Classical and quantum gravity in 1 + 1 dimensions. II: The universal coverings

Abstract

A set of simple rules for constructing the maximal (e.g. analytic) extensions for any metric with a Killing field in an (effectively) two-dimensional spacetime is formulated. The application of these rules is extremely straightforward, as is demonstrated at various examples and illustrated with numerous figures. Despite the resulting simplicity we also comment on some subtleties concerning the concept of Penrose diagrams. Most noteworthy among these, maybe, is that (smooth) spacetimes which have both degenerate and non-degenerate (Killing) horizons do not allow for globally smooth Penrose diagrams. Physically speaking this obstruction corresponds to an infinite relative red/blueshift between observers moving across the two horizons. -The present work provides a further step in the classification of all global solutions of the general class of two-dimensional gravity-Yang-Mills systems introduced in Part I [1], comprising, e.g., all generalized (linear and nonlinear) dilaton theories. In Part I we constructed the local solutions, which were found to always have a Killing field; in this paper we provide all universal covering solutions (the simply connected maximally extended spacetimes). A subsequent Part III [2] will treat the diffeomorphism inequivalent solutions for all other spacetime topologies.Part II is kept entirely self-contained; a prior reading of Part I is not necessary.

An action for the exact string black hole

“…We will focus exclusively on the ESBH, but again the results for the ESNS follow straightforwardly from switching to the second branch of the square root function. Eliminating the auxiliary fields X a and the spin connection ω yields 28) where g is the determinant of the metric g µν with respect to which the covariant derivative ∇ is torsion free and metric compatible. The curvature scalar r is multiplied by the auxiliary dilaton Φ which obeys the relations (3.2)-(3.4).…”

“…hypersurfaces). The global Carter-Penrose (CP) diagram may be obtained by well-known methods [28,31]: the basic idea is that by simple mirror flips (e.g. by assuming that X − = 0 instead of assuming X + = 0 in a given patch) one achieves almost a universal covering.…”

Dilaton gravity in two dimensions

“…In a simplified notation [264] the line element (3.26) is written for the outgoing case (f ∝ u, X ∝ r, k 2 (r) = ξ(r), ξ ∞ := +1)…”

“…We have glossed over several delicate points in this procedure [263,264]. As pointed out at the beginning of this section in a more complicated case a careful analysis of geodesics is necessary at external boundaries and, especially, at the corners of a diagram like Fig.…”

QUANTUM DILATONIC GRAVITY IN d=2,4 AND 5 DIMENSIONS

“…These terms come from the minimal scalar conformal anomaly (first term in (103)). [84,85,86,38,87,88,89,90] (for a review and more complete list of refs., see [91]). …”