PHYSICAL REVIEW LETTERS

VOLUME 77 11 NOVEMBER 1996 NUMBER 20

Asymptotic Conservation Laws in Classical Field Theory

Ian M. Anderson1 and Charles G. Torre2 1Department of Mathematics, Utah State University, Logan, Utah 84322-3900 2Department of Physics, Utah State University, Logan, Utah 84322-4415 (Received 24 June 1996) A new, general, field theoretic approach to the derivation of asymptotic conservation laws is pre- sented. In this approach asymptotic conservation laws are constructed directly from the field equations according to a universal prescription which does not rely upon the existence of Noether identities or any Lagrangian or Hamiltonian formalisms. The resulting general expressions of the conservation laws enjoy important invariance properties and synthesize all known asymptotic conservation laws, such as the Arnowitt-Deser-Misner energy in general relativity. [S0031-9007(96)01656-0]

PACS numbers: 03.50.–z, 04.20.Cv

In nonlinear gauge theories such as Yang-Mills theory The purpose of this Letter is to describe a new, general and general relativity, conserved quantities such as charge construction of asymptotic conservation laws for classical and energy-momentum are computed from the limiting field theories which provides a viable alternative to exist- values of two-dimensional surface integrals in asymptotic ing methods. Our construction is based upon a remark- regions. Such asymptotic conservation laws are most of- able new spacetime differential form C, which we derive ten derived by one of two, rather distinct, methods. One for any system of field equations. We show, by virtue method, applicable to gauge theories such as Yang-Mills of an identity involving the exterior derivative DC and and general relativity, relies upon the construction of iden- the linearized field equations, that C defines asymptotic tically conserved currents furnished by Noether’s theorem conservation laws (when they exist) for any field theory. [1–3] and subsequent extraction of a “superpotential” to Thus we are able to establish that asymptotic conserva- define a conserved surface integral. Unfortunately, there tion laws for field theories can be viewed as arising from is no general field-theoretic criterion to select the appro- (asymptotically) closed differential forms canonically as- priate current: For any field theory there are infinitely sociated to the field equations. We also show that C many currents that can be expressed as the divergence of possesses a number of attractive field theoretical prop- a skew-symmetric super-potential, modulo the field equa- erties. In particular, this differential form is constructed tions. This method of finding asymptotic conservation directly from the field equations according to a univer- laws is thus somewhat ad hoc. An alternative approach sal prescription which does not rely upon the existence of to finding asymptotic conservation laws in gauge theo- Noether identities or any Lagrangian or Hamiltonian for- ries is based upon the Hamiltonian formalism. In this malisms. The form enjoys important invariance properties approach, asymptotic conservation laws arise as surface and synthesizes all known asymptotic conservation laws; term contributions to symmetry generators [2,4–6]. The for example, it is easily seen to reduce to the Arnowitt- Hamiltonian approach to finding asymptotic conservation Deser-Misner (ADM) energy in general relativity. We laws lacks the ad hoc flavor of the superpotential formal- emphasize that in obtaining these results no a priori as- ism but is somewhat indirect: To construct asymptotic sumptions are made concerning the space of gauge sym- conservation laws using this method one must have the metries of the theory. The symmetries required for the Hamiltonian formalism well in hand and one must know existence of asymptotic conservation laws are automati- a priori the general form of the putative symmetry genera- cally derived as a consequence of our general formalism. tor in order to find the appropriate surface integral. Thus our formalism contains a number of distinct field

0031-9007͞96͞77(20)͞4109(5)$10.00 © 1996 The American Physical Society 4109 VOLUME 77, NUMBER 20 PHYSICAL REVIEW LETTERS 11NOVEMBER 1996 theoretical advantages. With regard to specific applica- and their formal linearization dhDB ෇ 0. Such linearized tions, we are able to extend existing results on asymptotic conservation laws play a pivotal role in the general theory conservation laws in a number of ways. For example, we of lower-degree conservation laws, as presented in [8] obtain covariant expressions for asymptotic conservation (see also [14]). In [8] it is shown that every closed laws which can be used to generalize the conservation p form v͓h͔, depending upon the fields wA and their laws in asymptotically flat general relativity and Yang- derivatives and linearly on the field variations hA and Mills theory to other asymptotic structures. We also their derivatives, can be cast into a universal normal form obtain a formula which generalizes the asymptotic conser- Cr͓h͔, which depends upon certain auxiliary fields r vation laws of the Einstein equations to arbitrary second- that are subjected to a set of algebraic and differential order metric theories. To illustrate this latter point, we constraints. This normal form, which forms the basis consider a class of “string-generated gravity models” [7]. for our construction of asymptotic conservation laws, is It follows immediately from our formalism that the stan- obtained as follows. Details, generalizations, and further dard ADM formulas still describe conservation of energy, examples of this construction will appear in [8]. momentum, and angular momentum for these theories, at To begin, it is helpful to write the formal linearization least in the asymptotically flat context. of the field equations (1) as To describe our results in more detail, let us fix ij A i A A an n-dimensional spacetime manifold M, with local dhDB ෇ sABh ,ij 1sABh ,i 1sABh . i coordinates x , i ෇ 0, 1, . . . , n 2 1, and let us label the We then define a linear lower-degree conservation law A totality of fields for our classical theory by w , A ෇ for the field equations (1) to be a p form v͓h͔, where 1, . . . , N. These fields are subject to a system of field p , n 2 1, of the type equations which we write as i i i ···i v͓h͔ ෇ M hA 1 M 1 hA 1 ··· 1 M 1 2 k hA , D ͑wA, wA , wA ͒ ෇ 0. (1) A A ,i1 A ,i1i2···ik B ,i ,ij (4) We have postulated that the field equations are of second order only for the sake of simplicity. The general theory which satisfies which we outline here is developed in [8] for field Dv͓h͔ ෇ rBd D 1rBi1 D d D 1 ··· equations of arbitrary order. To address the problem of h B i1 h B Bi1i2···ik finding asymptotic conservation laws for (1), we begin by 1r Di1i2···ik dhDB . (5) first broadening the usual notion of a local conservation i1i2···il In Eqs. (4) and (5) the coefficients M and rBi1i2···il law. We say that a conservation law of degree p for (1) A are spacetime p forms and ͑p 1 1͒ forms, respectively, is a spacetime p form which depend on the fields wA and their derivatives. A i A k1 kp Equation (5) is an identity in the field variations h and v ෇ vk1...kp ͓x , w ͔dx ^ ··· ^ dx , (2) their derivatives but is still subject to the field equations depending locally on the fields wA and their derivatives to DB ෇ 0. The next step is to derive equations for these some finite order, such that the exterior derivative multipliers r from the integrability condition D2v͓h͔ ෇ k k1 kp 0. It is a remarkable fact that the highest order multiplier Dv ෇ ͑Dkvk1...kp ͒dx ^ dx ^ ··· ^ dx , (3) rBi1i2···ik is thus constrained by purely algebraic condition vanishes on all solutions to the field equations. In (3), Dk jh B͑i1i2···ik l ͒ is the usual total derivative operator. When p ෇ n 2 1, r ^ dx sAB ෇ 0. (6) k k1 we may express (2) in the form v ෇ ´kk1···kn21 J dx ^ ··· ^ dxkn21 , in which case the vanishing of (3) coincides We call this fundamental equation the algebraic Spencer k equation for the linear conservation law v͓h͔ for the with the vanishing of the divergence DkJ of the current Jk . Accordingly, we shall say that the p-form conserva- field equations (1) [15]. For the field equations that one tion law (2) is an ordinary or classical conservation law typically considers, it is not too difficult to apply standard in the case p ෇ n 2 1 and a lower-degree conservation methods from tensor algebra to solve the Spencer equation law when p , n 2 1. (6) [8]. The solutions to (6) often allow one to simplify Recently ([8–13]) a number of methods have been the identity (5), by repeated “integration by parts,” to the developed for the systematic computation of all lower- reduced form degree conservation laws for field equations such as (1). B Dv͓h͔ ෇ r dhDB . (7) The central premise of this note is that the techniques introduced in [8] can be successfully adapted to the We remark that for Lagrangian theories, full knowledge analysis of asymptotic conservation laws. The principal of the gauge symmetries of the theory often allows one to ideas are as follows. If v is a lower-degree conservation pass directly to the reduced form (7). Bi i B B law for DB ෇ 0, then it is readily checked that the Define r ෇ dx ^ r and define mi to be the inte- A B i B variation dhv of v with respect to field variations h ෇ rior product mi ෇ ≠͞dx r . Assuming that the re- A dw is closed by virtue of the field equations DB ෇ 0 duced equation (7) holds, we then have the following

4110 VOLUME 77, NUMBER 20 PHYSICAL REVIEW LETTERS 11NOVEMBER 1996 complete characterization of all linear lower-degree con- To demonstrate the utility of our general theory of servation laws of degree p , n 2 1 [8]. asymptotic conservation laws, and to expose some of its Theorem (classification of linear lower degree conser- novel features, we consider the vacuum Einstein field A A A vation laws).—Let DB͑w , w ,i, w ,ij͒ ෇ 0 be a system equations. Here, as a consequence of (8a), the auxiliary of second-order field equations and suppose the linear p fields r are determined by a single vector field X and the form v͓h͔ satisfies the reduced equations (7). Then (i) normal form Cr͓h͔ becomes the conservation law multiplier rB satisfies B͑i jk͒ B͑i j͒ B͑i jk͒ 1 2 r sAB ෇ 0, r sAB 1 Dk r sAB ͔ ෇ 0, ti,rsuj CX͓h͔ ෇ ´ijhk 2 ͑᭞uhrs͒Xts Bi B͑i j͒ 16p " 3 r sAB 1 Dj͓r sAB͔ ෇ 0; (8a,b,c) 1 ti,rsuj (ii) the p form 1 hrs͑᭞uXt ͒s 3 # 1 A B i 2 B ij h k Cr͓h͔ ෇ h mi sAB 2 Dj͑mi sAB͒ 3 dx ^ dx , (10) q " q 1 1 #

2 A B ij where ᭞ is the covariant derivative defined by the 1 hj ͓mi sAB͔ , (9) ± rs,ijhk rs q 1 1 Chritoffel symbols of g and s ෇ ≠G ͞≠gij,hk͑g0͒ where q ෇ n 2 p, is closed by virtue of the equations is the symbol of the Einstein tensor. In expanded form, CX ͓h͔ becomes DB ෇ 0, dhDB ෇ 0, and (8); and (iii) every linear p form v͓h͔ satisfying (7) differs from Cr͓h͔ by an exact linear p 2 1 form. 1 li j 1 i j i lj ͑ ͒ C ͓h͔ ෇ ´ ͓h ᭞ X 2 h= X 2 X ᭞ h We remark that this theorem can be readily generalized X 32p ijhk l 2 l to the case where (5) holds, rather than (7). i jl j h k 1 X ͑᭞lh 2 ᭞ h͔͒dx ^ dx , (11) To use this theorem for computing asymptotic con- servation laws we proceed as follows. Suppose the ±ij where h ෇ g h , and indices are raised and lowered with M ij spacetime is non-compact and admits an asymptotic respect to the background metric. The differential form region which is diffeomorphic to R 3 C0, where C0 is n21 (11) recently appeared in [2] [see Eqs. (61), (63), and the complement of a compact set in R . Fix local (75)], where it was derived from the Noether identities t x1 xn21 coordinates ͑ , ,..., ͒ in the asymptotic region. and used in the study of black hole entropy. We consider a fixed solution w± A to the field equations and The spacetime form CX͓h͔ possesses the following then, given a second solution wA, we set hA ෇ wA 2w±A. ± desirable properties which reflect general features of our Let v͓w, h͔ be a spacetime ͑n 2 2͒ form depending formalism [8]. locally on the fields w± A and hA and their derivatives. (1) The form CX͓h͔ is closed by virtue of the Einstein We call v an asymptotic conservation law for the field ± ± equation for the background g, and the linearized Einstein equations DB ෇ 0 relative to the background w if, A equations for the variation h, provided that X is a Killing whenever h satisfies the appropriate asymptotic decay vector field of the background. By using the classification 1 2 2 2 n21 2 ! condition as r ෇ ͑x ͒ 1 ͑x ͒ 1 ··· 1 ͑x ͒ `, theorem above it can be shown that, modulo 2-forms the p form v͓w± , h͔ satisfies v ϳ O͑1͒ and Dv ϳ p which are exact by virtue of these equations, CX͓h͔ is the O͑1͞r͒. Under these conditions the limit lim 3 r!` only linear lower-degree conservation law for the vacuum ± v͓w, h͔, where S r,t is an ͑n 2 2͒-dimensional Einstein equations. S͑r,t0͒ ͑ ͒ sphere of radius r in the t t hypersurface, exists and (2) It is readily established that under the gauge trans- R ෇ 0 is independent of t0. We therefore can deduce from our formation hij ! hij 1D͑iYj͒the form CX͓h͔ changes by classification theorem that if the asymptotic boundary an exterior derivative of a canonically defined one-form. conditions are such that This implies that CX ͓h͔ is suitably “gauge invariant.” (i) Cr͓h͔ϳO͑1͒, (3) The form CX͓h͔ is constructed directly from ± (ii) the equations dhDB͑w͒ ෇ 0 hold asymptotically to the field equations in a covariant fashion and with B ± an order such that r dhDB͑w͒ϳO͑1͞r͒, and no reference made to the Bianchi identities or to any (iii) the conservation law multiplier equations (8) hold, Lagrangian or Hamiltonian. Indeed, most properties of ± not exactly and not for all field values, but only at w and CX ͓h͔ can be inferred directly from properties of the only to the appropriate asymptotic order, then the normal symbol of the Einstein tensor. form Cr͓h͔ will be an asymptotic conservation law for (4) If the vector field X is an asymptotic Killing ± ± the field equations DB ෇ 0 relative to the background vector field for g and if h ෇ g 2 g satisfies appropriate ± solution w. We note that the conditions (i), (ii), and (iii) decay conditions, then CX͓h͔ is always an asymptotic can be relaxed so long as the relevant integrals arising conservation law for the Einstein field equations. from the application of Stokes Theorem to (7) vanish (5) In terms of the ͑3 1 1͒ formalism, a lengthy but asymptotically. straightforward computation shows that the pullback v of

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CX ͓h͔ to a leaf of a foliation of spacetime by spacelike compute the asymptotic conservation laws for string gen- hypersurfaces becomes erated gravity in the asymptotically flat content from (10). Because the string-corrected symbol reduces to that of the 1 abcd Ќ Ќ v ෇ ͓G ͑X ᭞˜ bhcd 2 hcd᭞˜ bX ͒ Einstein equations on a flat background, it follows imme- 16p diately that the conservation laws (11), (12) still apply in b a a cd 1 2X dhpb 2 X hcdp the presence of the corrections to general relativity sug- a b b a 2 gested by string theory. 2 ᭞˜ b͑h X 2 h X ͔͒d Sa . (12) Ќ Ќ In summary, the theory of linearized conservation laws In this equation (i) XЌ and Xb are the normal and leads to a new, efficient, systematic, and covariant method tangential components of a Killing vector with respect for obtaining asymptotic conservation laws in field theory. a to the spacelike hypersurface and hЌ is the normal- We expect that this approach will prove useful for a tangential component of hij; (ii) the derivative operator wide range of field theories and asymptotic boundary ˜ ᭞b is compatible with the background metric gab induced conditions which have been heretofore unexplored and on the hypersurface; (iii) we have defined we anticipate that the spacetime form Cr͓h͔ will be a 1 valuable tool in the further study of conservation laws in Gabcd ෇ g1͞2͑gacgbd 1gadgbc 2 2gabgcd͒ ; field theory. 2 This research was supported, in part, by Grants a ac ac and (iv) pb ෇ gbcp , where p is the canonical field No. DMS94-03783 and No. PHY96-00616 from the momentum. With asymptotically flat or asymptotically National Science Foundation. Both authors gratefully anti-de Sitter boundary conditions the integral of (12) over acknowledge the hospitality of the Centre de Recherches the sphere at infinity coincides with the standard ADM- Mathématiques at the Université de Montréal where this type formulas ([4,6,16]) for the asymptotic conservation work was initiated. laws in this context Thus our construction is general enough to capture the usual conservation laws in general relativity. Moreover, CX provides a means of generalization of these conserva- tion laws to other asymptotic structures. In particular, the presence of the term Xah pcd in (12) coupling h to [1] J. N. Goldberg, in General Relativity and Gravitation: One cd ab Hundred Years after the Birth of Albert Einstein, edited by the canonical momentum of the background, which is dic- A. Held (Plenum, New York, 1980). tated by our derivation of v from a covariant spacetime [2] V. Iyer and R. M. Wald, Phys. Rev. D 50, 846–864 two-form CX͓h͔, has to our knowledge not appeared in (1994). previous ADM-type formulas. While this term does not [3] D. Bak, D. Cangemi, and R. Jackiw, Phys. Rev. D 50, contribute to the surface integrals at infinity for the 5173–5181 (1994). asymptotically flat and asymptotically anti-de Sitter [4] R. Beig and N. Ó. Murchadha, Ann. Phys. (N.Y.) 174, boundary conditions, we believe that its inclusion should 463–498 (1987). be necessary for other asymptotic conditions. [5] R. Benguria, P. Cordero, and C. Teitelboim, Nucl. Phys. Let us briefly describe other applications of our for- B 122, 61–99 (1977). malism. First of all, the classification theorem we have [6] T. Regge and C. Teitelboim, Ann. Phys. (N.Y.) 88, derived for linear lower-degree conservation laws is used 286–318 (1974). [7] D. Boulware and S. Deser, Phys. Rev. Lett. 55, in [8] to classify all conservation laws for the Einstein 2656–2660 (1985). equations, Yang-Mills equations, and similar equations. [8] I. M. Anderson and C. G. Torre, “Lower-Degree Conser- With regard to asymptotic conservation laws, in Yang- vation Laws in Field Theory” (to be published). Mills theory our classification theorem leads to the stan- [9] G. Barnich, F. Brandt, and M. Henneaux, Commun. Math. dard surface integral formulas in the asymptotically flat Phys. 174, 57–92 (1995); 174, 93–116 (1995). context [8]. Work is in progress to investigate Yang-Mills [10] R. L. Bryant and P. A. Griffiths, J. Am. Math. Soc. 8, conservation laws in the presence of other asymptotic con- 507–596 (1995). ditions. As another application, we consider the asymp- [11] T. Tsujishita, Differ. Geom. Appl. 1,3–34 (1991). totic conservation laws for any covariant metric theory [12] V. V. Zharinov, Math. USSR Sbornik 71, 319–329 (1992). described by second-order field equations. We can show [13] I. M. Anderson, in Proceedings of the AMS-IMS-SIAM that the normal form (10) generalizes without change Joint Summer Research Conference on Mathematical Aspects of Classical Field Theory, 1991, edited by to all such theories (provided, of course, that one uses M. Gotay, J. Marsden and V. Moncrief (American the symbol appropriate to the field equations of interest). Mathematical Society, Providence, 1992), pp. 51–73. To illustrate this point, we consider the string-generated [14] R. M. Wald, J. Math. Phys. 31, 2378–2385 (1993). gravity models, which are based upon the Lovelock [15] Historically, this equation first appeared in the mathemat- Lagrangian density [7]. Using the modified symbol of the ics literature in the mid 1950’s in the pioneering work of string-corrected Einstein equations, it is straightforward to D. Spencer on the theory of overdetermined systems of

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partial differential equations. But, to paraphase Bryant and to study lower-degree conservation laws in field theory. Griffiths [10](p. 578), one is lead inevitably to (6) and the [16] M. Henneaux and C. Teitelboim, Commun. Math. Phys. resulting theory of Spencer cohomology once one agrees 98, 391–424 (1985).

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