Inonu-Wigner Contractions of Kac-Moody Algebras

We discuss In\"on\"u-Wigner contractions of affine Kac-Moody algebras. We show that the Sugawara construction for the contracted affine algebra exists only for a fixed value of the level $k$, which is determined in terms of the dimension of the uncontracted part of the starting Lie algebra, and the quadratic Casimir in the adjoint representation. Further, we discuss contractions of $G/H$ coset spaces, and obtain an affine {\it translation} algebra, which yields a Virasoro algebra (via a GKO construction) with a central charge given by $dim(G/H)$.Comment: 11 pages, IMSc/92-2

Black Hole Entropy and Quantum Gravity

An elementary introduction is given to the problem of black hole entropy as formulated by Bekenstein and Hawking. The information theoretic basis of Bekenstein's formulation is briefly reviewed and compared with Hawking's approach. The issue of calculating the entropy by actual counting of microstates is taken up next within two currently popular approaches to quantum gravity, viz., string theory and canonical quantum gravity. The treatment of the former assay is confined to a few remarks, mainly of a critical nature, while some of the computational techniques of the latter approach are elaborated. We conclude by trying to find commonalities between these two rather disparate directions of work.Comment: 13 pages, Revtex, 5 eps figures. Invited talk at National Symposium on Trends and Perspectives in Theoretical Physics, IACS, Calcutta, India, April, 1998. One reference adde

Thermal stability of radiant black holes

Beginning with a brief sketch of the derivation of Hawking's theorem of horizon area increase, based on the Raychaudhuri equation, we go on to discuss the issue as to whether generic black holes, undergoing Hawking radiation, can ever remain in stable thermal equilibrium with that radiation. We derive a universal criterion for such a stability, which relates the black hole mass and microcanonical entropy, both of which are well-defined within the context of the Isolated Horizon, and in principle calculable within Loop Quantum Gravity. The criterion is argued to hold even when thermal fluctuations of electric charge are considered, within a {\it grand} canonical ensemble.Comment: Latex2e, 13 pages, 2 eps figures. Contribution to the volume to be published by the Indian Association for the Cultivation of Science, Kolkata, India in revered memory of Prof. A. K. Raychaudhur