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
Year: 1976
Type: dissertation
Abstract: The central objective of this work is to present an analysis of the \nasymptotic conformal Killing vectors in asymptotically-flat space-times \nof general relativity. This problem has been examined by two different \nmethods; in Chapter 5 the asymptotic expansion technique originated by \nNewman and Unti [31] leads to a solution for asymptotically-flat spacetimes \nwhich admit an asymptotically shear-free congruence of null \ngeodesics, and in Chapter 6 the conformal rescaling technique of Penrose \n[54] is used both to support the findings of the previous chapter and to \nset out a procedure for solution in the general case. It is pointed out \nthat Penrose's conformal technique is preferable to the use of asymptotic \nexpansion methods, since it can be established in a rigorous manner \nwithout leading to the possible convergence difficulties associated with \nasymptotic expansions. \nSince the asymptotic conformal symmetry groups of asymptotically flat \nspace-times Are generalisations of the conformal group of Minkowski \nspace-time we devote Chapters 3 and 4 to a study of the flat space case so \nthat the results of later chapters may receive an interpretation in terms \nof familiar concepts. These chapters fulfil a second, equally important, \nrole in establishing local isomorphisms between the Minkowski-space \nconformal group, 90(2,4) and SU(2,2). The SO(2,4) representation has been \nused by Kastrup [61] to give a physical interpretation using space-time \ngauge transformations. This appears as part of the survey of \ninterpretative work in Chapter 7. The SU(2,2) representation of the \nconformal group has assumed a theoretical prominence in recent years. \nthrough the work of Penrose [9-11] on twistors. In Chapter 4 we establish \ncontact with twistor ideas by showing that points in Minkowski space-time \ncorrespond to certain complex skew-symmetric rank two tensors on the \nSU(2,2) carrier space. These objects are, in Penrose's terminology [91, \nsimple skew-symmetric twistors of valence \n[J. \nA particularly interesting aspect of conformal objects in space-time is \nexplored in Chapter 8, where we extend the work of Geroch [16] on multipole \nmoments of the Laplace equation in 3-space to the consideration. of \nQ tý =0 in Minkowski space-time. This development hinges upon the fact \nthat multipole moment fields are also conformal Killing tensors. \nIn the final chapter some elementary applications of the results of \nChapters 3 and 5 are made to cosmological models which have conformal \nflatness or asymptotic conformal flatness. In the first class here we \nhave 'models of the Robertson-Walker type and in the second class we have \nthe asymptotically-Friedmann universes considered by Hawking [73].
Author: G. K. Griffin
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1499512043