Journal of the Arkansas Academy of Science
Volume 60 Article 13
2006 Variational Symmetries and Conservation Laws in Linearized Gravity Balraj Menon University of Central Arkansas, [email protected]
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Recommended Citation Menon, Balraj (2006) "Variational Symmetries and Conservation Laws in Linearized Gravity," Journal of the Arkansas Academy of Science: Vol. 60 , Article 13. Available at: http://scholarworks.uark.edu/jaas/vol60/iss1/13
This article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able to read, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the author. This Article is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Journal of the Arkansas Academy of Science by an authorized editor of ScholarWorks@UARK. For more information, please contact [email protected], [email protected]. Journal of the Arkansas Academy of Science, Vol. 60 [2006], Art. 13 ? BalrajMenon Department ofPhysics and Astronomy, 201 Donaghey Avenue, University ofCentral Arkansas, Conway, AR 72035 Correspondence: [email protected] — Abstract. The methods of symmetry group analysis are applied to the action functional oflinearized gravity to derive necessarj conditions for the existence of variational symmetries. Two classes of variational symmetries oflinearized gravity are discussed, anc the local conservation laws associated withthese variational symmetries are presented by applying Noether's theorem. — Key words. Gravity, variational symmetries, linearized, conservation laws, Noether's theorem. Introduction Methods Local conservation laws play a pivotal role in several Linearized gravity—In Einstein's general theory of branches of physics (Barrett and Grimes 1995, Goldberg relativity, spacetime is assumed tobe a 4-dimensional manifold 1958). The conserved quantities derived from conservation M endowed with a Lorentzian metric gab. The spacetime laws permit the characterization of a given physical system in metric g^ satisfies the Einstein equations (in units where = = terms of a relatively small number ofphysical quantities. For c 8jiG 1), example, quantities such as energy, linear momentum, angular y"-"f momentum, and charge, are often encountered ina wide range of rrt (1) classical and quantum systems because ofthe conservation laws with In quantum associated these quantities. the field theoretic where Gab is the Einstein tensor and Tab is the energy- description of the fundamental interactions, the existence of momentum tensor of the matter distributions (Wald 1984). In conservation laws is often the guiding principle that dictates the the absence ofany matter distributions (for example, outside a = correct choice of the field theory that describes the fundamental star or planet), Tab 0 and the metric gab solves the vacuum interactions (Kaku 1993). Ingeneral relativity, ithas often been Einstein equations ofgeneral relativity, argued that the existence of local conservation laws would lead (2) to the construction of observables of the gravitational field, Gab =0. which could play a significant role in any quantum theory of gravity (Torre 1993). Conservation laws also play an important Furthermore, ifwe restrict attention toregions of spacetime role ina variety of mathematical issues such as integrability, where the gravitational fieldis weak (forexample, very far away existence and uniqueness, and stability (Olver 1993). from a star or planet), we can always choose local coordinates spacetime In1918, the German mathematician Emmy Noether proved on such that the metric gab takes the form twoimportant theorems concerning the existence ofconservation +^ laws for physical systems that admit a Lagrangian formulation Sab tlab lab> (3) (Noether 1918). Her first theorem proved that ifthe Lagrangian admits a variational symmetry (a symmetry transformation that where hab can be viewed as a symmetric (0,2) -type tensor leaves the action functional invariant), then the system admits field propagating on a flat spacetime endowed with the a local conservation law. The second theorem states that ifa Minkowski metric Y)ab. The tensor hab is assumed to be a variational symmetry depends on arbitrary functions, then the small perturbation ofthe background Minkowski metric. This us to differential equations governing the system must satisfy an assumption allows restrict attention to terms linear inhab , identity. which in turn implies Inthis paper, Idiscuss the variationalsymmetries and local ab ab conservation laws admitted by the linearized, vacuum Einstein g =n -hab, (4) equations of general relativity. Ibegin with a brief review of ac = a ab ac bd the Lagrangian formulation of linearized gravity and then where g gcb db and h =r] r] hcd . proceed to apply methods ofsymmetry group analysis to derive The linearized, vacuum Einstein equations are obtained by necessary conditions for the existence ofvariational symmetries substituting equations (3) and (4)inequation (2) and expanding theorems, of linearized gravity. By applying Noether's the Gab to linear order in hab . This yields conservation laws associated with the variational symmetries c + c c =-(w -a m + *)=o> are derived. Iconclude with a discussion of two classes— of oA » W» a^-d ed'k^ -riji^h" r,0^ (5) variational symmetries admitted by linearized gravity the ¦=— cd = c Poincare group of symmetries and the gauge symmetry of where a a and h \=r)cdh h c (Carroll2004). linearized gravity. Journal of the Arkansas Academy of Science, Vol.60, 2006 80 Published by Arkansas Academy of Science, 2006 80 Journal of the Arkansas Academy of Science, Vol. 60 [2006], Art. 13 Variational Symmetries and Conservation Laws in Linearized Gravity linearized, vacuum Einstein equations do admit a The a Lagrangian formulation; they can be derived from a variational to hold for all symmetric tensor fields 8h satisfying the a principle. The action functional S[h ] of the variational boundary condition. The fundamental lemma inthe calculus of problem is variation (Courant and Hilbert 1989) implies that equation (15) holds in Q only ifthe linearized Einstein tensor G vanishes 4 ab S[h ab L(h,dh)d x, (6) in Q, proving the result that a necessary condition for the ]=fQ a a tensor field h to extremize the action functional S[h ] is where the Lagrangian density, that it satisfy the linearized, vacuum Einstein equations. — Variational symmetries and conservation laws The - - - £(*,ah) X*,A) O,a*X^a%) ) (7) linearized, vacuum Einstein equations are a system of ten jhaf +^"O^K^V = differential equations 0, involving 4 independent G^ a is defined on a compact region Q of the spacetime manifold variables, or spacetime coordinates x , and ten dependent M witha smooth boundary dQ (Carroll 2004). Let variables hab representing the components of a symmetric (0,2) -type tensor field. Let M represent the space of ab ab a hf=h +edh (8) independent variables (or coordinates) x and U represent the space of dependent variables with coordinates hab . Consider represent a one-parameter family of symmetric tensor fields on a one-parameter family of infinitesimal transformations on the a Q. The tensor field 8h satisfies the boundary condition product space MxU givenby «*1,0=0. (9) x a = xa +e% a (x,h) + O(e 2 ) and (16) = + + 2 In order to derive the linearized equations, we assumed the b hab eyab (x,h) O(e ), (i7) — a a K tensor field Aq h extremizes the action functional S[h ]. is, e«l, a That where and the functions £ (x,h) and Yab(x,h) are the components of a smooth vector field and a smooth, (10) symmetric, (0,2) -type tensor field, respectively, on MxU. de The infinitesimal transformations (16) and (17) transform the Equations (6) and (10) imply action functional giveninequation (6) to 4 4 = dS=fdLd x=f—I a Ld x 0. (11) ab 4 S[h L(h,dh)d x, (18) ]=JQ By virtue ofequations and ,equation can be writtenas where ab 4 = + i(A,aA)=i(a rf Gab dh d x dard\ o, (12) c A-)(a^)-i(acA*X^A%)+^ (d^Xa,V) fQ jQ i 09) where Gab is the linearized Einstein tensor given inequation (5), the vector field ~ d d ¦ a ~a represents respect to ° and d partial derivatives with the ¦ + ab a ci + a k a a J (d h )6h-2(d ch l,)dh {d h )6h -(dh)6h] (13) transformed coordinates x Expanding equations (18) and (19) b bc =. in a Taylor series about £ 0 and reorganizing the resulting ab ab and the trace of the variation 6h is dh=TJ abdh (Wald expression gives 1984). The divergence theorem implies a 4 3 ab = ab + + c 4 + 2 CdJ d n°J d x, (14) S[h ] S[h ] efjdL dc(g L)±l x O(e ),(20) Ja a x=fJea a a where n is the unit outward normal to the three-dimensional where boundary dQ . Equations (9)and (13) imply - <- * xa,*)-W.e*X3,*'.)-(3,**xa e ab 4 = Qab=Y*-M *chab (22) Gab dh d x O (15) JQ Journal of the Arkansas Academy of Science, Vol. 60, 2006 http://scholarworks.uark.edu/jaas/vol60/iss1/13 81 81 Journal of the Arkansas Academy of Science, Vol. 60 [2006], Art. 13 Balraj Menon a ac c a with =T] Y] and ab Inderiving equation Itis clear that on solutions of equation (5), if ab represents th Q Qcd Q=Tf Q . Q a (20) we have also used the fact that, to first order in £ , the characteristic ofa variational symmetry, then P represents i volume element d X transforms as conserved vector fieldof linearized gravity. We now explore twodistinct types of- variational symmetric j d4 x =(l+ e d£ )d\x+0{s2 ). that are significant inlinearized gravity the Poincare symmetric ; and gauge symmetries. Assume the vector field Z;" depend; a = a The infinitesimal transformations (16) and (17) represent onlyonthe coordinates on the manifold M(i.e., t~ % (x)). a variational symmetry of the action functional ifthey leave Consequently, the infinitesimal transformations (16) represents the action functional invariant up to an overall surface term for a one-parameter family of coordinate transformations on M. all symmetric tensor fields hab on M (Olver 1993). In other Since hab is a tensor field on M,it must transform according words, to the tensor transformation law,namely, ah ah 3 2 = a + c d c 2 S[h ] S[h ]+eC n A d x O(e ), (23) =~d x 'd (d + h O(e ). (29) JdQ a hab a bx hcd =hab -e arhcb d£ ac y a vector Equations imply where A are the components ofa fieldon M and na (17) and (29) is a one-form fieldnormal to the boundary dQ The divergence . = + theorem applied to the surface term in Equation (23) implies r* -G.r*,* a.r*J a represent that ifthe fields £ and yab variational symmetries oflinearized gravity then they must satisfy the condition and the characteristic Qab takes the form + c c (24) (30) 6L dc (g L)=dc A. Qab=-h hab, = Inequation ifthe vector field A° 0 ,the variational symmetry where Lthab is the Lie derivative of the tensor field hab with a is called a strict variational symmetry; otherwise the symmetry respect to the vector field £ (Wald 1984). Furthermore, let is referred to as a divergence symmetry (Olver 1993). us assume that the infinitesimal coordinate transformations Alocal conservation law ofthe linearized, vacuum Einstein generated by the vector field leave the Minkowski metric a £°a equations is a vector field P built from the coordinates X , Y]ab invariant. Inother words, £ is a Killingvector fieldof the Killingequation the tensor field hab ,and derivatives of the tensor field hab to Minkowski metric and hence satisfies the any arbitrary but finite order that satisfies the condition = + (31) a = V^ a^ d aP 0 Substituting equations (30) and (31) in equation (21) and on solutions of the field equations (5) of linearized gravity. In simplifying the resulting equation using the properties of the order to elicit the relationship between variational symmetries Lie derivative gives and local conservation laws established inNoether's theorems, ab (32) we rewrite equation (24) by integrating the term SL by parts. dL=-rdcL+F cda dhr, This yields, after some algebra, where = ab (25) SL Q Gab+ daS\ ah = a bJ + d ab hd hd a F c -$>c h ddh 6 c h ddh-h d ch°J -h ddh c where G is the linearized Einstein tensor defined inequation d ah ad b ab +h c d h h 'd h } (5)and -h'dd^ d +2n d ce - 1* h l* cb bc yields + h"r)Q +(d°h )Q -(a°h)Q] (26) Substituting equation (32) back into equation (24) S° (dhh" )Q-(dlh"h)Q -(dh bc ab c terms L+ F c(h,dh)d + c Substituting equation in equation and reorganizing the -rdc ad£ dc &L)=dc A. gives a Since % is a Killingvector field,it follows that a JT =0 and a ab (27) 5 c=0 (Crampin and Pirani 1994). Setting a = proves daP =-Q Gab , Al A 0 that all coordinate transformations on the spacetime manifold where M that leaves the Minkowski metric invariant are strict variational symmetries of the action functional of linearized a a a a P =A -S -% L. (28) gravity. These coordinate transformations are the ten-parameter Journal of the Arkansas Academy of Science, Vol.60, 2006 82 Published by Arkansas Academy of Science, 2006 82 Journal of the Arkansas Academy of Science, Vol. 60 [2006], Art. 13 Variational Symmetries and Conservation Laws inLinearized Gravity admit a ten-parameter family of local conservation laws, while group of Poincare symmetries. the gauge symmetry is a divergence symmetry admitting a To determine the conserved vector fieldassociated withthe trivialconservation law. Lookingahead, itwould be interesting i Poincare symmetries, we substitute equation (30) in equation a = to investigate the various conservation laws associated withthe (28) and set A 0 to obtain Poincare symmetries and explore their geometric and physical significance. Another interesting research direction is the - - classification of all local conservation laws of the linearized, vacuum Einstein equations. This is achieved by investigating +|«'f(»^ X«rf*)-(M*Xa<*'.)+|»i (*.*^X8rfV)-|l»''(MX«**)l solutions of equation , where the characteristic Qab depends on of the tensor field h to any arbitrary but The other variational symmetry is the infinitesimal derivatives the ab a = gauge transformation obtained by setting § 0 and finite order. = + a Yab da-^b m equati°ns (16) and (17), where X is an arbitrary vector field on the manifold M. Substituting = + (21) Literature Cited the characteristic Qab daXb dbXa in equation and recursively applying the integration by parts formula to the Grimes, resulting equation gives Barrett TW and DM editors. 1995. Advanced electomagnetism: foundations, theory and applications. = a ab Singapore: World Scientific. 560 p. 6L daA -2d a G Xb, (33) Carroll SM. 2004. Spacetime and geometry, San Francisco: Addison Wesley, 274-78. where p Courant R and D Hilbert. 1989. Methods of mathematical a a a b physics Volume 1. New York:John Wiley. 184 p. A =S +2G b X , (34) Crampin M and FAE Pirani. 1994. Applicable differential a geometry. London Mathematical Society Lecture Note i and S is the vector field defined in equation (26) with ~ + d linearized, Series 59. Cambridge: Cambridge University, p 188-96. Qab a Xb bXa . The vacuum Einstein equations satisfy the contracted Bianchi identity Goldberg JN. 1958. Conservation laws in general relativity. Physical Review 111:315—320. a Kaku M.1993. field theory: A modern introduction, d G b =O, (35) Quantum a New York:Oxford University Press, p33-58. Noether E. 1918. Invariante which reduces equation (33) to a form that clearly indicates that variationsprobleme. Nachrichten Konigliche the gauge transformation is not a strict variational symmetry von der Gesellschaft der Wissenschaften zu 235-7. For an English translation, see of the linearized theory, but instead a divergence symmetry. Gottingen Tavel MA. Transport Theory Physics 1:186. Equations (28) and (34) imply that the conserved vector fieldis 1971. and Statistical Olver PJ. 1993. Applications of Lie groups to differential a a b equations. 2nd Edition. New York: Springer-Verlag. p P =2G bX 242- . 85. a = Torre CG. 1993. However, note that the conserved vector field P 0 on Gravitational observables and local symmetries. Physical solutions ofequation (5)and hence defines a trivialconservation Review D 48:2373-6. Wald R. 1984. relativity. Chicago: law. This is because the gauge symmetry depends on an General University of a Chicago, p 437-44. arbitrary vector field X and hence falls under the purview of Noether's second theorem, which states that the linearized Einstein equations must satisfy a constraint equation. This constraint equation is the contracted Bianchi identity given in equation (35). Conclusions Iderived necessary conditions that must be satisfied by a variational symmetry of the linearized, vacuum Einstein equations and investigated two classes of variational symmetries: the ten-parameter group of Poincare symmetries and the gauge symmetry. Ishowed that the Poincare symmetries Journal of the Arkansas Academy ofScience, Vol.60, 2006 83 http://scholarworks.uark.edu/jaas/vol60/iss1/13 83