Variational Principles for Constrained Electromagnetic Field and Papapetrou Equation
aa r X i v : . [ g r- q c ] J un Variational Principles for ConstrainedElectromagnetic Field and PapapetrouEquation
Muminov A.T.Ulug-Beg Astronomy Institute, Astronomicheskaya 33, Tashkent 100052, [email protected]
Abstract
In our previous article [4] an approach to derive Papapetrou equations forconstrained electromagnetic field was demonstrated by use of field variationalprinciples. The aim of current work is to present more universal techniqueof deduction of the equations which could be applied to another types ofnon-scalar fields. It is based on Noether theorem formulated in terms ofCartan’ formalism of orthonormal frames. Under infinitesimal coordinatetransformation the one leads to equation which includes volume force of spin-gravitational interaction. Papapetrou equation for vector of propagation ofthe wave is derived on base of the equation. Such manner of deductionallows to formulate more accurately the constraints and clarify equations forthe potential and for spin.
Complete description of motion of a non scalar wave in gravitational fieldis given by its covariant field equations. Under quasi-classical considera-tion when wave length is sufficiently less than typical scales of observations,propagation of the wave is substituted by motion of a particle. A. Papa-petrou in early 50s showed that spin-gravitational interaction changes formof the trajectory of a particle with spin. Deduction equations of motion ofthe quantum particle with account of spin-gravitational interaction can beprovided by construction its classical relativistic Lagrangian. This problem1as chained a great interest of researchers for last decades. Nevertheless asatisfactory description was not obtained, see [1, 3].Evident progress in theoretical treatment of the problem for photons wasachieved in series of our recent articles [2, 3, 4, 5]. Particularly in thework [4] an approach to deduction Papapetrou equation for photon basedon field variational principle was demonstrated. Essence of that approachwas attachment the field of potential to congruence of 0-curves which spec-ify propagation of the wave. The congruence and the field were describedby frame { ~n ± , ~n , } which is orthonormal in sense that < ~n + , ~n − > = 1 ,< ~n α , ~n α > = − , α = 1 , ~n − is tangent to the 0-curves everywhere and vectors ~n , constitutepolarization tangent subspace [4]. Integrand of the action was presented asquadric form over derivatives of the potential. Under the constraints imposedthe one was reduced to form similar to action of classic particle with an ad-ditional term describing spin-gravitational interaction. Next it was shownthat varying this action under infinitesimal dragging changing shape of the0-curves leads to Papapetrou equation for shape of the 0-curves. Variationcomponents of the potential attached to the curves reduced equation for po-tential to D − A α = 0. That in turn led to equation for spin: D − S αβ = 0 , see[4]. In this article we are returning to consideration of the problem pursu-ing the aim to create more formal and universal approach to deduct quasi-classical equations for motion of photon under account of spin-gravitationalinteraction. At the beginning we explain technic of variation in framework oftheory of exterior differential forms. Next we develop and improve methoddemonstrated in paper [4]. In particular, conception that elements of the or-thonormal frame are variables of Lagrangian describing congruence of curvesof propagation of the wave fruited one of main ideas of the current work.That is to consider elements of the frame as variables of gravitational field.Then to apply infinitesimal coordinate transformations which transform or-thonormal frames, connection coefficients, potential of electromagnetic fieldbut leave invariant gravitational field. As usual Noether theorem gives differ-ential equations for the field observables like stress-energy tensor (SET) andcurrent of spin. However, this time the equations do not exhibit covariantconservation law for the SET, but express equation of dynamic of electro-magnetic wave. Left hand side (LHS) of the one is usual covariant divergenceof the SET, but in the right hand side (RHS) instead zero a term interpretedas volume force of spin gravitational interaction arises. It happens due to2ector character of electromagnetic field.Then we develop our quasi-classical approach. It is based on assumptionof existance a class of locally monochromatic (LM) electromagnetic waveswhich admit geometric optical description. It means that such the wavepropagates along congruency of worldlines satisfying an equation whose LHScoincides with equation of geodesic line in a canonical comoving frame [4].The equation should be derived from the equation of dynamic. Conditionswhich reduce the one to equation for shape of worldlines specify the class ofLM electromagnetic waves. Analysis LHS of the equation of dynamic givesconditions as follows. T ++ should be only non vanishing component of SET ofthe wave and equation of continuity for probability current had to be satisfied.In turn RHS of the equation of dynamic should include characteristics of theparticle but not the ones of the field. It suggests that current of spin canbe expressed via tensor of spin and vector of propagation: S ab ( c ) = δ + c S ab , besides S only non zero component of the tensor of spin due to definitehelicity of photon. By other words we demand that LM electromagneticwave should be transversal wave. Such the requirements are provided bytransversality of the potential and choice of suitable form of the Lagrangian.Analysis of form of the SET and restrictions put onto it gives constraintswhich should be imposed to the potential of the field. The constraints coin-cide with the ones were imposed in [4]. Condition of continuity of probabilitycurrent finds its application under deduction of equation for components ofthe potential. Now D − A α = 0 but proportional to A α , α = 1 , ∗ ν + componentof current of spin equation for spin leaves the same. Action of electromagnetic field in curved space-time usually expressed byintegral of Lagrangian density L multiplied to 4-form ǫ of unit volume. Foraims of our studies it is convenient to express the integrand as whole 4-formΛ which we call 4-form of Lagrangian: A = Z L ǫ = Z Λ , ǫ = √− g dx ∧ . . . ∧ dx = ν ∧ . . . ∧ ν , (1)3here { x i } some coordinates in considered domain of space-time and { ν a } is a field of orthonormal frames dual to { ~n b } . In general the 4-form ofLagrangian: Λ = Λ( A b , DA c , ν a ) , (2)depend on components of field potential A b ( x i ), their derivatives and ele-ments of orthonormal frame { ν b } which describes gravitational field. Besides,it is convenient to represent derivatives of the potential via covariant exteriorderivatives (CED): DA b = ( D a A b ) ν a = dA b − ω cb · A c , (3)where symbol d stands for exterior derivative and ω cb · is 1-form of Cartan’rotation coefficients. Notion of CED and rules of operating with it shouldbe briefly observed. We define CED of any exterior differential form with(exterior) tensor indexes as follows: D (cid:16) T ...b......a... (cid:17) = d T ...b......a... + . . . − ω ca · ∧ T ...b......c... + . . . − ω b · c ∧ T ...c......a... + . . . . Usual Leibnitz rules can be easily generalized for the CED under account ofantisymmetry of exterior product of the differential forms. We also defineCED of vector:
D~n a = ~n b ⊗ ω ba · ; D~ε = ~n a ⊗ D b ( ε a ) ν b , where symbol ⊗ stands for tensor product of elements of tangent and cotan-gent spaces. Thanks to definitions it is seen that rules as follows are valid: ~n a ⊗ D ( T a ) = D ( ~n a ) ∧ T a + ~n a ⊗ dT a . (4)In terms of 4-form of the Lagrangian the field equations have a form asfollows: ∂ Λ ∂ A b − D ∂ Λ ∂ DA b ! = 0 , (5)where ∂ Λ /∂ A b is a partial derivative of 4-form of the Lagrangian over fieldvariable A b ( x ). An algebraic derivative ∂ Λ /∂ DA b of 4-form Λ over 1-form DA b [6] reduces it to 3-form. CED of the algebraic derivative restores degreeof the differential form up to 4th. Notion of algebraic derivative provides4ompact view of formulae especially under consideration variations of the4-form of Lagrangian like follows:Λ( . . . , DA + δDA, . . . ) − Λ( . . . , DA, . . . ) = δ DA Λ = δDA ∧ ∂ Λ ∂ DA ,δ ν Λ = T ab < δν a , ν b > ǫ = T ab δν a ∧ ∗ ν b = δν a ∧ ∂ Λ ∂ ν a ; (6)where δDA is variation of CED of potential of the electromagnetic field, δν a expresses variation of gravitational field variables, T ab is stress-energytensor (SET) of electromagnetic field, asterisk stands for Hodge conjugationof exterior differential forms. Under infinitesimal rotations of field of the orthonormal frames { ν b } : δν a = − ε ab · ν b , ε ab + ε ba = 0 , (7)variation of CED of the potential becomes: δDA c = DδA c + δω ab ∂ DA c ∂ ω ab . (8)Due to field equations, only part of variation of the Lagrangian contributingto variation of the action is: δω ab ∂ DA c ∂ ω ab ∧ ∂ Λ ∂ DA c = 12 δω ab ∂ DA c ∂ ω [ ab ] ∧ ∂ Λ ∂ DA c = 12 δω ab ∧ S ab · · c ∗ ν c , (9)where square brackets means antisymmetrization and S ab · · c ∗ ν c stands for cur-rent of spin of the field. Variation of the connection 1-form under the con-sidered rotations is: δω ab = − Dε ab , where Dε ab = dε ab − ω ca · ε cb − ω cb · ε ac is CED of tensor of rotations. Substitutingit into (9) represents variation of (1) as follows: δ A = Z d [ . . . ] + Z ε ab D [ S ab · · c ∗ ν c ] = 0 . This gives us covariant equation of continuity of 3-form of current of spin: D h S ab · · c ∗ ν c i = 0 . (10)5 Coordinate variation
Infinitesimal vector field ~ε drags coordinate hyper-surfaces { x i } onto { y i } andelements of orthonormal covector and vector frames { ~n a } , { ν b } onto draggedframes { ′ ~n a } , { ′ ν b } . Applying this variation to action of Gravitational fieldgives equation of covariant continuity for Einstein tensor in natural vectorframe and in orthonormal one: G ji · | j = D b ( G ba · ) = 0 . The same procedure can be performed for (1) as follows: Z δ Λ = Z Λ( ′ ~n , ′ ~n , ′ ~n , ′ ~n ) ′ ν ∧ . . . ′ ν − Z Λ( ~n , . . . ) ν ∧ . . . ν , (11)where Λ( ~n , . . . ) means value of 4-form Λ on 4-vector [ ~n ~n ~n ~n ]. Underthis definition variations of variables of the Lagrangian are given by Liederivatives over vector field ~ε . Use of Cartan’ first and second structureequations expresses variations of ν a and ω ba · as follows: δν a = dν a ( ~ε ) + dε a = Dε a − ω ab · ( ~ε ) ν b , (12) δω ba · = dω ba · ( ~ε ) + d [ ω ba · ( ~ε )] ± [ ω bc · ∧ ω ca · ]( ~ε ) = Ω ba · ( ~ε ) + D [ ω ba · ( ~ε )]; (13)where Ω ba · = 1 / R ba · cd ν c ∧ ν d is 2-form of curvature. This way variation of4-form of Lagrangian (2) becomes: δ A Λ + δ ν Λ + δ DA Λ = δA b ∂ Λ ∂ A b + δDA b ∧ ∂ Λ ∂ DA b + δν a ∧ ∂ Λ ∂ ν a Noting that variation of CED as before is given by (8) where this time δω isgiven by (13) we after applying (5) and (9) rewrite the above expression asfollows: 12 { Ω ab ( ~ε ) + D [ ω ab ( ~ε )] } ∧ S ab · · c ∗ ν c + D ( ε a ) ∧ T ab ∗ ν b , (14)where only first term of variation of ν a (12) contributes owing to symmetry ofthe SET. Substituting the above expression into (11) we integrate by parts.Referring to (10) we obtain: Z d [ . . . ] + Z n / ab ( ~ε ) ∧ S ab · · c ∗ ν c − ε a D h T ab ∗ ν b io = 0 .
6t brings us equation as follows: ~n c ⊗ D h T c · b ∗ ν b i = 1 / ~n c ⊗ R cdab · · S ab · · d ǫ. (15)The one in contrast with analogous equations for scalar fields is not equationof covariant conservation of the SET due to non zero RHS appeared. TheRHS expresses volume force of spin-gravitational interaction while the LHSexhibits CED of current of stress-energy-momentum. Because of this wewill call (15) as equation of dynamic of the electromagnetic wave in curvedspace-time. Accordingly to rule (4) LHS of (15) can be represented as: ~n c ⊗ D h T c · b ∗ ν b i = D ( ~n c ) ∧ T c · b ∗ ν b + ~n c ⊗ d h T c · b ∗ ν b i . (16)It should be marked that use of the above mentioned rule becomes possibleowing to fact that all indexes in the above expression are dummy. Such formof presentation of covariant divergence of the SET is convenient for transitionto quasi classical consideration. Conception of our quasi classical description is a deduction of equation forvector of propagation ~n − from the equation of dynamic (15). Besides wedemand that obtained equation becomes coincide with equation of geodesicline under switch off the spin. This means that covariant divergence of SETshould be reduced to covariant derivative of ~n − . Equality (16) shows thatthis result is provided by constraints as follows: T ++ is only non vanishing component of the SET, (17) d h T ++ ∗ ν + i = 0 . (18)It should be distinguished that (17,18) characterize subspace of locally monochro-matic (LM) electromagnetic waves and class of canonical comoving frames[4] which allows most convenient form of quasi-classical equations for thewaves. The same form of SET has monochromatic electromagnetic wave inflat space-time propagating along ~n − : T ab = δ a − δ b − ω | ~A | , ω is frequency of the wave. Thus (18) expresses continuity of currentof probability in case of flat space-time. Under this conditions LHS of (15)becomes: D [ ~n − ] ∧ T −· + ∗ ν + = T −· + D − [ ~n − ] ǫ. This way we reduce (15) to sought form: T ++ D − [ ~n − ] = ~n c R cd · ab S abd . (19)LHS of the equation coincides with geodesic equation for vector field ~n − , however in the RHS stands a term including contraction of the curvaturewith current of spin of the field which we call force of spin-gravitationalinteraction. Besides the force still is written in terms of field theory. To passto complete quasi classical description we need to determine field Lagrangian.Under study process of propagation of LM electromagnetic wave it is rel-evant admit Schweber Lagrangian [7] which provides adequate quasi-classicaldescription of LM wave: L = 1 / D a A c D b A c < ν a , ν b > . (20)In terms of orthonormal frames we write 4-form of the Lagrangian as follows:Λ = 1 / DA c ( ~n a ) { DA c ∧ ∗ ν a } . (21)Now let’s calculate variation of the action under variation of elements of theframe: δ ν Λ = 1 / DA c ( δ ~n a ) DA c ∧ ∗ ν a + 1 / DA c ( ~n a ) DA c ∧ δ ∗ ν a . (22)It is evident that: δ ~n a = − δν b ( ~n a ) ~n b . So first term in (22) can be rewrittenas follows: − / < δν b , ν d > D b A c D d A c . During variation of the second term asterisk conjugation should be varied: δ ∗ ν a = 1 / ε a · bcd δ ( ν b ∧ ν c ∧ ν d ) = 1 / ε a · bcd δν b ∧ ν c ∧ ν d ⇒ ν e ∧ δ ∗ ν a = − / ε a · bcd ε ecd · · · f δν b ∧ ∗ ν f = ( η ae η bf − δ af δ eb ) < δν b , ν f > ǫ. This brings us: δ ν Λ = − (cid:16) D a A c D b A c − / D e A f D e A f η ab (cid:17) < δν a , ν b > ǫ
8y other words, accordingly to (6), we find: T ab = − (cid:16) D a A c D b A c − / D e A f D e A f η ab (cid:17) . (23)The next part of the variation contributing to RHS of (15) is variationover Cartan’ rotation 1-form. The one appears in expression of CED ofpotential (3). Evidently δ ω DA c = − δω dc · A d , so δ ω Λ = δ ω DA c ∧ ∂ Λ ∂ DA c = δω dc · ∧ A d D b A c ∗ ν b = 1 / δω cd ∧ A [ d D b A c ] ∗ ν b . Now equating this expression to (9) we obtain: S cdb = A [ d D b A c ] . (24)Since photons has definite helicity they are presented by electromagneticwaves with circular polarization [4]. Hence it is convenient to consider com-plex valued amplitudes of the potential which describe phase shift of circu-larly polarized waves. For this aim an ordinary procedure of redefinition theobservables being quadric forms of the potential components is used. Thatis A p A q → / A { p A q } , where curly brackets mean symmetrization over theindexes. After applying it to expressions for the SET and the current of spinwe obtain their form in complex amplitudes: T ab = − n D a ¯ A c D b A c + D a A c D b ¯ A c − η ab D c ¯ A d D c A d o , (25) S cdb = 12 (cid:16) ¯ A [ d D b A c ] + A [ d D b ¯ A c ] (cid:17) . (26)Now let us find constraints should be put to the potential. As we considerLM electromagnetic wave we expect that potential of the wave has a structureas follows: ~A = ~a e iφ , D + φ = ω, D − φ = 0 , D + ~a = 0 (27) ⇒ D + A α = iωA α . (28)Due to (25) it is seen that condition of dominance of T ++ demands vanishingderivatives of the potential along polarization vectors: D α ~A = 0 , α = 1 , . (29)9anishing the derivative along direction of propagation ~n − makes the con-dition surely satisfied, however it is not only possibility as it will be shownlater. Anyhow value of T ++ can be calculated now due to fact that η ++ = 0.Under constraints imposed it is reduced to the form of T ++ of monochromaticwave in flat space-time: T ++ = − ω < ¯ ~A, ~A > = ω | ~A | (30)Quasi classical picture assumes that spin of photon moves along field of vec-tors ~n − and has only S component due to fact of definite helicity of photon.Constraints (29,28) provide validity of the first condition which can be writ-ten as S cdb = S cd η b − , where S cd is tensor of spin. The second condition issupplied by additional constraints: A ± = 0 . (31) Under varying Lagrangian (20) over components of potential we obtain re-duced form of Maxwell equations: D h D b A c ∗ ν b i = 0 . Let’s consider equations for nonzero components of the potential. After im-posing constraints we obtain: D { + (cid:16) D −} A α (cid:17) ǫ + X c = ± D c A α d ∗ ν c = 0 . (32)As usual under quasi classical (geometric optical) consideration we assumethat values of second derivatives of amplitudes of the potential should beneglected: ω − | D a ( D b a c ) | = ω − O (cid:18) | ~a | − h D − | ~a | i (cid:19) = o (cid:16) | ~a | − D − | ~a | (cid:17) . It reduces (32) to:[2 iωD − + i ( D − ω )] A α ǫ + iω A α d ∗ ν + + ( D − A α ) d ∗ ν − = 0 . (33)10omplex conjugation of (33) gives equation for complex amplitude. Let’sconsider contraction of (33) with ¯ A α and contraction of the one complexconjugated with A α . Difference between them gives: i (cid:20) ωD − < ¯ ~A, ~A > +( D − ω ) < ¯ ~A, ~A > (cid:21) ǫ + iω < ¯ ~A, ~A > d ∗ ν + ++1 / h ¯ A α D − A α − A α D − ¯ A α i d ∗ ν − = 0 . Due to structure of potential and constraints we daresay that last term inthe above equation is zero. This manner we rewrite the equation as follows: ω − d [ ω < ¯ ~A, ~A > ∗ ν + ] − ( D − ω ) < ¯ ~A, ~A > = 0 . But owing to (30) and (18) first term in the equation is zero. It gives D − ω =0. By other words value of derivative of the frequency in chosen frame alongvector ~n − is vanishing in quasi classical approximation: D − ω = O " D − < ¯ ~a, ~a >< ¯ ~a, ~a > = o ω D − < ~a, ¯ ~a >< ~a, ¯ ~a > ! . (34)This result allows to rewrite (33) and (18) as follows: iω h D − A α ǫ + A α d ∗ ν + i + ( D − A α ) ∗ d ∗ ν − = 0 , (35) d (cid:20) < ~A, ¯ ~A > ∗ ν + (cid:21) = 0 . (36)Equating to zero factor at ω in (35) as it is used to do under geometricoptical approximation we obtain equation for potential components. In turn(36) expresses conservation of probability current of the wave and allows toexclude d ∗ ν + from the equation for the potential: D − A α = 1 / A α ∗ d ∗ ν + = A α D − ( ¯ A c A c )2 ¯ A c A c . (37)Equation (37) describes attenuation of the amplitude of potential caused bydivergence of worldlines of propagation of the wave.11 Explicit form of SET, current of spin andPapapetrou equation
Substituting (37) together with constraints into (25,26) allows us to obtainan explicit form of elements of SET: D a ¯ A b D a A b = D { + ¯ A b D −} A b = [ − iω ¯ A b A b / / A b iωA b ] D − ¯ A c A c ¯ A c A c = 0 ,T + − = − / D { + ¯ A c D −} A c = 0 , T −− = − D − ¯ A b D − A b ≪ T ++ . The calculations confirm assumptions about view of elements of SET of LMwave. Next step is to calculate elements of spin tensor. Thanks to constraintsall its elements with third index different from ” ± ” is surely zero. Butaccordingly to (26): S cd + = iω ¯ A [ d A c ] =: S cd ,S cd − = D − | ~A | | ~A | (cid:16) ¯ A [ d A c ] + A [ d ¯ A c ] (cid:17) = 0 , ⇒ S cdb = δ + b S cd . (38)It approves expected factorized form of current of spin. Now substituting(30) and (38) into (19) we obtain equation for vector ~n − being tangent toworldlines of propagation of the wave: ω | ~A | D − ~n − = 1 / ~n c R abc + S ab . (39)In fact the one coincides with Papapetrou equation for trajectory of photonobtained us in work [4], although combinatorial factor 1 / S ab in thearticles. Under practical calculations it is more convenient to introduce nor-malized tensor of spin [5]: σ ab = S ab ω | ~A | = ¯ A [ a A b ] i | ~A | . (40)As expected, the one is transporting parallel itself in polarization tangentsubspace: D − σ αβ = i D − ( ¯ A [ α ) A β ] + ¯ A [ α D − ( A β ] )¯ A c A c − ¯ A [ α A β ] D − ( ¯ A c A c )( ¯ A d A d ) == i D − ( ¯ A c A c )2( ¯ A d A d ) { ¯ A [ α A β ] + ¯ A [ α A β ] } − ¯ A [ α A β ] D − ( ¯ A c A c )( ¯ A d A d ) = 0 . D − σ αβ = 0 , ω D − ( ~n − ) = 1 / ~n c R cαβ · − σ αβ ; (41)where value of frequency ω leaves constant in canonical comoving frame alongthe trajectory with accuracy up to second order in accordance with [4]. It should be underlined that idea of this work is grounded on conception ofarticle [4]. Specifically, in the above cited paper we started by considerationgauge invariant Lagrangian L = 1 / D a A b D c A d < ν a ∧ ν b , ν c ∧ ν d > . Next weexpand scalar products as < ν a , ν [ c >< ν b , ν d ] > and put the constraints. Itwas guessed that only terms < ν a , ν c >< ν b , ν d > would contribute to variationof the action due to fact that they yields physically interpretable expressions.In present work that assumption finds its expression in choice of form ofLagrangian. From another point of view selection of Schweber Lagrangiantogether with the constraints serves transversality of LM electromagneticwave.We also note that variation of omitted scalar product in Schweber La-grangian D a A c D b A c → D a A c D b A d < ν c , ν d > may contribute only to theSET. Calculations with account of the constraints and symmetrization overcomplex amplitudes analogous to the ones performed in section 7 shows thatthe contribution vanishes.Equation for probability current (36) can be rewritten as: D − < ~A, ¯ ~A > + < ~A, ¯ ~A > Div ~n − = 0 , where Div stands for covariant divergence of the vector. Under our ap-proximation we daresay that both terms in the equation are small but notvanishing. But in our former work [4] it was implicitly assumed that (36) issatisfied by vanishing of both of its terms. Hence equation for amplitudes ofthe potential obtained us earlier is particular case of (37). The last is realizedwhen worldlines of propagation of the wave are parallel themselves locally,as it was demanded in the previous article.13 cknowledgment
Author thanks to Professor Z. Ya. Turakulov who involved the author to thestudies and provided the work by his help.
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