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On the Palatini Variation and Connection Theories of Gravity On The Pdatini Variation and Connection Theories of Gravity Howard Steven Burton A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 1998 @Howard Burton 1998 National Library Bibliothhue nationale du Canada Acquisitions and Acquisitions et Bibliographic Services services bibliographiques 395 Wellington Street 395. rue Wellington OttawaON KlAON4 Ottawa ON KIA ON4 Canada Canada The author has granted a non- L'auteur a accorde une licence non exclusive licence allowing the exclusive pennettant a la National Library of Canada to Bibliotheque nationale du Canzda de reproduce, loan, distriiute or sell reproduire, prgter, distribuer ou copies of this thesis in microform, vendre des copies de cette these sous paper or electronic formats. la forme de microfiche/^ de reproduction sur papier ou sur format eectronique. The author retains ownership of the L'auteur conserve la proprikte du copyright in this thesis. Neither the droit d'auteur qui protege cette these. thesis nor substantial extracts fiom it Ni la these ni des extraits substantiels may be printed or otherwise de celleci ne doivent &re imprimes reproduced without the author's ou autrement reproduits sans son permission. autorisation. The University of Waterloo requires the signatures of all persons using or photocopy- ing this thesis. Please sign below, and give address and date. Abstract This thesis involves a rigorous treatment of the Palatini Variational Principle of grav- itational actions in an attempt to fdy understand the r6le of the connection in such theories- After a brief geometrical review of afie connections, we examine N- dimensional dilatonic theories via the Standard Palatini principle in order to highlight the potential differences arising in the dynamics of theories obtained by utilizing the Hilbert and Palathi formalisms. We then develop a more generalized N-dimensional, torsion-heel Einstein-Hilber t-type action which is shown to give rise to Einsteinian dynamics but can be made, for certain choices of the associated arbitrary parameters, to yield either weak constraints or no constraints on the connection, I'. The latter case is referred to as a Kmaximallysymmetric" action. In the following Chapter this analysis is extended to the realm of a potentially non- vanishing torsion tensor, where it is seen that such actions do not, in general, lead to Einsteinian dynamics under a Palatini variation. Following another brief geometrical review, which highlights some elements of fibre bundle theory appropriate to our later analysis, we examine the secalled Palatini Tetrad formalism and show that it must be modified for a proper Palatini variation - i.e. to not assume anything a peoion about the relevant connection. We then analyze this modified approach &om a geometrical perspective and show that, for the torsion-kee case at least, a proper treatment of the Palatini Tetrad procedure is equivdent to the "maximally symmetric" case alluded to earlier. Furthermore, we recognize that the Palatini Tetrad approach should effectively be regarded as little more than a calculational technique resulting fiom analysis of the more generalized action S = J tr [RA t (P A /?)I + S,, where R,,tl are local ver- sions of the cunmture 2-form and solder 1-form fiom the GL(N,R) frill bundle of fkames respectively. Ow abovementioned modification of this approach not only renders treatment of this action geometrically consistent (i.e. by considering all of the terms in the action as pertaining to the same principal bunde), but also enables one to clearly see the manifest connection invariance of the fdl theory (given some connection independent matter action, S,) . Hence a rigorous Palatini analysis of Einstein-Hibert - like actions leads one to the rather unexpected conclusion that generalized Einsteinian actions of the type S = J tr [RA t (j? A P)]+ S, are connec- tion invariant and naturally give rise to Einsteinian dynamics if S, is independent of the connection, w; and otherarise only give rise to Einsteinian dynamics by an a pos- teriora' symmetry-breaking-type condition, T = 0 (for T the torsion tensor), identical to that of Einstein-Cartan theory. Acknowledgements I wodd like to thank Robb Mann for giving me the opportnnity to pursue my interests and being flexible and supportive enough to enable me to do so in my own way; Dan Brooks for kindly allowing me to be an "honorary biologistn for e-mail purposes; Ivan Booth for his much appreciated computer assistance; Bill Sajko for his frequently in- sightfid comments and constant willingness to talk about physics; Richard Sharpe for generously helping me with some geometrical matters and Chris Isham for inspiring me to look at mathematics in a completely new way. On a more personal level, I wodd like to thank my parents for always being there, my wife for her unflagging love and support, and my children, who show me what is truly important. To my parents, who set me on the path of learning and supported me every step of the way. Contents 1 Introduction 2 Affine Connections 2.1 Overview .................................. 2.2 Definition of an Afie Connection .................... 2.3 Metricity .................................. 2.4 Torsion ................................... 2.5 Levi-Civita Connections ......................... 3 Standard Pdatini Formalism (Torsion-Free) 3.1 Introduction ................................ 3.2 Hilbert and Palatini Dynamics of a Generalized N-Dimensional Dilaton Action ................................... 3 -3 Generalized N-Dim. Eins kin-IFilbert Action .............. 3.4 Extended Action Dynamics ....................... 3.5 Summary ................................. 4 Palatini Variation of Actions with Torsion 4.1 Overview .................................. 4-2 Einstein-Hilbert Action with Torsion .................. 4-3 Extended Einstein-Hilbert Action with Torsion ............. 41 4.3.1 Final Dynamics .......................... 46 4.4 Extra Torsion Terms to the Action ................... 47 4.5 Summary ................................. 52 5 Geometrical Divertimento 56 5.1 Fibre Bundle Review ........................... 56 5.2 Bundle of Frames ............................. 64 5.3 Metrics and The Bundle SO(N - 1, 1)(M) ............... 69 6 Palatini Tetrad Formalism 72 6.1 Overview .................................. 72 6.2 3+1 Palatini Tetrad Formalism ..................... 73 6.3 Generalized 3+1 Palatini Tetrad Formalism .............. 79 7 Andysis 84 7.1 Geometrical Picture of Generalized 3+1 Palatini Tefxad Formalism . 84 7.2 Relating Palatini Tetrad to Standard Palatini ............. 89 7.3 GravitationalActions ........................... 93 7.4 Matter actions ............................... 95 8 Conclusions and Discussion 97 A N = 2 Pdatini Dynamics From a Generalized Dilaton Action 102 B Variation of a (Torsion-Free) Extended Action Under a Deformation Transformation 105 C Explicit Calculation of The Connection for SBaB With Torsion For A of Rank 2 109 D Explicit Calculation of the Final Dynamics for SBHB(A of Rank 2) 114 E Moving &om (so(3,l)(M)R~, QN, M)to TM 118 List of Figures 2.1 Torsion as Non-Closure of an Infinitisemal Parallelogram ....... 14 5.1 Principal Fibre Bundle .......................... 57 5.2 Parallel Transport in Principal Fibre Bundles ............. 61 5.3 Parallel Transport in Associated Vector BnndIes ............ 65 5.4 Bundle of Frames with Solder Forms .................. 70 6.1 SO(3. l)(M). Induced Covariant Derivative on TM .......... 77 Glossary of Terms and Symbols v - Afbe Connection I', I', ', - Connection Coefficients T, ', - Torsion Tensor T, ', ', - I?, ', (in holonomic coordinates) RQBX,- Riemannian Curvature Tensor X X Rap = r, *,,, - r, + r, ^, r, ', - r, *, r, ., (in holonomic coordinates) RBc:= Rcfiuc- Ricci Tensor R := pRs, - Ricu Scalar f,',) := $9" [gWc,v + gm,rnt. - geYI.J- Christ0ffk.l Symbol V, - Covariant Derivative with I?, ', = {, ,) v2:= 'D*D~ z* := vpgAp g - Lie Group L(G) - Lie Algebra of 4 D - Exterior Covariant Derivative A - Principal Bundle Connection 1-form G - Principal Bundle Curvature 2-form a - Local CrossSection of Principal Bnndle o*A := w - Local Connection I-form a*G := R - Local Cunrature 2-form B (M) - Bundle of Ekames 8 - Solder 1-form on B (M) a'@ := p - Local Solder Form Chapter 1 Introduction Among the various schisms throughout the theoretical physics community, perhaps none is more contentious than that which separates researchers into the camps of metric-oriented or connection-oriented perspectives of gravity. Historically, of course, Einstein founded General Relativity as a dynamical theory of the metric tensor with the connection necessarily being that of the unique torsion- free metric-compatible Levi-Civita connection, {,',). This was more than a choice of mathematical convenience or simplicity as there are several important physicdy motivated reasons to opt for this connection, many of which were integral to Einstein's deep physical intuitions about the nature of space-time. Nonetheless, from the earliest days since the advent of General Relativity, at- tempts have been made to generalize it, sometimes explicitly with regards to the connection. One of the fist such attempts made in this regard is the secaUed Pda- tini variation [14,15, 361' where one subjects the generalized Einstein-Hilbat action, with I? no longer a priori regarded as any particular function of the metric, to a =An interesting irony is that
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