Two Dimensional Stringy Black Holes with One Asymptotically Flat Domain

The exact black hole solution of 2D closed string theory has, as any other maximally extended Schwarzschild-like geometry, two asymptotically flat spacetime domains. One can get rid of the second domain by gauging the discrete symmetry on the SL(2,R)/U(1) coset that interchanges the two asymptotic domains and preserves the Kruskal time orientation everywhere in the Kruskal plane. Here it is shown that upon performing this orbifold procedure, we obtain a theory of unoriented open and closed strings in a black hole background, with just one asymptotically flat domain and a time-like orbifold singularity at the origin. All of the open string states of the model are confined to the orbifold singularity. We also discuss various physical aspects of the truncated black hole, in particular its target duality – the model is dual to a conventional open string theory in the black hole geometry. ⋆ e-mail addresses: [email protected] or [email protected] † Robert R. McCormick Fellow; research also supported by the NSF under Grant No. PHY 90-00386; the DOE under Grant No. DEFG02-90ER40560; the Czechoslovak Chart 77 Foundation; and the ČSAV under Grant No. 91-11045. Black holes are spacetimes of amazingly rich geometrical and topological structure. The recent discovery of exact black hole solutions of 2D string theory in terms of an SL(2,R)/U(1) coset CFT ([1,2], see also [3]) allows one to study some properties of quantum theory in the geometry of a black hole quite explicitly [4,5,6,7], and in particular to search for stringy effects, such as duality [8,9,10]. Another important aspect of the black hole solution is its expected relation to c = 1 matter coupled to Liouville [1,5,11,12]. One of the many interesting issues of quantum black hole physics is the role of various asymptotically flat domains in the black hole geometry. As any other maximally extended Schwarzschild-like black hole, the 2D black hole has two distinct asymptotically flat exterior regions (denoted by I and II throughout the paper) which cannot influence each other causally. In this paper we will address the question of getting rid of the second asymptotically flat domain of the stringy black hole in a consistent way. ‡ Actually, we will point out that the truncated black hole with just one asymptotically flat exterior domain is naturally inhabited by unoriented open and closed strings. As any difference between closed and open strings gets lost in the point particle limit, this is a typical stringy effect in the black hole physics. After establishing this result, we will analyze more closely some aspects of the truncated black hole, in particular its spectrum, spacetime duality, and the description of the black hole singularity in terms of an effective topological field theory. (Some other aspects of black holes in 2D open string theory have been studied recently in [15,16].) In string theory, there is a particularly natural way of truncating spacetime manifolds, by using orbifold techniques. In fact, this strategy has been recommended for the 2D black holes quite explicitly by Witten [1], who has proposed to mod out the black hole coset by the Z2 symmetry that maps the two asymptotically flat domains to each other by mapping u to v in the Kruskal-Szekeres coordinate ‡ Considerable effort in this direction has been made in the 4D black hole physics since the middle of sixties [13] (for a review, see e.g. [14]). Most of the authors have identified points (u, v) ≡ (−u,−v) in the Kruskal plane, which is not the strategy accepted in this paper.