Time Contraction Within Lightweight Reference Frames

Braz J Phys (2017) 47:333–338 DOI 10.1007/s13538-017-0501-4

GENERAL AND APPLIED PHYSICS

Matheus F. Savi1 · Renato M. Angelo1

Received: 3 April 2017 / Published online: 20 April 2017 © Sociedade Brasileira de F´ısica 2017

Abstract The special theory of relativity teaches us that, its position, velocity, energy, mass, spin, charge, tempera- although distinct inertial frames perceive the same dynam- ture, and wave function? Some people would answer this ical laws, space and time intervals differ in value. We question in the affirmative by arguing that one can always revisit the problem of time contraction using the paradig- consider an immaterial reference system for which such matic model of a fast-moving laboratory within which a assertions would make sense. In contrast, we defend that photon is emitted and posteriorly absorbed. In our model, this position is not useful in physics because, within any sci- however, the laboratory is composed of two independent entific theory, we always look for a description that can be parallel plates, each of which allowed to be sufficiently light empirically verified by some observer. Now, the notion of so as to get kickbacks upon emission and absorption of “observer” does not need to be linked with a brain-endowed light. We show that the lightness of the laboratory accentu- organism. It suffices to think of a laboratory equipped with ates the time contraction. We also discuss how the photon apparatuses whose measurement outcomes make reference frequency shifts upon reflection in a light moving mirror. to a space-time structure, which is defined by rods and Although often imperceptible, these effects will inevitably clocks rigidly attached to this laboratory. Being a physi- exist whenever realistic finite-mass bodies are involved. cal entity, the laboratory itself is subjected to the laws of More fundamentally, they should necessarily permeate any physics. This means that the very act of observing a system eventual approach to the problem of relativistic quantum may imply a kickback to the laboratory, which then would frames of reference. become a non-inertial reference frame. This is the notion of reference frame we will keep in mind throughout this work: Keywords Time contraction · Lightweight reference a coordinate system rigidly attached to a real laboratory frames · Doppler effect · Special relativity which can physically interact with other systems. The concept of a physical reference frame is well known in classical mechanics. It appears in many textbooks [1, 1 Introduction 2], for instance in the form of the two-body problem, where the dynamics of two interacting point masses is anal- Think of a universe with only a single system, say an ysed while one of these particles is promoted to reference electron. Is it possible to make physical assertions about frame. As a result, Newton’s second law for the relative physics is shown to depend on the two-body reduced mass, which incorporates the effects of the reference frame lightness. As far as the quantum domain is concerned, it has Renato M. Angelo [email protected] been shown that nonrelativistic quantum mechanics can be consistently formulated in terms of genuine quantum Matheus F. Savi [email protected] reference frames [3–6], i.e. interacting quantum particles playing the role of reference frames. More recently, one of 1 Department of Physics, Federal University of Parana,´ us and a collaborator pointed out a framework, involving P.O. Box 19044, Curitiba, PR 81531-980, Brazil quantum reference frames, in which quantum theory is both 334 Braz J Phys (2017) 47:333–338 manifestly covariant under Galilei boosts and compatible (The time contraction1 is by now a well-established fact, with Einstein’s equivalence principle [7]. having been demonstrated in several experiments [10–19].) The question then arises as to whether relativistic quan- Here we will consider a sort of “microscopic elastic ver- tum theory as well, as described by Dirac’s equation, will sion” of this problem in which the light beam is replaced preserve its covariance upon Lorentz transformations in a with a single photon and the rigid train with two very light scenario where the moving frame itself is a relativistic quan- plates, which can move independently. In our model, the tum particle (i.e. a particle prepared in a quantum state with upper plate will be a mirror and the lower plate will play large mean momentum relative to a given inertial reference the role of moving reference frame. The motivation behind frame). While the answer to this query is expected to be this scheme is to understand how formula (1) changes in a given in the positive, it seems to us, in light of the efforts regime in which the moving system is allowed to get kick- demanded in the work [7], that the demonstration of such a backs upon emission and absorption of light, as would do point is a rather complicated task. In fact, to the best of our a quantum particle. Although, on the one hand, we may knowledge, no related study has been reported so far and, to suspect that any eventual correction must be negligible, on be honest, neither do we conceive by now a clear proposal the other, conservation laws ensure that it is fundamentally on how to conduct such an investigation. unavoidable. We then propose to assess here, as a first prototype, a In spite of all the idealizations of this model, specially much simpler situation in which the relativistic quantum the tacit use of Ehrenfest’s theorem to approximate quantum particles involved can be thought of as being prepared in systems by classical particles, thus avoiding the mathemat- minimal uncertainty wave packets with large mean momen- ical complications that would inevitably derive from the tum. As a further simplification, we may assume that uncertainty principle, we still expect to get some insight on the wave packets remain significantly localized during the the sort of phenomenon we should meet from the perspec- experiment and that no particle-antiparticle excitation is tive of relativistic quantum particles. After all, be localized activated throughout, so that we can apply the Ehrenfest as a classical particle or delocalized as a quantum wave, any theorem to the dynamics and thus effectively treat the quan- finite-mass system is compulsorily submitted to kickbacks tum systems as classical particles. We nevertheless use the deriving from the conservation laws. Lorentz transformations to look at the physics “seen” by one of these relativistic particles while the system evolves in time. All this makes the aforementioned complex problem 2Model involving relativistic quantum reference frames reduces to the problem of describing the physics from the perspective Let us consider the framework illustrated in Fig. 1.Twopar- U = of a finite-mass relativistic classical particle, a model that allel plates, each of mass M, move with velocities VS L = ˆ = has not been investigated so far either. VS u xS (u, 0) relative to an inertial reference frame ˆ Being more specific, we will phrase our approach in S,wherexS is a unit vector associated with the cartesian [ ] terms of a convenient adaptation of the famous problem in coordinate system xy S that defines S. The superscripts U which light travels within a fast-moving train [8, 9]. In the and L refer to points located at the upper and lower plates, usual version of this problem, light is emitted from the floor respectively. These indexes are also used to name the plates of the train, reflects in the roof, and is absorbed at the same themselves. Rigidly attached to the point L of the lower

[ ] point in the floor. An observer within the train measures, plate is the origin of a cartesian system xy S , which then using a single clock, a time interval tS between the two defines the moving reference frame S . The upper plate is events (emission and absorption in the floor), which occur an ideal mirror. For future convenience, we also consider an at the same position in his inertial reference frame S.An auxiliary reference frame A, equipped with a cartesian sys- [ ] ˆ observer in an external inertial reference frame S measures, tem xy A, that moves with constant velocity u xS relative

[ ] using two synchronized clocks placed in different locations, to S and is perfectly aligned with xy S . Hence, the initial a time interval tS . According to the laws of the special velocities of the lower plate and the mirror relative to A are L = U = relativity, these time intervals are related by the formula VA VA (0, 0). The velocity of the lower plate relative L = to its own coordinate system is VS (0, 0) and the velocity = U tS tS /γ , (1) = u of the mirror relative to the lower plate is VS (0, 0). − 1 1 2 2 2 Usually, Eq. (1) is expressed as t = γ t and referred to as a where γu = 1 − u /c , c is the speed of light in vacuum, S u S and u is the speed of the train relative to S. Given that u 1andt

To determine the velocity acquired by S due to the emission of the photon, we apply the relativistic conservation laws from the perspective of the inertial reference frame A. Energy and momentum conservation laws imply, respectively, that M c2 Mc2 = LA + hν (2a) − 2 2 A 1 VLA/c and hν M V A = LA LA , (2b) c − 2 2 1 VLA/c

where MLA (M) is the rest mass of the lower plate after

(before) the photon emission, h is the Planck constant, νA − ˆ is the photon frequency relative to A,and VLAyA is the velocity of the lower plate relative to A. Solving the above

equations for VLA and MLA yields V M √ LA = and LA = 1 − 2, (3a) c 1 − M with

ˆ hνA Fig. 1 a Two plates of mass M move independently with velocity u xS ≡ . (3b) relative to an external inertial reference frame S. The upper plate is a Mc2 mirror and the lower one, which is equipped with a source of photons Because the photon always carries a non-zero energy and and a cartesian system [xy]S , assumes the role of moving reference VLA

ˆ We are now ready to compute the time elapsed since the with defined by Eq. (3b). Here, VUAyA and MUA denote emission of the photon at the point L until its absorption at the velocity and the rest mass of the mirror after the absorp- the same point. For convenience, we divide the kinematics tion of the photon, respectively. From a quantum mechanical in two parts. viewpoint, the photon is absorbed by one atom of the mirror and is posteriorly emitted with a different frequency 2.1 Photon’s Rise (L → U) (as we will discuss latter). The time elapsed between these two events—the lifetime of the corresponding electronic LU We first calculate the time interval tA referring to the transition—is denoted here by τA. photon’s rise from L to U from the perspective of A.Inthis During a time interval that comprises the photon rise and case, the events to be considered are (i) the photon emission the atomic lifetime, the lower plate moves downwards a dis- LU + at the point L located in the lower plate and (ii) the photon tance VLA(tA τA) in A. By its turn, the mirror moves absorption at U, which is a point located in the upper mir- upwards a distance VUAτA. From this moment on, the two rored plate (see Fig. 1). From the discussion above, one has events to be considered are (i) photon emission at U and (ii) that the velocity of the lower plate after the photon emis- photon absorption at L. The time equations for the photon L = − = sion is VA (0, VLA), whereas the velocity of the mirror and the lower plate can be respectively written as yA(tA) U = + − L =− LU + + is VA (0, 0). Since the photon speed is the same in all (L VUAτA) ctA and YA (tA) VLA(tA τA tA), = reference frames, we have that vA (0,c). where tA is the time elapsed since the emission of the photon Concerning space-time intervals, for the events in ques- at U. Equating these expressions, we can easily determine LU = LU = LU = tion, it is clear that xA 0, yA L,andtA the time elapsed between the two events: L/c. Then, we can apply the transformations (4)and(5), L/c 2τA with pertinent adaptations, to obtain the frame conversions t UL = + . (9) A (1 − 2) (1 + )(1 − 2) A → S and S → S. The results can be expressed as Now, using the Lorentz transformations and the above LU γ ut LU LU u S results, we can compute the total time elapsed from the x = 0,x= , S S − 1 (1 − 2) 2 emission at L until the absorption at L in all reference LU LU frames: LU yA LU yS y = ,y= , S 1 S − 1 2L (1 − 2)2 (1 − 2) 2 = LU + + UL = + tA tA τA tA f() τA g(), LU LU c t γ t LU = A LU = u S t ,t . (6) t = γ t , S 1 S − 1 S u A (1 − 2)2 (1 − 2) 2 − ( VLA)yA = − = tS γ(−V ) tA tA/γV , (10) The last relation above shows that√ the usual dilation factor LA c2 LA − γu is reduced by a recoil factor 1 2. In particular, no =− wherewehaveusedyA VLAtA to derive the last dilation will occur when hν = Mu2/2, since in this case − √ A equality and introduced the functions f() ≡ 1 and − = 1−2 γu 1 2 1. Of course, this regime cannot be reached + − 2 g() ≡ 1 2 for the sake of notational compactness. To when low energy photons and heavy plates are involved. 1−−22 conclude, we can write √ 2.2 Photon’s Descent (U → L) − tS 1 2 = = tS tS /γu. (11) γuγV 1 − Before considering the descent of the photon from U to L, LA we need to look at the scattering of the photon by the mirror. It is interesting to notice that this relation does not depend From A’s perspective, the energy-momentum conservation on any information regarding the photon scattering in the in the absorption process at U demands that mirror (as for instance τA); such information is encoded

only on tA. In addition, the first equality above gives an M c2 2 UA hν + Mc = (7a) intuitive relation: tS connects with tS through Lorentz A − 2 2 1 VUA/c factors referring to both the horizontal motion and the recoil of S relative to S. and So far, our results have been expressed in terms of νA, hν M V A = UA UA , (7b) which is the frequency observed from the auxiliary ref- c − 2 2 1 VUA/c erence frame A during the photon rise. Now we want to abandon A and rewrite our results in terms of ν ,whichis whose solution is given by S the frequency measured in S. To this end, we apply the lon- V M √ gitudinal relativistic Doppler effect [8, 9]. It is clear that, UA = and UA = 1 + 2, (8) c 1 + M when the photon is rising, “the source” at L separates with Braz J Phys (2017) 47:333–338 337

→ speed VLA from “the detector”, which is fixed say at the ori- require that the frequency change νi νr upon reflection = gin yA 0ofA. It follows that the frequency values during be described, to the S perspective, as the photon rise as seen by A and S are related as 1 − β ν = ν , (16a) − √ r i + 1 VLA/c 1 β ν = ν = ν 1 − 2. (12) A S + S 1 VLA/c where This is still not the solution to the problem because the r.h.s. −1 1 − β term in the last equality depends on νA through the relation ≡ 1 + 2i , (16b) hν + = A defined in Eq. (3b). Using this relation, we can 1 β Mc2 solve Eq. (12)forνA so as to obtain = ≡ hνi → β v/c and i 2 . For infinite-mass mirrors, i 0 Mc 2 hνS → ν = ν + ε − ε ,ε≡ , and 1, in which case we recover the usual formula for A S 1 2 (13) Mc light reflection in a moving mirror [21]. Notice that even if −1 where ε has substituted in the role of “significant dimen- v = 0, a correction = (1 + 2i) will be present due to sionless parameter”. the lightness of the mirror. Of course, this correction could We are now in position to finish our calculations. By be implemented in the problem under scrutiny in this work, direct inspection√ of Eqs. (12)and(13), we learn how to if required. In our approach, however, this was not necessary express 1 − 2 as a function of ε. With that, we come back because the results have been exhibited in terms of the initial to Eq. (11) to derive, after some algebraic manipulations, frequency of the photon, the one defined in the very first our final result: emission at the lower plate.

tS /γu = √ tS . (14) 1 + ε2 3Conclusion In comparing this result with Eq. (1), which is obtained in the usual context of√ a rigid infinite-mass laboratory, one Once we admit that a laboratory is never rigorously isolated can readily regard 1 + ε2 as a correction factor deriving from the observed system, with which it physically inter- from considering a nonrigid finite-mass laboratory. Indeed, 2 acts, then we may wonder whether such a reference frame Eq. (14) reduces to Eq. (1) whenever Mc hν .Itis S can indeed be regarded as inertial. Usually, one can main- worth noticing that ε will be small even in extreme sce- tain this position only to some degree of approximation. To narios, as for example when the laboratory is thought of obtain a more reliable description in such scenarios and pos- as being formed by only two hydrogen atoms (e.g. a H 2 teriorly assess the quality of those approximations, we often molecule), one representing the lower plate and the other appeal to an absolute inertial reference frame from which the mirror. Taking M =∼ 1.0 u for the mass of each plate we describe the physics of both the system and the labora- gives Mc2 930 MeV. If we consider the highest energy tory. With this strategy, we then derive the pseudo physical photon that a hydrogen atom could eventually emit, we can −8 laws, the ones that govern the behaviour of the system estimate that hν 14 eV. It follows that ε 1.5 × 10 , S within the non-inertial laboratory. In the last decade, some which makes ε2 negligible in Eq. (14). authors reported work on Galilei boosts in quantum sys- Before concluding, a quick remark is opportune with tems, thus discussing the physics seen from the perspective respect to the phenomenon of frequency change upon reflec- of quantum reference frames. To our perception, however, tioninalight moving mirror. Consider a mirror plate of fundamental studies involving lightweight reference frames mass M moving with velocity vyˆ with respect to an inertial S still need to be done in the fields of special relativity and reference frame S. The plane of the mirror is always perpen- ˆ ˆ relativistic quantum mechanics. dicular to yS . A photon with velocity cyS and frequency νi The present work aims at contributing to this discussion impinges on the mirror and reflects with velocity −cyˆ and S by re-examining a well-established physical phenomenon as frequency νr . After the photon reflection, the mirror moves seen by a relativistic lightweight particle. We ask how the with speed v . The conservation laws for this scenario, usual formula governing the time contraction changes when Mc2 Mc2 light is emitted and absorbed within a finite-mass labora- hνi + = + hνr (15a) 2 2 tory. In our model, we replace the traditional rigid massive 1 − v 1 − v c2 c2 train with two fast-moving parallel light plates, the lower of and them playing the role of a reference frame S moving rela- tively to an inertial reference frame S. A single photon with hνi + Mv = Mv − hνr , (15b) 2 2 energy hνS is emitted from the lower plate, gets reflected c 1 − v 1 − v c c2 c2 in the upper plate (a mirror), and is finally absorbed at 338 Braz J Phys (2017) 47:333–338 the emission point in the lower plate. Because of the rela- 7. S.T. Pereira, R.M. Angelo, Galilei covariance and Einstein’s tivistic energy-momentum conservation law, upon emission equivalence principle in quantum reference frames. Phys. Rev. A or absorption of the photon, the lower plate gets a kick- 91, 022107 (2015) 8. D. 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