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Linearized Five Dimensional Kaluza-Klein Theory As a Gauge

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Linearized Five Dimensional Kaluza-Klein Theory As a Gauge INVESTIGACION´ REVISTA MEXICANA DE FISICA´ 54 (3) 188–193 JUNIO 2008 Linearized five dimensional Kaluza-Klein theory as a gauge theory G. Atondo-Rubio, J.A. Nieto*, L. Ruiz, and J. Silvas Facultad de Ciencias F´ısico-Matematicas,´ Universidad Autonoma´ de Sinaloa, 80000, Culiacan,´ Sinaloa, Mexico,´ *e-mail: [email protected] Recibido el 30 de julio de 2007; aceptado el 26 de marzo de 2008 We develop a linearized five-dimensional Kaluza-Klein theory as a gauge theory. By perturbing the metric around flat and de Sitter back- grounds, we first discuss linearized gravity as a gauge theory in any dimension. In the particular case of five dimensions, we show that in using the Kaluza-Klein mechanism, the field equations of our approach imply both linearized gauge gravity and Maxwell theory in flat and de Sitter scenarios. As a possible further development of our formalism, we also discuss an application in the context of gravitational polarization scattering by means of the analogue of the Mueller matrix in optical polarization algebra. We argue that this application can be of particular interest in gravitational wave experiments. Keywords: Kaluza-Klein theory; linearized gravity; Mueller matrix Desarrollamos una teor´ıa Kaluza-Klein linealizada en cinco dimensiones como una teor´ıa de norma. Discutimos primero gravedad linealizada como una teor´ıa de norma en cualquier dimension´ perturbando las metricas´ de fondo plana y la de de Sitter. Demostramos que usando el mecanismo de Kaluza-Klein en el caso particular de cinco dimensiones, las ecuaciones de campo de nuestra aproximacion´ implican tanto gravedad linealizada de norma como la teor´ıa de Maxwell en escenarios planos y de de Sitter. Como otro posible desarrollo de nuestro formalismo, tambien´ discutimos una aplicacion´ en el contexto de dispersion´ gravitacional polarizada, por medio de una matriz analoga´ a la de Mueller usada en el algebra de polarizacion´ optica.´ Argumentamos que esta aplicacion´ puede ser de interes´ particular en los experimentos de ondas gravitacionales. Descriptores: Teor´ıa de Kaluza-Klein; gravedad linealizada; matriz de Mueller. PACS: 04.50.+h, 04.30.-w, 98.80.-k, 42.15.-i 1. Introduction fiber-bundle M 4 £ B, with B as a properly chosen compact space. Thus, we have that this case can be summarized by the It is known that linearized gravity can be considered to be a heuristic picture gauge theory [1]. In this context, one may be interested in the idea of a unified theory of linearized gravity and Maxwell em à g: (2) theory. This idea is, however, not completely new since in fact the quest for a unified theory of gravity and electromag- Our aim in this paper is to combine the two scenarios (1) netism has a long history [2]. One can mention, for instance, and (2) in the form the five-dimensional Kaluza-Klein theory [3], which is per- haps one of the most interesting proposals. The central idea em Ã! g: (3) in this case is to incorporate electromagnetism into a geo- metrical five-dimensional gravitational scenario. The gauge Specifically, we start with linearized gravity in five- properties arise as a result of broken general covariance via a dimensions and apply the Kaluza-Klein compactification mechanism called “spontaneous compactification”. Symbol- mechanism. We probe that the resultant theory can be un- ically, one may describe this process through the transition derstood as a gauge theory of linearized gravity in five di- M 5 ! M 4 £ S1, where M 5 and M 4 are five- and four- mensions. Furthermore, we show that, by using this strategy, dimensional manifolds respectively and S1 is a circle. Thus, one can derive an unified theory of gravity and electromag- after compactification the fiber-bundle M 4 £S1 describes the netism with a generalized gauge field strength structure. As Kaluza-Klein scenario. Let us picture this attempt of unifica- an advantage of our formalism, we outline the possibility that tion as optical techniques can be applied to both gravity and electro- magnetic radiation in a unified context. Thus, we argue that em ! g; (1) our results may be of particular interest in the detection of where em means electromagnetism and g gravity. gravitational waves. In the case of linearized gravity theory, the scenario looks Technically this article is organized as follows. In Sec. 2, different because it can be understood as a gauge theory we develop linearized gravity in any dimension. In Sec. 3, we rather than a pure geometrical structure. Therefore, a unified discuss linearized gravity in a 5-dimensional Kaluza-Klein theory in this case may be understood as an idea for incorpo- theory and in Sec. 4 we generalize our procedure to a de Sitter rating linearized gravity in a Maxwell gauge context. In other background. In Sec. 5, we outline a possible application of words, one may start from the beginning with a generalized our formalism of unified framework of electromagnetic and LINEARIZED FIVE-DIMENSIONAL KALUZA-KLEIN THEORY AS A GAUGE THEORY 189 gravitational radiation to an optical geometry via the Mueller Let us now define the symbol matrix. In Appendix A we present a generalization to any dimension of the Novello and Neves work [4]. FADB ´ FADB + ´BAFD ¡ ´BDFA; (15) which has the property 2. Linearized gravity in any dimension F = ¡F : (16) Let us consider a 1 + d-dimensional manifold M 1+d; with ADB DAB the associated metric γ (xC ): We shall assume that γ AB AB Thus, by using expression (15) we find that field equa- can be written in the form tions (14) are simplified in the form γAB = ´AB + hAB; (4) A 8¼G1+d @ FABD = 2 TBD: (17) C c where ´AB = diag(¡1; 1;:::; 1) and hAB(x ) is a small perturbation, that is A A Since @ FADB = @ FABD; field equations (17) can also be written as jhABj ¿ 1: (5) A 16¼G1+d @ FA(BD) = TBD; (18) To first order in hAB, the inverse of the metric γAB becomes c2 γAB = ´AB ¡ hAB: (6) where the bracket (BD) means symmetrization of the indices B and D: It is worth mentioning that in a 1 + 3-dimensional Using (4) and (6) we find that the Christoffel symbols and spacetime field equations (18) are reduced to those proposed Riemann curvature tensor are by Novello and Neves [4]. 1 Furthermore, it is not difficult to show that field equa- ¡A = ´AB(h + h ¡ h ) (7) CD 2 BC;D BD;C CD;B tions (17) can be derived from the action Z and 1+d ADB A S = d xfF FADB ¡ 2F FA ¡ Lmatterg; (19) RABCD = @AFCDB ¡ @BFCDA; (8) where Lmatter denotes a Lagrangian associated with matter respectively. Here, the symbol FCDB means fields. In fact, by varying action (19) with respect to hAB and and assuming 1 FCDB = (hBC;D ¡ hBD;C ): (9) 2 ±Lmatter 32¼G1+d BD = 2 T ; ±hBD c Observe that FCDB is antisymmetric in the indices C and D: In terms of the quantity FA defined by one obtains the field equations (17). CB FA = ´ FACB; (10) 3. Linearized gravity in a five-dimensional and the symbol FADB, the Ricci tensor RBD reads Kaluza-Klein theory R = @AF + @ F : (11) BD ADB B D In a 5-dimensional spacetime the weak field metric tensor Thus, we find that the Ricci scalar R is given by γAB = ´AB + hAB; where ´AB = diag(¡1; 1; 1; 1; 1), can be written in the block-matrix form A R = 2@ FA: (12) µ ¶ ´¹º + h¹º h4¹ γAB = ; (20) Substituting (11) and (12) into the Einstein weak field h4º 1 + h44 equations in 1 + d dimensions with ¹; º = 0; 1; 2; 3. If one adopts the Kaluza-Klein ansatz, 1 8¼G1+d with RBD ¡ ´BDR = 2 TBD; (13) 2 c ® h¹º = h¹º (x ); (21) we find ® h4¹ = A¹(x ) (22) A A 8¼G1+d @ FADB + @BFD ¡ ´BD@ FA = TBD; (14) c2 and where G1+d is the Newton gravitational constant in 1 + d dimensions. h44 = 0; (23) Rev. Mex. F´ıs. 54 (3) (2008) 188–193 190 G. ATONDO-RUBIO, J.A. NIETO, L. RUIZ, AND J. SILVAS ® where A¹(x ) is identified with the electromagnetic poten- one discovers that F¹44 is identically equal to zero. tial, we discover that the only nonvanishing terms of FDAB For completeness let us observe that (37) leads to are h¹º ; = 0; 1 º F¹º® = (h®¹;º ¡ h®º;¹); (24) (38) 2 ¹º h = ´ h¹º = 0; 1 F = @ A ; (25) 4º® 2 º ® and the Lorentz gauge for A¹ ¹ and A ;¹ = 0: (39) 1 F = ¡ F ; (26) ¹º4 2 ¹º Consequently one finds that, in the gauge (38) and (39), the field equations (32) and (36) are reduced to where F¹º = Aº;¹ ¡ A¹;º is the electromagnetic field strength. Thus, from (15) we find that the nonvanishing com- 2 16¼G ¤ h¹º = ¡ 2 T¹º (40) ponents of FDAB are c and F¹º® = F¹º® + ´®¹Fº ¡ ´®º F¹; (27) 1 ¤2A¹ = ¡4¼J ¹; (41) F = (@ A ¡ ´ @¯A ); (28) 4º® 2 º ® ®º ¯ respectively, where ¤2 = ´¹º @ @ is the d’Alembertian op- 1 ¹ º F¹º4 = ¡ F¹º ; (29) erator. Thus, we have found a framework in which the grav- 2 itational and electromagnetic waves can be treated on the and same footing. F¹44 = ¡F¹: (30) 4. The de Sitter generalization Now, since all fields are independent of the coordinate x4, we see that field equations (17) can be written as In order to generalize the formalism described in the previ- ous section to a de Sitter scenario, we shall replace the flat ¹ 8¼G1+4 ® @ F = T : (31) metric ´¹º by the de Sitter metric f¹º (x ).
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