174 lines
11 KiB
Plaintext
174 lines
11 KiB
Plaintext
I f-I
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Phase Considerations in a Rotating System L o m e A. Page, Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 U.S.A.
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ABSTRACT A neutron interferometer in constant absolute rotation will exhibit a certain pnase-shift between its two beams, a Phenomenon shared with the classic Sagnac or Michelson-GalePearson experiments or with the modern laser-gyrocomoass composed of lasers in a ring. To first order in the rotational frequency it is oossible to understand by employing only rudimentary theory the essence of this phenomenon to any degree of relativisticness of the cartici^ating tjarticle. This naper is mainly caedagogical, noting the similarity anent rotation between Dhoton-, electronand neutron-interferometers. Future experimentation, aside from corroborating well believed tenets, may hope with improving nrecision to bring new approaches to measurement of fundamental effects.
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IntroductionThe sensitivity achieved in recent years with neutron
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interferometers (Bonse & Hart 1965; Bonse & Hart 1966; Rauch, Treimer and Bonse 197M (Overhauser & Colella 197^; Colella, Overhauser & Werner 1975J Werner et al 1975) recalls to mind the classic experiment (Sagnac 1913) in which an optical interferometer encompassing an area N 10-1 cm was rotated at several r_ evolutions per second resulting in a discernible phase-shift between two beams, the one rotating orogressively, the other retrogressively. Likewise Michelson, Gale and Pearson (Michelson 1925) employed an optical system embracing some ICp meter fixed to the earth and demonstrated the earth'^s rotation with respect to the fixed stars by means of the similar phase-shift. In this paper we examine such phase-shift in a paradigmatic interferometer letting the beam particle be alternatively non-relativistic, mediumly relativistic, or completely relativistic (as with the photon). The theory is especially simple if we restrict attention to a response linear in the rotation frequency,Jl0 rad/sec.
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The paradigm Arranged in a square for
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simplicity, we consider a sender of waves at position (1), the ultimate receiver at (*»•), with identical transceivers at positions (2) and (3) as depicted in Fig. 1. The discussion is Kinematic only; the dynamic details of eg how the sender nroduces two coherent beams, 'how the emitters (2) and (3) work, are not specified. The four active elements have only small extent with respect to dimension a. To obviate time dilation, Lorentz contraction and the like vie require the speed parameter /Ica/c to be negligible in second order; and to obviate direct consideration of aberration or transverse Dopnler shift we require the relevant nhase velocity to greatly exceed the speed fLQa . With relativistic Schroedinger waves or
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- 2-
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with electromagnetic waves in vacuum the first (mild) restriction imnlies the second. Nonetheless the seat of the phase-shift is in fact special relativity as will be explicitly shown. Looking forward to forming wave-packets we stipulate that the group velocity greatly exceed/loa to minimize centrifugal effects.
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We might term the 'Mach Lab1 that inertial system in which our interferometer (Fig. 1) is rotating anticlockwise at fixed /I0 •about its centre 0. We note the effective absence of a Doppler shifts thus in virtue of the constant angular velocity and the symmetry all four active elements send/receiAre at the same-frequency namely that at which (1) emits in its own comoving system.
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Photon, viewed in Mach Lab To first order in yiQa/c we see that
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the path from (1) to reflector (3) is increased from a to a(l + V~* ); and similarly the path from (3) to destination ( M . Inasmuch as (*O receives non Doppler shifted light via path 3 just as via oath 2 we end up with the phase-shift, path 3 minus path 2,
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^T3x~ * 7~ vc"*/4 ~^"°
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' a w e H known result.
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Notice that this same result would apply for an extremely fast
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oarticle of relativistic mass M = E/c^. In which case we can
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write
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Technically E is the particle's energy in a momentarily comoving system for any of the stations (1) through (U-) for that portion of trajectory with which the given station deals of course.
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Wave of any -phase velocity, viewed in certain comoving svstem(s) At the instant in the Mach Lab
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when sender (1) is moving just along x we make a Lorentz transformation"along x at ^ =-Aoa/ifrc. . which makes the speed of (1) to be zero but not its acceleration, which latter is to be ignored to first order inJlQ. This Lorentz system can be called CM-1 since momentarily at least it comoves with (1),
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••— (see Pig. 3 )
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In the system CM-1 we note that whatever the ohase velocity (be it
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c or larger) transverse Donpler effect and[aberration are to be
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ignored by our earlier stipulation. Since^the longitudinal Doppler
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effect is absent, both (3) and (2) receive a given wavefront at
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£ = 0 the same time in CM-1. However, and this is essential, such
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corresponds to a 4 t M a c h s +/3 —•ef1'which amounts to 4 t
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_ a.xJlo
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We may now send both reradiated sign&ls on their way to ( 4 ) . The
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time offset due to transforming from CM-1 to CM-4 amounts to 2. lil'
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The energy (frequency) received at (4) is in our approximation e*vd -We fitt+tts the. »-f- tfK*/ j/e\a*Ai „ . / .
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equal for both paths/^ consequently the ohase-shift E At/% is ^fk = ^ ^ J w h i e h is just expression 1 again. However the expression has just now been shown to be valid over therange from quite slow heavy oarticles to photons (or neutrinos!), since M stands for the relativistic mass.
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One contemplates of course assembling a wave-packet to be split somehow at (1) and sent along the two paths; understandably then A& is not completely sharp but is fuzzed according as M^k) reflects the distribution in k. In the extreme nonrelativistic domain
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does become sharp for the whole packet. It can be shown that for any polygon, regular or otherwise, expression 1 holddss with a2 replaced by ^feg^^EeS area. The factor allies when the loop is traversed but once, thus half-way round for each signal as in our paradigm. Expression 1}restricting to nonrelativistic particles, neutrons, was given earlier (Page 1975) on the basis of some simple hand-waving. A .more formal treatment (Anandan 1977) including gravitational as well as Rotational effects
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relativistic correctiorTXa first correction) to the
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rest-mass M Q as ffl P2/2MO insofar as^otation^fif^irst order
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is concerned. The factor two standing in extiression 1 is reproduced in Anandan's results.
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Before putting some samrsle numbers into expression 1 it might be
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r Reminiscent of other examples in Physics such as the Thomas Drecession
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we may not disregard this offset simply because we see c in the
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t ••
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denominator, even for extremely slow motion of apparatus or particle.
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compatible with the Dresent result.
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of some interest to run through an extremely simple hand-waving
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argument by which one arrives at the correct relativistic expression
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•n.--
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for ohase-shift.
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A free wave-packet in a slowly rotating system
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We picture an unconstrained
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essentially plane wave-oacket
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travelling in a quasi-inertial system
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•- 1
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which rotates very slowly at frequency
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JlQ with respect to the Mach system.
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We make the sensible requirement th^t
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if we were to follow the course of the
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nacket for a time it must become aware
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of a slanting angle §& (see Fig. k) which evolves with time as $& - 2/Le~£m .
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We relate nominal distance travelled s to time t via Z"/v s/$tw/t tfttJ ** ~¥lp~ ' W e attribute the required veering of a
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given component of the group (component of wave-number k) to the fact that the ohase difference Sta. k minus Sta. 3 viz yhi e x o e e d s the
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nhase difference f o l .
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Thus where K stands for the relativistic average mass for the racket. The olausibility of this argument seems best when (j^k Ijq c") is not too large.
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Some confidence in this kind of simple argument might be gained from this brief digression: Consider a charged particle bending
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mildly in a magnetic field B. The Larmor frequency is qB/2Mc --O-(, In the language of Fig. h, $&* 2/2.»t"as "fiefore. Finally
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This lories us to conclude, at least an this gently bending situation, that the wave-number (times ^" ) has to be the normal kinetic momentum pitils qA/c—-the result of course being no surprise since we have in affect simply supplied the Lorentz force; but we do note that no 'factor -^f two1 is lost or gained in this simple treatment. A corollary of this magnetic bending argument (running the argument backwards) is thax a fast charged particle traversing magnetic matter has to have its (gentle) bending governed by field B and not H; this is generally assumed to be borne out experimentally.— even though the particle may not enjoy physical access, so to say, to all or any of of the flux of 3.
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X:; .
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- 5-
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Size of the Rotational Phase-shift
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We have seen that the size of the rotational phase-shift is
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proportional to ^Relativistic Mass) times Area)times /angular veloai iyj'
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with respect to-the local Mach system^ We may-substitute nhoton
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mass in the same expression we use for particles. One may compare just roughly the product M A A 0 for
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several
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situations* Sagnac (S) who used 3 ev/c2 photons around an area
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& l<y cm with a reversible speed of /v 10 rad/sec obtained a nominal 0.1 fringe shift; Michelson et al (MGP) went to 106 times
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the area of (S), used the unreversible angular speed of the earth
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and with difficulty achieved about twice the fringe shiftj>£ [r
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the two experiments were compatible with each other and with
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known wavelengths, speed of light and known angular speed at the
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precision level of a couple per cent. (We can observe that the
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reciprocity betweejjarea A andjQ.0 has been well vindicated!) Neither
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experiment used Slanck^s constant explicitly of course. They
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could both be exnlained on the basis of a circulating "aether wind"
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if one so chose.
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Considering now the possibility of rotating a Coi/eila-overhauserWerner tyoe of experiment using slow neutrons, the mass of 10? ev/c^ is certainly favorable over the classic experiment (S) but thsre may be difficulties in rotating a spectrometer system substantially faster than the earth affords. Turning to electron diffraction the mass factor is some 10^ times better than an optical photon; if a system of area one or two cm2 could be rotated at a few revolutions per second one might hope for tens of fringe shifts. REFERENCES (Anandarr 1977) Anandan, J., Phys. Rev. D 15, 1448. (Bonse & Hart 1965) Bonse, U., and Hart, M., Appl. Phys. Lett. 6, 155. (Bonse & Hart 1966) Bonse, U., and Kart, M., Z. Phys. 194, 1. (Colella, Overhauser & Werner 1975) Colella, R., Cverhauser, A. W.,
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and Werner, S. A., Phys. Rev. Lett. 34, 1472. (Michelson, Gale & Pearson 1925) Michelson, A. A., Gale, H. G.,
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and Pearson, F., AstroDhys. J. 6l, 140. (Overhauser & Colella 1974) Overhauser, A. W. and Colella, R.,
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Phys. Rev. Lett, 33, 1237. (Page 1975) Page, L. A., Phys. Rev. Lett 35, 543,
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L o ••
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:•>-•
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- 6-
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(Rauch, Treimer & Bonse 197*0 Rauch, H., Treimer, W., and Bonse, U.
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Phys. Lett. 4?A, 369. : /•-. ,,.-x..-. r
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rr
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(Sagnac 1913) Sagnac, G., C. R. Acad.Sci. (Paris) 157;. 1^10.
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(Werner etal. 1975) Werner,..S. A.,_Colella, R., Overhauser, A. W.,::
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and Eagen, C. F., Phys. Rev. Lett. 35, 1053.
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'•'.""..
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