Chronology protection in two-dimensional dilaton gravity.

The global structure of 1 + 1 dimensional compact Universe is studied in two-dimensional model of dilaton gravity. First we give a classical solution corresponding to the spacetime in which a closed time-like curve appears, and show the instability of this spacetime due to the existence of matters. We also observe quantum version of such a spacetime having closed timelike curves never reappear unless the parameters are fine-tuned. † On leave of absence from Institute for Nuclear Study, University of Tokyo, Midori-cho, Tanashi-shi, Tokyo 188, Japan ∗ e-mail address: [email protected] Callan, Giddings, Harvey and Strominger[1] (abbreviated as CGHS) provided a useful toy model of (1+1)-dimensional gravity. They attempted to analyze the back-reaction of the Hawking radiation[2] in the two-dimensional analogue of black hole geometry in a consistent way by the use of this model. In their first paper, CGHS developed their original scenario as follows: First introducing 1+1 dimensional dilaton gravity, CGHS found the solutions corresponding to black hole formation, and showed occurrence of the Hawking radiation in the spacetime background they obtained. Next taking account of quantum effect of quantized conformal matters as the Polyakov term[3], they analyzed the back-reaction of the Hawking effect in a leading semi-classical approximation consistently. However their first scenario on the quantum black hole remains to be elaborated. Up to the present time, many people have studied the behavior of quantum black hole in this model[4,5]. On the other hand, in general relativity, there are other interesting problems which must be resolved taking account of quantum effects. For example, since Morris, Thorne and Yurtsever pointed out the possibility of making a time-machine[6], quantum effect on the spacetime in which closed time-like curves appear (abbreviated as CTC-spacetime) has been discussed by several authors [7-11]. One of the most important problems in this subject is whether CTC-spacetime continues to exist or not if the quantum effect is taken into account. Hawking suggested the possibility that the laws of physics may prevent the appearance of closed timelike curves (so called ‘Chronology Protection Conjecture’). In four dimensions the analysis of this problem so far have been done only in fixed background spacetimes. So it will be interesting to treat such spacetimes including back-reaction in closed form even in lower dimensions. In this paper we will investigate the quantum back-reaction problem on the stability of the CTC-spacetime in two-dimensional model of dilaton gravity. Following the CGHS scenario, we apply this model to a compact one dimensional universe. In the first half of this paper we give a classical solution corresponding to CTC-spacetime: an analogue of the Misner universe. Then we show the disappearance of this spacetime due to the existence of conformal matters even if the