BORN - INFELD - RIEMANN GRAVITY

Transcription

1 BORN - INFELD - RIEMANN GRAVITY A Thesis Submitted to the Grdute School of Engineering nd Sciences of İzmir Institute of Technology in Prtil Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Physics by Selin SOYSAL December 2012 İZMİR

2 We pprove the thesis of Selin SOYSAL Exmining Committee Members: Prof. Dr. Durmuş Ali DEMİR Deprtment of Physics, İzmir Institute of Technology Assoc. Prof. Dr. Tonguç RADOR Deprtment of Physics, Boğziçi University Assist. Prof. Dr. Ftih ERMAN Deprtment of Mthemtics, İzmir Institute of Technology 14 December 2012 Prof. Dr. Durmuş Ali DEMİR Deprtment of Physics İzmir Institute of Technology Prof. Dr. Nejt BULUT Hed of the Deprtment of Physics Prof. Dr. Rmzn Tuğrul SENGER Den of the Grdute School of Engineering nd Sciences

3 ACKNOWLEDGMENTS While I ws working on my thesis, there re some people round me who helps nd supports me nd hence deserves my specil thnks. Firstly, I would like to express my endless respect nd grtitude to my supervisor Prof. Dr. Durmuş Ali DEMİR. It s gret plesure for me to work with him. Beside my supervisor, I would like to thnk to the members of committee Assoc. Prof. Dr. Tonguç RADOR nd Assist. Prof. Dr. Ftih ERMAN for their contribution nd giving suggestions. I would like to thnk to my fmily for their endless support nd ptience insipite of my non-stop studies, they love me in ll situtions. I m indebted to them for everything in my life. I would like to thnk specilly to Aslı Altş for being more thn sister. It is wonderful feeling to discuss physics nd be in scientific world with her. I lso thnk to Hle Sert for helping me ll the time, even she is in Germny. Moreover, Sıl Övgü Korkut deserves my pprecition since I lwys ber up to her when I need. I would lso like to thnk to Brış Ateş nd Kory Sevim for their vluble friendship. Besides ll these, I would like to thnk to my nerest friends Zeynep İlgezer, Cndn Gelir Ağpınr, Burcu Gürbüz for their supports from the childhood nd my officemtes, Cnn Nurhn Krhn nd Ayşe Elçiboğ for their friendship, ll the disscussions nd fun. I thnk to ll my friends t Izmir Institute of Technology for their friendship.

4 ABSTRACT BORN - INFELD - RIEMANN GRAVITY The Universe we live in hs strted with bng, very big bng. Its evolution nd globl structure re controlled by grvittion nd its mtter nd rdition content. Grvittion, curving of spcetime, formulted in miniml or extended versions, provides different lyers of understnding bout the Universe. Einstein s Generl Reltivity GR gives description of grvity, nd there re vrious resons for extending it. One such extension refers to unifying the other forces in Nture with grvity in the frmework of GR. The very first pproch in this direction ws due to Born nd Infeld who hve tried to unify electromgnetism with grvity. It is generliztion of metric tensor to hve both symmetric nd ntisymmetric prts gives rise to merging of Mxwell s theory with Einstein s theory. In lter decdes, ttempts hve been mde to unify the other forces s well. In this thesis study, we extend Born-Infeld grvity to unify grvity with non- Abelin forces in nturl wy. This, which we cll Born-Infeld-Riemnn grvity, is ccomplished by devising grvity theory bsed on Riemnn tensor itself nd subsequently generlizing this tensor to nturlly involve guge degrees of freedom. With this method, preserving the successes of Born-Infeld grvity, we re ble to combine Yng- Mills fields W, Z bosons s well s gluons with grvity. We perform phenomenologicl test of our pproch by nlyzing cosmic infltion generted by non-abelin guge fields. iv

5 ÖZET BORN - INFELD - RIEMANN ÇEKİM KURAMI Evrenimiz büyük ptlmyl oluştu. Bilim insnlrı trfindn oluşturuln teoriler temelde evrenimizin oluşumunu nlmy yöneliktir. Bu mç için oluşturulmuş teoriler çok çeşitlidir ve frklı frklı ele ldığımızd çözmeye çlıştığımız bzı problemlere nck cevp verebildikleri görülmektedir. A. Einstein d dhil olmk uzere ortk mç bu teorileri tek bir teori ltınd toplybilmek ve bu teori ile evrenimizin oluşumu, ivmelenmesi, kr mdde gibi bir çok problem icin çözüm sunbilmektir. Anck bugüne kdr tm nlmıyl bir teori oluşturulmmıstır. Ess problem çekim kuvvetini diğer temel kuvvetlerle birleştiremememiz ve tum temel kuvvetleri tek bir teori ltind toplymmmizdir. Çekimi diğer temel kuvvetlerle birleştirme çblrındn en önemlisi Born- Infeld çekim teorisidir. Bu teoride cekim ve elektromnyetizm bir ry getirilmeye clışılmıştır. Ylnızc metrik tensörü modifiye edilerek bşrılbilmiştir. Metrik tensörü hem simetrik hem de ntisimetrik prçlrdn oluşturulmuş ve ntisimetrik prç elektromnyetizm ile bğdştırılmıştır. Teoriye W-Z lnlrını vey gluonu ktmk istediğimizde ne olur? Born-Infeld tipi çekim bunun için yeterli midir? Eldeki teoriler bunun için yeterli değildir. Bu yüzden tezimizde yeni bir teori oluşturduk ve bu teoriyi ylnizc Riemnn tensörü simetrilerini kullnrk geliştirdik. Teorimiz, 4. dereceden tensörleri icinde brındırdığı için non-abelin lnlr izin verir ve kozmik ptlmyı d sğlr. v

6 TABLE OF CONTENTS CHAPTER 1. INTRODUCTION CHAPTER 2. THEORIES OF GRAVITATION Metric Theory of Grvity Metric-Affine Theory of Grvity Sclr-Vector-Tensor Theory of Grvity CHAPTER 3. INFLATIONARY COSMOLOGY Energy-Momentum Tensor Friedmnn Equtions Infltion CHAPTER 4. BORN - INFELD GRAVITY CHAPTER 5. BORN - INFELD - EINSTEIN GRAVITY CHAPTER 6. BORN - INFELD - RIEMANN GRAVITY Born - Infeld - Riemnn Grvity Reltion to Vector Infltion CHAPTER 7. CONCLUSION REFERENCES APPENDICES APPENDIX A. INVARIANT VOLUME AND DETERMINANT OF TENSORS.. 50 APPENDIX B. VARIATIONAL APPROACH APPENDIX C. FIELD-STRENGTH TENSOR APPENDIX D. EXPANSION APPENDIX E. TRACE OF A MATRIX vi

7 CHAPTER 1 INTRODUCTION Our Universe, which hs strted with ultr high-energy bng, hs its entire history controlled by grvittion. Grvittion, s we understnd it fter Einstein, shows itself s the curving of the spcetime13, 15, 26, 28. The spcetime my lso twirl or hve torsion. All such properties, miniml or extended, provides different lyers of understnding bout the Universe. It is with strophysicl nd cosmologicl observtions tht correct picture of the Universe will eventully emerge. Since the very first phses of Einstein s Generl Reltivity GR, unifiction of grvity nd other forces hs lwys been something desired. The problem is to find nturl wy of embedding other forces in the relm of grvittion. To this end, the very first pproch ws due to Born nd Infeld 10 who generlized metric tensor to hve both symmetric nd ntisymmetric prts, nd identified its ntisymmetric prt with the field strength tensor of electromgnetism. This wy, one finds wy of combining Mxwell s theory with Einstein s theory in single pot. This gives complete description of electromgnetic phenomen in wy covering wek nd strong field limits together. One my wonder, if it is possible to unify other theories with grvity. For exmple, cn one unify the W nd Z bosons with grvity? How bout the gluon which holds qurks together in nucleus? In Born-Infeld grvity s well s Born-Infeld-Einstein 25 grvity this is not possible. The reson is tht these theories involve determinnts of the generlized metrics of the form C 1 g αβ + C 2 F αβ, nd if the field strength tensor of the vector field crries n index like Fαβ with being the index of genertors then the rgument of the determinnt breks the guge symmetry. The determinnt over the group genertors, on the other hnd, does not bring ny improvement. Hence, it is simply not possible to unify Yng-Mills fields with grvittionl theory. Wht cn be done? The first observtion to mke is tht, grvittionl dynmics cn be written in terms of the determinnt of the Ricci tensor. This is Eddington s pproch 29. However, it is not necessry to limit ourselves to Ricci tensor. The Riemnn tensor itself cn be used s well. Giving ll the necessry detils of such formlism in Appendix A, we switch from two-index tensor theory to four-index tensor theory by bringing the Riemnn tensor in the gme. This pproch opens up host of phenomen to be explored. One ppliction, pertining to the min topic of this thesis work, is the uni- 1

8 fiction of non-abelin guge fields with grvity. The point is tht, now theory includes double-determinnt of four-index tensor hving the symmetries of Riemnn tensor, nd one redily finds tht Fαβ F µν is such tensor field constructed from non-abelin guge fields. We cn this formlism s Born-Infeld-Riemnn grvity s it involves the Riemnn tensor24, 60. Aprt from this very feture of unifiction, the non-abelin fields, in homogeneous nd isotropic geometries like the Universe itself, possess the crucil spect tht they cn fcilitte cosmic infltion 15, 67. To this end, the Born-Infeld-Riemnn formlism provides nturl frmework to nlyze guge infltion 44. In Chpter 2, we give n overview of the GR nd its known extensions 27, 31, 53, 54, 56, 58, 59. We review there GR, metric-ffine nd sclr-vector-tensor theories. This mteril proves importnt in further chpters of the thesis. In Chpter 3, we discuss infltionry cosmology 2, 13, 15, 37, 52, 54, 67, in brief. We give here min properties of the Universe, nd the need to cosmic infltion. After the cosmologicl bckground, we try to exmine the min ide of combining grvity nd electromgnetism with Born-InfeldBI Grvity in Chpter 4. In Chpter 5, we discuss how determinnt of Ricci tensor cn be used for obtining grvittionl field equtions. It is Born-Infeld-Einstein grvity. The min ide is to construct new theory including both Eddington nd BI pproch. In Chpter 6 we discuss Born-Infeld-Riemnn grvity nd guge infltion in detil. There we find how rich the model to yield the requisite dynmics for infltion. Finlly, in Chpter 7 we conclude the thesis. 2

9 CHAPTER 2 THEORIES OF GRAVITATION Grvittion or dynmicl spcetime hs been formulted in different contexts with different purposes. Since Einstein s 1916 formultion of Generl Reltivity GR, there hs risen different generliztions. In this Chpter we briefly summrize them Metric Theory of Grvity According to Einstein s formultion of GR, spcetime is differentible mnifold, nd metric tensor g αβ governs curving nd twirling of the spcetime. It is the sole field on the mnifold 13. In generl, on smooth mnifold, there re two independent dynmicl objects: metric tensor collection of clocks nd rulers needed for mesuring distnces nd ngles nd connection guiding force for geodesic motion. In GR, connection is not n independent vrible; it depends on the metric tensor vi Γ λ αβ = 1 2 gλρ α g βρ + β g ρα ρ g αβ 2.1 which is known s the Levi-Civit connection. It is symmetric in lower indices since metric tensor is symmetric 13, 67. Since Levi-Civit connection is expressed in terms of the metric tensor, GR is described by single vrible; the metric tensor. The field equtions of GR re obtined by vrying the Einstein-Hilbert ction S g = { } 1 d 4 x g 1/2 2 M P 2 lrg + L mt g, ψ 2.2 with respect to the metric tensor See Appendix B for more generl fetures.. Here, M P l = 8πG N 1/2 is the Plnck scle or the fundmentl scle of grvity. The ction density depends on the curvture sclr Rg = g µν R µν Γ 2.3 3

10 where R µν Γ = R α µανγ 2.4 is the Ricci tensor, nd R α µβνγ = β Γ α µν ν Γ α µβ + Γ α βλγ λ µν Γ α νλγ λ µβ 2.5 is the Riemnn curvture tensor. The spcetime mnifold is perfectly flt in the close vicinity of x 0 µ if ll the components of Riemnn tensor vnishes t tht point; R α µβν Γx0 = 0. Vrition of the ction 2.2 gives the Einstein equtions of grvittion 4, 13, 15, 28, 41, 46, 49, 51, 52, 53, 54, 66, 67 Appendix B for detils. R µν 1 2 Rg µν = 8πG N T µν 2.6 whose right-hnd side T µν = 2 δl mt δg µν + g µν L mt g, ψ 2.7 is the energy-momentum tensor of mtter nd rdition. Here, S mt g = d 4 x g 1/2 L mt g, ψ 2.8 is the ction of the mtter nd rdition fields ψ. The curvture sclr nd mtter Lgrngin L mt both involve the sme metric tensor g µν. It is importnt tht the field equtions 2.6 rises from the grvittionl ction 2.2 by dding n extrinsic curvture term. The reson is tht, curvture sclr Rg involves second derivtives of the metric tensor, nd in pplying the vritionl equtions it is not sufficient to specify δg µν t the boundry. One must lso specify its derivtives δ α g µν t the boundry. This dditionl piece does not dmit construction of the Einstein- 4

11 Hilbert ction directly; one dds term extrinsic curvture to cncel the excess term. The metric formlism is the most common pproch to grvittion becuse equivlence principle is utomtic, geodesic eqution is simple, nd tensor lgebr is simplified metric tensor is covrintly constnt. Equivlence principle mens tht grvittionl force cting on point mss cn be ltered by choosing n ccelerted non-inertil coordinte system. In other words, there is lwys frme where connection cn be set to zero. This point corresponds to loclly-flt coordinte system. The curvture tensor involves both Levi-Civit connection nd its derivtives, nd mking the connection to vnish does not men tht curvture vnishes Metric-Affine Theory of Grvity As we mentioned in the previous section, in metricl theories of grvity, the metric tensor nd connection re not independent vribles. However, connection does not hve to be symmetric. It cn be ntisymmetric in lower indices, or it cn hve no symmetry condition which mens tht it hs both symmetric nd ntisymmetric prts. Such theory tht hs metric tensor nd ffine connection s priori independent vribles is clled the Metric-Affine theory of grvittion 8. Indeed, when the field equtions re formed for Einsten- Hilbert ction by tking metric tensor nd connection s priori independent dynmicl quntities, it is obviously shown tht the resulting connection turns out to be the sme s the Levi- Civit connection. This spect, known s Pltini formlism, overcomes the difficulties of metricl theory by eliminting the need to extrinsic curvture. This dynmicl equivlence between GR nd Metric-Affine pproch holds when mtter sector keeps involving only the Levi-Civit connection. Indeed, if the mtter Lgrngin only couples to the Levi-Civit connection i.e. it does not couple to generl connection, Metric-Affine formultion lwys reduces to metric formultion 17. For pure Einstein- Hilbert ction the Pltini nd Metric-Affine formultions re the sme while for the other theories they split s explined. Metric-Affine theory of grvity we do not refer to Pltini formultion involves the metric tensor g µν nd generl connection ˆΓ α µν where ˆΓ α µν Γ α µν. It is non- Riemnnin grvittionl theory. Indeed, generlity of connection enbles one to define the new geometro-dynmicl structures which re obtined by using generl connection definition. These new geometro-dynmicl structures re clled torsion nd non-metricity tensors. Torsion tensor rises from the ntisymmetric prt of connection. The nonmetricity tensor, conversely to the metric theory of grvity which is metric-comptible, is 5

12 the covrint derivtive of metric tensor ccording to the generl connection ˆΓ α µν. Mthemticlly, T α µν = ˆΓ α µν ˆΓ α νµ nd Q αµν = α g µν. 2.9 As dynmicl theory, Metric-Affine grvity is non-trivil theory does not reduce to GR dynmiclly if the mtter Lgrngin involves the generl connection ˆΓ α µν explicitly. The Metric-Affine grvity hs one more brnch in ddition to the Pltini formlism: It is the Einstein-Crtn theory 39. This theory is metric-comptible nd torsion tensor is non-vnishing. The geometry which Einstein-Crtn theory defined is clled the Riemnn-Crtn geometry Sclr-Vector-Tensor Theory of Grvity As we mentioned in Introduction, GR is grvittionl theory which hs mgnificent success to explin the universe. In this section, we re going to exmine such theories whose im is to nswer some contrdictions tht could not be explined by GR. Grvittionl force is medited by spin-2 tensor field coming from metric tensor in given bckground in the GR. However, the grvittionl sector cn be expnded to include other spin multiplets s well 17. Those other fields cn be sclr spin 0 field, vector spin 1 field or both. From this point of view, one figures out tht there re extended grvittionl theories which include dditionl spin multiplets nd hence dditionl degrees of freedom. The first type of extended theories, which re well-studied lterntive theories of GR, re sclr-nd-tensor theories. These theories consist sclr field φ in ddition to the metric tensor 16, 17, nd their generl ction is given by S = d 4 x g 1/2 R ωφ φ 2 L mt Ψ, φ 1 g where g µν is Einstein metric becuse the ction given in 2.2 is written in Einstein frme theory hs well-defined Newton s constnt. In fct, the eqution 2.10 stnds like in ddition to Einstein-Hilbert ction there re some dditionl terms referring to sclr fields 16, 43. Einstein frme is useful to discuss generl chrcteristic of such theories. However, it is esily shown tht sclr-tensor theory which is defined in Einstein frme is not 6

13 metricl theory since its Mtter Lgrngin couple to φ 1 in ddition to the metric tensor. To work on metricl theory one cn define Einstein metric in terms of physicl metric s in the given form 43; g µν = φg µν 2.11 nd then the ction 2.10 tkes the form 43. S = d 4 x g 1/2 Rφ ωφ φ gµν µ φ ν φ + L mt Ψ, g 2.12 This ction of the sclr-tensor theory is metricl theory, nd its frme is clled Jordn frme the theory needs well-defined Newton s constnt. A well-known exmple of sclr-tensor theories is Brns-Dicke theory 43. If ω in the ction 2.10 does not depend on φ, then the ction tkes the form of Brns- Dicke theory S = d 4 x g Rφ 1/2 ω φ gµν µ φ ν φ + L mt Ψ, g 2.13 As metricl theory of grvity, the Brns-Dicke theory is purely dynmicl one since the field equtions for the metric tensor involve sclr fields nd vice vers 16. The second type of extended theories constructed by using dditionl fields re vector-tensor theories. In this kind of theory, there is dynmicl 4-vector field u µ in ddition to metric tensor. Generl ction of these theories given s 16 S = 16πG 1 d 4 x g 1/2 R 1 + ωu µ u µ K µν αβ µu α ν u β + λ 1 + u µ u µ +L mt Ψ, g 2.14 where K µν αβ = c 1g µν g αβ + c 2 δ µ αδ ν β + c 3 δ µ β δν α c 4 u µ u ν g αβ

14 If the vector field is time-like i.e ds 2 < 0, vector tensor theories become constrined theory where u µ u µ = 1 nd vector field hs unit norm. If there is no condition on vector fields, it is unconstrined theory. The well-known vector tensor theory is Einsten-Aether theory which is constrined one i.e. u µ is time-like, u µ u µ = 1 nd hs unit norm 16. vector-tensor theory with given ction is lso metricl theory since mtter Lgrngin only couples to metric tensor. The lst type is the sclr-vector-tensor theory TeVeS. TeVeS is the reltivistic generliztion of Modified Newtonin Dynmics MOND 9, 45. It is proposed to eliminte the drk mtter prdigm by explining glxy rottionl curves vi modified Newtonin dynmics. Even if it could not replce drk mtter prdigm without ny unknown mtter, it hs gret success to explin glxy rottionl curves 17. The ction for the TeVeS theory is given s S = S g + S A + S φ + S m 2.16 where g is clled s Bekenstein metric g µν = e 2φ g µν sinh2φa µ A ν 2.17 which is time-like for g µν A µ A ν = The vrious prts of the ction 2.16 re given by S g = 16πG 1 d 4 x g 1/2 R 2.19 S A = 32πG 1 d 4 x g 1/2 KF µν F µν 2λA µ A µ

15 S φ = 16πG 1 d 4 x g µĝ 1/2 µν µ φ ν φ + V µ 2.21 where 17 ĝ µν = g µν A µ A ν There is one more wy to obtin extended grvittionl theories. As we know, generl reltivity hs t most second order derivtives. Therefore, we cn obtin nother type of extended theory by using higher derivtives. The well-known theory is fr grvity. fr is the function of sclr curvture R thus this theory generlizes Einstein s GR. Their ction is given s 17, 63. S = d 4 x g 1/2 fr Finlly, s nother extended grvittionl theory we recll the Born-Infeld theory of grvittion. In Born-Infeld grvity, the min ide is tht in constructing invrint ctions ll we need is determinnt of tensor see Appendix A nd this tensor does not need to be the metric tensor s in ll the grvittionl theories we hve mentioned bove. Any other rnk-2 tensor, for exmple, the very Ricci curvture tensor, cn well do the job. Since this theory is the min topic of this thesis work, it will be detiled in the following chpters. 9

16 CHAPTER 3 INFLATIONARY COSMOLOGY Wht is cosmology nd why is cosmology so importnt? Cn one explin some problems which re unsolved from the viewpoint of GR? People hve sked these type of questions for yers. When Einstein discovered the field equtions 2.6, which equtes the curvture of spce-time to directly the energy-momentum source of mtter nd rdition, he thought tht the universe must be sttic. In other words, it ws necessry to dd n extr term to hve sttic solutions of G µν = 8πG N T µν 3.1 Tht extr term is constnt curvture source known s the cosmologicl constnt CC, Λ 21, 23. This dditionl Λ term is lso geometricl modifiction to the field equtions. To blnce the grvittionl ttrction this term must yield repulsive effect. G µν + Λg µν = 8πG N T µν 3.2 where Λ denotes vcuum energy which mens tht it is constnt energy density nd hs negtive pressure. At this point we should discuss the mening of negtive pressure. Why does vcuum energy hve negtive pressure? There is bsic physicl explntion to the notion of negtive pressure. For prticulr conserved system, energy per unit volume is constnt 66. When the energy of the system is chnged, volume chnges directly in proportion to the rte of energy chnge. This ctully mens tht the blnce between the ttrction by grvity nd the needed repulsion is supplied by vcuum energy. Considering universe without CC mens tht the universe is empty. On the other hnd, if we consider vcuum stte, it corresponds to vnishing energy-momentum tensor T µν = 0. Rerrnging eqution 3.2 G µν = Λg µν = 8πG N T Λ µν

17 Here Tµν Λ is energy-momentum tensor of the vcuum stte nd CC becomes new source term. Actully, when we interpret grvity theoreticlly, sttic universe seems impossible. Becuse of the fct tht not only reltivistic grvity but lso non-reltivistic grvity is ttrctive. Therefore, becuse of the ttrction sttionrity of the universe cn not be ttined. At the beginning of the twentieth century, with the help of new genertion telescopes lots of glxies nd glxy sets begn to be discovered in the visible universe 53. As conclusion of the discoveries, the distribution of the glxies ws interpreted nd it is understood tht the universe is homogeneous nd isotropic. Homogeneity mens tht the universe looks the sme from every point. Isotropy lso mens tht there is no centre of the universe. We re going to exmine these notions, homogeneity nd isotropy, with Friedmnn equtions 2, 13, 15, gin. In 1929, with Hubble s gret discovery it s understood tht the universe is not sttic. On the contrry, the universe is expnding. The min clue ws the red-shifting of the light spred by nerby glxies which concludes tht the glxies re moving wy from us rdilly -nd of course from ech other-. One my expect tht the expnsion slows down becuse the glxies ttrct ech other due to grvittion. But ginst the expecttion, expnsion ccelertes. Mgiclly the observed ccelertion is comptible with the vcuum stte energy density ide which ws not even liked by Einstein t first. How cn the dynmics of sptilly homogeneous nd isotropic universe be determined? Are Einstein s field equtions sufficient for explntion? We re going to try to nswer these questions. As consequence of isotropy one my write metric tht is sphericlly symmetric. Beside this, generlly the metric cn tke the form s; ds 2 = dt t 1 kr 2 dr2 + r 2 dω which is the Friedmnn-Robertson-Wlker FRW metric 15, 66, 67. In eqution 3.4 dr is rdil prt nd dω is zimuthl prt of sphericlly symmetric spces. There k denotes constnt sptil curvture. Due to the geometry of spce-time vlues of k differs: k { 1, 0, +1} corresponds to open, flt nd closed geometries, respectively. Hyperboloid geometries, k = 1, which corresponds constnt negtive curvture expnd foreveropen. Flt geometries, k = 0, which cuses the curvture term to vnish expnd forever. Sphericl geometries, k = +1, which corresponds constnt positive curvture 11

18 expnsion will be cesed nd it turns into singulr stte. In our thesis work we re going to ssume k = 0 nd modify the field equtions s flt spce. Then our metric tkes the form ds 2 = dt t dr 2 + r 2 dω or in crtesin coordinte system ds 2 = dt t dx 2 + dy 2 + dz The mtrix form of the metric tensor g µν = t t t 3.7 eses the identifiction of the pressure nd energy densities for given cosmologicl fluid. In the bove, the function t is the scle fctor tht depends only on time. In other words t is function tht is responsible for time evolution nd the expnsion of the universe. It is the rdius of the evolving sphere. To construct the dynmics of the homogeneous nd isotropic universe we should substitute our chosen metric into Einstein s field equtions. To do tht we should clculte the Ricci Tensor components nd Ricci Sclr: As seen in eqution 3.7 g tt = 1, g ti = 0, g ij = 2 tδ ij. Inserting these components into Levi-Civit connection, surviving terms re Γ t ij = ȧδ ij Γ i tj = ȧ δi j

19 from which the Ricci tensor cn be directly clculted: ä R tt = 3 R ti = 0 R ij = ä ȧ Contrcting with metric tensor one gets the Ricci sclr: R = R tt + 3R ij = 6 ä + ȧ Here, we should interpret the mening of ȧ. Assume tht there re two isotropic observers ny time t nd the distnce between these observers is D. The rte of chnging of D is υ dd dt = dd d d dt = D d dt = D ȧ 3.11 so one cn write dd dt = HD where we defined H = ȧ 3.12 s the Hubble prmeters. Hubble prmeter hving the dimension of inverse time or mss mesures the frctionl growth rte of the Universe s size Energy-Momentum Tensor First of ll let us exmine the mening of electromgnetic current nd its components. One my consider 4-electromgnetic current; j µ = ρ, j where ρ is electric chrge density nd j is current density. As we know electric chrge density is the electric chrge per volume, we cn demonstrte mthemticlly s ρ = q x y z nd current density is chrge flow per unit time nd per re, we cn demonstrte mthemticlly s j = If we combine them s 4-vector electromgnetic current; j µ = q V µ q. t y z where V µ is vol- 13

20 ume. With the help of definition of 4-vector electromgnetic current one cn construct energy momentum tensor of spce-time vi 4-momentum which is p µ = p 0, p = E, p. Mthemticlly, T µν = pµ V ν 3.13 As seen directly in Einstein s field equtions 2.6, energy momentum tensor is the source of grvity. One of the importnt fetures of energy momentum tensor for conservtive system is its conservtion. If there is n dditionl source, then energy momentum tensor does not conserve. Wht is the physicl mening of the components of energy momentum tensor? One cn directly clculte them from the generl definition of energy momentum tensor 3.13 s the following: T 00 = p0 V 0 = E x y z 3.14 which mens tht energy per unit volume energy density. T 0i = p0 V i = E t y z 3.15 which mens tht energy per unit time per re energy flux. T i0 = pi V 0 = p i x y z 3.16 With little help of lgebr of reltivity one cn esily see tht eqution 3.15 nd eqution 3.16 re equl to ech other T 0i = T i0. It mens tht energy momentum tensor T µν is symmetricl rnk 2,0 tensor. For digonl components of energy momentum tensor; T ii = pi V i = p i t y z } {{ } re = normlforce re

21 According to eqution 3.17 bove momentum flux is directly proportionl to pressure. The off-digonl terms does not supply this feture. In the presence of this section we cn sy tht the energy momentum tensor of prticulr system includes ll the fetures of the system such s energy density, momentum flux, energy flux etc. We cn clssify prticles ccording to their energy or momentum flux. The simplest type is dust. Dust includes non-intercting prticles. If the prticles do not interct with ech other, pressure is not mentioned for this kind of systems. Such systems hve the energy momentum tensor s the following form; T µν = ρ nd 4-velocity is of the form; U µ = c, 0 To exmine the systems which hve huge number of prticles, for simplicity, we try to nlyse these type of systems s doing pproximtions. One of the useful pproximtion is perfect fluid pproximtion. In our work we lso use perfect fluid pproximtion. It s continuous system whose elements interct only through norml force. As mentioned bove intercting through norml force llows energy momentum tensor only hs digonl components. T µν = ρ p p p ρ + p U µ U ν + g µν p 3.19 For n rbitrry coordinte frme we cn use the expression Friedmnn Equtions We exmined energy momentum tensor nd perfect fluid form in Section 1 in detil. Now, we hve sufficient bckground to derive the Friedmnn Equtions. 15

22 Inserting 4-velocity U t = 1 nd U t = 1 U i = 0 nd U i = 0 in 3.19 time component tkes the form s; T tt = ρ + p U t U t + g tt p = ρ G tt = 8πG N ρ 3.20 nd sptil components tke the form s; T ij = ρ + p U i U j + g ij p = pδ ij G ij = 8πG N δ ij p 3.21 Combining 3.20, 3.21 nd Einstein field equtions for empty spce together, G tt = R tt 1 ȧ 2 2 g ttr = 3 = 8πG N ρ 3.22 nd G ij = R ij 1 2 g ijr = 2ä ȧ 2 = 8πG N δ ijp 3.23 From eqution 3.22; ρ = 3 8πG N ȧ

23 From eqution 3.23; 3ä = 4πG N 3p + ρ 3.25 The equtions 3.24 nd 3.25 re clled Friedmnn Equtions. One cn extrct importnt informtion from Friedmnn Equtions. First of ll eqution 3.25 indictes tht ä < 0 which is n exct proof of non-sttionry of the universe. In n other words the universe either expnds ȧ > 0 or contrcts ȧ < 0. Hubble s discovery bout redshift shows tht the universe expnds ȧ > 0 which mens tht the universe expnds fster nd fster. Under the homogeneous nd isotropic universe ssumption, GR predicts tht t time pproximtely less thn H 1 when =0 the universe ws in singulr stte. The zero size spce is conclusion of homogeneity. In n other words t this t time the distnce between the points in the spce ws zero nd the curvture of spcetime ws infinite. Essentilly, this is the definition of -Big Bng-. The extension of spce-time mnifold beyond the big bng mkes no sense. This interprettion of GR concludes tht the universe begn with the big bng. These interprettions is comptible with cosmologicl solutions since the singulrities come from the cosmology. For the mss density evolution, multiplying eqution 3.24 by 2 nd derivting with respect to t, one finds tht ρ + ȧ 2 + ρ + 3p = For dust p = 0; ρ + ȧ 2 + ρ = 0 ρ 3 = constnt ρ

24 For rdition p = ρ 3 ; ρ + ȧ 2 + 3ρ = 0 ρ 4 = constnt ρ These rtios show tht when increses, energy density of rdition decreses fster thn energy density of dust. Hence, the rdition content of the present universe my be neglected. But s t goes to zero for the pst process rdition should hve been dominted over the ordinry mtter Infltion As seen in Section 3.2, lthough the erly universe is rdition dominted; the present universe is mtter dominted 67. In Friedmnn eqution vcuum energy does not exist tht is clled fltness problem. Beside tht for FRW cosmologies horizon problem exists15. Actully, the motivtion comes from the Spontneously Symmetry Breking SSB for the infltionry models. Bsiclly, the question is tht how did the expnsion begin? If it is cused by the CC, Λ, then which sort of physics cuses this lrge energy? People hve tried to interpret this infltionry phse in the erly universe for yers. The most common view is tht the huge vcuum energy Λ is cused by the potentil of sclr field which is clled inflton. Suppose tht theory which includes sclr field is constructed. Then, one my derive the dynmicl eqution for this sclr field in FRW metric s follows φ + 3H φ + V φ = where V φ is derivtive of potentil with respect to φ nd H 2 = N 1 2 φ 2 + V φ

25 For infltion models t the erly universe potentil energy domintes kinetic energy. So the second derivtive of φ is very smll. It shown mthemticlly s the following; φ 2 V φ 3.31 φ 3H φ, V φ 3.32 In the presence of tht one my define prmeters which re clled slow-roll prmeters tht yield conditions of infltion: V ɛ = 1 2 V nd η = V V 3.34 We cn lso define the prmeters in terms of the Hubble prmeter. The only difference between these two nottions is tht 3.33 nd 3.34 cuse rolling slowly for while. If we define the prmeters vi Hubble constnt it cuses the field to roll slower. The min point is tht, the Universe must strt with n exceedingly flt inflton potentil, evolve slowly, nd finlly lnd to minimum of the potentil where its oscilltions rehet the universe to give tody s structure. 19

26 CHAPTER 4 BORN - INFELD GRAVITY In 1934 Born nd Infeld 10 extended Mxwell s theory of electrodynmics from liner nture to non-liner nture. For the wek field limit, Mxwell s Theory works well but for strong field limit it fils. The min ide of B-I Grvity is to modify Lgrngin density. New terms would be dded, if one ensures tht Lgrngin is still sclr density with ll of the terms in it. See Appendix A To modify Lgrngin density, we consider rnk-2 tensor field which is neither symmetric nor ntisymmetric clled αβ. As we know from Appendix A, the Lgrngin density is of the form L = µν 1/2 4.1 where µν is the determinnt of the tensor field µν. An rbitrry tensor field is formed of symmetricl field nd n nti-symmetricl field. According to this let µν = g µν + f µν. It mens tht the symmetricl prt of µν is metric tensor nd the nti-symmetricl prt of µν is totlly ntisymmetric n it is tken to be proportionl to the field strength tensor of electromgnetic field F EM µν = µ A ν ν A µ. With this identifiction, L g µν + f µν 1/2 4.2 where for the purpose of constructing the Mxwell ction we write L = g µν + f µν 1/2 g µν 1/2 4.3 s the complete ction. We re going to expnd our expression for smll f µν : S = d 4 x g µα 1/2 δ α ν + f α ν 1/

27 by using the identity See Appendix D 1 + A 1/2 = T ra T ra2 1 4 T ra2 + OA Then the second term in ction 4.4 expnds s I + f α ν 1/2 = T rf T rf2 1 4 T rf2 + Of Since f α ν is totlly nti- symmetric tensor, both T rf nd T rf 2 give vnishing contribution to the expnsion. Only one term survives: T rf 2 = T rf α ν f ν β = f α ν f ν α = f µν f µν 4.7 Then, our ction tkes the form S = d 4 x g αβ 1/2 1 4 f µν f µν 4.8 which is known ction for electromgnetic field. From this ction we re ble to derive Mxwell s equtions for electromgnetic field. Since its proposl by Born nd Infeld, this theory of electromgnetism hs been developed nd pplied to different problems. As consequence there occured different type of theories with different properties. These theories re clled Born-Infeld type theories 5, 6, 11, 19, 20, 33, 36, 40, 47, 68. The common point of Born-Infeld grvity theories re tht when we exmine the ction 4.4 functionl, we relize tht it seems impossible to embed non-abelin guge fields into the theory. If we force to embed them into the theory, guge invrince of theory is broken. The min topic of this thesis work is Born-Infeld-Riemnn grvity, nd it will be discussed in Chpter 6. 21

28 CHAPTER 5 BORN - INFELD - EINSTEIN GRAVITY Deser nd Gibbons 25 suggested to modify BI grvity. Insted of dding field strength tensor of electromgnetic field, it is lso good ide to dd Eddington term in ction. In other words, the ction cn contin the curvture tensor nd the metric tensor. It mkes ction purely grvittionl nd geometricl: S D G = d 4 x g µν + br µν 1/2 5.1 where nd b re constnts. Applying the procedure in Chpter 4, one gets: S D G = = = d 4 x g µα δν α + b Rα ν 1/2 d 4 x g µα 1/2 δ αν + b Rαν 1/2 d 4 x 2 g µα 1/2 δ αν + b Rαν 1/2 5.2 Now, the second term bove cn be expnded s δν α + b Rα ν 1/2 = b 2 T rr + 1 b 8 T rr2 1 b 4 T rr2 + OR in the smll curvture limit. As we know T rr 2 nd T rr 2 terms cuse ghosts. To brek wy ghosty terms we should dd n rbitrry tensor field X µν to our theory. In the presence of tht, ction tkes the form; S D G = d 4 x g µν + br µν + cx µν 1/

29 Here,, b nd c re coupling constnts. The sme expnsion procedure gives now: S D G = = = d 4 x g µα δν α + b Rα ν + c Xα ν 1/2 d 4 x g µα 1/2 δ αν + b Rαν + c Xαν 1/2 d 4 x 2 g µα 1/2 δ αν + b Rαν + c Xαν 1/2 5.5 whose second term in rdicl sign is expnded s See Appendix D δν α + b Rα ν + c Xα ν 1/2 = b 2 T rr + 1 c 2 T rx b T rr + c 2 T rx 1 4 T r b Rα ν + c Xα ν 2 + O3 5.6 Inserting expnsion 5.6 in the ction 5.5, we get: S D G = d 4 x 2 g µα 1/ b 2 T rr + 1 c 2 T rx + 1 b 2 8 T 2 rr2 + c2 T 2 rx2 + 1 bc T rrt rx b 2 4 T r R µνr µν + c2 2 X µνx µν + bc Xµν R µν S D G = d 4 x 2 g µα 1/ b 2 T rr + 1 c 2 T rx + 1 b 2 T rr c 2 8 T 2 rx2 + 1 bc 8 4 T rrt rx 1 b T rr µνr µν 2 + c2 T rx µνx µν 1 bc 2 2 T 2 rxµν R µν S D G = d 4 x 2 g µα 1/ b 2 T rr + 1 c 2 T rx + 1 b T rr2 T rr µν R µν + 1 c T rx2 T rx µν X µν + bc T rrt rx 22 1 c 2 2 T 2 rxµν R µν 5.7 To cncel out the ghost term 1 2 T rr2 T rr µν R µν we mke use of the tensor field X µν 23

30 to set 1 4 b T rr2 T rr µν R µν = 1 c T rx or b 2 T rx = 1 2 c 1 2 T rr2 T rr µν R µν 5.9 Letting X µν = 1 4 g µνx α α we get X µν = b2 1 8c 2 T rr2 T rr αβ R αβ g µν 5.10 Since X µν is order of R 2, the terms which hve X µν R µν mixed terms cncel out. Becuse these terms re in cubic or higher contributions. Then our ction tkes the form; S D G = d 4 x 2 g µα 1/ b 2 R 5.11 This theory is nothing but the GR ction with cosmologicl constnt. It is importnt to note tht, if we re to construct consistent theory we need to introduce some extr tensor field X µν to cncel the ghost-giving higher curvture terms. The min dvntge of the Born-Infeld-Einstein grvity is tht it provides us with cler rtionle for Einstein-Hilbert term. Tht term rises s the smll curvture limit of generl determinnt theory. The Eddington theory 7, 29 corresponds to tking = 0, tht is, killing the metric tensor. Tht theory resides in complete ffine spce where there is no notion of distnce. One reclls here tht, the Ricci tensor does not involve the metric tensor but Ricci sclr does. The disdvntge of the Born-Infeld-Einstein grvity is tht it does not llow us to embed non-abelin guge fields into the theory due to the ction functionl 5.1. Otherwise the guge symmetry is broken. Although the disdvntge of the theory, it is n ccomplished theory for Abelin guge fields included 48, 64,

31 CHAPTER 6 BORN - INFELD - RIEMANN GRAVITY In this chpter we give generliztion of Born-Infeld-Einstein grvity. We try to construct theory which includes directly the Riemnn tensor i. e. rnk 0,4 tensor field not 0,2 one. We re going to see tht, constructing theory from rnk - 0,4 tensors llows us to expnd the determinnt to include non-abelin guge fields in ddition to the electromgnetic field. This method hs not been discovered before, nd proves highly importnt in unifying non-abelin guge fields nd grvity 24. Also importnt is tht, in this theory vector infltion comes out nturlly 60. This hppens with no need to extr degrees of freedom. All these fetures re going to be the subject mtter of this chpter. From Appendix A, we know tht the notion of determinnt cn be generlized to higher-rnk tensors. For instnce, if T αβ nd F αβµν re two tensor fields, one cn form n invrint volume s d 4 x T αβ 1/2 or s d 4 x DDetF αβµν 1/4 where DDet stnds doubledeterminnt s needed by rnk-4 tensor 61, 62. The detils cn be found in Appendix A. Also the reference 22 gives more generl description Born - Infeld - Riemnn Grvity For D dimensions we write the effective ction s S eff = d D xm D/2 D DDet κ 2 g µανβ + λ RR µανβ + λ F F µανβ + λ F Fµα F νβ + X µανβ 1/4 6.1 where M D is mss prmeter. The prmeter κ is curvture constnt which hs mss dimension 2. Constnt curvture term g µανβ equls g µανβ = g µν g αβ g µβ g αν

32 which is nothing but the curvture tensor of spcetime with constnt sclr curvture proportionl to κ 2. This term hs different importnce. When we expnd the doubledeterminnt, it is going to be expnded round this constnt-curvture spcetime configurtion. In 6.1 λ R is dimensionless constnt nd the mss dimensions of λ F nd λ F re λ F = λ F = 2. Finlly, the field X µανβ is n rbitrry tensor field introduced for cnceling ghosts. Now, let us discuss the term F µανβ which hs the sme symmetries s g µν g αβ g µβ g αν nd the Riemnn Curvture Tensor R µανβ. It is esy to see tht, for electromgnetic field, this tensor field ttins the unique form F µανβ = F µα F νβ. 6.3 This term unifies Mxwell theory nd grvittion in the wy Born-Infeld grvity does. In other words, insted of generlizing metric to g µν g µν + F µν s in Born-Infeld grvity, we cn construct the tensor field 6.3 which hs the sme symmetries s the Riemnn tensor. The results of the two pproches will be the sme. However, the min novelty is not the inclusion of Mxwell theory; the novelty is tht the Yng-Mills theories cn be unified with grvity 24. Indeed, the tensor field 6.3 directly generlizes to F µανβ = F µαf νβ 6.4 where runs over the djoint of the group. For SUN = 1,..., N 2 1. The min novelty is tht the non-abelin guge fields cnnot be included in the Born-Infeld formlism which is not guge invrint becuse of Fµα ppering with index not contrcted. Here, in the Born-Infeld-Riemnn formlism, s we cll it, it is utomtic. This generliztion of Born-Infeld theory thus opens new venue where one cn embed the Yng-Mills dynmics in grvittionl dynmics, which ws not possible before. As usul, the dul of Fνβ is defined s F νβ τκ = ɛνβ Fτκ. 6.5 In Born-Infeld-Riemnn grvity, guge fields enter the double-determinnt in 26

33 guge-invrint wy, nd hence, both Abelin electromgnetism nd non-abelin Yng-Mills theories like electrowek theory nd chromo-dynmics theories re nturlly included in the formlism 24. In our work we exmine our theory for homogeneous nd isotropic Universe FRW bckground which is explined in Chpter 3, in detil. We rewrite the effective ction for FRW bckground nd clculte the equtions of motion. Specilizing to D = 4 we get, S eff = d 4 xm 2 D DDet κ 2 g µανβ + λ R R µανβ + X µανβ + λ F F µανβ S eff = d 4 xm 2 D + λ F F µα F νβ 1/4 DDet κ 2 g µν g αβ g µβ g αν + λ R R µανβ + X µανβ + λ F F µανβ + λ F F µα F νβ 1/4 6.6 Now, pplying the sme procedure we did in Born-Infeld nd Born-Infeld-Einstein grvities we get S eff = d 4 xm 2 D DDet κ 2 g µν g αβ g µβ g αν 1/4 1 DDet 2 δν ν δ β β δν β δν β + λ R α ν β Invµ R κ2 µ α νβ + 1 κ 2 Invµ α ν β X µ α νβ + λ F κ 2 F µ α νβ + λ F κ 2 F µ α F νβ 1/4 6.7 Let us first compute the first fctor in the integrl. Using the notion of double-determinnt 27

34 See Appendix A we obtin DDet κ 2 g µν g αβ g µβ g αν 1/4 = κ 2 DDet g µν g αβ g µβ g αν 1/4 2! 2 = κ 2 DDet 1 2! 2 ɛ µ 1µ 2µα ɛ µ1 µ 2 ν β 1/4 = κ 2 1 2! 2 DDet ɛ µ 1 µ 2µα ɛ µ1 µ 2 ν β 1/4 = κ 2 g ρσ 1/2 1 ɛ µ 1 µ 2 µ 3 µ 4 ɛ α 1 α 2 α 3 α 4 ɛ ν 1 ν 2 ν 3 ν 4 ɛ β 1 β 2 β 3 β µ 4 ɛ 1 1/4 1 µ1 2 µ 1 α 1 ɛ µ2 1 µ2 2 µ 2 α 2 1/4 ɛ µ3 1 µ3 2 µ 3 α 3 ɛ µ4 1 µ4 2 µ 4 α 4 ɛ µ 1 1 µ 2 ɛ 1 ν 1 β 1 µ 1 ɛ 2 µ2 1 ν 2 β 2 µ 3 ɛ 1 µ3 2 ν 3 β 3 µ 4 1 µ4 2 ν β For simplicity let us cll the coefficient of the term κ 2 g ρσ 1/2 s C DD. It reds explicitly C DD = 1 ɛ µ 2! 2 1 µ 2 µ 3 µ 4 ɛ α 1 α 2 α 3 α 4 ɛ ν 1 ν 2 ν 3 ν 4 ɛ β 1 β 2 β 3 β µ 4 ɛ 1 1/4 1 µ1 2 µ 1 α 1 ɛ µ2 1 µ2 2 µ 2 α 2 1/4 ɛ µ3 1 µ3 2 µ 3 α 3 ɛ µ4 1 µ4 2 µ 4 α 4 ɛ µ 1 1 µ 2 ɛ 1 ν 1 β 1 µ 1 ɛ 2 µ2 1 ν 2 β 2 µ 3 ɛ 1 µ3 2 ν 3 β 3 µ 4 1 µ4 2 ν β As result, we get: DDet κ 2 g µν g αβ g µβ g αν 1/4 = CDD κ 2 g ρσ 1/ Hence, we hve clculted the first prt of 6.8. It is expected tht the double-determinnt of the Riemnn curvture tensor of constnt-curvture spcetime yields the squre of the determinnt of the metric tensor. Now, we proceed with the remining clcultion. First of ll, Inv µ α ν β is nothing but the 4-index identity tensor. Its explicit expression s well s vrious other detils re given in Appendix B. With the clculted pieces replced in, the effective ction tkes the form S eff = 1 d 4 xc DD MDκ 2 2 g DDet 2 1/2 δν ν δ β β δν β δν β + λ R β Rν κ2 νβ + 1 β κ 2 Xν νβ + λ F ν β F F κ2 νβ + λ 1/4 F ν β τκ F ɛ κ2 νβ Fτκ

35 which is of the form S eff = d 4 xc DD M 2 D κ2 g 1/2 DDet I + A 1/ From Appendix D, we know the expnsion of the term DDet I + A. Let us now pply the expnsion to our tensor fields. DDetI + A = 1 + λ R R κ T r ν β 2 νβ + 1 κ T r X ν β 2 νβ + λ F κ 2 T r F ν β F νβ + λ F κ 2 T r F ν β ɛ τκ νβ Fτκ 1 λr 2 T r β Rν κ2 νβ + 1 β κ 2 Xν νβ + λ F κ + λ 2 F ν β τκ F ɛ κ2 νβ Fτκ T r λr + λ F ν β F ɛ κ2 β Rν κ2 τκ νβ Fτκ νβ + 1 β κ 2 Xν νβ + λ F κ 2 ν β F F 2 νβ ν β F F 2 νβ 6.13 where ech of the pieces of which cn be clculted s follows: λr T r R ν β κ 2 νβ + 1 X ν β κ 2 νβ + λ F κ 2 F ν β Fνβ + λ F κ 2 F ν β ɛ τκ νβ Fτκ 2 = T r λ 2 R R ν β σ κ 4 νβ Rρ ρσ + λ2 F F ν β F κ 4 νβ F bρ σ Fρσ b κ 4 F ν β τκ ɛνβ FτκF bρ σ τ ɛ κ ρσ Fτ b κ + λ 2 F 1 +T r X ν β σ κ 4 νβ Xρ ρσ + 2 λ R λ F R ν β κ 4 νβ F ρ σ Fρσ +2 λ R λ F R ν β κ 4 νβ F ρ σ τ ɛ κ ρσ Fτ κ +T r 2 λ R R ν β σ κ 4 νβ Xρ ρσ + 2 λ F λ F ρσ F b τ κ F ν β F κ 4 νβ F bρ σ τ ɛ κ +2 λ F F ν β F σ κ 4 νβ Xρ ρσ +T r X ρ σ ρσ F ν β τκ ɛ νβ F τκ

36 = λ2 R T r R ν β σ κ 4 νβ Rρ ρσ + λ2 F T r F ν β F κ 4 νβ F bρ σ Fρσ b + λ 2 F κ 4 T r F ν β τκ ɛνβ FτκF bρ σ τ ɛ κ ρσ Fτ b κ + 1 T r X ν β σ κ 4 νβ Xρ ρσ + 2 λ R λ F T r R ν β κ 4 νβ F ρ σ Fρσ +2 λ R λ F T r R ν β κ 4 νβ F ρ σ τ ɛ κ ρσ Fτ κ +2 λ R T r R ν β σ κ 4 νβ Xρ ρσ + 2 λ F λ F T r F ν β F κ 4 νβ F bρ σ τ ɛ κ ρσ Fτ b κ +2 λ F T r F ν β F σ κ 4 νβ Xρ ρσ + λ F κ 4 T r X ρ σ ρσ F ν β τκ ɛνβ Fτκ 6.15 Clcultion of the lst term; λr T r R ν β κ 2 νβ + 1 X ν β κ 2 νβ + λ F κ 2 F ν β Fνβ + λ F κ 2 F ν β ɛνβ 2 = λ2 R T r R ν β λ κ 4 νβ + 2 F κ T r F ν β F 2 4 νβ + λ F κ 4 T r F ν β τκ ɛνβ Fτκ + 1 T r X ν β κ 4 νβ +2 λ R λ F T r R ν β κ 4 νβ T r F ρ σ Fρσ +2 λ R λ F T r R ρ σ κ 4 ρσ T r sf ν β τκ ɛνβ Fτκ +2 λ R T r R ν β κ 4 νβ T r X ρ σ ρσ +2 λ F λ F T r F ν β F κ 4 νβ T r F bρ σ τ ɛ κ ρσ Fτ b κ +2 λ F T r F ν β F κ 4 νβ T r X ρ σ ρσ +2 λ 2 F κ 4 T r F bρ σ ɛ τ κ ρσ F b τ κ T r X ν β νβ τκ Fτκ

37 With ll off the clculted terms eqution 6.13 tkes the form; DDetI + A = 1 + λ R R κ T r ν β 2 νβ + 1 κ T r X ν β 2 νβ + λ F κ T r F ν β F 2 νβ + λ F κ T r F ν β τκ ɛ 2 νβ Fτκ λ2 R R 2κ T r ν β 4 νβ Rρ σ ρσ λ2 F 2κ T r F ν β FνβF bρ σ F b 4 ρσ λ 2 F 2κ T r F ν β τκ ɛ 4 νβ FτκF bρ σ τ ɛ κ ρσ Fτ b κ λ 2 F 2κ T r F ν β τκ ɛ 4 νβ FτκF bρ σ τ ɛ κ ρσ Fτ b κ 1 X 2κ T r ν β 4 νβ Xρ σ ρσ λ R λ F T r R ν β κ 4 νβ F ρ σ Fρσ λ R λf T r R ν β κ 4 νβ F ρ σ τ ɛ κ ρσ Fτ κ λ R R κ T r ν β 4 νβ Xρ σ ρσ λ F λf T r F ν β F κ νβf bρ σ τ ɛ κ 4 ρσ Fτ b κ λ F λ F X ρ σ ρσ F ν β τκ ɛ F κ T r ν β FνβX ρ σ 4 ρσ κ T r 4 + λ2 2 R T r R ν β λ 2 2κ 4 νβ + F T r 2κ 4 + λ 2 2 F T r F ν β τκ ɛ 2κ 4 νβ Fτκ 1 + 2κ 4 + λ R λ F T r R ν β κ 4 νβ T r F ρ σ Fρσ + λ R λf κ 4 T r R ρ σ ρσ T r + λ R R κ T r ν β 4 νβ T r + λ F λf T r F ν β F κ 4 νβ + λ F κ 4 T r F ν β F νβ + λ 2 F κ 4 T r τ κ F ν β ɛ X ρ σ ρσ T r T r F ν β F νβ T r τκ νβ Fτκ 2 X ν β νβ F bρ σ τ ɛ κ ρσ Fτ b κ X ρ σ ρσ F bρ σ ɛρσ Fτ b κ T r X ν β νβ νβ 2 F τκ 6.17 Tking cre of binomil expnsion; 1 + x 1/4 = 1 + x x2 + O

38 Applying binomil expnsion to 6.17; DDet I + A 1/4 = 1 + λ R T r R ν β 4κ 2 νβ + 1 T r X ν β 4κ 2 νβ + λ F T r F ν β F 4κ 2 νβ + λf T r F ν β τκ ɛ 4κ 2 νβ Fτκ λ2 R T r R ν β σ 8κ 4 νβ Rρ ρσ λ2 F T r F ν β F 8κ 4 νβ F bρ σ Fρσ b λ2 F T r F ν β τκ ɛ 8κ 4 νβ FτκF bρ σ τ ɛ κ ρσ Fτ b κ λ 2 F T r F ν β τκ ɛ 8κ 4 νβ FτκF bρ σ τ ɛ κ ρσ Fτ b κ 1 T r X ν β σ 8κ 4 νβ Xρ ρσ λ R λ F T r R ν β 4κ 4 νβ F ρ σ Fρσ λ R λ F T r R ν β 4κ 4 νβ F ρ σ τ ɛ κ ρσ Fτ κ λ R T r R ν β σ 4κ 4 νβ Xρ ρσ λ F λ F T r F ν β F 4κ 4 νβ F bρ σ τ ɛ κ ρσ Fτ b λ κ F T r F ν β F σ 4κ 4 νβ Xρ ρσ λ 2 F T r X ρ σ 4κ 4 ρσ F ν β τκ ɛνβ Fτκ + λ2 R T r R ν β 32κ 4 νβ + λ2 F 32κ T r F ν β F 2 2 λ2 4 νβ + F T r F ν β τκ ɛ 32κ 4 νβ Fτκ 2 + T 1 r X ν β λ 32κ 4 νβ + R λ F T r R ν β 16κ 4 νβ T r F ρ σ Fρσ T r R ρ σ ρσ T r F ν β τκ ɛνβ Fτκ + λ R T r R ν β 16κ 4 νβ T r X ρ σ ρσ + λ F λ F T r F ν β F 16κ 4 νβ T r F bρ σ τ ɛ κ ρσ Fτ b κ + λ F T r F ν β F 16κ 4 νβ T r X ρ σ ρσ + λ R λ F 16κ 4 + λ 2 F 16κ 4 T r F bρ σ ɛ τ κ ρσ F b τ κ T r X ν β νβ

39 In the presence of these, our effective ction 6.7 tkes the form; S eff = d 4 xc DD MDκ 2 2 g 1/2 1 + λ R R 4κ T r ν β 2 νβ + 1 X 4κ T r ν β 2 νβ + λ F 4κ T r F ν β F 2 νβ + λ F 4κ T r F ν β τκ ɛ 2 νβ Fτκ λ2 R R 8κ T r ν β 4 νβ Rρ σ ρσ λ F 4κ 4 T r + λ2 F 32κ κ 4 T r T r X ν β νβ λ 2 F 8κ 4 T r λ2 F 8κ 4 T r F ν β F νβf bρ σ F b ρσ F ν β ɛ τκ νβ FτκF bρ σ τ ɛ κ ρσ Fτ b κ λ 2 F 8κ T r F ν β τκ ɛ 4 νβ FτκF bρ σ τ ɛ κ ρσ Fτ b κ 1 X 8κ T r ν β 4 νβ Xρ σ ρσ λ R λ F T r R ν β 4κ 4 νβ F ρ σ Fρσ λ R λf 4κ T r R ν β 4 νβ F ρ σ τ ɛ κ ρσ Fτ κ λ R R 4κ T r ν β 4 νβ Xρ σ ρσ λ F λf 4κ T r F ν β FνβF bρ σ τ ɛ κ 4 ρσ Fτ b κ λ F F 4κ T r ν β FνβX ρ σ 4 ρσ X ρ σ ρσ F ν β τκ ɛνβ Fτκ + λ2 2 R T r R ν β 32κ 4 νβ 2 F ν β Fνβ λ2 2 + F T r F ν β τκ ɛ 32κ 4 νβ Fτκ 2 λ R λ F + R 16κ T r ν β 4 νβ T r F ρ σ Fρσ + λ R R λf 16κ T r ρ σ 4 ρσ T r F ν β τκ ɛνβ Fτκ + λ R R 16κ T r ν β 4 νβ T r X ρ σ ρσ T r F bρ σ τ ɛ κ ρσ Fτ b κ + λ F 16κ T r F ν β F 4 νβ T r X ρ σ ρσ + λ F λf 16κ 4 T r F ν β F νβ + λ 2 F 16κ 4 T r τ κ F bρ σ ɛρσ Fτ b κ T r X ν β νβ 6.20 We should cncel out the ghosty terms s in previous chpter vi our rnk-4 rbitrry 33

40 tensor field X µανβ. 1 4κ X = 1 R ν β 2 8κ 4 νβ R X = 1 R ν β 2κ 2 νβ R νβ ν β νβ ν β R2 4 R Assume tht; X ν β νβ = A λ R R ν β 2κ 2 νβ R νβ ν β R2 δν ν δ β β 4 δβ ν δβ ν 6.22 The trce of X µανβ ; X = 1A λ R R ν β 2 2κ 2 νβ R X = 1 2 Xν β νβ νβ ν β R2 4 δν ν δ β β δβ ν δβ ν δν ν δ β β δβ ν δβ ν δ νν δββ δ νβ δβν A = 1 DD D=4 A = Hence, X ν β νβ = λ R R ν β 24κ 2 νβ R νβ ν β R After inserting eqution 6.24 into effective ction 6.20 nd cutting our expnsion t 34

41 second order, the effective ction tkes the form s; λ 2 F λ F λ F S eff = d 4 xc DD M 2 D κ2 g 1/2 1 + λ R 4κ 2 R + λ F F νβ F 4κ 2 νβ + λ F F νβ ɛ 4κ 2 τκ νβ Fτκ λ2 F F ν β F 8κ 4 νβ F bν β F bνβ 1 4 F νβ F νβ 2 2 F ν β τκ ɛ 8κ 4 νβ FτκF bν β ɛ νβ ρσ F bρσ 1 F νβ ɛ νβ 4 τκ F τκ λ R λ F R ν β 4κ 4 νβ F τ κ ɛ νβ τκ F τκ RF νβ τκ ɛ 4 νβ Fτκ F ν β F 4κ 4 νβ F bν β ɛνβ τκ F bτκ 1F ηθ F 4 ηθ F bνβ τκ ɛνβ Fτκ b + λ R λ F R ν β 4κ 4 νβ F ν β F νβ Fν β RF νβ F 4 νβ +OA There re two possibilities of grvittionl constnt nd constnt curvture. When they re equl to ech other M D = κ; S eff = d 4 xc DD g 1/2 κ 4 + κ 2 λ R 4 R λ2 F 8 +κ 2 λ F 4 F νβ Fνβ + κ2 λ F4 F νβ ɛνβ Fτκ F ν β Fνβ F bν β F bνβ 1 4 F νβ F νβ 2 λ 2 2 F 8 F ν β τκ ɛνβ FτκF bν β ɛ νβ ρσ F bρσ 1 F νβ ɛ νβ 4 τκ F τκ λ R λ F R ν β 4κ 4 νβ F τ κ ɛ νβ τκ F τκ RF νβ τκ ɛ 4 νβ Fτκ λ F λ F F ν β F 4κ 4 νβ F bν β ɛνβ τκ F bτκ 1F ηθ F 4 ηθ F bνβ τκ ɛνβ Fτκ b + λ R λ F R ν β 4κ 4 νβ F ν β F νβ Fν β RF νβ F 4 νβ τκ +OA

42 When they do not equl to ech other M D κ; S eff = d 4 xc DD M 2 D g 1/2 κ 2 + λ R 4 R τκ Fτκ + λ F 4 F νβ Fνβ + λ F4 F νβ ɛνβ λ2 F F ν β F 8κ 2 νβ F bν β F bνβ 1 4 F νβ F νβ 2 2 F ν β τκ ɛ 8κ 2 νβ FτκF bν β ɛ νβ ρσ F bρσ 1 F νβ ɛ νβ 4 τκ F τκ λ R λ F R ν β 4κ 2 νβ F τ κ ɛ νβ τκ F τκ RF νβ τκ ɛ 4 νβ Fτκ F ν β F 4κ 2 νβ F bν β ɛνβ τκ F bτκ 1F ηθ F 4 ηθ F bνβ τκ ɛνβ Fτκ b + λ R λ F R ν β 4κ 2 νβ F ν β F νβ Fν β RF νβ F 4 νβ λ 2 F λ F λ F +OA We re going to consider 6.27 in our work Reltion to Vector Infltion Our gol is to derive dynmicl equtions for homogeneous nd isotropic universe, for FRW bckground nd to demonstrte infltion comes nturlly from our effective ction. To do tht we should clculte trces of tensor fields in FRW bckground 44 s 36

43 second step. λ 2 F 8κ 2 λ F λ F 4κ 2 S eff = d 4 xc DD M 2 D g 1/2 κ 2 + λ R 4 R λ2 F 8κ 2 τκ Fτκ + λ F 4 F νβ Fνβ + λ F4 F νβ ɛνβ } {{ } } {{ } 1 2 F ν β FνβF bνβ bν β F F νβ F νβ 2 } {{ } F ν β τκ ɛνβ FτκF bν β ɛ νβ ρσ F bρσ } {{ } 4 λ R λ F 4κ 2 + λ R λ F 4κ R ν β νβ F τ κ ɛ νβ τκ F τκ R } {{ } F νβ ɛ F νβ ɛ νβ τκ F τκ τκ νβ Fτκ F ν β FνβF bν β ɛνβ τκ F bτκ 1 } {{ } F ηθ F 4 ηθ F bνβ τκ ɛνβ F b τκ 7 R ν β νβ F ν νβ β F R 4 } {{ } F νβ F νβ + OA Clcultion of the first term; F νβ F 0β = F 0β F 0β + F iβ F iβ = F 00 F 00 + F 0i F 0i + F i0 F i0 + F ij F ij = 2 φδ j δji φδ i + 2 φδ j δji φδ i + 4 gφ 2 δkiδljɛ klgφ 2 ɛ ij = 2 φ2 δ 2 φ2 δ + 4 g 2 φ 2 ɛ klɛ kl = 6 2 φ g 2 φ

44 Clcultion of the second term; F νβ τκ ɛνβ Fτκ = F 0β ɛ τκ 0β Fτκ + F iβ ɛ τκ iβ = F 0i ɛ τκ 0i + F i0 ɛ τκ i0 F τκ + F 0i ɛ τκ 0i F τκ + F ij ɛ τκ ij F τκ F τκ F τκ = 2 φδ i ɛ jκ 0i Fjκ + 2 φδ i ɛ jκ i0 Fjκ 4 gφ 2 ɛ ij ɛ 0κ ij F0κ + ɛ kκ ij Fkκ = 2 φδ i ɛ jk 0i Fjk ɛ jk i0 Fjk 4 gφ 2 ɛ ij ɛ 0k ij F0k + ɛ k0 ij Fk0 + ɛ kl ij = 2 2 g φφ 2 ɛ jk i 2 ɛ jk 0i + 4 ɛ i0 jk = 2g φφ 2 ɛ jk i 6 ɛ 0ijk + 6 ɛ jki0 F kl = 24g φφ Clcultion of the third term; F ν β F νβf bν β F bνβ = F ν 0 Fν b 0 + F ν i Fν b i F bνβ Fνβ = F νβf bνβ F i0 F b i0 + F 0i F b 0i + F ji F b ji = FνβF 2 bνβ 2 g φ 2 δ b + 4 gφ 4 ɛ jiɛ ji F = 2δ b 4 2 φ2 + g 2 φ 4 0β F b0β + FiβF biβ = 2δ b 4 2 φ2 + g 2 φ φ2 δ b + g 2 φ 4 4 2δ b = 12 8 g 2 φ 4 2 φ

45 Clcultion of the fourth term; τκ F ν β ɛνβ FτκF ν b β ɛνβ ρσ F bρσ = F 0β ɛ + F iβ ɛ τκ νβ = F 0i F b 0iɛ F τκf b 0β ɛνβ ρσ F bρσ τκ νβ FτκF iβ b ɛνβ ρσ F bρσ τκ νβ F τκɛ νβ ρσ F bρσ + F i0 Fi0 b + F ij Fij b = 2δ b 4 2 φ2 + g 2 φ 4 ɛ ɛ νβ0σf b0σ + ɛ νβiσf biσ ɛ τκ νβ Fτκɛ νβ ρσ F bρσ τκ νβ Fτκ = 2δ b 4 2 φ2 + g 2 φ b φδ i ɛ νβ 0i gφ2 4 ɛ b ijɛ νβ ij ɛ νβ 0k F 0k + ɛ νβ k0 F k0 + ɛ νβ kl F kl = 2 4 g 2 φ 4 2 φ φ g 2 φ Clcultion of the fifth term; R ν β νβ F ν β F νβ = R 0β νβ F 0 β νβ F + R iβ νβ F iβ νβ F = 4 φ 2 2 δ ij R 0i 0j + g 2 φ 4 4 ɛ ijɛ klr ij kl ȧ 2 = 6 4 ä + 2 φ g 2 φ

46 Clcultion of the sixth term; R ν β νβ F ν β ɛνβ τκ F τκ = R 0β νβ F 0β + Riβ νβ F iβ ɛ νβ0κf 0κ + ɛ νβjκf jκ = 2 φδ i R 0i νβ gφ 2 ɛ ikr ik νβ 2 2 φδ j ɛ νβ 0j 4 gφ 2 ɛ jlɛ νβ jl ä = 4 6 g φφ 2 2ȧ2 + ɛ i jlɛ 2 0ijl ä g φφ 2 2ȧ2 + ɛ j 2 ik ɛ ik0j ä = 36 6 g φφ 2 2ȧ Clcultion of the seventh term; F ν β F νβf b ν β ɛνβ τκ F bτκ = F ν β F νβf b ν β ɛ νβ 0κ F b0κ + ɛ νβ iκ F biκ = Inserting the terms into the eqution 6.28; S eff = d 4 xc DD M 2 D g 1/2 κ 2 + λ R 4 R + λ F φ g 2 φ 4 + λ F4 24g φφ λ2 F 12 g 8 2 φ 4 φ2 2 8κ F νβ F νβ 2 λ 2 2 F 48 g 8 2 φ 4 φ2 2 8κ λ R λ F 36 6 g φφ 2 ä + 2 ȧ2 4κ 2 F νβ ɛ νβ τκ F τκ 2 RF νβ τκ ɛ 2 4 νβ Fτκ λ F λ F 1F ηθ F 4κ 2 4 ηθ F bνβ τκ ɛνβ Fτκ b 6 4 ä + ȧ 2 2 φ g 2 φ 4 4κ 2 λ R λ F R F νβ F 4κ 2 4 νβ + λ R λ F

47 We cn rerrnge the ction; S eff = d 4 xc DD M 2 D g 1/2 κ 2 + 3λ R 2 + 3λ F φ2 + g 2 φ λ F g φφ λ2 F 12 g 8 2 φ 4 φ2 2 8κ g 8 2 ä + ȧ 2 2 g 2 φ 4 2 φ2 λ 2 2 F 48 g 8 2 φ 4 2 φ2 8κ g 2 φ2 φ 4 2 λ R λ F 36 6 g φφ 2 ä + 2 ȧ g φφ 2 ä + ȧ2 4κ λ F λ F g φφ 2 g 2 φ 4 2 φ2 4κ Action tkes the form s; 3λR λ R λ F S eff = d 4 x4md κ 2 g 1/ g 2 φ 4 + 3λ 4κ 2 R λ F 4κ 2 3λR 2 2 3λ R λ F 6 g 2 φ λ 4κ 2 R λ F 4 φ2 ȧ 2 4κ 2 3λ F 4 g 2 φ 4 3λ F φ2 + 9 λ F λ F 8 g φ 3 φ 2 κ 2 3 λ2 F 8 g 2 φ 4 6 λ2 8κ 2 F 8 g 2 φ 4 3 λ2 κ 2 F 6 φ2 8κ 2 3 φ2 ä 6.38 To find the dynmics; ρ = L red φ φ L red p = 3 L red L red φ φ = 3λ R λ F 2κ 2 3 φä + 15λ R λ F 4 9λ R λf φ 2κ 2 3λ F φ 2 27λ F λf κ 2 gφ 2 φ λ F 4κ 2 κ 2 8 gφ λ F λf κ 2 ȧ 2 6 φ2 + 6 λ F g φφ λ F λf κ 2 g 3 φ 6 φ λ F κ 2 gφ 4 φ

48 According to eqution 6.39; ρ = 4MD 2 κ 2 3λR λ R λ F 5 g 2 φ 4 3λ R λ F 3 4κ 2 4κ 2 φ2 ä 3λR 2 2 3λ R λ F 6 g 2 φ λ R λ F 4 4κ 2 4κ 2 φ2 ȧ 2 3λ F 2 4 g 2 φ 4 + 3λ F 2 2 φ2 18λ F λf 2 gφ 2 κ 2 φ3 3λ 2 F + 8κ + 6 λ 2 F 8 g 2 φ κ 2 φ λ 2 F 12 κ 2 φ2 g 2 φ nd p = 4M 2 D + κ 2 + λ R 1 3λ R λ F 5 g 2 φ 4 ä 2κ 2 λ R λ R λ F 6 g 2 φ 4 5λ R λ F 4κ 2 4κ 2 4 φ2 + 15λ F λf κ 2 8 gφ 2 φ ȧ 2 λ F 2 4 g 2 φ 4 + λ F 2 2 φ2 21λ F λf 10 g 3 φ 6 15λ F λf φ 8 gφ 2 κ 2 κ 2 φ3 5λ 2 F λ 2 F 8 g 2 φ 4 3λ 2 F κ 2 κ 2 8κ + 6 λ 2 F 6 2 κ 2 φ2 6 λ F 6 g φφ λ 2 F κ 2 12 φ2 g 2 φ As mentioned in Chpter 3; to determine infltion conditions we should derive slow-roll prmeters. First step is to write down the Hubble Constnt H, Ḣnd H 2 ; H 2 = 4MD 2 κ2 3 λr λ R λ F 5 g 2 φ 4 λ R λ F 4κ 2 4κ 2 λr 2 2 λ R λ F 6 g 2 φ 4 + 5λ R λ F 4 4κ 2 4κ 2 φ2 ȧ 2 λ F 2 4 g 2 φ 4 λ F 2 2 φ2 + 6λ F λf 2 gφ 2 κ 2 φ3 λ 2 F + 8κ + 2 λ 2 F 8 g 2 φ κ 2 φ2 + 6 λ 2 F κ 2 12 φ2 g 2 φ 4 3 φ2 ä

49 Ḣ = 4MD 2 λ R λ R λ F 5 g 2 φ 4 3λ R λ F 3 4κ 2 4κ 2 φ2 ä + λ R 2 3λ R λ F 6 g 2 φ 4 + 5λ R λ F 4 8κ 2 κ 2 φ2 15λ R λ F 8 g κ φφ λ F κ 2 33λ F λf κ 2 4 g 2 φ φ2 8 gφ 2 φ3 λ 2 F κ λ F λf 10 g 3 φ 6 φ κ λ 2 F κ λ F 6 g φφ λ 2 F κ 2 12 φ2 g 2 φ 4 8 g 2 φ 4 ȧ To supply infltion conditions, we should interpret the rtio of Ḣnd H2 since ɛ = Ḣ H 2 Ḣ H Then in view of bsolute vlue; Ḣ H 2 The Friedmnn equtions bove my seem too complicted to drw conclusion bout slow roll behvior. However, their t dependence lredy gives some clues on their evolutionry chrcter. Both H 2 nd Ḣ hve two pieces; one piece tht depends on φt nd nother piece tht does not. By simply exmining those terms up to O 2 one sees tht Ḣ/H 2 1. Though numericl computtion might give better view of the solutions, still one concludes tht the slow-roll conditions re stisfied for wide rnge of prmeter vlues. 43

50 CHAPTER 7 CONCLUSION In this work, we hve estblished new grvittionl theory tht we nme s Born- Infeld-Riemnn grvity. This theory is bsed on the Riemnn tensor in the ction. Among other fetures, it hs the importnt property tht, it llows for unifiction of Yng-Mills fields nd grvity in one single formlism. This feture is completely new, nd hs not been found in other grvittionl theories. For both extrcting the physics implictions of the model nd performing n ppliction to physicl phenomenon, we hve discussed, fter building the model, the guge field infltion. In this scenrio, cosmic infltion is cused by non-abelin guge field in homogeneous nd isotropic bckground. This thesis work, supplemented by number of ppendices, concludes tht Born- Infeld-Riemnn grvity is physiclly consistent extension of the GR, nd it brings striking novelty in the tretment of non-abelin guge fields. Moreover, it covers infltionry epoch for wide rnge of prmeters. This theory is n extension of the Born-Infeld theory to non-abelin fields, nd it cn explin number of cosmologicl phenomen. 44

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53 rxiv: gr-qc 32 Ferrris M. et l. 1982: Vritionl Formultion of Generl Reltivity from 1915 to Pltini s Method- Discovered by Einstein in Generl Reltivity nd Grvittion, Vol. 14, Issue 3, Fiorini F. et l. 2009: A Type of Born-Infeld Regulr Grvity And Its Cosmologicl Consequences. Interntionl Journl of Modern Physics A, Vol. 24, Issue 08n09 34 Gltsov D. V. 2008: Non-Abelin Condnstes As Alterntive For Drk Energy. rxiv: gr-qc 35 Go, C. 2010: When Sclr Field is Kineticlly Coupled to the Einstein Tensor. rxiv: gr-qc 36 Gullu, I. et l. 2010: Unitrity Anlysis of Generl Born-Infeld Grvity Theories. Phys. Rev. D 82, Guth A. H. 1981: Infltionry Universe: A Possible Solution to Horizon nd Fltness Problem. Phys. Rev. D 23, 347â Ho J. G. et l. 2004: An Anlyticl Infltion From Born-Infeld Type Sclr Field. Chinese Physicl Letters, Vol:21, No:3, Hehl, F. W. et l. 2008: 4th WE-Hereus-Seminr Spce nd Time 100 Yers fter Minkowski Metric-Affine Grvity MAG: Generl Reltivity nd Its Guge- Theoreticl Extension. 40 Kruglov, S. I. 2010: On Generlized Born-Infeld Electrodynmics. rxiv: hep-th 41 Lndu L. D. et l. 1973: The Clssicl Theory of Fields: Course of Theoreticl Physics Volume 2. Fourth edition, 428 p. 42 Li, B. et l. 2007: The Cosmology of fr Grvity in the Metric Vritionl Approch. rxiv: gr-qc 43 Mgnno, G. 2005: Are there metric theories of grvity other thn Generl Reltivity? rxiv: gr-qc 44 Mleknejd, A. et l. 2011: Guge Fltion: Infltion from Non-Abelin Guge Fields. rxiv: hep-ph 45 Milgrom, M. 1983: A Modifiction of the Newtonin Dynmics s Possible Alterntive to the Hidden Mss Hypothesis. Astrophysicl Journl, Prt 1 vol. 270, 47

54 MIT Deprtment of Mthemtics. 2005: Geometry of Mnifolds Lecture Notes: Levi-Civit Connection nd Its Curvture. 47 Nieto, J. A. 2004: Born-Infeld Grvity in Any Dimension. Phys. Rev. D 70, Pni, P. et l. 2012: Eddington-Inspired Born-Infeld Grvity. Phenomenology of non-liner Grvity-Mtter Coupling. rxiv: gr-qc 49 Peldn P. 1993: Actions for Grvity, with Generliztions: A Review. rxiv: gr-qc 50 Petersen, K. B. et l. 2008: The Mtrix Cookbook, Version: November 14, Plebnski, J. et l. 2006: An Introduction to Generl Reltivity nd Cosmology. Cmbridge University Press, 534 p. 52 Rychudhuri A. K. et l. 1992: Generl Reltivity, Astrophysics, nd Cosmology. Springer-Verlg New York, 296 p. 53 Ryder, L. 2009: Introduction to Generl Reltivity. Cmbridge University Press, 441 p. 54 Schutz B. 1985: A First Course in Generl Reltivity. Cmbridge University Press, 376 p. 55 Sotiriou T. P. 2006: Constrining fr Grvity in the Pltini Formlism. Clss. Qunt. Grv. vol.23 no:4, Sotiriou T. P. et l. 2007: Metric-Affine fr Theories of Grvity. Annls of Physics Vol. 322, Issue 4, 935â Sotiriou T. P. 2009: fr Grvity, Torsion nd Non-Metricity. Clss. Qunt. Grv. vol. 26, Sotiriou T. P. 2009: 6+1 Lessons from fr Grvity. Journl of Physics: Conference Series Sotiriou T. P. et l. 2010: fr Theories of Grvity. Rev. Mod. Phys, vol. 82, 451â Soysl, S. et l. 2012: Born-Infeld-Riemnn Grvity. Submitted 48

55 61 Tpi, V. 1993: Integrble Conforml Field Theory in Four Dimensions nd Fourth-Rnk Geometry. Interntionl Journl of Modern Physics D, Vol. 02, Issue Tpi, V. 2004: Differentil Invrints for Higher Rnk Tensors. A Progress Report. rxiv: gr-qc 63 Vitglino, V. et l. 2012: The Dynmics of Metric-Affine Grvity. Annls of Physics, vol. 329, Vollick, D. N. 2004: Pltini Approch to Born-Infeld-Einstein Theory nd Geometric Description of Electrodynmics. Phys. Rev. D 69, Vollick, D. N. 2005: Born-Infeld-Einstein Theory with Mtter. rxiv: gr-qc 66 Wld R. M. 1984: Generl Reltivity. The University of Chicgo Press, 491 p. 67 Weinberg, S. 1972: Grvittion nd Cosmology: Principles nd Applictions of the Generl Theory of Reltivity. New York, USA: Wiley, 657 p. 68 Wohlfrth, M. 2004: Grvity l Born-Infeld. Clss. Qunt. Grv. vol. 21 no. 8,

56 APPENDIX A INVARIANT VOLUME AND DETERMINANT OF TENSORS Einstein s Specil Reltivity Theory shows tht we live in four dimensionl world which is clled spce-time. With the Theory of Generl Reltivity, spce-time is considered in geometricl nture. Bsiclly the theory is bsed on geometricl structure which is clled mnifold. A mnifold is smooth nd loclly flt 13, 66, 67. Becuse of its smoothness, mnifolds re differentible. We re going to mention bout tensor fields which re defined on mnifold, briefly. First of ll is the metric tensor. Metric tensor is purely mthemticl object nd benefits to mesure spce-time intervls. Coordintes cn be choosed s x µ x 0, x 1, x 2, x 3. Here Greek indices lbel spce-time coordintes. For ny point of spce-time we cn find point tht loclly inertil: ζ µ ζ 0, ζ 1, ζ 2, ζ 3. The line element is; ds 2 = η µν dζ µ dζ ν A.1 where η µν is Minkowski metric of flt spce-time. ds 2 = η µν dζ µ dx τ dζ ν dx ρ dxτ dx ρ ds 2 = g τρ dx τ dx ρ A.2 As seen from eqution 2 g τρ is n n digonl, symmetric mtrix. The symmetric prt of the metric tensor is different from zero so tht norm cn be mesured. If the metric tensor is defined in theory, it is lso used for rising or lowering indices..the discussions still go on bout metric tensor: Should theory include metric tensor, is it necessity? We re going to explin these type of theories in Chpter 2. In curved spce-time how cn we prllel trnsport 13, 41, 53, 67 vector long curve? Clerly the trnsporttion is not going to be the sme with flt spce. Riemnn 50

57 Curvture Tensor comes from this rgument. R µ ανβ = νγ µ βα βγ µ να + Γ µ νλ Γλ βα Γ µ βλ Γλ να A.3 Riemnn curvture tensor 46 contins everything bout the curvture of spce-time. We mention flt spce if nd only if curvture tensor equls zero. The lst structure is connection. It ppers in geodesic eqution which mens tht the pth of moving object is determined by connection. It is not tensoril structure tht chnges with respect to the chosen coordinte system. The symmetriztion or ntisymmetriztion in lower indices of connection gives us torsion of the spce-time. As mthemticlly we know tht the difference between two non-tensoril structure gives tensoril structure tht mens torsion is tensoril structure. Therefore if determined connection is nti-symmetric in its lower indices; Γ λ αβ Γ λ βα = S λ αβ A.4 If determined connection is symmetric in its lower indices, torsion of the curved spcetime is zero. In clssicl mechnicl systems 41 we use ction functionl, in generl. To understnd fetures of motion, we tke the vrition of the ction with respect to dynmicl vribles of motion. In Generl ReltivtyGR Theory ction plys n importnt role, too. We construct our theories upon Lgrngin Density which is symboliclly clled L. Beside the gret success of GR, it is unfortuntely n incomplete theory. It cn not explin such big problems tht cosmologicl constnt problem, infltion etc. Becuse of tht for yers people hve tried to modify Einstein s GR Theory in severl wys. An pproprite ction functionl is constructed due to our theory nd then equtions of motion re derived. Finlly due to the equtions of motion the results re interpreted. One wy to check theory whether it survives or not, it should be exmined tht does the theory include GR for limiting cses. The other wy, for limiting cses, is to serch the results re comptible with the cosmologicl observtions. First of ll the question is tht how we cn form n invrint ction. Bsiclly we define 51

58 the ction; S = dtlt A.5 where L is clled Lgrnge function. If it is pplied on spce-time mnifold for four dimensions; S = d 4 xlg, Γ, φ, φ... A.6 Here L is clled Lgrngin density nd g is metric tensor, Γ is connection, φ is n rbitrry sclr field, φ is the derivtion of sclr field with respect to time. As long s we obey some bsic rules, we cn dd inifinite number of terms to Lgrngin density. Now we re going to mention bout these bsic rules. As we know ction is clled sclrquntity which mens it is n invrint under ll of the chnges. For exmple; ction should be n invrint under coordinte trnsformtions, guge trnsformtions, conforml trnsformtions.. etc. It mkes sense becuse we don t wnt the equtions of motion to chnge when the system is in different frme. Our primry im is to mke ction Lorentz invrint. Suppose tht Lgrngin depends only on connection nd prtil differentition of connection. On D dimensionl spce-time mnifold; S = d D xlγ, Γ A.7 where d D x = d µ 0 x d µ 1 x d µ D 1 x A.8 Chnging coordinte system x x 3, 13, 67; d D x = dx dx dd x A.9 52

59 Since d D x d D x ;d D x is not tensor! We cll these type of quntities s tensor density. The coefficient of trnsformtion in the form of determinnt dx dx is clled the Jcobin. According to power of the Jcobin the weight of tensor density is defined. d D x is tensor density of weight +1. In this sitution ction is obviously not n invrint. To persist invrince of ction, Lgrngin must involve tensor density of weight -1. The best wy of succeeding tht is to use the notion of determinnt. Lorentz coordinte trnsformtion of n rbitrry rnk 0,2 tensor field R µν ; R µ ν = xµ x ν R µν x µ x ν A.10 x µ x µ x ν x ν R µ ν = R µν A.11 Tking determinnt of both sides; R µν = x x 2 R µ ν A.12 where R µν = Det R µν. Since the coefficient of R µ ν is x 2, then the determinnt of rnk 0,2 tensor field is tensor density of weight -2. In the sme wy Lorentz coordinte trnsformtion of n rbitrry rnk 1,3 tensor field Q µανβ ; x Q µ α ν β = xµ x µ x α x α x ν x ν x β x β Q µ ανβ A.13 x µ x µ x α x α x ν x ν x β x β Qµ α ν β = Q µ ανβ A.14 53

60 Tking determinnt of both sides; Q µ ανβ x 4 = Q µ x α ν β A.15 where Q µ ανβ = DDet Q ανβ µ. Then the determinnt of rnk 1,3 tensor field is tensor density of weight -4.Double Determinnt is going to be clled DDet in the rest of thesis. As we see in eqution, the notion of determinnt differs from rnk0,2 tensor field. The reson lies under the definition of determinnt. R µν = 1 D! µ 0µ 1 µ 2 µ 3 ν 0ν 1 ν 2 ν 3 R µ0 ν 0 R µ1 ν 1 R µ2 ν 2 R µ3 ν 3 A.16 nd Q µ ανβ = 1 D! 2 µ0 µ 1 µ 2 µ 3 α 0α 1 α 2 α 3 ν 0ν 1 ν 2 ν 3 β 0β 1 β 2 β 3 Q µ 0 α0 ν 0 β 0 Q µ 1 α1 ν 1 β 1 Q µ 2 α2 ν 2 β 2 Q µ 3 α3 ν 3 β 3 A.17 where µ0 µ 1 µ 2 µ 3 is totlly nti-symmetric Levi-Civit Symbol 3, 13. It s not tensor or tensor density. Becuse under the coordinte trnsformtions, components sty the sme. In ny coordinte system; If µ 0 µ 1 µ 2 µ 3 is n even permuttion of 0,1,2,3; then µ0 µ 1 µ 2 µ 3 = 1 If µ 0 µ 1 µ 2 µ 3 is n odd permuttion of 0,1,2,3; then µ0 µ 1 µ 2 µ 3 = 1 Otherwise µ0 µ 1 µ 2 µ 3 = 0 Using the definition of determinnt 61, 62; ɛ µ0 µ 1...µ D = R 1/2 µ0 µ 1...µ D A.18 or ɛ µ0 µ 1...µ D = Q 1/4 µ0 µ 1...µ D A.19 54

61 nd ɛ µ 0µ 1...µ D = R 1/2 µ 0µ 1...µ D A.20 or ɛ µ 0µ 1...µ D = Q 1/4 µ 0µ 1...µ D A.21 Since ɛ µ 0µ 1...µ D is no longer symbol, indices cn be rised or lowered. Our im ws to mke ction sty invrint. As mentioned before ccording to eqution 5 L tensor density of weight -1. After ll these discussions; L R 1/2, Q 1/4 A.22 In our work, we use these fundmentl concepts nd construct our theory bsed on determinnt of rnk 0,4 tensor fields nd our metric convention is Finlly, here re some properties used in this work of completely nti-symmetric Levi-Civit Tensor which re exemplified for 3D: δ δi δ j ɛ ij ɛ ij = δ i δi i δj i = 6 A.23 δ j δ j i δ j j δb δi δ j ɛ ij ɛ b ij = δb i δi i δj i = 2δb A.24 δ j b δj i δ j j 55

62 nd some of them re exmplified for 4D: ɛ 0i kl ɛ kl 0i = g 0k g bl ɛ b0i ɛ kl 0i = ɛ b0i ɛ b0i δk k δl k δ0 k δ k i 3 δl k 0 δi k δk l δl l δ0 l δi l δk l 3 0 δi l = = δk 0 δl 0 δ0 0 δi δk i δl i δ0 i δi i δk i δl i 0 3 0i ɛlm ɛ lm kj = g l g mb ɛ b0i ɛ lm kj = ɛ b0i ɛ bkj δk l δm l δk l δ l j 3 δl k 0 δi k δl m δm m δk m δj m δk l 3 0 δi l = = δl 0 δm 0 δk 0 δj δl i δm i δk i δj i δk i δl i 0 3 = 6 A.25 = 0 A.26 56

63 APPENDIX B VARIATIONAL APPROACH In the previous chpter, we explined the mening of grvity, nd the fundmentl notions. By using knowledge obtined from introduction, let us continue with explining vritionl procedures, to obtin field equtions, nd extended theories of grvity. Firstly we explin vritionl method 1, 13, 39, 41, 43, 49. In clssicl field theory, the wy of finding equtions of motion is clled vritionl method. To understnd wht this method is, let us consider Lgrngin density Lφ, φ nd the ction is given s; S = dtl = d 4 xl φ i, µ φ i B.1 By considering smll vrition in this field; φ i φ i + δφ i µ φ i µ φ i + δ µ φ i = µ φ i + µ δφ i B.2 Lgrngin Density vries; Lφ i, µ φ i Lφ i + δφ i, µ φ i + µ δφ i B.3 thus ction vries by virtue of this smll vrition S S + δs. Hence; δs = L L d 4 xδφ i φ i µ µ φ i B.4 We ssume tht φ i is the sme t the end points of the integrl. Thus, by using this ssumption, we conclude tht δs should be zero. This leds to field equtions 13. Therefore, by using vritionl principle, we obtin the eqution of motion for Lgrngin which 57

64 hs one dynmicl vrible. In generl reltivity, we use the sme procedure to obtin field equtions. However, there re some different methods ccording to dynmicl vrible in the Lgrngin. We exmined these methods in three prts in Chpter 2: Metric formultion, Metric Affine formultion nd Affine formultion 7, 12, 13, 15, 30, 43,

65 APPENDIX C FIELD-STRENGTH TENSOR Field Strength Tensor which is n ntisymmetric, trceless rnk 0,2 tensor field depends on the electromgnetic field of prticles. For 4 dimensionl spce-time mnifold covrint field strength tensor tkes the form s; F µν = 0 E x E y E z E x 0 B z B y E y B z 0 B x E z B y B x 0 C.1 nd contrvrint field strength tensor which is constructed vi two contrctions of field strength tensor by metric tkes the form s F µν = 0 E x E y E z E x 0 B z B y E y B z 0 B x E z B y B x 0 C.2 Field strength tensor includes ll the informtion bout electromgnetic fields. Since it is n ntisymmetric tensor, there re six independent components. The other importnt notion is the dul of Field Strength Tensor which is; Fµν = 0 B x B y B z B x 0 E z E y B y E z 0 E x B z E y E x 0 C.3 Dul field strength tensor is n tisymmetric tensor field, too. One my ssocite the dulity with rottion. Electric field nd Mgnetic field re trnsformtive quntities. Dul 59

66 field strength tensor is good wy to show this trnsformtion. We obtin dul field strength tensor with the help of totlly ntisymmetric tensor field. F µν = 1 2 ɛ µναβf αβ C.4 As seen in Appendix A, it is lso importnt to construct invrint volume elements in GR. Dul field strength tensor is one of the wys to construct. Both F µν F µν nd F µν Fµν re Lorentz invrint quntities. With this nlysis we understnd the field strength tensor nd its dul. The next step is to exmine Field Strength Tensor in the view of Prticle Physics. Field Strength Tensor is built up from the four-potentil which is A µ = A 0, A. Here, A 0 is electric potentil nda is 3D vector potentil. Under Guge Trnsformtion field strength tensor remins unchnged. For belin theories field strength tensor is F µν = µ A ν ν A µ C.5 If we consider consider non-belin theories, wht does field strength tensor rects? Would it be remin unchnged? To nswer this question, from C.5 under locl guge trnsformtion A µ i g Aµ field strength tensor tkes the form s; F µν = GF µν G 1 = µ GA ν G 1 + i g νgg 1 ν GA µ G 1 + i g µgg 1 = G µ A ν ν A µ G 1 + ν GA µ µ GA ν G 1 + G A µ ν G 1 + A ν µ G 1 + i µ G ν G 1 ν G µ G 1 g GF µν G 1 C.6 Field Strength Tensor chnges under locl guge trnsformtions. It mens tht n dditionl term should be dded to field strength tensor. This term comes from the non-belin group structure. Since F µν = 1 iq D µ, D ν with D µ = µ + iqa µ Consequently for non- 60

67 belin group structures, Field Strength Tensor tkes the form s; F µν = µ A ν ν A µ + iq A µ, A ν C.7 Field strength tensor is no longer chngble in locl guge trnsformtions. If we rerrnge equtionc.8, considering group index, F µν = µ A ν ν A µ gɛ bca b µ, A c ν C.8 In our work, we choose the condition 44 A 0 = 0 nd µ = i A µ = φtδ µ µ = 0 A µ = 0 Then infltion is cused only by sptil terms which depend only on t. In the precense of tht Field Sterngth Tensor components re; F 00 = 0 C.9 F 0i = φδ i C.10 F 0i = 2 φδ i C.11 61

68 APPENDIX D EXPANSION In this ppendix, we re going to exmine the expnsion of Determinnt expression. It is going to be concluded tht the expnsion of determinnt nd the expnsion of double determinnt expressions re the sme. The best wy is to exmine determinnt expression, firstly 3, 67. DetM = e T rlnm D.1 k 1 1 j DetI + A = T ra j k! j k=0 j=1 1 j = exp T ra j j j=1 1 j = exp T r T ra j = exp T rlni + A D.2 j j=1 We re going to express the terms of summtion on j; = k 1 1 j DetI + A = T ra j k! j k=0 j=1 1 T ra 1 k! 2 T ra T ra3 1 4 T ra4 + 1 k 5 T ra D.3 k=0 62

69 Now we re going to express the terms of summtion on k; DetI + A = 1 }{{} k=0 + T ra 1 2 T ra2 } {{ } k=1 + 1 T ra 2 T rat ra T ra D.4 } {{ } k=2 If we cut the expnsion in the order of cubic level; DetI + A = 1 + T ra 1 2 T ra T ra2 + OA 3 D.5 We know binomil series expnsion Arfken, 1 + x α = k=0 α x k k D.6 DetI + A 1/2 = 1 + T ra 1 2 T ra2 + 1 T ra2 } {{ 2 } x 1/2 D.7 Due to expnsion in eqution D.6, one cn expnd eqution D.7; DetI + A 1/2 = T ra 1 T 2 ra2 + 1 T ra T ra 1 T 2 ra2 + 1T ra2 2 2 DetI + A 1/2 = T ra 1 4 T ra T ra2 + Ox 3 D.8 Since Double Determinnt expression is the sme with Determinnt expression s in equ- 63

70 tion D.1, the form of expnsion must be the sme. According to D.2; DDetI + A = k=0 1 k! 1 j T ra j j j=1 = 1 + T ra 1 2 T ra T ra D.9 k Due to eqution D.6; 1 + x 1/4 = x 3 32 x2 + Ox 3 D.10 nd rrnging eqution D.11 DDetI + A 1/4 = 1 + T ra 1 2 T ra2 + 1 T ra2 } {{ 2 } x DDetI + A 1/4 = T ra 1 8 T ra T ra2 D.11 1/4 We re going to use these expnsions during our work. 64

71 APPENDIX E TRACE OF A MATRIX In this thesis, we try to construct new theory. Expnsions mke us to clculte trces of the rnk-4 tensors. In this Appendix, we try to formulte wht trce is 3, 50. Bsicly, trce of mtrix is the sum of digonl elements of mtrix. Consider n rbitrry mtrix M b. Trce of M; T rm = M E.1 Mthemticlly we use some fetures of trce such s: T rm N = T rm T rn E.2 nd T rmn T rmt rn E.3 On the contrry, it should be pointed tht T r MN = MN ii = m ij n ji i i j = n ji m ij = MN jj j i j = T r NM E.4 It indictes tht we should displce the mtrices wit ech other in this wy. For tensoril structures, the contrction is cused by metric tensor. For exmple, 65

72 trce of n rbitrry rnk-2 tensor field, Aµν, is T ra = A µ µ E.5 It should be used either metric tensor g µν or especilly for ffine theories δ µ ν T ra = A µ µ = g µν A µν E.6 or T ra = δ ν µa µ ν E.7 In this thesis, we write down the trces of terms in this wy: T rf ν β Fνβ = I νβ ν β F ν β F νβ = 1 δ νν 2 δββ δ νβ δβν F ν β F νβ = 1 2 F νβ Fνβ F}{{} βν F νβ F νβ = T rf νβ F νβ E.8 nd τκ τκ Fτκ T rf ν β ɛνβ Fτκ = I νβ ν β F ν β ɛ νβ = 1 δ νν 2 δββ δ νβ δβν F ν β ɛ = 1 F νβ ɛ 2 τκ τκ νβ Fτκ F}{{} βν F νβ ɛ τκ νβ Fτκ τκ νβ F τκ = F νβ ɛνβ Fτκ E.9 66