The geometrical origin of the Doppler factor in the Lienard-Wiechert potentials
aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n The geometrical origin of the Doppler factor inthe Li´enard-Wiechert potentials
C˘alin GaleriuFebruary 1, 2021
Abstract
We present an in depth analysis and a new derivation of the Doppler factorin the Li´enard-Wiechert potentials, based on geometrical considerations inMinkowski space. We argue that, contrary to a common assumption, themethods used for deriving the Doppler factor in the case of an electricallycharged extended particle are not applicable in the case of a point particle. Ageometrical interpretation of the Doppler factor is nonetheless found in thelater case, based on the electromagnetic interaction model of Fokker. In thismodel the interaction takes place between infinitesimal worldline segmentswith end points connected by light signals, just like the points where lightpulses are emitted and detected in general. This analogy reveals that therelativistic Doppler effect is the missing link between the classical Dopplereffect and the Doppler factor in the Li´enard-Wiechert potentials.
The Li´enard-Wiechert (LW) potentials of a moving point charge [1, 2] canbe derived from the potentials of a stationary electric charge. The transitionfrom the static to the mobile case can happen in different ways, dependingon how we look at the electrically charged particle. The particle can be amaterial point with an electric charge Q , or it can extend in space over asmall volume, which in the limit collapses into a point. In the later casethe particle is described by an electric charge density ρ ( −→ r ) whose integral inspace gives the total electric charge Q . Z ρ ( −→ r ) dx dy dz = Q. (1)1or an electrically charged point particle at rest at position −→ r = ( x , y , z ),the electric (scalar) potential (in Gaussian units) at −→ r = ( x , y , z ) is φ ( −→ r ) = Q/R , where −→ R = −→ r − −→ r = ( x − x , y − y , z − z ). In theelectrostatic case the magnetic (vector) potential is −→ A ( −→ r ) = 0, and theelectromagnetic four-potential at −→ r is Φ = ( −→ A , iφ ) = (0 , , , i QR ) . (2)In Minkowski space the field point has a position four-vector X =( x , y , z , ict ), while the stationary source charge has a retarded positionfour-vector X = ( x , y , z , ict ) and a four-velocity V = (0 , , , ic ). Sincethe two points are connected by a light signal, ( X − X ) · ( X − X ) = 0,and therefore R = c ( t − t ). It follows that ( X − X ) · V = − cR . Theelectromagnetic four-potential (2) at X becomes Φ = − Q V ( X − X ) · V . (3)For an electrically charged point particle in motion, with a retarded ve-locity −→ v = ( v x , v y , v z ) and a retarded four-velocity V = ( γv x , γv y , γv z , iγc ),where γ = (1 − v /c ) − / , the electromagnetic four-potential (3) at X be-comes Φ = QR ( v x /c, v y /c, v z /c, i )1 − −→ R · −→ v / ( Rc ) . (4)The denominator of the Doppler factor may also be written as 1 − ˆ R · −→ v /c [3](where ˆ R = −→ R /R is a radial unit vector), or as 1 − v r /c [4] (where v r = −→ v · ˆ R is the radial component of the retarded velocity, directed toward the fieldpoint), or as 1 − β cos( θ ) [5] (where β = v/c and θ is the angle between −→ v and −→ R ).The electrostatic potential φ ( −→ r ) produced by a continous electric chargedistribution at rest is φ ( −→ r ) = Z ρ ( −→ r ) R dx dy dz, (5)where −→ R = −→ r − −→ r . When ρ ( −→ r ) = 0 only in a very small neighborhoodcentered on −→ r , R gets out of the integral and, because of (1), we get thesame electromagnetic four-potential (2).When the continuous electric charge distribution is in motion, the retar-dation condition changes (5) into φ ( −→ r , t ) = Z ρ ( −→ r ret , t ret ) R dx dy dz, (6)2here −→ R = −→ r − −→ r ret and t ret = t − R/c . When ρ ( −→ r ret , t ret ) = 0 only in avery small neighborhood centered on −→ r , R gets out of the integral and weget the same electric potential as in (4), provided that Z ρ ( −→ r ret , t ret ) dx dy dz = Q − v r /c . (7)When comparing (7) with (1), “most undergraduate students have majordifficulties understanding where the additional factor [...] comes from.”[5]The correct explanation is that the integral in (7) “does not represent thetotal particle charge because the charge density in the integrand is taken atdifferent retarded times”. We have to recognize the fact that the integral in(7) “is not a simultaneous integral.”[5]No wonder this step in the derivation of the LW potentials is counterin-tuitive, since in Newtonian physics a volume integration is always done atthe same time. In relativistic physics, due to the relativity of simultaneity,things change a little. Often a 3D volume integration is done at the sametime, but only in a specific reference frame. This does not apply here, be-cause there is no reference frame in which the retarded electric charges onthe lightcone are simultaneous. There is no such a reference frame becausewe cannot have a Lorentz boost with the speed of light. For a Lorentz boostwith the speed of light, the spatial volume element would decrease to zero,while the electric charge density would increase to infinity. Agurregabiria et al. [5] mention that “one can change variables to have a simultaneousintegral. This gives the factor (1 − βcosθ ) − as the Jacobian of the transfor-mation, but the actual computation is a bit cumbersome”, probably referringto a derivation by O’Rahilly that makes use of curvilinear coordinates. [4]Sometimes a 3D volume integration is done on an invariant hypersurface, a3D subspace of the 4D Minkowski space. This does not apply here either,because the infinitesimal volume element on the lightcone is zero, and as aresult the volume element in the integral would be zero.New concepts are needed, and Aguirregabiria et al. [5] describe “theeffective integration region”as the region where the retarded electric chargedensity is different from zero, different from “the spatial region occupied bythe charge at a single instant of time”. Griffiths [3] uses an equivalent term,“the apparent volume”of the moving extended object, different from “theactual volume”of the same object.Here we show that a very clear understanding of the concept of effectiveintegration region emerges when the integral in (7) is evaluated in Minkowskispace. We use geometrical operations, intersections and projections, in orderto justify the apparition of the Doppler factor in the LW potentials. As statedby Griffiths, “this is a purely geometrical effect.”[3] A similar geometrical3pproach (based on intersections) was used by Aguirregabiria et al. [5] inorder to find the retarded shape of a moving sphere. A first question to ask, when looking at (7), is: What is the domain ofintegration? To what 3D space does the dx dy dz volume element reallybelong to? The domain of integration is the 3D space simultaneous with thefield point where the LW potentials are calculated. It is the hyperplane ofconstant time t = t , a 3D spatial slice through the 4D Minkowski space.This fundamental fact is hidden from plain sight by many authors wholook at a volume element at the retarded time t = t . Morse and Feshbach[6] consider a volume element sandwiched between two spherical boundariesdrawn at two different retarded times, and mention that ”in the integration,the amount of charge dq inside the volume element dA dr r is not ρ dA dr r , asit would be if the charge were not moving, but is [1 + ( u/c ) cosβ ] ρ dA dr r ”, asif the electrically charged matter could move in or out of this multi-temporalvolume element. This construction of the two spherical boundaries closelymirrors that of Li´enard [1] and Wiechert[2], who compare the volume ofthis retarded volume element as seen from the stationary reference frameof the observer with the volume of the same retarded volume element asseen from the moving reference frame of the electrically charged extendedparticle. Li´enard mentions that during the integration in (7) the contentof this retarded volume element is swept (”balay´e”) by a spherical surfacecentered on the field point. Panofsky and Phillips [7] replace the static pic-ture of the two spherical boundaries drawn at two different retarded timeswith the dynamic picture of one spherical surface collapsing with the speedof light c toward the observation point. They mention that ”during the timethe information-collecting sphere [...] sweeps over the charge distribution thecharges may move so as to appear more or less dense” and then conclude that”the retarded potential of an approaching charge will be larger than that of areceding charge at the same distance from the observer, since the approachingcharge stays longer within the information-collecting sphere”. The Dopplerfactor is the result of a multi-temporal electric charge density. In the samespirit Feynman writes that “there is a correction term which comes aboutbecause the charge is moving as our integral ‘sweeps over the charge.’ ”[8] Inyet another derivation, again based on a retarded multi-temporal volume ele-ment, Page and Adams [9] describe how, due to the velocity of the electricallycharged matter, the retarded rectangular volume element changes shape inorder to accommodate the retardation condition on a pair of opposing sides,4hus becoming a slanted prism. When contemplating all these derivations ofthe Doppler factor, one may very easily become confused.A second question to ask, when looking at (7), is: What do we get whenwe replace the retarded electric charge density in the integral with the instan-taneous electric charge density? In this case the integrand is non-zero onlyinside the volume where the electric charge is present at the actual moment t = t . This space region is the intersection of the worldtube of the chargedparticle with the Minkowski hyperplane of constant time t = t . When thecharged particle is at rest, this intersection is a sphere. When the chargedparticle is in motion, this intersection is a Lorentz ellipsoid. In both casesthe total electric charge of the particle is the same.A third question to ask, when looking at (7), is: How do we calculatethis non-simultaneous integral? At time t = t , for each point ( x, y, z ) atthe center of a volume element dx dy dz , we need to find out the valueof the retarded electric charge density ρ ( x, y, z, t ret ). We have to start atthat point, and then go back in time, keeping the same x, y, z coordinates,until we meet the retarded lightcone drawn through the field point. Whatis the retarded electric charge density at this place? In other words, is theintersection point that we get in this way also a point inside the worldtubeof the electrically charged particle? In order to better answer this question,we realize that we can travel up and down along this path in both ways.In Minkowski space, we may start from the ( x, y, z, ict ) point, go down tothe ( x, y, z, ict ret ) point on the retarded lightcone, and then see whether thispoint belongs to the worldtube of the electrically charged particle. Or wemay start with the intersection of the retarded lightcone with the worldtubeof the electrically charged particle, and then project this intersection on theMinkowski hyperplane of constant time t = t . We will use this last methodin order to find the volume of the region with non-zero retarded electriccharge density.In conclusion, the effective integration region of Aguirregabiria et al. [5],same as the apparent volume of Griffiths [3], is the projection on the fieldpoint’s 3D space of the intersection of the retarded light cone with the world-tube of the electrically charged particle. Since the electric charge density isassumed uniform, the integral (7) is proportional to this apparent volume,the volume of the effective integration region.5 The actual volume of a spherical particlein motion
Consider a very small spherical particle of radius a and uniform electriccharge density ρ ′ , at rest in the reference frame K ′ . The particle is at adistance d from the origin, on the Ox ′ axis. The volume of the sphere is4 πa ′ /
3, and the total electric charge of the particle is Q = ρ ′ πa ′ /
3. Thecenter of the sphere, with coordinates( x ′ α , y ′ α , z ′ α ) = ( d, , , (8)belongs to worldline α . Inside this particle we consider a second point, alsoat rest in the reference frame K ′ , with coordinates( x ′ β , y ′ β , z ′ β ) = ( d + ξ, η, ζ ) , (9)that belongs to worldline β . The region with a non-zero electric chargedensity is given by ( x ′ − d ) + y ′ + z ′ ≤ a ′ , (10)which in Minkowski space is a worldtube parallel to the Oict ′ axis.Consider now that the particle is in motion along the Ox axis of anotherreference frame K , with a uniform velocity −→ v = ( v, , K ′ moves with the same velocity relative to this sta-tionary reference frame K . The origins of the two reference frames coincidewhen t = t ′ = 0. We use the Lorentz transformation x ′ = x − vt p − v /c , y ′ = y, z ′ = z, t ′ = t − vx/c p − v /c , (11)to make substitutions into (8), (9), and (10).In the stationary reference frame K the α worldline is given by( x α , y α , z α ) = ( d p − v /c + vt, , , (12)and the β worldline is given by( x β , y β , z β ) = (( d + ξ ) p − v /c + vt, η, ζ ) . (13)The region with a non-zero electric charge density is given by( x − vt − d p − v /c ) − v /c + y + z ≤ a ′ . (14)6hrough the field point at ( x , y , z , ict ) we draw the hyperplane ofconstant time t = t . The intersection of this hyperplane with the worldtubeof the particle (14) produces a space region described by( x − vt − d p − v /c ) − v /c + y + z ≤ a ′ . (15)This is the actual (instantaneous) integration volume, a Lorentz ellipsoidwith axes a ′ p − v /c , a ′ , and a ′ . The center of the ellipsoid, the point withcoordinates ( vt + d p − v /c , , α . The volumeof the ellipsoid is 4 πa ′ p − v /c /
3, and the total electric charge of theparticle is Q = ρ πa ′ p − v /c / ρ ′ = ρ p − v /c , is consistent with the fact that ”total charge could bemeasured by a counting operation which is presumably also an invariant”.[7] Unlike the invariance of electric charge, the Doppler factor in the LWpotentials ”has nothing whatever to do with special relativity or Lorentzcontraction”. [3] Through the field point at ( x , y , z , ict ) we draw the retarded lightcone, ahypersurface described by the equation( x − x ) + ( y − y ) + ( z − z ) − c ( t − t ) = 0 , (16)which, since t < t for a retarded time, we can also write as c ( t − t ) = p ( x − x ) + ( y − y ) + ( z − z ) . (17)The intersection of the retarded lightcone (17) with the worldtube of theparticle (14) is a spacetime region whose projection on the hyperplane ofconstant time t = t gives the apparent integration volume.In order to find the ratio of the apparent volume to the actual volume, wecompare the spatial separation of two points from the apparent volume withthe spatial separation of two corresponding points from the actual volume.Worldline α intersects the hyperplane of constant time t = t at A ( x A , y A , z A , ict A ) = ( d p − v /c + vt , , , ict ) , (18)7nd worldline β intersects the same hyperplane at B ( x B , y B , z B , ict B ) = (( d + ξ ) p − v /c + vt , η, ζ , ict ) . (19)Worldline α intersects the the retarded lightcone (17) at C ( x C , y C , z C , ict C ) = ( d p − v /c + vt C , , , ict C ) , (20)and worldline β intersects the same lightcone at D ( x D , y D , z D , ict D ) = (( d + ξ ) p − v /c + vt D , η, ζ , ict D ) . (21)The retarded point C is projected on the hyperplane of constant time t = t at E ( x E , y E , z E , ict E ) = ( d p − v /c + vt C , , , ict ) , (22)and the retarded point D is projected on the hyperplane at F ( x F , y F , z F , ict F ) = (( d + ξ ) p − v /c + vt D , η, ζ , ict ) . (23)As shown in Figure 1, the spatial separation of points A and B , insidethe actual volume, is( x B , y B , z B ) − ( x A , y A , z A ) = ( ξ p − v /c , η, ζ ) , (24)while the spatial separation of the corresponding points E and F , inside theapparent volume, is( x F , y F , z F ) − ( x E , y E , z E ) = ( ξ p − v /c + v ( t D − t C ) , η, ζ ) . (25)The ratio ( x F − x E ) / ( x B − x A ), equal to the ratio of the apparent to theactual volume, is responsible for the apparition of the Doppler factor in theLW potentials. From a pedagogical point of view, it is helpful to start by analyzing thesimpler case of an extended particle in radial motion. [3, 8, 10] In thissituation the fieldpoint is on the Ox axis, and therefore y = z = 0. Thecenter of the retarded electric charge density is at point C , and −→ R = −→ r −−→ r C =( x − x C , , x C > x , the unit vector ˆ R points in the negative x direction, ˆ R = ( − , , − ˆ R · −→ v /c = 1 + v/c . When x C < x ,the unit vector ˆ R points in the positive x direction, ˆ R = (1 , , − ˆ R · −→ v /c = 1 − v/c . 8 ctx y z b O α b C b A β b D b B b ict b b E b F Figure 1: An electrically charged extended particle in motion (gray area).From (20) we know that y C = z C = 0, and equation (17) becomes c ( t − t C ) = p ( x C − x ) , (26)where x C = d p − v /c + vt C . (27)When x C > x , from (26) we get c ( t − t C ) = x C − x , (28)9hich, together with (27), allows us to find t C = x + ct − d p − v /c c + v , (29) x C = d p − v /c + vt + x v/c v/c = x A + x v/c v/c . (30)When x C < x , from (26) we get c ( t − t C ) = x − x C , (31)which, together with (27), allows us to find t C = − x + ct + d p − v /c c − v , (32) x C = d p − v /c + vt − x v/c − v/c = x A − x v/c − v/c . (33)From (21) we know that y D = η , z D = ζ , and equation (17) becomes c ( t − t D ) = p ( x D − x ) + η + ζ ≈ p ( x D − x ) (1 + η + ζ x D − x ) ) , (34)where x D = ( d + ξ ) p − v /c + vt D . (35)Since the extended particle is of very small size, in order to calculate thespatial separations (24) and (25) we need only the first order contributionsin ξ , η , and ζ . In this approximation (34) becomes c ( t − t D ) = p ( x D − x ) . (36)When x D > x (implying x C > x , since points C and D are very closeto each other), from (36) we get c ( t − t D ) = x D − x , (37)which, together with (35), allows us to find t D = x + ct − ( d + ξ ) p − v /c c + v , (38) x D = ( d + ξ ) p − v /c + vt + x v/c v/c = x B + x v/c v/c . (39)10hen x D < x (implying x C < x ), from (36) we get c ( t − t D ) = x − x D , (40)which, together with (35), allows us to find t D = − x + ct + ( d + ξ ) p − v /c c − v , (41) x D = ( d + ξ ) p − v /c + vt − x v/c − v/c = x B − x v/c − v/c . (42)Since x E = x C and x F = x D , it is now clear that in both cases ( x C > x or x C < x ) we arrive at the Doppler factorapparent volumeactual volume = x F − x E x B − x A = x D − x C x B − x A = 11 − ˆ R · −→ v /c . (43) In this general situation the fieldpoint can be anywhere. The retarded electriccharge density is non-zero in a very small neighborhood of point C , and −→ R = −→ r − −→ r C = ( x − x C , y , z ). The radial unit vector isˆ R = ( x − x C , y , z ) p ( x − x C ) + y + z , (44)and the denominator of the Doppler factor is1 − ˆ R · −→ vc = 1 − v ( x − x C ) c p ( x − x C ) + y + z . (45)For point C equation (17) becomes c ( t − t C ) = q ( x C − x ) + y + z , (46)and if we substitute (27) into (46) we obtain a quadratic equation in t C . Onesolution is the retarded time that we keep, the other is the advanced timethat we discard. However, unlike in the simpler case of radial motion, in thegeneral case the discriminant of the quadratic equation is no longer a perfectsquare, and the solution does not look very nice.For point D equation (17) becomes c ( t − t D ) = p ( x D − x ) + ( η − y ) + ( ζ − z ) , (47)11nd if we substitute (35) into (47) we obtain a quadratic equation in t D . Wenotice that we don’t have to actually find the exact solution of this quadraticequation, since we are only interested in the first order approximation in ξ , η , ζ . For this reason we decide to consider ξ , η , ζ as free parameters, andwe look at the exact solutions t D ( ξ, η, ζ ) and x D ( ξ, η, ζ ) as functions of theseparameters. Since x D (0 , ,
0) = x C and t D (0 , ,
0) = t C , we can expand x D in a Taylor series around x C , keeping only the first order contributions in ξ , η , ζ , and we obtain x D ( ξ, η, ζ ) = x C + ∂x D ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) C ξ + ∂x D ∂η (cid:12)(cid:12)(cid:12)(cid:12) C η + ∂x D ∂ζ (cid:12)(cid:12)(cid:12)(cid:12) C ζ . (48)By applying partial derivatives to (35) we get ∂x D ∂ξ = r − v c + v ∂t D ∂ξ , (49) ∂x D ∂η = v ∂t D ∂η , (50) ∂x D ∂ζ = v ∂t D ∂ζ , (51)and by applying partial derivatives to (47) we get − c ∂t D ∂ξ = ( x D − x ) ∂x D ∂ξ p ( x D − x ) + ( η − y ) + ( ζ − z ) , (52) − c ∂t D ∂η = ( x D − x ) ∂x D ∂η + η − y p ( x D − x ) + ( η − y ) + ( ζ − z ) , (53) − c ∂t D ∂ζ = ( x D − x ) ∂x D ∂ζ + ζ − z p ( x D − x ) + ( η − y ) + ( ζ − z ) . (54)From (49) and (52) we find that ∂x D ∂ξ = cv q − v c p ( x D − x ) + ( η − y ) + ( ζ − z ) x D − x + cv p ( x D − x ) + ( η − y ) + ( ζ − z ) , (55)which gives ∂x D ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) C = cv q − v c p ( x C − x ) + y + z x C − x + cv p ( x C − x ) + y + z . (56)12rom (50) and (53) we find that ∂x D ∂η = y − ηx D − x + cv p ( x D − x ) + ( η − y ) + ( ζ − z ) , (57)which gives ∂x D ∂η (cid:12)(cid:12)(cid:12)(cid:12) C = y x C − x + cv p ( x C − x ) + y + z . (58)From (51) and (54) we find that ∂x D ∂ζ = z − ζx D − x + cv p ( x D − x ) + ( η − y ) + ( ζ − z ) , (59)which gives ∂x D ∂ζ (cid:12)(cid:12)(cid:12)(cid:12) C = z x C − x + cv p ( x C − x ) + y + z . (60)With these partial derivatives, equation (48) becomes x D = x C + cv q − v c p ( x C − x ) + y + z ξ + y η + z ζx C − x + cv p ( x C − x ) + y + z , (61)and since x B − x A = ξ p − v /c , we have x D − x C x B − x A = cv q − v c p ( x C − x ) + y + z ξ + y η + z ζξ q − v c (cid:16) x C − x + cv p ( x C − x ) + y + z (cid:17) . (62)We now make use of the infinitesimal nature of η and ζ , and we evaluate(62) in the limit η → ζ →
0. Surprisingly, ξ cancels out and we don’thave to worry about its limit. From this perspective, the Doppler factor inthe LW potentials is seen as ”a zeroth-order contribution”. [5] We havelim η → ζ → x D − x C x B − x A = cv p ( x C − x ) + y + z x C − x + cv p ( x C − x ) + y + z . (63)Based on (45) and (63), we finally arrive at the Doppler factorapparent volumeactual volume = 11 − vc x − x C √ ( x C − x ) + y + z = 11 − ˆ R · −→ v /c . (64)13 The transition to an electrically chargedpoint particle
It is generally assumed that the derivation of the Doppler factor in the LWpotentials, given in the case of a continuous electric charge distribution, alsoapplies in the case of an electrically charged point particle. This assumptionseems natural, since the LW potentials of a point charge are exactly the sameas those of a continuous charge distribution localized in a vanishingly small3D volume, as long as the point particle and the extended particle have thesame electric charge and velocity. The widespread nature of the assumptionis demonstrated by its apparition in some important physics textbooks. Forexample, Griffiths writes: ”Because this correction factor makes no referenceto the size of the particle, it is every bit as significant for a point charge asfor an extended charge.” [3] and Feynman writes: ”Finally, since the “size”of the charge q doesn’t enter into the final result, the same result holds whenwe let the charge shrink to any size - even to a point.” [8]We respectfully disagree with these statements. We believe that the as-sumption is not justified, and this is the reason: While in the case on anextended particle we can have two worldlines α and β inside the particle,and the derivation of the Doppler factor in the LW potentials proceeds asshown, in the case of a point particle we only have one worldline. The pre-vious reasoning no longer applies, the ratios (43) and (64) no longer exist.Other authors have also voiced concerns. Joel Franklin calls the assumption”a hard sell” [11], and Vesselin Petkov writes: ”If this were the case, the ex-planation of the physical origin of the Li´enard-Wiechert potentials would notmake sense.” [12] The above mentioned assumption has survived for so longonly because the conceptual difference between a diameter of infinitesimallength and a point of zero size is quite elusive. We have seen that, for an extended particle, the Doppler factor in the LWpotentials has a geometrical origin, this factor being the result of a geometri-cal derivation performed in Minkowski space. We cannot use the same proof,based on the two worldlines α and β , for a point particle. Nonetheless, thequestion is, can we still find the geometrical origin of the Doppler factor inthis later case? 14n this quest, we start by noticing that [13] ∂t ret ∂t = 11 − ˆ R · −→ v /c , (65)where t is the time at the field point ( t in our notation), and t ret is theretarded time at the source charge ( t or t C in our notation). Equation (4)becomes Φ = QR (cid:16) v x c , v y c , v z c , i (cid:17) ∂t ret ∂t . (66)In order to understand the geometrical origin of the Doppler factor (65)we need a geometrical representation of (66). Consider an electrically chargedpoint particle in motion, with a retarded velocity −→ v = ( v x , v y , v z ), and a fieldpoint A with coordinates ( x A , y A , z A , ict A ). As shown in Figure 2, throughthe field point A we draw the retarded lightcone that intersects the worldlineof the source particle at point C with coordinates ( x C , y C , z C , ict C ). Point C is projected on the time axis at E . In order to calculate the partialderivative of t ret with respect to t we keep the x A , y A , z A coordinates ofthe field point A fixed, and we increase the time t A by an infinitesimalamount δt , obtaining in this way the new field point B with coordinates( x B , y B , z B , ict B ) = ( x A , y A , z A , ict A + icδt ). Through the field point B wedraw the retarded lightcone that intersects the worldline of the source par-ticle at point D with coordinates ( x D , y D , z D , ict D ). Point D is projected onthe time axis at F . The variation of the retarded time is δt ret = t D − t C , andthe Doppler factor (65) is ∂t ret ∂t = δt ret δt = t D − t C t B − t A . (67)It is as if the point particle, devoid of a spatial volume, has instead gaineda temporal extension, and now the ratio of these temporal dimensions isresponsible for the apparition of the Doppler factor.We notice that the right side of (67) is the temporal component of aMinkowski four-vector −−→ CDAB = ( x D − x C , y D − y C , z D − z C , ict D − ict C ) ict B − ict A = ( v x δt ret , v y δt ret , v z δt ret , icδt ret ) icδt = 1 i (cid:16) v x c , v y c , v z c , i (cid:17) ∂t ret ∂t . (68)This allows us to write (66) as Φ = iQR −−→ CDAB . (69)15 ctx y z b O b C b D b A b B b Eict C b Fict D b ict A b ict B Figure 2: An electrically charged point particle in motion (worldline CD).At the same time, −→ CA = ( −→ R , iR ) and −→ AB = ( −→ , icδt ), where the length of −→ AB is AB = icδt . Since −→ CA · −→ AB = iR AB , we can write (69) as Φ = − Q −−→ CD −→ CA · −→ AB . (70)We recognize that the displacement four-vector −−→ CD from (70) is closelyrelated to the four-velocity V from (3), because V = −−→ CD/δτ ret , where δτ ret is an infinitesimal retarded proper time interval and the length of −−→ CD is CD = icδτ ret . Since X − X = −→ CA , we can write (3) as Φ = − Q −−→ CD −→ CA · −−→ CD . (71)Comparing (71) with (70), a very important equation emerges −→ CA · −→ AB = −→ CA · −−→ CD. (72)16his equation is true whenever two infinitesimal segments (in our case AB and CD ) have their endpoints connected by light signals. The same equa-tion (72) was used by Fokker [14] who, in his variational method, assumedthat the electromagnetic interaction takes place between such correspondingeffective elements (”entsprechended effektiven Elementen”), segments of in-finitesimal length on the worldlines of the electrically charged particles, withnull spacetime intervals between their corresponding endpoints. In Fokker’snotation, the relation (72) is written as ( R · dx ) = ( R · dy ).Next assume that at the field point A we have an electrically chargedparticle at rest, with electric charge q . The infinitesimal segment AB lies onthe worldline of this test charge. The electromagnetic interaction is describedby the term ( q/c ) p − v /c Φ · V in the relativistic Lagrangian [15], andby the term ( q/c ) Φ · V dτ = ( q/c ) Φ · dX in the integral of the action,where dτ = p − v /c dt is an infinitesimal proper time interval and dX is the corresponding infinitesimal variation of the position four-vector X .Jackson has an additional minus sign in his expression, because he is using the( − , − , − , +) metric, while we are using the ( x, y, z, ict ) Minkowski formalism,equivalent to the (+ , + , + , − ) metric. Since in our case dX = −→ AB , we endup with − Q qc −−→ CD · −→ AB −→ CA · −−→ CD , (73)in the relativistic action. The expression (73) is a Lorentz invariant, andit describes the electromagnetic interaction even when the test charge AB is not at rest. Due to symmetry considerations (related to the action andreaction principle), Fokker has also added to (73) the contribution of theadvanced potential.One way to reveal the geometrical origin of the Doppler factor in the LWpotentials is to say that electrically charged point particles are points onlyin 3D space, while in Minkowski space they are not points, but infinitesimallength elements along the worldlines of the particles. The electromagneticinteraction, as shown by the expression of the relativistic action, happensbetween infinitesimal worldline segments that have their end points connectedby light signals.This interaction model, first proposed by Fokker, was also independentlydiscovered by Galeriu [16], based on the following argument: Consider asource charge at rest and a test charge in uniform motion. In the expres-sion of the electromagnetic four-force we have an explicit dependence on thevelocity of the test particle, and an implicit dependence on the velocity ofthe source particle (through the chosen reference frame). However, ”from a geometrical point of view, a point in Minkowski space is just a fixed point17 it does not have a velocity!” [16] The four-force acting on a point particlemust be replaced by a four-force linear density acting on an infinitesimal seg-ment on the particle’s worldline. [17] This paradigm shift is mathematicallypossible because, as noticed by Costa de Beauregard [18], we have a perfectisomorphism between the equation of motion for a relativistic point particleand the static equilibrium condition for an elastic string.It is very likely that Minkowski himself was on the verge of discoveringthe possibility of replacing the interacting material point particles with corre-sponding infinitesimal segments on the particles’ worldlines, given the words”Let BC be an infinitely small element of the worldline of F; further let B*be the light point of B, C* be the light point of C on the worldline of F*[...]” that he used when describing his first theory of gravitational interaction[19]. These words would have easily escaped our attention were it not for amatching diagram drawn by Scott Walter. [20]Another way to reveal the geometrical origin of the Doppler factor inthe LW potentials is to talk about the ”thickness” of the lightcone. Thereare two possible ways of drawing the thick lightcones. One could hold the AB segment of the test charge constant, draw the lightcones with verticesat A and B , and notice that the velocity of the source charge will changethe length of the intersection segment CD that the source charge has insidethe thick lightcone. Or one could hold the CD segment of the source chargeconstant, draw the lightcones with vertices at C and D , and notice that thevelocity of the test charge will change the length of the intersection segment AB that the test charge has inside the thick lightcone.The second method was used by Nicholas Wheeler and Kevin Brown.After deriving the LW potentials, Nicholas Wheeler draws a thick lightconeand writes: ”If the lightcone had ”thickness” then the presence of the Dopplerfactor in [...] could be understood qualitatively to result from the relatively”longer look” that the field point gets at approaching charges, the relatively”briefer look” at receding charges.” [21] Analyzing the LW potentials in thesimpler radial case, Kevin Brown draws a thick lightcone and writes: ”Thelight cone is shown with a non-zero thickness to illustrate that the duration oftime spent by each particle as it passes through the light cone depends on thespeed of the particle. [...] In general, the duration of coordinate time spentby a point-like particle in the light cone shell is proportional to 1 / (1 + v/c ).”[22] 18 The relativistic Doppler effect factor
The algebraic structure of the Doppler factor (65) closely mirrors the alge-braic structure of the classical Doppler effect factor [23], and this was mostlikely the reason for the name given to this factor in the LW potentials.However, historically, the exact relationship between the two factors was notvery well understood. For example, O’Rahilly mentions that “Many writersregard the factor 1 / (1 − v ′ R /c ) as somehow connected with Doppler’s prin-ciple. Heaviside [...] says the Li´enard potential is ‘dopplerised.’ ”[4] Now,based on our geometrical interpretation, we can understand the relationshipbetween the Doppler factor in the LW potentials and the classical Dopplereffect factor. The missing link is the relativistic Doppler effect factor.Indeed, we can use the Minkowski diagram shown in Figure 2 in orderto discuss the relativistic Doppler effect [24, 25]. Worldline CD representsthe source of the electromagnetic wave, and worldline AB represents thedetector. Suppose that flashes of light are emitted once per period. Theflashes emitted at C and D are received at A and B . The spacetime interval CD = icT ′ gives the period of the wave in the proper reference frame of thesource, while the spacetime interval AB = icT gives the period of the wavein the reference frame of the detector. The relativistic Doppler effect factoris the ratio of the frequency f measured by the detector to the frequency f ′ produced by the source: ff ′ = T ′ T = CDAB = CDEF EFAB = r − v c t D − t C t B − t A . (74)The only difference between the relativistic Doppler effect factor (74) andthe Doppler factor in the LW potentials (67) is the p − v /c factor. Thisadditional factor, responsible for the transverse Doppler effect, represents therelativistic time dilation. For an electrically charged extended particle, we have derived the Dopplerfactor in the LW potentials with the help of intersections and projections.In Minkowski space nothing moves, nothing changes shape, and as a resultour geometrical method is very intuitive. For an electrically charged pointparticle, we have discussed how the Doppler factor in the LW potentials isrelated to the electromagnetic interaction model of Fokker. In this way wehave elucidated not only the geometrical origin of the Doppler factor, butalso the origin of the name, showing that the relativistic Doppler effect is19he missing link between the Doppler factor in the LW potentials and theclassical Doppler effect.
The author is very much indebted to David H. Delphenich for translatingthe article by Fokker, upon personal request.
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