as an obserable physical effect. This is the ientical to how relatie motion ilates time, an how graitational fiels ilate (reshift) or contract (blueshift) time. In this way, it becomes possible to hae a spacetime metric which although a function of the electrical charge an inertial mass of test particles also remains inariant with respect to those charges an masses. This preseres the integrity of the fiel theory, an it establishes that electroynamic motion is in fact 6

7 Jay R. Yablon, May 4, 16 geoesic motion that satisfies the minimize property time = δ τ from (1.1). s a result, it becomes possible to lay an entirely geometroynamic founation for electroynamics. References [1]. Einstein, The Founation of the General Theory of Relatiity, nnalen er Physik (ser. 4), 49, (1916) [] [3] [4] H. Weyl, Graitation an Electricity, Sitzungsber. Preuss. ka.wiss., (1918). [5] H. Weyl, Space-Time-Matter (1918) [6] H. Weyl, Electron un Graitation, Zeit. f. Physik, 56, 33 (199) 7