Neutron interference in the Earth's gravitational field
NNeutron interference in the Earth’s gravitational field
Andrei Galiautdinov and Lewis H. Ryder Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA School of Physical Sciences, University of Kent, Canterbury, Kent, CT2 7NH, United Kingdom (Dated: October 4, 2018)This work relates to the famous experiments, performed in 1975 and 1979 by Werner et al., mea-suring neutron interference and neutron Sagnac effects in the earth’s gravitational field. Employingthe method of Stodolsky in its weak field approximation, explicit expressions are derived for thetwo phase shifts, which turn out to be in agreement with the experiments and with the previouslyobtained expressions derived from semi-classical arguments: these expressions are simply modifiedby relativistic correction factors.
I. INTRODUCTION
It is now several decades since the ground-breakingwork by Werner and his co-workers showed that gravita-tional [1, 2] and rotational [3] effects were to be foundin neutron interference experiments performed on theearth’s surface [4–6]. The predicted and experimentallyconfirmed gravitational phase shift is the only expressionin physics to feature both Newton’s constant of gravita-tion G and Planck’s quantum of action (cid:126) , which surelymakes these experiments particularly noteworthy. Thetwo experiments are referred to hereafter as the COWexperiment and the neutron Sagnac effect.Straightforward, semi-classical derivations of these ef-fects have already appeared in the literature (see for ex-ample [7–9]) and in abbreviated form are summarizedin Sections II and III below. What is very clear, how-ever, is that a proper account of this topic should reallybe sought in General Relativity (GR) — which is, afterall, a theory of gravity! — and indeed numerous papershave been written using this approach (see for example[10–14]). Some of these explore rather sophisticated no-tions, for example a possible parallel between the COWexperiment and the Aharonov-Bohm effect, based on theintegrated curvature of an enclosed path, on the one handin parameter space and on the other hand in field space[13]. We do not aim to explore these higher-flown top-ics, but rather to present a simple demonstration of howGR can account for the findings in neutron interferom-etry, and, at an introductory level suitable for instancefor inclusion in an introductory course, demonstrate thatgeneral relativity has an application in quantum physics[15] — a notion which might still cause some surprise!We use the Kerr solution of GR [16, 17], since thisincludes the rotation of the earth through the angularmomentum parameter a , as well as ω , the angular veloc-ity of the earth, and r s its Schwarzschild radius. Theseare all small parameters, and we calculate the relevanteffects to second order in all these quantities (mixed andunmixed). The general method of procedure is the weakfield approximation, adopted by Stodolsky [10].The next Section describes a standard, elementaryderivation of the COW effect, and in Section III is asimilarly elementary derivation of the Sagnac effect for neutrons. In Section IV the general relativistic settingfor more realistic derivations of these effects is presented.The Kerr metric is displayed as well as a coordinate trans-formation to a Cartesian system relevant to our prob-lem. In the final Section our results are derived. Useis made of the weak field approximation in conjunctionwith a specific assumption which allows the calculationsto be preformed. It is found that the resulting phaseshifts are, in both cases, those predicted by the simplemodels in Sections II and III, with correction factors of γ = (1 − v /c ) − / , and additional small terms involving ω , the angular velocity of the earth. II. SIMPLE DERIVATION OF COW EFFECT
The setup described in reference [2] (see also [9]) isbased on the splitting of the neutron beam by Braggdiffraction from perfect crystals, as first implemented forX rays by Bonse and Hart [18]. Rauch and Werner [9]point out that when the desired degree of crystal cut-ting is achieved, the resulting interferometry ”exhibitsthe fundamentals of quantum mechanics in a very directand obvious way”. The interference involved is ”topo-logically equivalent to a ring”, which we represent as arectangle, of macroscopic dimensions (centimetres). Theneutrons enter at the bottom left corner where the beamsplits into two, and the beams recombine at the top rightcorner, where the interference takes place.The spatial part of a plane matter wave describing aneutron beam is given by e i k · r , where k is the wave vectorand k ≡ | k | = 2 π/λ is the wave number, with λ being thede Broglie wavelength, so the phase accumulated over apath from r to r isΦ( r ) = (cid:90) rr k · d r , (1)or, since λ = h/p , where p is particle’s momentum,Φ( r ) = 1 (cid:126) (cid:90) rr p · d r . (2)This refers to a particular path , so the phase difference a r X i v : . [ g r- q c ] J a n between neutron beams along two distinct paths is∆Φ = 1 (cid:126) (cid:90) rr ( p I − p II ) · d r . (3)In our case the path I is the lower route and path II theupper route. The contributions to ∆Φ from the verticalparts of these two routes cancel, since the relevant mo-menta are equal and opposite; and putting p I = mv and p II = mu along the horizontal lower and upper routes re-spectively (with v and u being the corresponding particlespeeds), we find ∆Φ = 1 (cid:126) m ( v − u ) L, (4)where L is the length of the interferometer. Conservationof energy now gives us12 mu = 12 mv − mgH (5)where g is the acceleration due to gravity and H theheight of the interferometer. Since gH is of the order of10 − m s − and v ≈ × m s − for thermal neutrons,then gH (cid:28) v , and v − u ≈ gHv , (6)giving finally ∆Φ = mgA (cid:126) v , (7)where A = LH is the area of the interferometer. Thisphase shift was first predicted and observed in 1975 byColella, Overhauser and Werner [2].It is pertinent to note that the above expression forthe phase shift may alternatively be obtained by startingfrom a Lagrangian L given by L = p m + m g · r , (8)with p (= m v = m ˙r ) defined by p = ∂ L ∂ ˙r . (9)Equation (5) then yields the expected result (7). III. SIMPLE DERIVATION OF THE NEUTRONSAGNAC EFFECT
The experiment, first performed by Werner, Stauden-mann and Colella [3], measured the effect of the earth’srotation on the neutron phase. To take account of a ro-tating frame the Lagrangian (8) should be modified to L = p m + m g · r + ω · (cid:96) , (10)where ω is the angular velocity of the frame and (cid:96) theangular momentum of the particle. Then the momentum(9) becomes p = m v + m ω × r . (11)The phase coming from the term in ω is∆ α = 1 (cid:126) (cid:73) m [ ω × r ] · d r = 2 m ω · A (cid:126) , (12)where A is again the area of the interferometer. ThisSagnac phase is typically of the order of 10 − of the grav-itational COW phase, so to detect it means setting up theapparatus in such a way that the COW contribution tothe phase is zero. This is achieved by having the interfer-ometer in a vertical plane — say in the rθ or rφ plane —and then rotating it about a vertical axis. The observa-tions of the phase shift of the neutron due to the earth’srotation were found to be in good agreement with thetheory [3]. IV. KERR METRIC
We now turn to a general relativistic derivation of theCOW and neutron Sagnac effects. To describe the grav-itational field of the rotating earth, we first write downthe Kerr metric [16] in its standard Boyer-Lindquist form[17], ds = (cid:32) − r s r
11 + (cid:0) ar (cid:1) cos θ (cid:33) c dt + 2 r s r ar sin θ (cid:0) ar (cid:1) cos θ r ( cdt ) dϕ − (cid:0) ar (cid:1) cos θ − r s r + (cid:0) ar (cid:1) dr − (cid:18) (cid:16) ar (cid:17) cos θ (cid:19) r dθ − (cid:32) (cid:16) ar (cid:17) + r s r (cid:16) ar (cid:17) sin θ (cid:0) ar (cid:1) cos θ (cid:33) r sin θdϕ , (13)where a = (2 / R ω/c is the angular momentum param-eter, r s = 2 GM/c is the Schwarzschild radius, M is themass, and R is the radius of the earth. This metric de-scribes the rotating earth as seen from an inertial frame.The experiments we are considering, however, take placeon the earth, and therefore in a rotating frame, so to find the appropriate metric we must replace ϕ by ϕ (cid:48) given by ϕ = ϕ (cid:48) + ωt, (14)in which the metric becomes ds = (cid:40) − r s r
11 + (cid:0) ar (cid:1) cos θ + 2 r s r ar rωc sin θ (cid:0) ar (cid:1) cos θ − r ω c (cid:34) (cid:16) ar (cid:17) + r s r (cid:16) ar (cid:17) sin θ (cid:0) ar (cid:1) cos θ (cid:35) sin θ (cid:41) c dt + 2 (cid:40) r s r ar sin θ (cid:0) ar (cid:1) cos θ − rωc (cid:34) (cid:16) ar (cid:17) + r s r (cid:16) ar (cid:17) sin θ (cid:0) ar (cid:1) cos θ (cid:35) sin θ (cid:41) r ( cdt ) dϕ (cid:48) − (cid:0) ar (cid:1) cos θ − r s r + (cid:0) ar (cid:1) dr − (cid:20) (cid:16) ar (cid:17) cos θ (cid:21) r dθ − (cid:34) (cid:16) ar (cid:17) + r s r (cid:16) ar (cid:17) sin θ (cid:0) ar (cid:1) cos θ (cid:35) r sin θdϕ (cid:48) . (15)This expression is exact. Taking into account that for r ≈ R (radius of the earth), r s /r ∼ − , ωr/c ∼ − , a/r ∼ − , aω/c ∼ − , we expand to order 10 − and get ds = (cid:18) − r s r − r ω c sin θ (cid:19) c dt + 2 (cid:16) r s r ar − rωc (cid:17) r sin θdϕ (cid:48) ( cdt ) − (cid:20) r s r − (cid:16) ar (cid:17) sin θ (cid:21) dr − (cid:20) (cid:16) ar (cid:17) cos θ (cid:21) r dθ − (cid:20) (cid:16) ar (cid:17) (cid:21) r sin θdϕ (cid:48) . (16)It is convenient to rewrite (16) in terms of the “shifted”Cartesian coordinates erected on the surface of the earth,by analogy with how it was done in the Schwarzschildcase in Ref. [19]. The idea is to work in a coordinatesystem whose origin is “in the laboratory”, on the earth’ssurface, and also that this should be a Cartesian system,since this simplifies the calculation. We first introduce the “usual” Cartesian coordinates ( x, y, z ) defined by r = (cid:0) x + y + z (cid:1) / , ϕ (cid:48) = arctan yx , θ = arccos zr , (17)and get ds = (cid:20) − r s r − ω c ( x + y ) (cid:21) c dt + 2 (cid:16) r s r ar − rωc (cid:17) xdy − ydxr ( cdt ) − (cid:20) r s r − (cid:16) ar (cid:17) x + y r (cid:21) ( xdx + ydy + zdz ) r − (cid:20) (cid:16) ar (cid:17) z r (cid:21) (cid:0) zxdx + zydy − ( x + y ) dz (cid:1) r ( x + y ) − (cid:20) (cid:16) ar (cid:17) (cid:21) ( xdy − ydx ) x + y . (18)We next perform the rotation around the y -axis by anangle θ (the co-latitude of interferometer location onthe earth’s surface; see Fig. 1) and shift the origin by R along the new z -axis in accordance with x = x (cid:48) cos θ + ( R + z (cid:48) ) sin θ , (19) y = y (cid:48) , (20) y x z y ’ θ R x ’ z ’ y ’ x ’ θ FIG. 1: (Color online.) Local Cartesian coordinates on thesurface of the rotating Earth. z = − x (cid:48) sin θ + ( R + z (cid:48) ) cos θ , (21)and dx = dx (cid:48) cos θ + dz (cid:48) sin θ , (22) dy = dy (cid:48) , (23) dz = − dx (cid:48) sin θ + dz (cid:48) cos θ , (24)where x (cid:48) , y (cid:48) and z (cid:48) are the Cartesian coordinates whoseorigin is on the earth’s surface. We now restrict theexperimental region to the neighborhood of this shiftedorigin and introduce the weak field approximation, inwhich g µν = η µν + h µν and | h µν | (cid:28)
1, with η µν =diag(1 , − , − , − x (cid:48) /R , y (cid:48) /R and z (cid:48) /R , h = − r s R (cid:18) − z (cid:48) R (cid:19) − ω R c (cid:20)(cid:18) z (cid:48) R (cid:19) sin θ + x (cid:48) R sin (2 θ ) (cid:21) , (25) h = (cid:18) ωRc − ar s R (cid:19) y (cid:48) R cos θ , (26) h = − ωRc (cid:20)(cid:18) z (cid:48) R (cid:19) sin θ + x (cid:48) R cos θ (cid:21) + ar s R (cid:20)(cid:18) − z (cid:48) R (cid:19) sin θ + x (cid:48) R cos θ (cid:21) , (27) h = (cid:18) ω Rc − ar s R (cid:19) y (cid:48) R sin θ , (28) h = − a R (cid:20)(cid:18) − z (cid:48) R (cid:19) cos θ − x (cid:48) R sin (2 θ ) (cid:21) , (29) h = 12 a R y (cid:48) R sin (2 θ ) , (30) h = − (cid:18) r s R − a R (cid:19) x (cid:48) R , (31) h = − a R (cid:18) − z (cid:48) R (cid:19) , (32) h = − (cid:18) r s R − a R (1 + sin θ ) (cid:19) y (cid:48) R , (33) h = − r s R (cid:18) − z (cid:48) R (cid:19) + a R (cid:20)(cid:18) − z (cid:48) R (cid:19) sin θ + x (cid:48) R sin (2 θ ) (cid:21) . (34) V. RELATIVISTIC DERIVATION OF THE COWAND SAGNAC EFFECTS
We are now in a position to give a relativistic accountof the COW and neutron Sagnac effects. To do so, weneed a relativistic expression for the phase shift, whichcomes from the Feynman-Dirac formula exp ( iS/ (cid:126) ), theamplitude for a particle to travel along a path, with S = (cid:82) L dt being the action along the path. The relativisticexpression for S is − mc (cid:82) ds , with ds = g µν dx µ dx ν = c dτ , τ being proper time. Dividing the expression for ds by ds gives ds = g µν dx µ dτ dx ν = 1 c g µν dx µ dτ dx ν , (35)so S = − m (cid:90) g µν dx µ dτ dx ν = − (cid:90) g µν p µ dx ν = − (cid:90) p µ dx µ , (36)consistent with equation (2) above.We may now proceed, following Stodolsky [10], by stat-ing that the phase Φ AB accumulated by a particle movingfrom spacetime event A to event B is, invoking the weakfield approximation,Φ AB = − mc (cid:126) (cid:90) BA ds ≈ − mc (cid:126) (cid:90) BA (cid:18) ds M + 12 h µν dx µ ds M dx ν (cid:19) , (37)where h µν is the deviation of the metric g µν fromits Minkowskian form η µν = diag(1 , − , − , − ds M = η ρσ dx ρ dx σ . Eq. (37) represents the action, nor-malized to Planck’s constant, of a freely falling gravita-tional probe. We assume, as an additional hypothesis,that (37) can also be applied to a probe whose worldlineis shaped by, say, a collection of ideally reflecting mirrorsthat are at rest relative to the chosen coordinate system.(A mirror is regarded as ideal if on reflection there is nochange of particle’s energy and of the tangential com-ponent of its momentum, while the normal componentof the momentum changes sign.) A similar assumptionfor calculating gravitational effects, though in a differ-ent context, was made in Ref. [20]. The gravitationallyinduced phase is then given byΦ AB = − mc (cid:126) (cid:90) BA h µν u µ M dx ν , (38)where u µM = dx µ /ds M = ( γ, γ v /c ) is the usual relativis-tic four-velocity of the particle, v is its three-velocity,and γ is the corresponding gamma-factor. The phasedifference between the two interfering paths is then∆Φ = − mc (cid:126) (cid:73) h µν u µ M dx ν , (39)where the line integral is taken around the loop formedby the paths.We now make an important observation that, in thelinearized approximation, ∼ O ( h µν ), used in Eq. (38),neutron’s speed, v ≡ | v | , should be treated as constant .Any change in the speed acquired due to gravity, etc.,had already been taken into account when we made the linearized approximation (38). Thus, the gravitationallyinduced phase difference between the interfering pathsmay be found from the formula∆Φ = − γmc (cid:126) (cid:73) (cid:20)(cid:18) h + h i v i c (cid:19) dt + (cid:18) h j + h ij v i c (cid:19) dx j c (cid:21) , (40)where v = (cid:112) δ ij v i v j is regarded as constant.Eq. (40) represents the accumulated phase differencefor a single orientation of the loop. This phase difference,which we call intrinsic, is not directly observable. In anactual experiment, at least two orientations are involved,and it is the shift in the intrinsic phase difference duringthe rotation of the loop from one position to the otherthat is experimentally measurable.Assuming that the loop is a rectangle placed in the x (cid:48) y (cid:48) -plane, with the sides parallel to the x (cid:48) and y (cid:48) axes,we have z (cid:48) = 0 and v = ( v x , v y , x (cid:48) y (cid:48) = − γmc (cid:126) (cid:40)(cid:90) (∆ x (cid:48) , , , , − (cid:90) (∆ x (cid:48) , ∆ y (cid:48) , , ∆ y (cid:48) , (cid:41) (cid:20)(cid:18) h + h vc (cid:19) dt + (cid:18) h + h vc (cid:19) dx (cid:48) c (cid:21) − γmc (cid:126) (cid:40)(cid:90) (∆ x (cid:48) , ∆ y (cid:48) , x (cid:48) , , − (cid:90) (0 , ∆ y (cid:48) , , , (cid:41) (cid:20)(cid:18) h + h vc (cid:19) dt + (cid:18) h + h vc (cid:19) dy (cid:48) c (cid:21) = − γmc (cid:126) (cid:40)(cid:90) (∆ x (cid:48) , , , , − (cid:90) (∆ x (cid:48) , ∆ y (cid:48) , , ∆ y (cid:48) , (cid:41) (cid:18) h v + 2 h c + h vc (cid:19) dx (cid:48) − γmc (cid:126) (cid:40)(cid:90) (∆ x (cid:48) , ∆ y (cid:48) , x, , − (cid:90) (0 , ∆ y, , , (cid:41) (cid:18) h v + 2 h c + h vc (cid:19) dy (cid:48) = + γmc (cid:126) v (cid:26) vc (cid:18) ωRc − ar s R (cid:19) cos θ + ω R c sin (2 θ ) (cid:27) ∆ x (cid:48) ∆ y (cid:48) R , (41)which vanishes in the a, ω → z (cid:48) x (cid:48) and y (cid:48) z (cid:48) orien- tations, we get(∆Φ) z (cid:48) x (cid:48) = γmc (cid:126) v (cid:20) r s R − ω R c (cid:0) θ − sin (2 θ ) (cid:1) + v c a R (cid:0) θ − sin (2 θ ) (cid:1)(cid:21) ∆ z (cid:48) ∆ x (cid:48) R , (42)(∆Φ) y (cid:48) z (cid:48) = γmc (cid:126) v (cid:20) r s R − ω R c sin θ − vc (cid:18) ωRc + ar s R (cid:19) sin θ + v c a R (cid:21) ∆ y (cid:48) ∆ z (cid:48) R . (43)Combining Eqs. (41) and (42), and assuming that the loop is now rotated around the x (cid:48) -axis from horizontal x (cid:48) y (cid:48) to vertical z (cid:48) x (cid:48) position, we get, using ∆ x (cid:48) ≡ L and∆ y (cid:48) = ∆ z (cid:48) ≡ H , the experimentally observable COW change of phase,(∆Φ) COW ≡ (∆Φ) z (cid:48) x (cid:48) − (∆Φ) x (cid:48) y (cid:48) = γmc (cid:126) v LHR (cid:26) r s R − vc (cid:18) ωRc − ar s R (cid:19) cos θ − ω R c sin θ + v c a R (cid:0) θ − sin (2 θ ) (cid:1)(cid:27) = γ mgA (cid:126) v + (terms in ω and ω ) , (44)where we have made the identification r s R ≡ gc , (45)with g being the acceleration due to gravity at the earth’ssurface. It therefore turns out that in this relativisticformulation the COW phase shift is merely the simpleresult (7), corrected by the factor γ , and beyond that,further corrected (slightly surprisingly!) by terms result- ing from the rotation of the earth. These terms are twoor more orders of magnitude smaller than the first termsin (44): r s /R = 1 . × − , 4 vωR/c = 4 . × − , ωR/c = 1 . × − , ar s /R = 0 . × − , where wehave taken v = 2 . × m/s for thermal neutrons.On the other hand, combining Eqs. (42) and (43), andassuming that the loop is rotated around the z (cid:48) -axis fromvertical z (cid:48) x (cid:48) to vertical y (cid:48) z (cid:48) position, we get, using ∆ x (cid:48) =∆ y (cid:48) ≡ L and ∆ z (cid:48) ≡ H , the phase shift(∆Φ) Sagnac ≡ (∆ α ) y (cid:48) z (cid:48) − (∆ α ) z (cid:48) x (cid:48) = + γmc (cid:126) v LHR (cid:26) − vc (cid:18) ωRc + ar s R (cid:19) sin θ − ω R c sin (2 θ ) + v c a R (cid:0) θ + sin (2 θ ) (cid:1)(cid:27) = − γ mωA (cid:126) + (terms in a, a , ω ) . (46)We see, similarly to the COW case, that the magnitudeof the Sagnac effect is the same as obtained in the simplederivation, corrected by γ , and modified by considerablysmaller terms. VI. SUMMARY
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