GRG Vol. 5, No.5 (1974), pp. 593- 597. A CLASS OF SINGULAR SPACE-TIMEST E. IHRIG and D.K. SEN Department of Mathematics, University of Toronto, Toronto, Canada M5S IAI Revised version received 2 August 1973 ABSTRACT We present a singularity theorem for a certain class of space-times. The theorem contains an 'energy' condition stronger than Hawking's, but does not require any con- ditionabout Cauch9 surfaces, normals or time orientab ~ ility. w INTRODUCTION In recent times much interest has arisen in singularities in general relativity. Part of this interest has been due to theorems of Hawking ([1,2,3]) which show that singularities occur under very general circumstances. Hawking's theorems cover a very large por- tion of the 'physically interesting' space-times, but they do not cover all such. If in fact one allows the cosmological constant to be non-zero, one can find a complete space-time which satisfies all the desirable global properties and which locally can be made the same as any Robertson-Walker space-time metric. We need only take a time function R(t) which matches observations as far as we can observe them, and then make it periodic after that (assuming a big bang has not been observed). Thus it would be interesting to see if singularity or non-singularity theorems can be made for some general classes of space-times which contain some of the Robertson- Walker space-times that Hawking's theorems exclude. There are three basic classes of Robertson-Walker metrics. The first contains those metrics for which R(t) goes to zero as t ap- proaches tO. These are the castrophic models. All geodesics are singular at t = to; space collapses to or arises from a point. The second class consists of the singular space-times in which R(~) is defined for all real values of t. This second class is not so dras- tic since it has at least one past and one future directed geodesic which is complete; that is, t = s, the affine parameter and r,O,r constant. An example of such a space-time is the model where R(t)= e~0 t. The third class is that of the complete Robertson-Walker % Supported in part by National Research Council (Canada) Research Grant No. A 4054. Copyright 9 1974 Plenum Publishing Company Limited. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photo-copying, microfilming, recording or otherwise, without written permission of Plenum Publishing Company Limited.