View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Massey Research Online Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author. SCALAR-'i'ENSOR THEORIES O:E' GRAVITATION A thesis pI·esEmted in partial fulfilment of the requirements for the degr ee of Master of Science in Na thematicG at Massey University Ian Paul Gibbs 1973 CO NTENTS Chnptor One . I nt r oduction page Cha pte r Two . B:-tclcgr ound 3 Cha ptGr 'rhrcG . Tho sca b.r- t e nsor nod.ol of gr nvitc. tion 14 3. 1 Spa c e-tino thoorie s 14 3. 2 Gra vi t<1tion t heorios 19 3. 3 Tho sca. l:1r -tenscr nodol of gr ,?..vi t :L tion 22 3. 4 Ezp-c: r i 1.1o nt :1l t ost s ;1nd tho s c 'J. l .'.l r-tensor IJ.o cl e l 27 3. 5 Coni'or r:1al i nv ..,_ rio.11c 8 cmd tho sca J ar-tons or aode l 29 3. 6 Doser' s sca l a.r- t ;,, nsor theory 35 Chapter Fou r Tl10 Bra ns-Dicke sc.'.ll '.lr-tensor t h oor y 38 4.1 'I'ho ED s cnl nr-tcnsor t h eory 38 4. 2 V.'.lcu~~ oolutions 50 4. 3 Stand.'.lr d BD cos nol og,J 58 4.4 Units tr:rnsfor r:nti :. ns nnd tho BD t heor y 63 4. 5 St'.'ln·.:lard BD cosr.:o l ogy ( ctd) 6G 1). 6 Hon- sbm u::..r c'. BD cos:,cl cgy 69 , n ' r • I Scnl -:J. r-n::i. t tcr fiol d e-o.ugo f r o _; doc 71 4.8 Nordstro.::- RG i m::mer typo solutions 80 4. 9 :wE f crnula tion in t he BD theor y 8 2 4 . 10 BD Theor y nnd Ma ch I s Principl o 8!1- Cha ptor Five. Spccia l sca l a r - t ensor theories 89 5. 1 Gursey ' s theor y 89 5.2 H:1s s i v e BD t heoriGs 92 5.3 Othor s ca l a r-tenso r theories 94 Chapter Si x . The s ca l a r-tensor nodol of gr avitation (ctd) 99 6 . 1 Linoa r connections 102 6. 2 Weyl spo.ce- tin e 104 6. 3 Grnvita tion in Weyl spo.ce-tin e 107 -6.4 Dirac' s theory 113 6.5 Lyra space- time 117 606 Gravita tion in Lyra space-tine 120 Chapt or Seven. Conclusion 126 Appendix 123 Bibliogr aphy 1 - hapter One I NTRODUCTION The problem the writer wishes to consider here , is essentially one rela ted to the classical field description of I~a ture. The fram ew ork o f General Relativity provides a t heory for the geometry of t h e f our dimensional space­ time ma nifold and a t t he same time g ives a description of t h e gravitatio nal field i n terms o f the metric tensor, wh ile t he e lectromagnetic field can be interpre ted in t erms o f a par ticul a r second r ank , skew-symme tric tensor -- t he covariant curl of a vector field defined on the manifold. Howe ver t he sca l a r field, the simplest geo­ me tric object tha t could be defined on t he manifo ld, does not s eem t o be experimenta l ly e vident wh en it is interpret­ ed a s a third, classica l long r ange field. I n spite of t his l a ck of expe rimental evidence and as there appears to be no theoretica l ob j ection to the exi stenc e of such a long range field, the problem i s to i ntroduce the s ca l a r field - into th e clas sical scheme of t h ings a nd t o construct a viable theory c ont a ining a ll three long range fields. It is interesting to compare the physical descriptions involved with these fields. Both the gravi­ tationa l and the electromagnetic fields have gauge-like degrees of freedom a nd before a situation could be physic­ ally relevant these degrees of fre e dom must be fixed - for the g7avitational field by imposing coordinate condi- · tions while for the electromagnetic field, after co9rdin­ ate ·conditions have been imposed t h e gauge of the field potential must be chosen . As a consequence of these gauge freedoms, ih order that the fields couple consist­ ently with matter sources, th_e energy momentum tensor of the souree must be covariantly conserved and the electro- 2 magnetic current density of t he source must be c o nserved. The scalar field on t he other hand has no gauge-lik e degree o f fr ~edom and c onsequently has no c onserved 11 charge" as a source. Thus for e xamp l e , i n contrast to the other two fields, no c onstrain ts exist by which the scalar field c ould be separated i nt o a sourc e 11 bound" part and a free " wave 11 part. In rec ent y e ars the pr oblem of i ncorporating t he sca lar field i n to the description of gravitati on has led t o the investigation of a speci a l class of gr avita­ tion theories -- the scala r-tensor gravitation theories. With the previously mentioned pr obl em in mi nd , the goals o f this the sis are ( i ) t o review work that has been done on these theories and (ii) t o discuss them i n a way t ha t compares them t o t he the ory of gr a vitation g iven i n Gener al Relativity. Chapter Two basically gives an historical ba ck­ ground and introduces mor e specific mo tives for cons idering the scalar f i e ld as a fundamental physica l fieldo Chapter Three consider s the i mportan t class of scalar-tensor gravitation t heories based on a Riemann space-time and Chapt er Four continues this t heme by looking at the "most developed" and perhaps simp l est member - the Brans-Dicke t heory. For c ompleteness the 11 massive Brans-Dicke" theories and some special scalar-tensor theories are looked at briefly in Chapter Five. Chapter Six retnrns to the scalar-tensor model of gravitation developed in Chapter Three and looks at the implications for the model in more general space-times. ACKNOWLEDGEMENTS I wish to express my gratitude to Dean Halford for supervising work for this thesis. Also I would like to thank the typists who have helped in the completion of the thesis. 3 Chapter Two BACKGROUND The electromagnetic end gravi t a tional fields were described in t he introduction aG classical long r ange fields . These fields are r esponsible for forces that fall off i nversely proportional t o the square of the dis­ tance apart of the intera c ting bodies (sources) ; in contrast to short r ange forc es which show an exponential hehaviour. Einstein (1916) ( 1 ) , a ttribu ted to t he space- time manifold a Riemann :::: tructure ,nd gave "mea ning" t o the gr a vitational field in termP. of curvature through h i s ~ravi­ t a tional f i e ld ~qu~t io~n G­µv , wher e G i s Newton ' s gra vita tio~~l constant . The notation est abli shed here is used in most sections. Units of length and time ar& c hosen such t hat c 5 1, a lthough with this understanding some formulae may-still c ontai-n c • Greek indices r ange over the VD.lue-s : • } ) lo> 11 1, 3 , the coordinates xJ and xi., (t - 1 ' 2 , ,3 , a r e assumed time -like and space-like r espectively and the signature of the spac e -time me tric, g !'.\/ is _ + + + The Ri emann and Ricc i t ensors have the res p e cti~e forms The close relation between the Riemann curvature tensor and gravitational effects is further illustra ted, for example, in t he equations of geodesic devia tion, (2) = , 2.2 4 de fi n e t he p at h s o f a D pair o f Drii ghb o uri11g , fr e ely f a lling particles and. I),r denot e s t he abso lut e deri va t ive a long t he curve xa(T). A fr eely f a l ling partic l e i s a t r e st i n a c o or d i nate frame f a l l i ng with it, wher eas a pair of nei ghbouring fr e ely f a lling particle s will s how a r e l a tive accelera- tion given b y eq. 2 .2. Tc a n observer tra v elling with t he frame t h i s mo tion will i n dica te t he pr esenc e of a gr a vita tional fi e lio The e l e ctromagne t i c fi e ld on t he o t her hand , a ppear s i n t h i s p i c ture as a fi e l d " embedded" i n spac e -time , the geome try o f whi c h i s d e t er mi ne d by g r a vita tion.