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A gravitational analogue of the Aharonov-Bohm effect

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1981 J. Phys. A: Math. Gen. 14 2353

(http://iopscience.iop.org/0305-4470/14/9/030)

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J. Phys. A: M ath. Gen. 14 1981) 2353-2357. Printed in Grea t Britain

gravitational analogue of the Aharonov-Bohm effectL H Ford and Alexan der VilenkinPhysics Depa rtm ent, Tufts University, Medford, Massachusetts 02155, U SA

Received 26 January 1 981

Abstract. It is shown that particles constrained to m ove in a region where th e Riem anntensor va nishes may nonetheless e xhibit physical effects arising from non-zer o c urva ture ina region from which they are excluded. This is a gravitational analogue of the Aharonov-Bohm effect.

Th e A haro nov -Bo hm effect in electrodynamics can be interpr eted as an il lustration ofnon-locality in quantum theory. This effect, which was predicted theoretically byAharo nov and Bohm (1959) and confirmed experimental ly by Cham bers (1960 ), ariseswhen a cohe ren t bea m of electrons is directed a rou nd eith er side of a solenoid withinwhich a non-zero mag netic field is pr ese nt. If the electron b eams are allowed tointerfere, i t is found that th e phase difference de pen ds upo n the m agnetic flux passingthrough the surface bou nd ed by the beam p ath s. This is still tru e even if th e electronsare excluded from the region wh ere the m agnetic field is non-zero; hence th e inter -pre tatio n of non-locality. This effect can equally well be interp rete d as illustrating th eneed to couple the electron to the electromagnetic vector potential. Th e phaseshift ispropo rtional to the line integral of th e vector potential a rou nd the path of th e electronbeam which by Stokes theor em is equa l to the enclosed magnetic flux. T he vectorpotential in the region traversed by the electrons is pu re gaug e in the sen se that in a localneighbo urhoo d it is possible to find a gauge in which the vector potential is identicallyzero. How ever, the re do es not exist any gauge choice which mak es vanish every-where a long the path.T he qu estion natura lly arises as to whe the r an analogou s effect exists in the theory ofgravitation. T he metric g and Riem ann curvature tensor R play roles analogou s tothose of th e potentials and field strengths respectively in electromag netism. A gen uinegravitational field is associated with a non-va nishing Rie m an n tensor, wh ereas a metricwhose Riemann tensor vanishes may be transformed to the Minkowski metric by acoo rdina te transformation. How ever, the Aha ron ov-B ohm effect suggests that parti-cles constrained to move in a region whe re the R iem ann tensor vanishes may no nethe -less exhibit physical effects arising from no n-z ero curvature in a region fro m which theyar e excluded. It is the pu rpo se of this pap er to show tha t this is indeed th e case.T he Sagnac effect in general relativity has bee n investigated by A shtek ar andMagnon (1975),who point ou t that th e gravitational field of a rotat ing mass distributionproduces effects analogous to the Aharanov-Bohm effect. T heir work has beenextended by Anandan (1977). In the present paper, we shal l be concerned with asomewhat different gravitational analogue to the Aharonov-Bohm effect which canarise even for a static mass distribution. A particular case was recently considered byVilenkin (19 81) in the context of a string model.0305-4470 /81 /092353 +05 01.50 @ 19 81 T he Institute of Physics 23 53

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2354 L H Ford and A VilenkinLe t us consider a space-time associated with a tube-lik e distribution of m atte r. W eassum e the ex istence of two Killing vector fields, t which is time-like and z*which isspace-like and genera tes translations along the direction of th e tube. Thu s the matterdistribution is stationary and uniform along z*.We also assume that the space is

asymptotically flat in a direction perpendicular to z* (i.e. at large distances from thetube) . Let S den ote a 2-surface which is orthogo nal to both nd z p . Because of thetime and spatial translational symm etry, all such 2-surfaces are equivalent. W e ar eassuming that the Riemann tensor approaches zero as one moves in S in directions awayfrom the tube . Le t C be a closed curve in S which lies entirely in the asym ptotically flatregion bu t which encloses the tub e whe re the curv ature is non-zero. If one t ransports avector parallel to C arou nd t he closed curve, it will, in general, not retu rn t o itself bu twill undergo a rotation by an angle r which is expressible as the ar ea integral of theGaussian curvatu re over So, he subsurface of S enclosed by C (Stoker 196 9) :r = Is K d a .

This is one form of the Gauss--Bonnet theorem . Thus even though the curva ture mayvanish along C, the effects of non-zero curvature in the interior region are still felt.Geometrically, this m eans that S is asymptotically a conical surface rath er th an a plane.Physically, it mea ns tha t particles may b e gravitationally deflected by the tu be withoutever entering a region of no n-zero cu rvature.To understand this effect more explicitly, let us consider a static space-timedescribed by th e metricd s 2 = -D2(r,19 d t 2 + A 2 ( r , ) d r 2 + B 2 ( r , d B Z + E 2 r , d z 2 (2)

which is translationally invariant in the t and z directions. (This form of th e metriccontains m ore functions tha n ar e essential; for exam ple, by a suitabl e choice of r and 8one could require that A = B . ) The 2-surface S is a surface defined by t and z equalconstants and has the me tricd s i = A2(r, d r 2 +B 2 ( r , d e 2 . (3)

W e now assume that the coo rdinates can be chosen so that A =1and B - as r 0 andthat A 1and B br as r CO. Fu rth erm ore , for all values of r the points 8 and 6 +2.rrar e identified. Thu s, in a neigh bou rhoo d of th e origin, S is flat (i.e. free of conicalsingularities) a nd f or large r it becomes a cone.Th e Gaussian curva ture of S may be expressed in terms of A and B and their p artialderivatives:K = A-3B-3(A2A, ,J? ,e+B Z A , , B , ,A B 2 B , , - A 2 B A , e e )

= -A- [ ( ) ,e + ,I 4)If we integrate K over S , using th e a bov e asymp totic form s of A and B , we find that

5 )

where Jg = AB. This is equivalent to 1)with = 2 ~ ( l - ) . Th e surface S is, at large r,a co ne with conical angle A = r b . T he conical angle of a con e is he re defined as the an glesubtended at the apex so A = for a plane).

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A n analogue of the Aharonov-B ohm effect 2355Th e Gaussian curvature of S may be readily expressed in term s of th e Riem anntensor 4)REYPf the four-d ime nsion al space-time within which S is em bedd ed, Firstconsider a 3-surface which is orthogonal to t. The Riemann tensor (3 )Rjk lf isexpressible in terms of (4)REuPnd the extrinsic curvature Kii of by me ans of th e

Gauss-Codazzi equations. How ever, because t is a Killing vector, Z has zero extrinsiccurvature. This follows because (Misner et a1 1973)( 6 )..= -t. .1:l

and is symmetric Kji= Kii. Killings equation requires tha t t i i i= o r Kii= -Kii andhence Kii=O. In this case 3)Rjkl s obtained by projecting ( 4 ) R L pnto Z or incoordinates where Z is defined by t = constant,(7)3 )R i 4)RiI k l I k l .

Similarly, on e obtains tha t the R iemann tensor of the two-dimensional spac e S is8)2)Ri - 4)RfIkl - I k l .

Th e Gaussian curvature is on e half of th e scalar curva ture ( 2 ) R f S , and hence

where (2 )g j is the metric on S . This result, combined with equation (5), enables us to findthe conical angle in terms of the Rieman n tensor of th e four-dimensional space-time.In general it does not seem to be possible to express K in terms of the four-dimensiona l Ricci tenso r, and henc e of th e energy-m omentum tenso r of the source,TwYHowever, in the case that the gravitational field is sufficiently weak that thelinearised theory may be app lied, such an expression can be given. Let th e metric be

g,, = q h,, where q = diag(-1, 1 ,1 ,1and let

(10)

K y = 0 11)

OK = - 1 6 ? ~ T , , (12)

1h,, = h,, - 2qwuhwith h = h,. With t he gauge condition

the linearised field equations becom e

and the Riemann tensor is13)( 4 )RDLLL~uZ(hau,fiP hwP,au heu,DLp ap,,u).

(Our notation is that of Misner et a1 (1973) with G = c = 1.) s before, th e source isassumed to be independ ent of the x o and x 3 coordinates. Then14)K = (4)R1212= dh12,21+ h21,21- h22,11.- h11.22).

Using 11)and (12 ), this may be rew ritten asK = 8 ~ I + T22 - T )where T = T:. This inserted into (5) yields b in terms of the source. Of particular

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2356 L H F o rd a n d Vilenkininterest is the case where p = Toos the only non-zero c om ponen t of Twv hen we have

l - b = 4 7 r p d 2 X .ITh us fo r a dus t sourc e, the conical angle is given in term s of the m ass pe r unit length ofthe tube.Le t us now consider th e physical consequences of the fact that S s asymptotically acone rathe r than a plane. O ne consequence follows immediately from 1). If a spinningparticle travels around a closed curve in such a way that the spin vector undergoesparallel transport, the spin vector will be rotated by an angle a upon return to thestarting point. This will effect the interference properties of a coherent beam ofparticles such as neutro ns. (Fo r a discussion of exp eriments using the interferenceproperties of neutrons t o me asure gravitational and rotational effects, see An and an(1977) and G reen berg er and O verhau ser (1979 ).) N eutrons which traverse a region ofspa ce wh ere th e curv ature is identically zero a re nonetheless ca pab le of detecting th eeffects of curvature in other regions of space-time.Th er e is a seco nd type of in terfe renc e effect by me ans of which on e could dete ct theregion of non-zero curvature. Consider a circular loop of proper radius R which iscen tred a bo ut a region containing a so urc e, with locally flat spac e outside of the sou rce.If R is much larger th an the dimensions of the sou rce, then th e circumference of thecircle is 27rbR. If on e wer e to vary the characteristics of t he so urce in such a way that bchanges, but require that R remains fixed, the proper length of such a circular pathwould chan ge. He nce , a coh eren t beam of light or othe r particles which travels arou ndthis path could be m ad e to exhibit interfe renc e which reflects this varying path length.In th e case tha t th e space-time is identically flat (rathe r than asymptotically flat)outs ide of the s ourc e, on e can rela te the conical angle to th e deflection of a b eam ofparticles (Vilenkin 1981). The metric in the flat region can be written as

d s 2 = - d t 2 + d r 2 + b 2 r 2 d q 5 2 + d z 2 ( 1 7 )

d s 2 = - d t 2 + d r 2 + r 2 d 4 I 2 + d z 2 18)where 0< q5

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A n a na lo gu e of the Aha rono v-B ohm effect 2357Light beams propaga ting on diffe ren t sides of t he source can intersect, and thus th erelative deflection of two beams cannot, in general, be tra nsf orm ed away. Th e situationhere is analogous to th e electromagnetic Aharonov-Bohm effect, where the vectorpotential can be transform ed to zero in any region containing no closed loops enclosingthe solenoid.A difference between the electromag netic Aharonov-Bohm effect and the gravita-tional analogue discussed here is that the former is a quantum interference effect,whereas th e lat ter is classical i.e. ndependent of ti). This is attributable to th e fact tha tthe classical equations of motion in the electromagnetic case involve only the fieldstrength FFy o the re can be no classical deflection in a region w here the fields are zero.O n the oth er han d, the equation of m otion for a particle in a gravitational field, thegeodesic equation, involves the metric rather than t he Riemann tenso r; consequentlynon-local effects of curv ature can arise a t the classical level.Exam ples of space-times which are flat outsid e of th e sourc e include the vacuumstrings (Vilenkin 1981). It is necessary for th e source to exhibit eith er negative press ureor negative energy density, which do not occur in most familiar types of m atte r but d oarise in both classical and qu ant um field theories.

ReferencesAharonov Y and Bohm D 1959 Phys. Rev. 115 485Anandan J 1977 Phys. Rev . D 15 1448Ashtekar A and Magnon A 1975 J Math. Phys. 16 34Chambers R G 1960 Phys. Rev . Lett. 3Greenberger D M and Overhauser A W 1979 Re v . Mod. Phys. 51 43Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (San Francisco: Fre ema n) 5 21.5Stoker J J 1969 Differentia l Geometry (New York: Wiley Interscience) p 191Vilenkin 1981 Phys. Rev. D 23 852