sign in sign up
<<
Home , Graviton, Gravity, Linearized gravity

INVESTIGACION´ REVISTA MEXICANA DE FISICA´ 54 (3) 188–193 JUNIO 2008

Linearized five dimensional Kaluza-Klein as a

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 as a gauge theory in any . In the particular case of five , we show that in using the Kaluza-Klein , 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 experiments.

Keywords: Kaluza-Klein theory; ; 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 . 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 ← 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 ←→ g. (3) in this case is to incorporate 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 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 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 [4]. FADB ≡ FADB + ηBAFD − ηBDFA, (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 = ηAB + hAB, (4) A 8πG1+d ∂ FABD = 2 TBD. (17) C c where ηAB = diag(−1, 1,..., 1) and hAB(x ) is a small , that is A A Since ∂ FADB = ∂ FABD, field equations (17) can also be written as |hAB| ¿ 1. (5) A 16πG1+d ∂ FA(BD) = TBD, (18) To first order in hAB, the inverse of the metric γAB becomes c2 γAB = ηAB − 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 field equations (18) are reduced to those proposed Riemann are by Novello and Neves [4]. 1 Furthermore, it is not difficult to show that field equa- ΓA = ηAB(h + h − h ) (7) CD 2 BC,D BD,C CD,B tions (17) can be derived from the Z and 1+d ADB A S = d x{F FADB − 2F FA − Lmatter}, (19)

RABCD = ∂AFCDB − ∂BFCDA, (8) where Lmatter denotes a Lagrangian associated with 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 δLmatter 32πG1+d BD = 2 T , δhBD 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 = η 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 Thus, we find that the Ricci R is given by γAB = ηAB + hAB, where ηAB = diag(−1, 1, 1, 1, 1), can be written in the block-matrix form A R = 2∂ FA. (12) µ ¶ ηµν + hµν h4µ γAB = , (20) Substituting (11) and (12) into the Einstein weak field h4ν 1 + h44 equations in 1 + d dimensions with µ, ν = 0, 1, 2, 3. If one adopts the Kaluza-Klein ansatz,

1 8πG1+d with RBD − ηBDR = 2 TBD, (13) 2 c α hµν = hµν (x ), (21) we find α h4µ = Aµ(x ) (22) A A 8πG1+d ∂ FADB + ∂BFD − ηBD∂ FA = TBD, (14) c2 and where G1+d is the 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

α where Aµ(x ) is identified with the electromagnetic poten- one discovers that Fµ44 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 hµν , = 0, 1 ν Fµνα = (hαµ,ν − hαν,µ), (24) (38) 2 µν h = η hµν = 0, 1 F = ∂ A , (25) 4να 2 ν α and the Lorentz gauge for Aµ µ and A ,µ = 0. (39) 1 F = − F , (26) µν4 2 µν Consequently one finds that, in the gauge (38) and (39), the field equations (32) and (36) are reduced to where Fµν = Aν,µ − Aµ,ν is the electromagnetic field strength. Thus, from (15) we find that the nonvanishing com- 2 16πG ¤ hµν = − 2 Tµν (40) ponents of FDAB are c and Fµνα = Fµνα + ηαµFν − ηαν Fµ, (27) 1 ¤2Aµ = −4πJ µ, (41) F = (∂ A − η ∂βA ), (28) 4να 2 ν α αν β respectively, where ¤2 = ηµν ∂ ∂ is the d’Alembertian op- 1 µ ν Fµν4 = − Fµν , (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.

Fµ44 = −Fµ. (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 µ 8πG1+4 α ∂ F = T . (31) metric ηµν by the de Sitter metric fµν (x ). In this case the µBD c2 BD perturbed Kaluza-Klein metric γAB takes the form These field equations can be separated as follows: µ ¶ fµν + hµν Aµ 8πG γAB = , (42) ∂µF = 1+4 T , (32) Aν 1 µνα c2 να whose inverse is given by µ 8πG1+4 ∂ Fµ4ν = T4ν , (33) µ ¶ c2 f µν − hµν −Aµ γAB = . (43) 8πG −Aν 1 ∂µF = 1+4 T , (34) µν4 c2 ν4 By combining the results from Sec. 2 and Appendix A it and is not difficult to obtain the generalized field equations

µ 8πG1+4 ∂ Fµ44 = T44. (35) A 3 8πG1+d c2 D FABD − (hBD − hfBD) = TBD. (44) 2l2 c2 Using (28) and (29), we find that (33) and (34) lead to exactly Thus, using (42) and considering that in five dimensions the same field equations, namely d = 4 and Λ = 6/l2 we find µ ∂ Fνµ = 4πJν , (36) α Λ 8πG1+4 D Fαµν − (hµν − hfµν ) = Tµν , (45) 2 2 where Jν =(4G1+4/c )Tν4 is the electromagnetic current. 4 c Of course, field equations (32) and (36) correspond to lin- and earized gravity and Maxwell field equations, respectively. If ν Λ we set T44 = 0, then one can see that field equation (35) is a D Fµν − Aµ = 4πJµ. (46) pure gauge expression. In fact, if one assumes the transverse 4 4 traceless gauge in five dimensions Here, we have used the fact that D F4AB = 0 and f4D = 0. Observe that in this case the electromagnetic field strength hAB, = 0, B F is given by (37) µν AB h = η hAB = 0, Fµν = DµAν − Dν Aµ.

Rev. Mex. F´ıs. 54 (3) (2008) 188–193 LINEARIZED FIVE-DIMENSIONAL KALUZA-KLEIN THEORY AS A GAUGE THEORY 191

The field equations (45) and (46) are remarkable because if and the vector we set   g0 2 Λ  g  m = , (47) g =  1  , (52) 4   one discovers that, up to factors, both the and the g3 have the same m and even more intriguing is where the fact that such a mass is proportional to the root of the . ∗ ∗ g0 = ExEx + EyEy , (53) g = E E∗ − E E∗, (54) 5. Final remarks 1 x x y y ∗ ∗ g2 = ExEy + ExEy , (55) The present work may have a number of interesting develop- ments. In particular, as Novello and Neves [4] have shown, and

Eq. (15) can be derived from the formula ∗ ∗ g3 = i(ExEy − ExEy ). (56) A∗ ∂ FA(BD) = 0, (48) The Jones matrix J and the Mueller matrix M apply to where C and the vector g, respectively, as follows:

∗ A(BD) ABEF D ADEF B 0 F = ε FEF + ε FEF , (49) C = JC (57) ABEF with ε the completely antisymmetric symbol. Thus, and one may consider an alternative approach [5] for duality as- pects of linearized gravity [6] (see Refs. 7 to 11) as in the g0= Mg. (58) case of Maxwell theory (see Refs. 12 and 13). Another source of physical interest of the present for- It turns out that J can be identified with a SU(2) matrix, while malism is a possible connection with the Randall-Sundrum M is a 4×4 augmented matrix form of O3. Of course the ma- world scenario [14,15], with gravitational wave formal- trices J and M must be related: ism (see Refs. 16 and references therein) and with 1 M = T rJ ∗T σJσ, (59) linearized gravity [17,18]. Moreover, our work may also be 2 useful in clarifying some aspects regarding the relation be- where σ denotes the four Pauli matrices (see Ref. 22 for de- tween the mass of the graviton and the cosmological constant, tails). which has been the subject of some controversy [19,20]. Let us make the identification Aside from theoretical developments, the present work also opens the possibility of making a number of applica- F0iα = Eiα. (60) tions of linearized gravity arising from the Maxwell theory itself. Let us outline just one possibility. In Maxwell theory Here F0iα denotes some of the components of FADB accord- the concept of polarization scattering is of considerable in- ing to the expression (15). The idea is now to consider gen- terest in optical physics (see Ref. 21 and references therein). eralization of (50) µ ¶ The subject of interest in this arena is to describe the inter- Exα action of polarized waves with a target in a complex setting. (61) Eyα It turns out that the useful mathematical tool in the scattering radiation process is the so-called Mueller matrix [22] (see and to consider the analogue of (53)-(56), namely also Ref. 23 and references therein). What appears interest- G = EαE∗ + EαE∗ , (62) ing about such a matrix is that its elements refer to intensity 0 x xα y yα α ∗ α ∗ measurements only. Let us recall the main ideas for the con- G1 = Ex Exα − Ey Eyα, (63) struction of the Mueller matrix. ∗α α ∗ A polarized radiation field may be represented by a 2- G2 = Ex Eyα + Ex Eyα (64) dimensional complex field µ ¶ and Ex ∗α α ∗ . (50) G3 = i(E Eyα − E E ). (65) Ey x x yα From the components of this vector, one may define the Her- Thus, we can apply the Mueller matrix M as in (58): mitian coherency matrix G0= MG. In fact, since M contains the information of the µ ¶ intensities of both gravitational and electromagnetic radiation ∗ ∗ ExEx ExEy via the quantity F one should expect a broad range of ap- C = ∗ ∗ (51) 0iα ExEy EyEy plications of M in a gravitational wave context.

Rev. Mex. F´ıs. 54 (3) (2008) 188–193 192 G. ATONDO-RUBIO, J.A. NIETO, L. RUIZ, AND J. SILVAS

Appendix A Substituting (74) and (76) into the Einstein weak field equations in 1 + d dimensions, with cosmological constant In this appendix we shall study the linearized gravity with f 1 AB as a metric background. Consider the perturbed metric R − (f + h )R + (f + h )Λ BD 2 BD BD BD BD γAB = fAB + hAB, (66) 8πG1+d = 2 TBD, (77) where hAB is a small perturbation, that is |hAB| ¿ 1. To c first order in hAB one finds we obtain γAB = f AB − hAB. (67) 1 DAF + D F − f DAF − R hAC ADB B D BD A 2 ABCD The Christoffel symbols can be written in the interesting 1 1 1 form + R hE + f R hEF − h R+h Λ 2 EB D 2 BD EF 2 BD BD A A A ΓCD = ΩCD + HCD, (68) 8πG1+d = 2 TBD. (78) A c where ΩCD are the Christoffel symbols associated with fAB A Here, we used the fact that and HCD is defined by 1 A 1 AE H = f (D h + D h − D h ). (69) RBD − fBDR+fBDΛ = 0. (79) CD 2 C DE D CE E CD 2 A Here, the symbol DA denotes covariant derivatives with ΩCD Since we have as a connection. 1 By using (68) it is straightforward to check that the Rie- RABCD = (fAC fBD − fADfBC ) (80) l2 mann tensor can be written in the form and A A A A RBCD = RBCD + DC HBD − DDHBC , (70) d(d − 1) A A Λ = 2 , (81) where RBCD is the Riemann tensor associated with ΩCD. 2l With the help of the commutation relation we discover that (78) is reduced to D D h −D D h =−RE h −RE h (71) C D AB D C AB ACD EB BCD AE A A D FADB + DBFD − fBDD FA we find that the (70) can also be 1 8πG written as − (d − 1)(h − hf ) = 1+d T . (82) 2l2 BD BD c2 BD RABCD = DAFCDB − DBFCDA + RABCD Thus, by defining the symbol 1 1 + RE h − RE h . (72) 2 DAB EC 2 CAB ED FADB ≡ FADB + fBAFD − fBDFA, (83)

Here, the symbol FCDB takes the form we find that the field equations (82) are simplified in the form 1 1 8πG FCDB = (DDhBC − DC hBD). (73) DAF − (d−1)(h −hf )= 1+d T . (84) 2 ABD 2l2 BD BD c2 BD F Observe that CDB can be obtained from (9) by replacing or- 2 dinary partial derivatives by covariant derivatives. From (72) In four dimensions d = 3 and Λ = 3/l . Therefore (84) we get the Ricci tensor becomes

Λ 8πG1+3 R = DAF + D F + R DAF − (h − hf ) = T , (85) BD ADB B D BD ABD 3 BD BD c2 BD 1 1 − R hAC + R hE (74) which are the field equations obtained by Novello and 2 ABCD 2 EB D Neves [4]. where

CB FA = f FACB. (75) Acknowledgments

Thus, we find that the Ricci scalar R is given by We would like to thank C.M. Yee and E.A. Leon´ for their helpful comments. This work was supported in part by the A BD R = 2D FA + R − RBDh . (76) UAS under the program PROFAPI-2006.

Rev. Mex. F´ıs. 54 (3) (2008) 188–193 LINEARIZED FIVE-DIMENSIONAL KALUZA-KLEIN THEORY AS A GAUGE THEORY 193

1. J.A. Nieto, Mod. Phys. Lett. A 20 (2005) 135; hep-th/0311083. 12. E. Witten, Selecta Math.1 (1995) 383; hep-th/9505186. 2. G.’t Hooft et al., Nature 433 (2005) 257. 13. Y. Lozano, Phys. Lett. B 364 (1995) 19; hep-th/9508021. 3. M.J. Duff, “Kaluza-Klein theory in perspective”, talk given at 14. L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690; The Oskar Klein Centenary Symposium, Stockholm, Sweden, hep-th/9906064. 19-21 Sep 1994. In *Stockholm 1994, The Oskar Klein cente- nary* 22-35, and Cambridge U. - NI-94-015 (94/10,rec.Oct.) 15. S.B. Giddings, E. Katz, and L. Randall, JHEP 0003 (2000) 023; 38 p. Texas A&M U. College Station - CTP-TAMU-94-022 hep-th/0002091; H. Collins and B. Holdom, Phys. Rev. D 62 (94/10) 38 p; hep-th/9410046. (2000) 124008; hep-th/0006158; N. Deruelle and T. Dolezel, Phys. Rev. D 64 (2001) 103506: gr-qc/0105118 4. M. Novello and R.P. Neves, Class. Quant. Grav. 20 (2003) L67; “Apparent mass of the graviton in a de Sitter background”, gr- 16. J.A. Nieto, J. Saucedo, and V.M. Villanueva, Phys. Lett. A 312 qc/0210058. (2003) 175: hep-th/0303123. 5. E.A. Leon´ and J.A. Nieto, work in progress (2008). 17. T. Fukai and K. Okano, Prog. Theor. Phys.73 (1985) 790. 6. J.A. Nieto, Phys. Lett. A 262 (1999) 274; hep-th/9910049. 18. J.B. Hartle, Phys. Rev. D 29 (1984) 2730. 7. M. Henneaux and C. Teitelboim, Phys. Rev. D 71 (2005) 19. M. Novello and R.P. Neves, Class. Quant. Grav. 20 (2003) 024018; gr-qc/0408101. L67; M. Novello, S.E. Perez Bergliaffa, and R.P. Neves, “Re- 8. C.M. Hull, JHEP 0109 (2001) 027; hep-th/0107149. ply to ’Aacausality of massive charged -2 fields’ ”; gr- 9. S. Cnockaert, “Higher spin gauge field : Aspects of du- qc/0304041. alities and ”, (Brussels U., PTM), Jun 2006. 193pp., 20. S. Deser and A. Waldron, “Acausality of massive charged spin- Ph.D. Thesis, e-Print Archive: hep-th/0606121. 2 field”, hep-th/0304050. 10. A.J. Nurmagambetov, “Duality-symmetric approach to general 21. S.R. Cloude, Optik 75 (1986) 26. relativity and ”, To memory Vladimir G. Zima and Anatoly I. Published in SIGMA 2 (2006) 020; hep-th/0602145. 22. S. Huard, Polarization of (John Wiley & Sons 1997). 11. B. Julia, J. Levie, and S.S. Ray, JHEP 0511 (2005) 025; hep- 23. G. Atondo-Rubio, R. Espinosa-Luna, and A. Mendoza-Suarez,´ th/0507262. Opt. Commun. 244 (2005) 7.

Rev. Mex. F´ıs. 54 (3) (2008) 188–193