General relativity in the large

Abstract

in the ea:ly years of general relativity, the most pressing questions ~vcre local in character, e.g., the suitability of de~'ribing the gra~ itational ticld by a metric, the structure and consequences of Einstein's eqttalions, etc. It was presumably felt that global questions, while possibly of some physical interest, could safely be deferred until a litter stage. We are now involved in this later stage. Perhaps the most important reason for the rcceilt emphitsis on global properties was tile realization that to establish certain results ~ i n particular, the theorems on singularities ~it is neces,;ary at the outset to exercise some control over the admissible global bchavi()r of the spacetime. The .scope of global work his gradually I~'en extended beyond attempts to control misbehavior until t t~ay global m,:thods comprise a small but vigorous brat.oh of relativity. "What have we gained from these efforts ? We h:|ve becolne nlorc sensitive to glob:d possibilities and pitfalls. (When will a global condition be needed? Which intuitive ideas have been formulalcd precisely? What i, re the consequences of imposing various conditions ? Ilow restrictive are the conditions ?j We hitve reached a lx.ttcr understanding of the various levels of structure of a space-time and their interaction. (Which properties of a .~pacetintc involve only the c:.tusal .,,|ructurc, which only the toi~t~logy, etc. ? Do the causal relations determine the topology?) We have becll led to deeper and more r trealments of kno~.~, n result.,;. (]'lie IIOlit)ll Ol,"l 'generic' property of space-times, for example, offers a potentially v;,luablc approach to singularities.) Finally. ~e h:,ve come to appreciate that the structure of a space-time in tile large may have tme,~pectcd physical signiticante. (I)o the homoh)gy or homot,~py groups, the Siicfcl Whitney cL,sscs, or the spinor structure have implic:ttions for clcmclltary particle physics ?) This pap~;r con,,ists of a collecti,m r162 ;tl~otit the present status of glob;,I work plus ;l fi:w remarks about ~hcre t~c might go from here. (]'he usual caution is in force: many of the impt~rt;mt ildvallccs ill physics have been a result of asking new tlue.~tions which did not appear on ii~ts of