Conformal symmetries in special and general relativity.The derivation and interpretation of conformal symmetries and asymptotic conformal symmetries in Minkowski space-time and in some space-times of general relativity.

摘要

The central objective of this work is to present an analysis of theudasymptotic conformal Killing vectors in asymptotically-flat space-timesudof general relativity. This problem has been examined by two differentudmethods; in Chapter 5 the asymptotic expansion technique originated byudNewman and Unti [31] leads to a solution for asymptotically-flat spacetimesudwhich admit an asymptotically shear-free congruence of nulludgeodesics, and in Chapter 6 the conformal rescaling technique of Penroseud[54] is used both to support the findings of the previous chapter and toudset out a procedure for solution in the general case. It is pointed outudthat Penrose's conformal technique is preferable to the use of asymptoticudexpansion methods, since it can be established in a rigorous mannerudwithout leading to the possible convergence difficulties associated withudasymptotic expansions.udSince the asymptotic conformal symmetry groups of asymptotically flatudspace-times Are generalisations of the conformal group of Minkowskiudspace-time we devote Chapters 3 and 4 to a study of the flat space case soudthat the results of later chapters may receive an interpretation in termsudof familiar concepts. These chapters fulfil a second, equally important,udrole in establishing local isomorphisms between the Minkowski-spaceudconformal group, 90(2,4) and SU(2,2). The SO(2,4) representation has beenudused by Kastrup [61] to give a physical interpretation using space-timeudgauge transformations. This appears as part of the survey ofudinterpretative work in Chapter 7. The SU(2,2) representation of theudconformal group has assumed a theoretical prominence in recent years.udthrough the work of Penrose [9-11] on twistors. In Chapter 4 we establishudcontact with twistor ideas by showing that points in Minkowski space-timeudcorrespond to certain complex skew-symmetric rank two tensors on theudSU(2,2) carrier space. These objects are, in Penrose's terminology [91,udsimple skew-symmetric twistors of valenceud[J.udA particularly interesting aspect of conformal objects in space-time isudexplored in Chapter 8, where we extend the work of Geroch [16] on multipoleudmoments of the Laplace equation in 3-space to the consideration. ofudQ tý =0 in Minkowski space-time. This development hinges upon the factudthat multipole moment fields are also conformal Killing tensors.udIn the final chapter some elementary applications of the results ofudChapters 3 and 5 are made to cosmological models which have conformaludflatness or asymptotic conformal flatness. In the first class here weudhave 'models of the Robertson-Walker type and in the second class we haveudthe asymptotically-Friedmann universes considered by Hawking [73].