The mass-energy relation and the Doppler shift of a relativistic light source
TThe mass-energy relation andthe Doppler shift of a relativistic light source ∗ Ademir Xavier Jr † .Brazilian Space Agency,Bras´ılia, DF - Brazil Abstract
This work considers the cospectral and arbitrary light emission of a moving source. Theobserved wavelengths of the emitted photons are described in term of kinematic and dynamicalDoppler shifts in which the mass-energy relation plays a fundamental role. The presentation isan alternative way of emphasizing the importance of the concept of proper mass as a conservedquantity and the implications of the mass-energy relation when a source emits radiation. Thephysical contexts in which the source changes velocity after emission are discussed and a set ofadditional problems is presented.
Keywords : Doppler shift, light emission, frequency shift, relativistic source, mass-energy rela-tion.
Relativity has changed the way we understand the dynamics of bodies interacting via elec-tromagnetic radiation. In fact, the development of relativity can be seen as an attempt to unifyelectromagnetism and mechanics [1]. Since mechanics provided a wide range of applicationsin the two centuries that followed Newton’s work and therefore was seen as a solid theoreticalframework, relativity and its new world view were deep revolutionary steps after their first pre-dictions were confirmed. Such revolution represented the incorporation of electromagnetic lawsinto our understanding of the mechanical world.The historical context of relativity coincides with the downfall of the Ether hypothesis [1]as an all pervading medium responsible for the propagation of light much in the same way asthe air is the medium in which sound waves propagate. The existence of a frequency shift inthe light emitted by a moving source was seen both as an evidence of this medium [1] as of themechanism of light propagation through it. However, the Doppler shift is not the only effectproduced by sources in motion, light aberration was recognized as an important astronomicalcorrection in the position of stars since the XVIIIth century [7], because it was soon realizedthe Earth moves itself in relation to the alleged Ether. Disagreements between theoretical andexperimental predictions for both effects soon became unsolvable [8] by the turn of the XXthcentury, giving rise to the new paradigm of special relativity.One of the basic concepts in which difficulties of understanding often arise is the idea of masswhich remains constant during the temporal evolution of a system in a non-invariant descrip-tion. With the idea of mass, a conservation law is associated, the so-called mass conservationlaw. However, in relativity, mass is neither ordinarily constant nor can be added [2] in the same ∗ Pre-print version. Submitted to Physics Education, February 3 2020. † E-mail:[email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] A ug ay as in non-relativistic mechanics. The primary conserved quantities in the new paradigmare momentum and energy, which are intrinsically linked through the definition of a new fun-damental entity: the invariant 4-momentum. On the other hand, special relativity can be seennaively as a dynamic of bodies described by distinct reference systems moving in relation toeach other at speeds approaching c (the velocity of light). Most relativistic effects (in the lengthof objects, their velocities and clock rates) will therefore show up only when high velocities areinvolved. While relativity is understood as a relevant matter for the physicist curriculum, thehigh velocity limit imposes serious limitations in the practical appreciation of its effects.The clash between past and modern ways of understanding this world reverberates untiltoday when students have to learn the basics of relativity after being taught many concepts ofnon-relativistic mechanics. There is a way however of demonstrating the impact of relativity onlow velocity systems which is the aim of this work. It involves radiation and the reformed conceptof mass which is presented to students only later. Apart from past and modern controversies[3, 4] involving the definition of rest mass, relativistic mass and proper mass [4, 5, 6], light hasa special role to play in such demonstration which is even more paradoxical since light and itsassociated photons are considered massless. In particular, we will be interested in the dynamicsof a light source as described from two distinct reference frames and the role played by thesource mass in the Doppler shift of the emitted photons.To illustrate the main concepts and avoid complications arising from more complex move-ments, we restrict the analysis to the motion of a source emitting radiation along the line ofmotion (forward and backward emissions). Limiting the analysis to 1-d motion further allowsto illustrate the radiation process on the space-time diagram as discussed in Section 4.1. The derivation of Doppler shift relations according to the principles of relativity is presentedin a variety of ways in the literature [8, 9], always with the fundamental Lorentz transformationrelations as the starting point. Thus, [9] considers a constraint which would render invariant thephase of a light wave emitted by a source at rest in the reference system S as seen from anothersystem S moving with velocity v (say, along x -axis) in relation to S . If ν is the frequency ofthe source in its reference frame, the frequency measured by the observer is ν = ν γ (1 + β cos θ ) , (1)with β = v/c , γ = (1 − β ) − / and θ the angle between the line of motion of the source andthe position of the observer. If the source moves toward the observer ( θ = 0), Eq. 1 reduces to ν = ν (cid:115) β − β . (2)In these derivations, mass plays no role. The Doppler shift is seen as a kinematic effectarising from a relative state of motion between source and observer. Light emission in factinvolves a change of state by the source. Since the energy is conserved for the integral system(source+observer), a change in the source mass ∆ m is expected as the ratio∆ m = ∆ E/c , (3)with ∆ E the total amount of energy of the emitted radiation. Eq. 3 follows from the famousmass-energy equivalent relation E = M c which establish a correspondence between the source‘rest mass’ (also called ‘proper mass’) M before the emission, and the rest energy E . Themass-energy equivalence is obtained [10] from an integral of motion (total energy) based ona generalization of Newton’s law F = dp/dt with F an external force applied to the sourceand p the relativistic momentum p = γβM c . However, the truly conserved quantity is the4-momentum P = (cid:18) piE/c (cid:19) = (cid:18) γβM ciγM c (cid:19) , (4) Cospectral emission: 1(a) a source at rest in a laboratory frameemits two identical photons in opposite directions; 1(b) the same situationas seen from a reference system in which the source moves toward +ˆ x .whose squared norm P · P = − M c is an invariant and proportional to the system total mass.Since c is large, the correction given by Eq. 3 is considered too small to have any relation tothe Doppler shift: it is a ‘side effect’ of the internal process of light emission. Only in the limitof small masses (particles) or high-energy photons and source velocities (hence, in the works ofhigh energy particle physics) such effects would play a relevant role. Restricting the description to 1-d motion for simplicity (see Fig. 1), we consider two distinctproblems:1. (a) A source with proper mass M at rest in the laboratory frame emits two counterpropagating photon pulses simultaneously with the same frequency ν . Find the finalproper mass M (cid:48) of the source after such ‘cospectral emission’.2. (a) The same as above as described by an inertial frame moving to the left with velocity v .Using conservation of energy (C. E.), problem 1(a) is solved easily with M (cid:48) c + 2 (cid:15) = M c , (5)and (cid:15) = hν . Because both photons have the same frequency, they carry the same momentaand the final source velocity is v = 0. Conservation of momentum (C. M.) is implicit in suchsymmetrical system at rest. Therefore, the answer of problem 1(a) is M (cid:48) = M (1 − z ) , (6)with z = hν /M c , the ratio between the one photon energy and the source rest energy.Therefore, the final proper mass of the source is reduced by the amount 2 z . For macroscopicbodies and low energy photons z (cid:28) v (cid:54) = 0, and the photon frequenciesare ν + (to the right) and ν − (to the left) due to the Doppler shift. Before moving on, let usdefine the dimensionless quantities: z + = hν + /M c ,z − = hν − /M c ,µ = M (cid:48) /M. (7) ow, C. E. demands γM c = γ (cid:48) M (cid:48) c + hν + + hν − , which, in view of Eqs. 7, can be written as γ = µγ (cid:48) + z + + z − , (8)where again primed quantities correspond to the state after the photon emission. In the sameway, C. M. requires that γM v = γ (cid:48) M (cid:48) v (cid:48) + hν + c − hν − c , which in dimensionless variables may be rewritten as γβ = µγ (cid:48) β (cid:48) + z + − z − . (9)The two fundamental equations, Eqs. 8 and 9, can be be solved for µ and β (cid:48) or z + and z − .However, since the context of problem (1) is given, the last option reads z ± = 12 (cid:34)(cid:115) ± β ∓ β − µ (cid:115) ± β (cid:48) ∓ β (cid:48) (cid:35) (10)where the two equations are written in a single line using the ± symbol. Because the observerknows that β (cid:48) = β (in fact, the observer sees no velocity change) and, in view of Eq. 6 or µ = 1 − z , we find from Eqs. 10 z ± = z (cid:115) ± β ∓ β , (11)corresponding exactly to the kinetic Doppler shift, Eq. 2, for each photon. According to theseequations z + > z > z − , because one photon is moving toward the observer while the otherone is moving away from him. Thus the mass-energy relation is of fundamental importancein the origin of the kinetic Doppler effect. The mass change of Eq. 6 is related directly to afundamental parameter of the emitted radiation. In principle, the context of problem 2(a) is completely general. An observer would have noway to know that the two emitted photons have the same frequency in the source referencesystem - if he sees a moving source. The only thing the observer could do is to measure thephoton frequencies and the final source velocity β (cid:48) . If β (cid:48) = β , the source proper frequency couldbe inferred by the observer from z + and z − as z = √ z + z − . Hence, a new set of problems canbe enunciated (Fig. 2):1. (b) A source with proper mass M at rest in the laboratory frame emits two counterpropagating photon pulses simultaneously with the distinct frequencies ν +0 and ν − to theright and to the left, respectively. Find the final proper mass M (cid:48) of the source after theemission.2. (b) The same as above as described by an inertial frame moving to the left with velocity v , given that the observed photon frequencies are ν + and ν − .The transition from problems (a) to (b) corresponds to a dynamical change of state whichdistinguishes itself in principle from the purely kinematic description. The photon frequenciesare a function of an internal process of radiation generation which becomes accessible externallyby measuring ν ± . Although no force actuates on the source, there is a velocity change, v (cid:48) (cid:54) = v , Arbitrary emission: 1(b) a source at rest in a laboratory frameemits distinct photons in opposite directions; 1(b) the same situation asseen from a reference system in which the source moves toward +ˆ x .that is, the source experiences a recoil . If v = 0, it may be set in motion after the emission. Asolution for µ and β (cid:48) from Eqs. 8 and 9 in terms of z ± and β are obtained easily by defining α = (cid:115) β − β ,α (cid:48) = (cid:115) β (cid:48) − β (cid:48) . (12)in terms of which, Eq. 8 and Eq. 9 are written as α = µα (cid:48) + 2 z + ,α (cid:48) = α ( µ + 2 z − α ) (cid:48) . (13)It is straightforward to eliminate µ from Eqs. 13 and find the new source mass µ = (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) − z + (cid:115) − β β (cid:33) (cid:32) − z − (cid:115) β − β (cid:33) . (14)The answer to problem 1(b) (source at rest) is calculated by setting β = 0 in Eq. 14 and notingthat, in this reference frame, z ± = z ± or µ β =0 = (cid:112) (1 − z + )(1 − z − ) . (15)If moreover z +0 = z − as in problem 1(a), Eq. 6 is retrieved.In order to study the source recoil, we should calculate β (cid:48) . Eqs. 12 can be rewritten as β = α − α + 1 ,β (cid:48) = α (cid:48) − α (cid:48) + 1 , and from Eqs. 13, an expression for β (cid:48) is found as β (cid:48) = γβ − ( z + − z − ) γ − ( z + + z − ) , (16) n terms of β and the measured ‘Doppler shifts’. The recoil expression in the source initialreference frame is therefore β (cid:48) β =0 = − ( z +0 − z − )1 − ( z +0 + z − ) . (17)If z +0 > z − , β (cid:48) <
0, that is, if the right propagating photon is more energetic than the leftpropagating one, the source will recoil to the left as intuitively expected. The opposite happensif z +0 < z − . This is the principle of the photon rocket [11, 12, 13]. The denominator of Eq. 17is positive because it implies in the inequality M c ≥ hν +0 + hν − or the total photon emissionenergy is exhausted potentially by the total energy content of the source represented by its restenergy. An interesting fact about Eq. 16 is that there is velocity change for β (cid:54) = 0 even though z + = z − because the mass content of the source has changed. In relativistic terms, given themomentum conservation equation γ (cid:48) M (cid:48) v (cid:48) = γM v , if M (cid:48) is reduced by isotropic and cospestralemission, v (cid:48) has to increase in modulus. For β (cid:28)
1, to first order in z + + z − , the source velocityafter emission is written explicitly as v (cid:48) ≈ v (cid:20) h ( ν + + ν − ) M c (cid:21) . (18)The situation seems paradoxical because, had the observer chosen originally a reference systemcomoving with the source, that is, a reference frame for which v = 0, no velocity change wouldbe observed! However, a little bit of analysis shows that the paradox is only apparent, and arisefrom the point of view of non-relativistic mechanics. For, in the relativistic case, if the observerchooses a frame in which v = 0, the two photons would not be cospectral, and the velocitychange would be compatible with the one calculated in Eq. 18.To see this exactly, using the Lorentz transformation for velocities [8, 10], the new velocityin the original source reference system (for z ± = z ) will be given by β (cid:48) = 2 zβγ − zγ . (19)In the source reference system, in accordance to Eqs. 10, the photon dimensionless energies arethen expressed as z ± = 12 (cid:34) − µ (cid:115) ± β (cid:48) ∓ β (cid:48) (cid:35) , (20)because β = 0. Substituting Eq. 19 into the equation above and writing everything in termsof the original reference frame velocity β we find z ± = z (cid:115) ∓ β ± β . (21)Therefore, under special circumstances, a moving source can emit photons exhibiting no Dopplershift. The photons of the source, if observed in its proper frame, will show different frequenciesin accordance to Eq. 21 whose velocity ratios are inversely proportional to the ones of theoriginal Doppler shift, Eq. 11. Moreover, the source will show a small velocity change (as givenby Eq. 19) in its proper system, which is expected because the photons have different momenta.Given Eq. 16, it is possible to find the condition on radiation emission for which no velocitychange is observed. Calling the special constant velocity ¯ β we find¯ β = z + − z − z + + z − . (22) hus, for a co-moving frame with the source, ¯ β = 0 if z +0 = z − . Given a reference frame inwhich the source moves with an arbitrary velocity β , z ± in this frame will be such that Eq. 22is obeyed and β (cid:48) = ¯ β . From this equation, one immediately obtains the mass ratio that keepsthe velocity constant, ¯ µ , or ¯ µ = 1 − √ z + z − . (23) Since in practical cases z ± (cid:28) β (cid:28)
1, all relations obtained here suggest expansions interms of these coefficients. An example is Eq. 14 which relates the mass change to z ± and β .Considering that in practical cases the values of z ± are very small, one can expand preferablyEq. 14 in powers of z (see the Appendix) and obtain a much simpler expression to manipulate.We should be careful, however, in using such new expansions because they may imply in mixingup concepts pertaining to distinct physical theories (e. g., classical versus relativistic dynamicsbecause the expanded versions represent “corrections” to an ordinary non-relativistic behavior).Because z ± is so small for current photonic propulsion systems, a similar expansion of Eq. 16leads to an approximate relation for the velocity gain (or loss). The resulting equations areeasier to manipulate because they are linear in z .To second order in z ± , including the velocity dependent terms, such simpler relations are µ = 1 − (cid:32) z + (cid:115) − β β + z − (cid:115) β − β (cid:33) − (cid:32) z + (cid:115) − β β − z − (cid:115) β − β (cid:33) , (24) β (cid:48) = β [1 + (cid:112) − β ( z + + z − )] − (cid:112) − β ( z + − z − ) − (25)(1 − β )( z + − z − )[ z + (1 − β ) + z − (1 + β )] . The energy of the two emitted photons cannot be arbitrary. The emission is source-dependent and, as such, it is established by a constraint among ν + , ν − , β and the sourcerest mass. Eqs. 14 and 16 can be used to extract the energy restriction relations on the rangeof possible photon energies hν + and hν − . To begin with, Eq. 14 is constrained by 0 ≤ µ ≤ z + ≤ α ,z − ≤ α ,z + ≥ z − α z − α − . (26)A constraint in the total energy is represented by the denominator of Eq. 16, or γ − ( z + + z − ) > z + < γ − z − . (27)These energy frontiers are graphically represented in Fig. (3) where traced lines are the zonesdefined by the inequalities Eq. 26 and Eq. 27. The third relation in Eq. 26 defines two zonesabove z + = z − α / (2 z − α −
1) on the ( z + , z − ) plane. The interception implies simply that0 ≤ z + ≤ α/ ≤ z − ≤ / α , or the squared region shown in Fig. (3).After determining the feasible region for the radiation emission energy, contour plots of thesource mass relation, Eq. 14, are as shown in Fig. (4). In this plot, the axes are expressedin terms of re-scaled values z + /α and αz − to provide a general view of the mass dependenceon the emitted radiation. If, for example, β = 0, the mass ratio µ → z ± → / β , and the constrain β = β (cid:48) is shown in Fig. (5).The energies depicted in this plot correspond to the classical Doppler shift ratios per unit oflost mass fraction of the source, (1 − µ ) /
2, as given by Eq. 10 for a cospectral emission inthe source rest frame. In the low velocity limit, the photon energies are proportional to β , Graphical representation of the feasibility region on the ( z + , z − )plane for the photon emission energies as determined by Eqs. 26 and 27. Figure 4:
Contour plot of the mass ratio Eq. 14 as a function of z + /α and αz − .that is z ± ≈ / − µ )(1 ± β ) while the high velocity limit with β = 1 − (cid:15) , (cid:15) (cid:28)
1, are z + ≈ / − µ ) (cid:112) /(cid:15) and z − ≈ / − µ ) (cid:112) (cid:15)/ In reality a source can emit a bunch of photons (or a beam) with arbitrary frequency dis-tributions. The emission may involve unequal photon numbers and be called ‘anisotropic’.Anisotropic emissions may be responsible for unexplained behavior of spacecraft as observed inthe anomalous acceleration in the ’Pioneer anomaly’ [14]. Similarly, the emission may not besimultaneous in the source rest frame. A possible generalization of the dynamical Doppler shiftwith arbitrary intensities but still monochromatic beams is to assume energies n ± hν ± , with n ± Right (black) and left (red) radiated photon energies normalizedby source mass lost 2 z ± / (1 − µ ) as a function of the velocity β of a referencesystem.the number of emitted photons in each beam. These quantities are invariant upon a change ofreference frame, or n ± = n ± . It is straightforward to show that for such a case, instead of Eqs.14 and 16, the following relations should be used µ = (cid:112) (1 − n + z + /α ) (1 − n − z − α ) , (28)and β (cid:48) = γβ − ( n + z + − n − z − ) γ − ( n + z + + n − z − ) . (29)The new feasible mass domain is now dependent on the total number of photons in each beam,but essentially remains the same: 0 ≤ n + z + ≤ α/ ≤ n − z − ≤ / α as suggested by Eqs.26. It is instructive to apply the equations to a real system. Consider for example two 525nm laser pens attached to each other. Each pen has M = 50 g, and emits, for 7 days at themaximum power of 5 mW, two counter propagating laser beams. The system total mass is 100 gand the equivalent total energy released is 3.91 × J. Each light beam contains n = 8 . × photons carrying 3 . × − J. The 7-days light beam stretches for 1212 A.U. (AstronomicalUnits) or 0.02 light-years from the source initial position. The dimensionless energies of eachbeam is therefore z ± = 6 . × − and deplete the source mass by ∆ µ = 1 . × − %.Such small numbers make evident how large M c is in relation to typical emission powers ofcommercially available sources. In order to be effective, the radiation sources cannot be basedon chemical processes, but on much more powerful ones - like nuclear reactors [15]. The process of light emission by a moving source in 1-d may be illustrated on the Minkowskispace-time diagram of Fig. (6) with a source resting in the S system. In this frame, the twocounter propagating photons with momenta ± z ± M c (with z +0 = z − ) will spread out on a lightcone at 45 ◦ in relation to the orthogonal axis x and ct . Sensors placed at A and B on the x -axis with OA = OB will detect the tip of the beam simultaneously (at time T A = T B ). Minkowski diagram for cospectral emission in the source restframe, hence the energies are such that z +0 = z − = z . In this diagramtan φ = β , and δT A and δT B are the projections of each beam’s lengthsonto the time axis of the moving frame.A moving frame is represented by a set of non-orthogonal axis sharing the same origin O . Thesource world-line will be represented by the segment Oct forming an angle φ with the ct -axisso that tan φ = β . As it is clearly seen, the two space-time events will be first detected by asensor located at B and then at A . Both sensors are not equally spaced in relation to the pointof emission. The beam heads will be detected at distinct times T A and T B on the ct -axis with T B < T A . The time interval between successive wave crests or troughs will be distorted, so thatwaves moving toward + x will have higher frequencies than those going to − x . Figure 7:
Minkowski diagram for asymmetrical emission in the source restframe, but, as observed from the right reference system, z + = z − . Heretan φ = β . δT A and δT B are the projections of each beam’s lengths ontothe time axis of the moving frame.Given the time transformation between reference time frames, S → S , t = γ [ t + βx /c ][8, 9], so that the intervals calculated in each frame will be related by δt = γ [ δt + βδx /c ]. InFig. 6, δt may be taken as δT A or δT B representing the projection of a given crest count onthe moving frame time-axis. Dividing both sides by N or the total number of crests counted bythe sensors in each frame (which is invariant) we find δt/N = γ [ δt /N + βδx / ( cN )]. However, ν − = δt/N and ν − = δt /N . Moreover, δx /cN = ( δt /N )[ δx / ( cδt )] = δt /N = ν − since δx = cδt , the total length of N crests during the interval δt . Therefore, ν + = ν α is thefrequency measured for the right propagating photon by the sensor at point B . For the left ropagating photon at point A the same relations can be applied and we get ν − = ν /α . Therelations Eqs. 10 for problem (a), Fig. 1, are then graphically explained.On the other hand, Fig. 7 represents problem 2(b), Fig. 2 when z + = z − . In the source restframe, the two counter-propagating beams should be asymmetrically distributed in frequencyso that their projections onto a particular moving frame time-axis becomes cospectral. As givenby Eq. 16, this is only possible if the source velocity changes. Such a velocity change would berepresented in Fig. 7 as a change in the inclination of the both time and space axis (to a new φ (cid:48) with tan φ (cid:48) = β (cid:48) ). The approach used to calculate the mass and velocity variations is completelygeneral and carries an implicit assumption that the time interval of radiation emission is muchshorter than any typical propagation times of the source as seen withing the observer timeframe. In order to further strengthen the concepts, this section suggests six problems based on thepresented discussion.1. Calculate the velocity of the reference frame ˆ β for which the source will be at rest afterthe emission of two photons with z ± asˆ β = ± ( z + − z − ) (cid:112) z + + z − ) . (30)2. Show that, to first order in z ± , the mass ratio can be written in terms of α (cid:48) as µ ≈ (cid:20) − (cid:0) z − α (cid:48) (cid:1) (cid:18) z + α (cid:48) (cid:19)(cid:21) . (31)3. Write the squared norm of the system 4-momentum, Eq. 4, after the photon emission interms of the final source velocity, showing that it can be written compactly as P (cid:48) · P (cid:48) = − M c (cid:18) µ + 2 z + α (cid:48) (cid:19) (cid:0) µ + 2 z − α (cid:48) (cid:1) . (32)4. Show that, in Problem (3), P (cid:48) · P (cid:48) = − M c , or that mass is strictly conserved in theprocess. Discuss the meaning of this conservation in face of the reduction in the sourcemass (1 − µ ) M .5. A moving source emits two cospectral counter propagating beams with ν ± = ν and doesnot change its velocity after the emission. Show that this is only possible if the sourcevelocity ¯ β is ¯ β = n + − n − n + + n − , (33)with n ± the number of emitted photons in each beam.6. Show that, in the source reference system of Problem 5, the two beams have differentfrequencies given by ν +0 = ν (cid:114) n − n + ,ν − = ν (cid:114) n + n − . (34)In this case, no velocity change is observed in the source proper frame as well. Comparethis situation with the one described in Section 2.3. Conclusion
The Doppler shift is an intuitive phenomenon apprehended easily when an approachingsiren is heard at distance. In ordinary optics, the Doppler shift is presented formally as relationinvolving the velocity of the radiation source and its proper frequency. Such relationship mightgive the impression that it is a purely kinematic expression as suggested by the sound equivalent.So, a question worth discussing with students is on the fate of the Doppler shift, because,according to Eqs. 10, the emission frequencies depend on the final source mass and velocitywhich is true even in the cospectral case. In fact, by imposing z + = 0 and z − = 0 on Eqs. 10,the only possible solution leads to µ = 1 or no mass change.Using conservation of energy and momentum, which are central concepts in special relativity,this work emphasized the role of the mass-energy equivalence and mass conservation. But howis mass conserved? The suggested problem (4) clarifies the question. Problem (5) explores thedynamical case, when two counter-propagating beams are emitted with distinct frequencies inthe source rest frame; however, they are detected as cospectral from a reference frame movingwith velocity ¯ β .Some interesting pedagogical consequences can be drawn from this study. For an isotropicsource, the asymmetry in the forward and backward photon frequencies observed by a referenceframe moving at velocity v in relation to the source is not associated with any velocity change.However, it is possible to have a moving source emitting isotropic and cospectral radiationfollowed by an apparent velocity change which is very small, or of the order zβ as expressedin Eq. 19. In the source reference frame however the emitted photons do not share the samefrequency nor are emitted simultaneously as illustrated by the Minkowski diagrams of Fig. 6.Notice however that the kinematic aspects of measurement process in distinct reference framesare bypassed by the Lorentz invariance of Eqs. 8 and 9, which imply in relations among initialand final source velocities and photon energies only. This is an important pedagogical advantageof using conservation equations.The examples discussed here show the internal coherence of the relativity theory. In it, allconcepts are interrelated: the necessary 4-vector invariance upon a change of reference systemthrough the Lorentz transformation implies in the conservation of the new fundamental quantity,the 4-momentum. Mass is in fact conserved, but should be properly substituted or reinterpretedby the concept of energy which characterizes radiation while mass does not. Acknowledgments
The author would like to thank Christine F. Xavier for the help with the work.
First and second order derivatives of µ (mass ratio), Eq. 14: ∂µ∂z + = − α (cid:115) − z − α − z + /α∂µ∂z − = − α (cid:114) − z + /α − z − α , (35) µ∂z +2 = − α (cid:115) − z − α (1 − z + /α ) ,∂ µ∂z − = − α (cid:115) − z + /α (1 − z − α ) ,∂ µ∂z + ∂z − = 1 (cid:112) (1 − z + /α )(1 − z − α ) . (36)First and second order derivatives of β (cid:48) (final velocity), Eq. 16: ∂β (cid:48) ∂z + = (cid:18) α (cid:19) (2 z − α − γ − ( z + + z − )] ,∂β (cid:48) ∂z − = α (1 − z + /α )[ γ − ( z + + z − )] , (37) ∂ β (cid:48) ∂z +2 = (cid:18) α (cid:19) (2 z − α − γ − ( z + + z − )] ,∂ β (cid:48) ∂z − = 2 α (1 − z + /α )[ γ − ( z + + z − )] ,∂ β (cid:48) ∂z + ∂z − = 2[ βγ − ( z + − z − )][ γ − ( z + + z − )] . (38) References [1] E. Whittaker. A History of the Theories of Aether and Electricity (Vol. II: The ModernTheories, 1900-1926. Courier Dover Publications, 1989)[2] A. M. Gabovich & N. A. Gabovich, N. A. (2007). Eur. J. of Phys., 28(4), 649.[3] L. J. Wang (2017). Physics Essays, 30(1), 75-87.[4] E. Hecht (2006). The Physics Teacher, 44(1), 40-45.[5] L. B. Okun (1989). Physics today, 42(6), 31-36.[6] T. R. Sandin (1991). American Journal of Physics, 59(11), 1032-1036.[7] A. B. Stewart (1964). Scientific American, 210(3), 100-109.[8] A. P. French. Special Relativity (W. W. Norton & Co, 1968)[9] R. W. Ditchburn. Light. (Black & Son Limited, 1958).[10] R. P. Feynman, R. B Leighton and M. Sands. The Feynman lectures on physics (Vol. I:The new millennium edition: mainly mechanics, radiation, and heat. Basic books, 2011).[11] M. M. Michaelis & A. Forbes (2006). South African journal of science, 102(7-8), 289-295.[12] S. Datta (2018). Physics Education. 34(4).[13] T. Singal & A. K. Singal (2019). Physics Education. 35(4).[14] S. G. Turyshev, V. T. Toth, G. Kinsella, S. C. Lee, S. M. Lok & J. Ellis (2012). Physicalreview letters, 108(24), 241101.[15] J. Huth (1960) ARS Journal, 30(3), 250-253[1] E. Whittaker. A History of the Theories of Aether and Electricity (Vol. II: The ModernTheories, 1900-1926. Courier Dover Publications, 1989)[2] A. M. Gabovich & N. A. Gabovich, N. A. (2007). Eur. J. of Phys., 28(4), 649.[3] L. J. Wang (2017). Physics Essays, 30(1), 75-87.[4] E. Hecht (2006). The Physics Teacher, 44(1), 40-45.[5] L. B. Okun (1989). Physics today, 42(6), 31-36.[6] T. R. Sandin (1991). American Journal of Physics, 59(11), 1032-1036.[7] A. B. Stewart (1964). Scientific American, 210(3), 100-109.[8] A. P. French. Special Relativity (W. W. Norton & Co, 1968)[9] R. W. Ditchburn. Light. (Black & Son Limited, 1958).[10] R. P. Feynman, R. B Leighton and M. Sands. The Feynman lectures on physics (Vol. I:The new millennium edition: mainly mechanics, radiation, and heat. Basic books, 2011).[11] M. M. Michaelis & A. Forbes (2006). South African journal of science, 102(7-8), 289-295.[12] S. Datta (2018). Physics Education. 34(4).[13] T. Singal & A. K. Singal (2019). Physics Education. 35(4).[14] S. G. Turyshev, V. T. Toth, G. Kinsella, S. C. Lee, S. M. Lok & J. Ellis (2012). Physicalreview letters, 108(24), 241101.[15] J. Huth (1960) ARS Journal, 30(3), 250-253