On the Doppler Effect of Light, consequences of a quantum behaviour

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1 On the Doppler Effect of Light, consequences of a quantum behaviour Stefano Quattrini Via Jesi 54 ANCONA, ITALY stequat@libero.it (Dated: June 5, 05) The matter-radiation interaction regarding the Relativistic Doppler Effect (RDE) is re-examined. A method is applied in order to find the actual energies and momenta involved in the effect. The RDE presents a non-vanishing energy and momentum shift of Electro-Magnetic (EM) radiation confirmed also by experimental evidences. By applying the conservation laws a matter-radiation interplay emerges. It is responsible of a net radiation force due to the quantum pulses exchanged with consequences on the dynamics of the bodies involved in the interaction. More complex relations emerge which show how the derivation of the RDE from the Lorentz Transformations represents a limit. I. INTRODUCTION The attention has been casted recently,3,4,5 on the derivation of the Doppler Effect based on the energies and momenta involved in the matter-radiation interaction 4. It is an alternative to the derivation based on the invariance of the EM wave phase factor under the Lorentz Transformations (LT) which describes how the same radiation appears from relative moving Inertial Reference Frames 4 (IRFs). Quantum mechanically the Doppler effect results from the recoil momentum, changing the translational energy of the radiating atom. It would be very difficult to reconcile the recoil of the source of radiation with the classical wave theory 4 according to Schrödinger. Einstein in a letter to von Laue in 95, described what he called a second type of radiation pressure made of indivisible point-like localized quanta of energy which are reflected undivided. The renewed interest related to the RDE 6,,3,4,5 and the different viewpoints illustrated, together with some results of experimental gravitation, motivated the present revisitation. The Analysis of the moving mirror case 6, based on the energy transfers and its derivation, performed with the conservation laws, have a central role in the present re-examination of the RDE. This paper follows three key points: (i) account for energies and momenta involved in the RDE, (ii) apply the conservation laws, (iii) draw the relevant consequences for the phenomenon. The key points are developed according to the following steps. The one dimensional case of the RDE is applied in a closed path (CP) of radiation between two moving RFs, in order to report the frequencies in the same RF 6 and to account for the actual energies and momenta involved. Energy and momentum shifts are then determined for the RDE and confirmed also by the results and interpretations of the Harvard Tower experiments 9,,3. The conservation laws are applied and show that the RDE transforms kinetic energy of bodies to/from EM energy, as it also was shown to occur in the moving mirror case 6,0. The RDE, re-derived from the conservation laws and the relativity of motion, is provided with a definition based on the interplay of energies and momenta between moving bodies exchanging radiation. A quantum effect emerges, based on pulses which act as a brake or accelerator between moving bodies exchanging radiation. Some limits are set to previous relations,6 by performing an analisys of the moving mirror configuration. The phenomenon at the base of the RDE is shown to go beyond the neutrality implied by its original derivation from the LT. II. THE RDE AND THE CP OF RADIATION A. Frequencies in the RDE The RDE relations between the frequencies of the radiation emitted from a RF o and observed in a RF, which moves at relative speed V o referred to RF o, in scalar form, for the one dimensional configuration, are: ν =ν o (+β)γ=ν o [(-β)γ] -. () with β=v o /c, -<β<, γ=/(-β ) / and V o >0 is oriented from right to the left as in Fig., so that β>0 stands for the approaching (frequency increases at the observer), β<0 for the moving away (frequency decreases at the observer) configurations. The relations about energies E = E o /[(-β)γ] = E o (+β)γ 3,5,4 descend directly from Eq. (). It could be argued that such energies are already what is needed to get the energy differences, being one of the purposes of this reexamination. But since E and E o are referred to distinct RFs, the difference E -E o would be referred to two RFs at once, making such quantities not useful. B. Frequencies in the Closed Path of radiation A method to refer quantities to the same RF (the one of the emitter) is necessary in order to be prepared for the application of the conservation laws. For this reason a closed path is applied as in Fig. to make the radiation go

2 back to the same RF. The forward path, RF o to RF, is represented by the Eq.() of the RDE between the frequencies. The backward path, RF to RF o, is represented also by the Eq.(), which takes the same form due to the principle of the relativity of motion. The frequency ratio of the closed path of radiation results in the following relation: ν' o =ν [(+β)γ]=[ν o (+β)γ](+β)γ=ν o [(+β)γ] = (+β) γ ; ν' o /ν o = (+β) γ = (+β) /(-β ) = (+β)/(-β). () ν o ν' o ν = ν o (+β)γ ν' o = ν (+β)γ Fig.. The closed path of radiation, of RFs moving at constant speed. The Eq.() can be found in other cases 8,5-p., it provides also the relation between frequencies for the Einstein s moving mirror,0 (in the one dimensional case, the versus of the speed is opposite) or the double Doppler shift 6. It is also the Doppler RADAR 6 frequency ratio, used in real applications for measuring the speed of moving objects. From the Eq. (): Δν CP = ν' o -ν o = ν o β/(-β) β ν o Δν CP /ν o = β/(- β) β (β <<). (3) C. Energies, momenta, forces in the CP of radiation In the CP of radiation the attention is mainly focused on the reference frame of the generator/final observer RF o 8,5-p. as in the case of a Doppler RADAR 6. In some other cases,6,0 the attention was casted mainly on the reflecting surface RF, only mentioning the presence of the source and the final observer at rest in the lab frame. Given E o =nhν o as the energy at the emission of n photons as measured at rest in RF 0 and assuming perfect surfaces, which preserve the number of the incident photons, from Eq. () it results that the energy absorbed in RF 0 after the CP of radiation is E o =nhν' o =hν o (+β)/(-β)=e o (+β)/(-β). The energies found can now be compared in the same RF 0 : ΔE RFo ΔE CP =E o -E o =E o [(+β)/(-β)-]= E o β/(-β). (4) The Eq.(4) is present also in Ref. 6-Eqs.(3) and Ref. 6- Eqs.(b) (with the opposite versus of the speed) and from Eqs.(3)-(4) it is ΔE CP = nhδν CP, in agreement with the Ref.6-Eq. 9. In the case of approaching RFs it is β > 0 and from Eq.(4) ΔE CP > 0, a positive energy shift is detected, as also results in Ref.6. Considering the surface represented by S in Fig. the net flux of energy is positive across it, in the case of the approaching configuration. For radiant energy or photons P o = E o /c and from Eq.(4): ΔP CP = P o β/(- β); ΔP CP / P o = ΔE CP /E o = Δν CP /ν o = β/(- β) β (β<<). (5) ν RF o RF V S o The radiation force of a source is 0 F 0 =ε o A R, if the radiation bounces on a moving lossless slab it is 0 F=F 0 (- β )/(+β ), being β =-β. By setting ΔF CP =F-F 0, it results that ΔF CP /F 0 =β/(-β), also in agreement with Eq.(5). For the CP of radiation the relevant quantities have been computed. The point of view of the CP of radiation allows the RDE or the single path of radiation to be considered as follows. III. ENERGY SHIFTS AND THE EXPERIMENTAL EVIDENCE. The two paths involved in the CP of radiation should have an identical behavior. It is assumed that for each path it is valid the Eq.(), hence from Eq.(4) it results ΔE RDE = ΔE (forth) = ΔE (back) = ½*ΔE CP = E o β/(-β), and the energy shift involved in the RDE is determined as ΔE RDE =E o β/(- β) and from Eq.(5) it results also: Δν RDE /ν o = ΔE RDE /E o = ΔP RDE /P o = β/(-β) β (β<<) (6) The meaning of Δν RDE /ν o is the variation of the frequency in the RDE as if observed in the RF of the emitter. The energy shift of radiation had a key role in the Harvard Tower Experiments 9,. In order to test the presence of the gravitational redshift, the source and the observer were separated by the height H, at rest on the surface of Earth. A certain speed Vo=gH/c towards the ground was given to the upper absorber, in order to restore the resonant absorption, implementing a Doppler compensation. From Eq. (6) being β=vo/c, β=gh/c an energy shift would result as ΔE RDE E o gh/c (7) The gravitational redshift has at most two interpretations 3. ) An energy loss of photons in a gravitational field as E ph (H)=E ph (0)(+gH/c )->E ph (H)-E ph (0)=E ph (0)gH/c, compensated with the same amount of energy. ) 3 An energy shift of atomic/nuclear levels. The difference between energies of atomic/nuclear levels E lab depends on the position of an atom situated at height H relative to an identical atom emitting a photon, ΔE lab =E lab (H)-E lab (0). The relative energy difference of levels, gh/c due to gravitation equals the relevant increment, of the relative energy difference of atomic/nuclear levels : ΔE lab /E lab =gh/c. Both interpretations require an energy supply of ΔE EM =E o gh/c >0 provided by nothing else but the RDE compensation above declared. Such energy shift is in agreement with the predicted energy shift of Eq.(7) hence Eq.(6) finds an experimental confirmation. IV. ENERGY SHIFTS AND CONSERVATION LAWS From the Eq. (6) with β > 0 it is ΔE RDE > 0 and ΔP RDE >0, a net increase of EM energy and momentum emerges in the approaching configuration, the detector receives more EM energy than the amount it would receive if it had no relative motion with the source as in Fig..

3 If RF o and RF in Fig., identify massive objects which move reciprocally and exchange radiation exclusively with each other, no other interaction is involved, they define an isolated system (IS). It is possible then to apply the conservation laws, at rest with the RF o. Being ΔK = K - K where K is the kinetic energy of the IS right before the emission and K right after the absorption of radiation, measured in RF o it has to be ΔK= -ΔE EM according to the conservation laws. EM beam ν o eam ν RF o V o RF Fig.. The isolated system (IS) of RF 0 and RF, moving relatively at constant speed, gains EM-energy by losing K-energy. From the Eq. (6): For 0<β<; ΔE EM = E o β/(-β)>0 ΔK<0 the reciprocal motion is slowed down, increasing the internal EM energy. For -<β<0; ΔE EM = E o β/(-β) <0 ΔK>0 the reciprocal motion is speeded up, decreasing the internal EM energy. Defining ΔP as the global variation of the momentum of the IS, it is ΔP EM =-ΔP for the momentum conservation, hence: [ΔE RDE /E o =ΔP RDE /P o =β/(-β)] (-ΔK, -ΔP) RDE In the approaching configuration according to ΔE EM Δmc, the overall EM mass-energy within the IS increases at the expenses of its macro kinetic energy ΔK. What so far shown demonstrates that the RDE implies a matter-radiation energy and momentum exchange, transforming energy and momentum of photons to/from energy and momentum of objects, according to the conservation laws. The inverse implication is briefly illustrated. V. DERIVATION OF THE RDE AND ITS INTERPLAY The RDE was derived also from the conservation laws and special relativity 4, so that the inverse implication previously mentioned is actually already proven. The moving mirror relations were derived from the conservation laws only. The nonrelativistic relations for the moving source and moving observer ν A =ν o (-β) -, ν B =ν o (+β) in the one dimensional case, originally found by Doppler from his observations of the binary stars, whose derivation from the conservation laws is omitted, are derivable one by one similarly to the moving mirror case. By increasing β, the values of the two nonrelativistic expressions diverge rapidly, but they should be equal according to the principle of the relativity of motion. In fact it does not make sense to state who is approaching to whom in the absence of a material medium, there has to be an expression Q : ν /ν o = Q. A term X has to be found such that Q = X (+β) = X - (-β) -. By solving in X: X =/(- β ) ; X=(-β ) -/ =γ the Lorentz factor emerges and Q=(+β)γ=[(-β)γ] - so that ν /ν o =(+β)γ= [(-β)γ] -, and the Eq. () is re-derived. The conservation laws and the principle of relativity of motion are sufficient to obtain the relations of the RDE. The energy and momenta shift of bodies and radiation with the conservation laws define the matter-radiation interplay by providing a definition for the phenomenon. A double implication characterizes the RDE (at least in the one dimensional case) according to its relative shifts. [ΔE EM /E o =ΔP EM /P o =-ΔK/E o =-ΔP/P o = β/(-β)] [RDE]. VI. CONSEQUENCES OF A QUANTUM BEHAVIOUR A. The effect of the quantum pulses The basic energy and momentum shift, at the base of the interplay matter-radiation which characterizes the RDE, consist also in an exchange of two pulses given within a finite interval of time Δt elapsed within the emission and absorption instants of the same quanta: ΔP RDE /Δt=Po/Δt*β/(-β) (8) Such quantum effect is dimensionally a force, it performs a work because it alters the kinetic energy of objects. For a radiation flux, ΔP RDE /Δt =ΔF RDE =Fo β/(-β) 0 ε o A β, or flux of photons sufficiently continuous, the quantum effect consists in a net radiation force. The approaching configuration can be viewed as a pre-scattering situation involving objects which converge to the same point but slow down their motion, due to the effect of the quantum pulses. The departing configuration can be viewed as a post-scattering situation where the objects escape faster from the same point, pushed away by the reciprocally exchanged quantum pulses. B. Extensions of the previous formulation and limits When the momentum imparted by the quantum pulses is significant, the observer s speed may not remain constant if not sufficiently massive. It will be shown that there are cases, better representing real situations, when the Eq.() should not be used for both paths to form the Eq.(). By considering the moving mirror problem,6 and applying the conservation laws in the one dimensional case, the relation of Eq.() between frequencies is obtained and then the dependence of the relations on the mass of the mirror m 0 will be investigated as follows. The energies of the photons before and after the impact against the reflective surface are nhν o and nhν, from Ref. Eqs.()-(3) the two relevant equations of energy and momentum conservation (according to the chosen positive direction of the speed in Ref.) become: nhν o /c + m 0 V = -nhν /c -m 0 (V+ΔV) (9a) nhν o +/m 0 V = nhν +/m 0 (V+ΔV) (9b) from Eq.(9a) m 0 ΔV = nh(ν o + ν )/c (9c) From Eq.(9b) nhν o =nhν +Vm 0 ΔV+/ m 0 ΔV ; (9d) By replacing (9a) in (9d) it is: nhν o =nhν +V nh(ν o +ν )/c+ / m 0 ΔV ; (9e) 3

4 for (nhν o /c )/m 0 << the mass of the mirror is considered very large so that /m 0 ΔV is negligible : ν o =ν +V(ν o +ν )/c ; ν +V/c ν = ν o -V/c ν o ; (by setting β = - V/c) ν =ν o (+β)/(-β), the Eq.() is obtained from the conservation laws where ν has to be intended as ν o in the Eq.(). From Eq.(9c) it is ΔV=nh(ν o +ν )/(m 0 c) hence /m 0 ΔV = m 0 ΔV/* nh(ν o +ν )/(m 0 c) = ΔV/*nh(ν o +ν )/c; by setting Δβ=-ΔV/c, and considering also the squared term in Eq.(9e) which accounts for the finite mass of the mirror ν o =ν -β(ν o +ν )-Δβ/(ν o +ν ), with Δβ= -nh(ν o +ν )/(m 0 c ) -nhν o /(m 0 c ) < 0 (9f) where ν o +ν ν o ν =ν o (+β+δβ/)/(-β-δβ/) (0) The variation of the frequencies after a CP of radiation from Eq.(0) is Δν CP /ν o =(β+δβ/)/(-β-δβ/), since Δβ < 0, the variation of the frequency is smaller respect to the case of the Eq.(). The more general case described by the Eq.(0) poses limits to the Eqs.()-(5) and for Ref.6,Eqs.(0)-(3). In Eqs.(6)-(8) the attempt to replace β/(-β) with (β+δβ/)/(-β-δβ/), in order to find the relation for a single path, may not be correct, since the speed cβ of the RF is slowed down by cδβ. The slowdown occurs at the impact with the mirror and does not guarantee equal influence on the two paths. Also the Eqs.(6)-(8) should not hold in the case of a non-negligible Δβ/= - nhν o /(m 0 c ). More complex relations emerge in the case of relativistic momenta for high speeds, where m 0 γδv = nh(ν o +ν )/c and further complexity arises if the approximation of the Eq. (9f) does not hold, due to a remarkable difference in the frequencies after the CP of radiation. In addition to what just mentioned, the effect of the recoil of radiation on the emitter RF o has not been considered, performed instead in the analysis of the RDE for a radiating atom 4. The Δβ represents also the deviation of the behavior of the bodies, relevant to the RF and RF 0, from the inertial motion β, in the Eq. (). RFs relevant to real objects which reciprocally exchange radiation can be considered inertial only to a certain approximation. The RDE relation from the LT relies on the abstraction of the constancy of the speed of the sources and the observers, 4 giving the connection between energies of a single photon in two IRFs. Due to the presence of the energy shifts the RDE transforms Kinetic to/from EM energy, same thing for momenta. The derivation from the LT represents a limit for the behavior of the Doppler effect, an accurate model only in the case of massive objects and low power radiation, where the energy and momentum shifts involved have a negligible impact on the inertial motion of the material objects relevant to the RFs. VII. SUMMARY AND DISCUSSION The application of a closed path of radiation between reciprocally moving RFs allowed to account correctly for the energies and momenta shifts involved in the Relativistic Doppler effect, confirmed also by experimental evidences. The conservation laws permitted to find the interplay of matter-radiation at the base of the effect which alters the dynamics of bodies exchanging radiation. [ΔE EM /E o =ΔP EM /P o =-ΔK/E o =-ΔP/P o =β/(-β)] [RDE]. The RDE includes a quantum effect: differences of couples of pulses, given at different times at the emission and the absorption, ΔP RDE /Δt = P o /Δt β/(-β) where P o = nhν o /c. It is a net radiation force if the pulses are sufficiently frequent, it acts as a brake in the case of approaching objects and as an accelerator otherwise. A more general relation for the moving mirror frequency ratio in the one dimensional case has been found. Nonaccelerating bodies possessing an initial reciprocal speed, while exchanging radiation, preserve their inertiality only under a certain approximation. The neutrality of the RDE as derived from the Lorentz transformations has been investigated, the derivation from the LT consists in a limitation of a more sophisticated model of the phenomenon of the radiation exchanged by moving bodies, subject of further research. R. Feynman : The Feynman Lectures on Physics, Vol I, 34 6 The Doppler effect, 34-9 The momentum of light (963). A. Gjurchinovski: Reflection from a moving mirror-a simple derivation using the photon model of light Eur. J. Phys. 34, L-L4 (03). 3 G. Giuliani: Experiment and theory: the case of the Doppler effect of photons, Eur. J. Phys. 34, (03). 4 G. Giuliani: On the Doppler effect for photons in rotating systems, Eur. J. Phys. 35, (04). 5 D. Redzic: The case of the Doppler effect for photons revisited, (Leipzig) Eur. J. Phys. 34, (03). 6 G. Goedeke, V.Toussaint, C. Cooper: On energy transfers in reflection of light by a moving mirror Am. J. Phys. 80, 684 (0). 7 C. M. Will: The confrontation between general relativity and experiment, Living Rev. Relativity 7, 4 (04). 8 R. Vessot, M. Levine: Test of Relativistic Gravitation with a Space- Borne Hydrogen Maser, Phys. Rev. Lett. 45, 08 (980). 9 R. Pound, G. Rebka: Apparent weight of photons, Phys. Rev. Lett. 4, 337 (960). 0 D. Censor: Energy balance and radiation forces for arbitrary moving objects, Radio Science 6 (0), (97). R. Dicke: The Effects of Collisions upon the Doppler width of spectral lines, Phys. Rev 89, 47 (953). R. Pound, J. Snider: Effect of gravity on gamma radiation, Phys. Rev. B 40, B788 (965). 3 L. Okun, K. Selivanov, V. Telegdi: Gravitation, photons, clocks, Phys.-Usp (999). 4 D. Redzic: The Doppler effect and conservation laws revisited, Am. J. Phys. 58,, (990). 5 R. Vessot, M. Levine: Gravitational Redshift space-probe experiment NASA Center for Aerospace Info., Tech. Rep. n (979). 6 R. Ditchburn : Light, Dover publications Inc., pp (99) 4

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English CPH E-Book Theory of CPH Section 2 Experimental Foundation of CPH Theory Hossein Javadi

English CPH E-Book Theory of CPH Section 2 Experimental Foundation of CPH Theory Hossein Javadi Javadi_hossein@hotmail.com Contains: Introduction Gravitational Red Shift Gravity and the Photon Mossbauer