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Variational Symmetries and Conservation Laws in Linearized Gravity Balraj Menon University of Central Arkansas, [email protected]

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Variational Symmetries and Conservation Laws in Linearized Gravity Balraj Menon University of Central Arkansas, Menonb@Uca.Edu 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] Follow this and additional works at: http://scholarworks.uark.edu/jaas Part of the Cosmology, Relativity, and Gravity Commons 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 ? <iri<iiioii<iik>y iiiifiLirits <«oq v^oiiSGi v<irion j_i«iWa illJ_iiDC3rizcu vji<iviij' 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<ec.) * ee J- =0' a </ < ' dl '(d )o v)-'j''(3cex^/i)]' W +»7 ee l/ which (12) to vanish, forces the second integral in equation and the characteristic Qab is defined as leaving 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 .
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