We present an action principle formulation for the study of motion of an
extended body in General Relativity in the limit of weak gravitational field.
This gives the classical equations of motion for multipole moments of arbitrary
order coupling to the gravitational field. In particular, a new force due to
the octupole moment is obtained. The action also yields the gravitationally
induced phase shifts in quantum interference experiments due to the coupling of
all multipole moments.Comment: Revised version derives Octupole moment force. Some clarifications
  and a reference added. To appear in Phys. Rev. 

A. Einstein

A. Papapetrou

A.P. Balachandran

A.W. Overhauser

B. Brezger

D.V. Ahluwalia

E.W. Hagley

I. Bailey

J. Anandan

J. Baudon

J. Madore

J.E. Simsarian

Jeeva Anandan

K. Yee

M. Mathisson

M.S. Chapman

Naresh Dadhich

Parampreet Singh

R. Colella

T. Damour

W.G. Dixon

Y. Aharanov

English

arXiv

Crossref

Action principle formulation for the motion of extended bodies in general relativity

We present an action principle formulation for the study of the motion or an extended body in general relativity in the limit of a weak gravitational field. This gives the classical equations of motion for multipole moments of arbitrary order coupling to the gravitational field. In particular, a new force due to the octupole moment is obtained. The action also yields the gravitationally induced phase shifts in quantum interference experiments due to the coupling of all multipole moments. ©2003 The American Physical Society

Anandan, Jeeva

Dadhich, Naresh

Singh, Parampreet

LSU Scholarly Repository (Louisiana State Univ.)

Louisiana State University LSU Scholarly Repository Faculty Publications Department of Physics & Astronomy 1-1-2003 Action principle formulation for the motion of extended bodies in general relativity Jeeva Anandan University of South Carolina Naresh Dadhich Inter-University Centre for Astronomy and Astrophysics India Parampreet Singh Inter-University Centre for Astronomy and Astrophysics India Follow this and additional works at: https://repository.lsu.edu/physics_astronomy_pubs Recommended Citation Anandan, J., Dadhich, N., & Singh, P. (2003). Action principle formulation for the motion of extended bodies in general relativity. Physical Review D, 68 (12) https://doi.org/10.1103/PhysRevD.68.124014 This Article is brought to you for free and open access by the Department of Physics & Astronomy at LSU Scholarly Repository. It has been accepted for inclusion in Faculty Publications by an authorized administrator of LSU Scholarly Repository. For more information, please contact ir@lsu.edu. arXiv:gr-qc/0212130v3  6 Oct 2003Action principle formulation for motion of extended bodies inGeneral RelativityJeeva Anandan⋆ a, Naresh Dadhich† b, Parampreet Singh† c⋆ Department of Physics and Astronomy, University of South Carolina,Columbia, SC 29208, USA.† Inter-University Centre for Astronomy and Astrophysics,Post Bag 4, Ganeshkhind, Pune 411 007, India.AbstractWe present an action principle formulation for the study of motion of anextended body in General Relativity in the limit of weak gravitational field.This gives the classical equations of motion for multipole moments of arbitraryorder coupling to the gravitational field. In particular, a new force due tothe octupole moment is obtained. The action also yields the gravitationallyinduced phase shifts in quantum interference experiments due to the couplingof all multipole moments.PACS numbers: 04.20.Cv, 04.25.-g, 04.20.FyTypeset using REVTEX1The study of motion of extended bodies possessing multipole moments in the gravita-tional field has a long history [1]. The starting point was the Einstein - Infeld - Hoffman[2] derivation of the geodesic equation for a point test particle from the gravitational fieldequation and the conservation law for stress energy tensor. The test particle approxima-tion breaks down if body’s extension in space is non negligible compared to the radius ofcurvature of the background field and secondly when the back reaction due to the body onthe background field is non ignorable. In this letter, we shall be concerned with the formeraspect. This is particularly motivated by the fact that astrophysical bodies like planets andstars are extended and should in a realistic analysis be treated as such. The interaction ofcovariant generalization of Newtonian multipole moments with the gravitational field willbe given by their coupling to Riemann curvature and its derivatives. This would appear asmodification to the geodesic equation.The modification to the geodesic for a spinning body is given by the Mathisson-Papapetrou equation [3,4], which may be extended to a particle with intrinsic spin [5].Subsequent to the treatment of spinning bodies, various authors have obtained the correc-tions up to the covariant generalization of Newtonian quadrupole moment [6–8]. A com-prehensive study of the problem including comparison of various approaches and results iscarried out by Dixon [8], but, to our knowledge, no one has obtained corrections to geodesicequation arising due to coupling of covariantized higher order Newtonian multipole momentswith gravitational field. More importantly a procedure to derive equations of motion of ex-tended bodies, with arbitrary multipole moments, through an action principle has not beenobtained during the past 65 years in which the equations of motion in a gravitational fieldhave been studied [9]. In the absence of a general principle to obtain the action made up ofterms uniquely attributed to couplings of all multipole moments, such a task is very difficult.This is precisely what we wish to do in the following.While the action is used in classical physics only as a tool to obtain the equations ofmotion, the action is directly observable in quantum physics as the phase of the wavefunc-tion. Therefore, the phase shift produced by the coupling of multipole moments with the2gravitational field can in principle be measured (action giving an algorithm to calculatevarious multipole phase shifts). Neutron interferometry provided the first instance whereeffect of the earth’s gravitational field on the phase of the neutron wavefunction was ob-served [10]. Interesting gravitational analogs [11] of the topological Aharanov-Bohm [12]and Aharanov-Casher [13,14] phase shifts have also been proposed.In this paper we present a formalism which yields in a simple and elegant way thecorrections to the geodesic equation up to all multipoles of the extended body. The equationsof motion for multipoles simply follow from variation of our action. As a demonstration of ourformalism we obtain for the first time corrections to the geodesic equation till the covariantgeneralization of Newtonian octupole moment. Moreover, our action gives the quantumphase shifts in interferometry due to the coupling of all multiple moments to the gravitationalfield. Our formulation of action principle for extended body may facilitate and promptfurther investigations in the important and emerging area of interface between quantum andgravitational realms [15]. Particularly, the ongoing experiments in atomic [16], molecular [17]and Bose-Einstein condensate interferometry [18] hold promise for experimentally testing thenew gravitational phase shifts that will be obtained in the present letter.We envision an extended rigid body as a thin world tube in spacetime and its thicknessis small compared to the scale over which curvature varies. We would further assume thatthere are no external or internal forces acting on this body apart from the gravitational fieldin which it is propagating. In the thin world tube we choose a reference world-line (zµ)having 4-velocity uµ = dzµ/ds = (1, 0, 0, 0) and define multipole moments with respect to iton a spacelike hypersurface. The multipole moments of order 2n are defined as [8]tκ1...κnµν =∫δxκ1 ...δxκn√−g T µν wα dΣα (1)where δxµ = xµ − zµ, T µν is the energy-momentum tensor, and the integration is over thespacelike hypersurface identified by the unit normal vector field wα. The above multipole mo-ments are defined in a class of coordinate systems that are related by linear transformationsin order that the expression (1) is covariant. But once they are defined this way, tκ1...κnµν can3be transformed to any arbitrary coordinate system as a tensor. All the relativistic equationsin the present letter are covariant with respect to the above linear transformations, if notwith respect to general coordinate transformations.Since what we do must be consistent with the Newtonian theory in the appropriatelimit, we shall now establish the relation between the covariant multipole moments of order2n and the covariant generalizations of the anti-symmetric spin tensor Sij, the symmetricquadrupole moment tensor I ij and the symmetric octupole moment tensor Oijk in Newtoniangravity. We Write the Newtonian potential energy U of the body with mass density ρ(x) interms of the potential φ(x) expanded in a Taylor series around the central world-line,U =∫ρ(x)φ(x) d3x = mφ(z) + di ∂iφ|z +12I ij ∂i ∂j φ|z +16Oijk ∂i ∂j ∂k φ|z + ... (2)where the mass m =∫ρ(x) d3x, the dipole moment di =∫ρ(x) δxi d3x, the quadrupolemoment I ij =∫ρ(x) δxi δxjd3x and the octupole moment Oijk =∫ρ(x) δxi δxj δxkd3x, withδxi = xi− zi. In view of eq.(1), we identify ti00 = di, tij00 = I ij , tijk00 = Oijk. The Spintensor(orbital angular momentum) in the Newtonian limit is defined as Sij = 2∫ρ δx[ivj]d3xwhere vi = dxi/dt. The Spin tensor then satisfies dSij/dt = 2 p[iuj]−2 ∫ ρ δx[i∂j]φd3x, whereui = dzi/dt and the momentum pi =∫ρ(x) vid3x. Using Taylor expansion of the potential,and choosing zi to be the center of mass so that di = 0, the spin propagation equation upto octupole term becomesddtSij = 2 p[iuj] − 2 Ik[i∂j]∂kφ|z −Okr[i∂j] ∂k∂rφ|z (3)The covariantization of this spin tensor leads toSµν = tµν0 − tνµ0. (4)In the weak field limit, the metric is gµν = ηµν + hµν with hµν ≪ 1. In the Newtonian limitg00 = 1+ 2φ which implies φ = h00/2. Thus the dipole term in eq.(2) leads to the covariantform tµαβ hαβ,µ|z, where tµαβ includes the spin tensor. We shall now choose the referenceworld line to be the center of mass so that di = 0. We consider only matter distributionsfor which [19]4tµνα = Sµ(νuα), (5)whereSµν satisfies Sµνuν = 0.The quadrupole term leads to ∂i ∂jφ|z I ij = 12 ∂i ∂jh00|zI ij = −R0i0j |z I ij. We thus have∂i ∂jh00|z tij00 = −2R0i0j |z I ij which covariantizes tohαβ,µν |z tµναβ = −2Rαµβν |z Iµν uα uβ (6)where the covariant quadrupole tensor, Iµν = Iνµ. Similarly, the octupole term leads tohαβ,µνσ|z tµνσαβ = −2Rαµβν,σ|z Oµνσ uα uβ (7)where Oµνσ is the fully symmetric covariant octupole tensor. Eqs.(5), (6) and (7) are the keyrelations which would unambiguously provide the connection between covariant multipolemoments with their Newtonian analogs.In the Newtonian approximation the phase shift in quantum mechanical interference dueto the gravitational field isΦ = −1h̄∫Udt = −1h̄(∫mφdt+∫di ∂iφdt+∫12I ij ∂i ∂j φdt+∫16Oijk ∂i ∂j ∂k φdt+ ...)(8)where U is given by the expansion in using eq.(2). The first term of eq.(8) correspondsto the phase shift observed in the COW experiment [10], and the subsequent dipole andquadrupole terms are corrections to it. The higher order multipole contributions to thephase shift may also be obtained from this expansion. In General Relativity this phase shiftis obtained by letting the path ordered operator resulting from the covariant generalizationof this Newtonian phase shift act on the initial wavefunction. Using eqs.(2, 5, 6, 7) andnoting that in the linear field limit ωab βSba = hαβ,µSαµ, where ωab β are Ricci rotationcoefficients,this path ordered operator is given byg = Pexp[− ih̄∫(− m+ 12ωabβSab uβ − 12RαµβνIαβ uµuν − 16∫Rαµβν;ρ Oαβρ uµ uν + ...)ds].(9)5In the special case of Iαβ and Oijk being zero, this result is in agreement with the gravita-tional phase shift for intrinsic spin [5].We shall now obtain this expression in the weak field limit of General Relativity startingfrom the action principle. Choose a coordinate system such that along the reference world-line gµν = ηµν . We would here like to recall that the extended body under considerationis rigid and subject to only external gravitational force and no other forces (external orinternal). In that case the monopole moment corresponds to the mass (m) of the body withwhole matter concentrated on the reference world-line and ignoring the back reaction on thebackground gravitational field we write m = pα uα. The higher order multipole momentsare defined by considering the metric perturbations as one moves away from the referenceworld-line zµ. For simplicity we restrict perturbations of the metric to the first order [20],and later covariantize the equation of motion by changing ordinary derivative to covariantderivative. The actionS =∫ √−g L d4x =∫ √−g L|gµν=ηµν d4x+(∫ √−gL d4x−∫ √−gL|gµν=ηµν d4x). (10)The first term is the kinetic energy term,∫ √−g L|gµν=ηµν d4x = −∫pα uα ds. So,S = −∫pα uα ds+∫δgµνδδgµν(√−gL) d4x+ ... = −∫pα uα ds+12∫δgµν√−g T µν d4x+ ...(11)Note δgµν = gµν − ηµν = hµν and hence it can be written as δgµν = hµν,σ|z δxσ +12hµν,σρ|z δxσ δxρ + ...where we have hµν(z) = 0. In the present weak field limit, we ne-glect all terms that are quadratic or higher order in metric perturbations. Substituting forδgµν in eq.(11) and using eqs.(1,5,6,7), we finally obtainS = −∫pα uα ds+12∫hαβ,µ Sµα uβ ds− 12∫Rαµβν Iαβ uµ uν ds− 16∫Rαµβν,ρ Oαβρ uµ uν ds(12)up to octupole (all derivatives are evaluated on zµ). In the linear field limit, since hαβ,µ Sµα =ωabβSab, hence the accumulation of infinitesimal phases arising from eq.(12) is the same aswhat is obtained from the path ordered operator eq.(9) in this limit.6We now obtain the equation of motion by extremizing the action (δS = 0) and requiringthat coordinate variations vanish at end points of the path. This leads to the equation ofmotiondpσds= Rσλµν(12uλ Sµν +dds(Iλµ uν))+12hνσ,µddsSµν−Rσλµν,ρ uµ Iν(ρ uλ) +13Rαµβσ,νρ uαOµρ(ν uβ) − 13Rαµσν,ρdds(Oµνρ uα) (13)In a coordinate system in which hαβ,µ = 0, (13) givesDpσDs= Rσλµν(12uλ Sµν +DDs(Iλµ uν))− uµ Iν(ρ uλ)∇ρRσλµν+13∇ν∇ρRαµβσ uαOµρ(ν uβ) −13∇ρ RαµσνDDs(Oµνρ uα) . (14)Eq.(14) is generally covariant and thus is valid in every coordinate system. Since the dipolemoment couples to Riemann curvature, it is expected that quadrupole should couple toits first derivative and octupole to its second derivative. However there is a coupling ofquadrupole with Riemann curvature in the second term and a coupling of octupole withfirst derivative of Riemann curvature in the last term which suggests that pα should besuitably redefined [21] as, p∗σ = pσ − Rσλµν Iλµ uν − 13∇ρRσµναOµνρuα, which would thenyield the expected formDp∗σDs=12Rσλµν uλ Sµν +12∇σRαµβν u[µIα][βuν] +16∇ρ∇σ Rαµβν u[αOµ]ρ[νuβ] . (15)This is the equation of motion for a body possessing dipole, quadrupole and octupole mo-ments in a gravitational field. The spin propagation equation, eq.(3), can also be covariantlygeneralized toDDsSαβ = 2 p[α uβ] − 2R[αµνσ u[µIβ]][νuσ] −R[αµνσ;ρ u[µOβ]]ρ[ν uσ] (16)which on the redefinition of momentum vector form pα to p∗αmodifies as follows:DDsSαβ = 2 p∗[α uβ] + 2(Rαµνσ u[βIµ][νuσ] − Rβµνσ u[αIµ][νuσ])−2R[αµνσ u[µIβ]][νuσ] −53R[αµνσ;ρ u[µOβ]]ρ[ν uσ] . (17)7To simplify the notation we would further define Jµαβν := −3 u[µIα][βuν] and Gµαβσν :=−u[µOα]β[σuν] and then finally obtainDp∗σDs=12Rσλµν uλ Sµν +16Jµαβν ∇σRµαβν +16∇ρ∇σ Rαµβν Gαµρνβ . (18)DDsSαβ = 2 p∗[α uβ] − 43R[αµνσ Jβ]µνσ − 53R[αµνσ;ρ Gβ]µρνσ . (19)Thus we have obtained for the first time the correction to propagation equations till thecoupling of covariant generalization of Newtonian octupole moment with the backgroundgravitational field. These equations agree with the earlier results obtained till quadrupole[8] but the force due to the octupole moment is a new result. Our procedure can be easilygeneralized easily extended to obtain further corrections from due to all higher multipolemoments.This is a very simple and elegant method of deriving the equation of motion for anextended body incorporating coupling of multipole moments of arbitrary order with thegravitational field. And above all we have found the action for such a body which we havealso used to compute the gravitational phase shifts in its quantum wavefunction due to themultipoles. We hope that the new gravitationally induced phase shifts would be measuredin future interferometry experiments based on atomic, molecular and Bose-Einstein conden-sates [16–18]. Our novel algorithm also yields modifications to the geodesic equation which,with present day high precision astronomical observations yielding multipole moments ofplanetary or stellar bodies,can be applied to obtain aberrations in their orbits. It would beinteresting to relax the other aspect of test body character; i.e. non namely ignorability ofback reaction, and then study the motion in full generality.Acknowledgements: We thank J. Ehlers and C. W. Misner for informative discussions.PS thanks Council for Scientific and Industrial Research for grant number: 2-34/98(ii)E.U-II.8REFERENCESa since then deceasedb nkd@iucaa.ernet.inc param@iucaa.ernet.in[1] W. G. Dixon, Proc. International school of Physics “Enrico Fermi” LXVII, pp.156(1979).[2] A. Einstein, L. Infeld, B. Hoffman, Ann. Math. 39, 65(1938).[3] M. Mathisson, Acta. Phys. Polon. 6, 167 (1937).[4] A. Papapetrou, Proc. R. Soc. Lond. A 209, 248 (1951).[5] J. Anandan, Il. Nuovo. Cim. A 53, 221 (1979).[6] A. H. Taub, Proc. Galileo IV Centenary Conference, Florence, pp.77 (1964).[7] J. Madore, Ann. Inst. Henri Poincare 11, 221 (1969).[8] W. G. Dixon, Phil. Trans. R. Soc. Lond. A 277, 59 (1974).[9] For a particle possessing intrinsic spin, however, a Lagrangian was proposed by A. P.Balachandran, G. Marmo, B. S. Skagerstam, A. Stern, Phys. Lett. B 89, 199 (1980).Lagrangian approach has also been used to study dynamics of polarized media by I.Bailey and W. Israel,Ann. Phys. 130, 188 (1980) and for motion of spinning particles inexternal electromagnetic and gravitational fields by K. Yee and M. Bander, Phys. Rev.D 48, 2797 (1993). But the present paper obtains an action principle for all multipoles,not just spin. The problem of multipole couplings has also been addressed using 1 PNformalism, T. Damour et al., Phys. Rev. D 47, 3124 (1993).[10] A. W. Overhauser, R. Colella, Phys. Rev. Lett. 33, 1237 (1974); R. Colella, A. W.Overhauser, S. A. Werner, Phys. Rev. Lett. 34, 1472 (1975).9[11] J. Anandan, Quantum Coherence and Reality, proceedings of the Conference on Funda-mental Aspects of Quantum Theory, Columbia, SC, Dec. 1992, eds. J. S. Anandan andJ. L. Safko (World Scientific, 1994), gr-qc/9504002; J. Anandan, Phys. Lett. A 195,284 (1994).[12] Y. Aharanov, D. Bohm, Phys. Rev. 115, 485 (1959).[13] Y. Aharanov, A. Casher, Phys. Rev. Lett. 53, 319 (1984).[14] J. Anandan, Phys. Rev. Lett. 48, 1660 (1982).[15] For a recent work in this field, see the Proceedings of First IUCAA Meeting on theinterface of gravitational and quantum realms,eds. D. V. Ahluwalia and N. K. Dadhich,Mod. Phys. Lett. A 17, 899 (2002).[16] J Baudon, R Mathevet, J Robert, J. Phys. B 32, R 173 (1999); P. R. Berman (ed.),Atom Interferometry, Academic Press, NY (1997).[17] M. S. Chapman et al., Phys. Rev. Lett. 74, 4783 (1995);B. Brezger et al., Phys. Rev.Lett. 88, 100404 (2002).[18] E. W. Hagley et al., Phys. Rev. Lett. 83, 3112 (1999);J. E. Simsarian et al., Phys. Rev.Lett. 85, 2040 (2000).[19] See eq.(8.31) in Ref. [8].[20] If the size of the body is L and the radius of curvature of the background spacetime is R,restricting to the first order would imply that we can go up to octupole. This is becausethe metric perturbation goes as L2/R2 while the expansion in multipole coupling goesas L/R. That means octupole would go as L3/R3 which would be fine. However, if onewishes to go to higher multipole expansion, one would have to include the second orderterm in the metric perturbation.[21] To compare his results with earlier work dealing till quadrupole correction, Dixon defined10p∗σ = pσ − Rσλµν Iλµ uν .11