THERMODYNAMICS AND/OF HORIZONS: A COMPARISION OF SCHWARZSCHILD, RINDER AND de SITTER SPACETIMES

Abstract

The notions of temperature, entropy and ‘evaporation’, usually associated with space- times with horizons, are analyzed using general approach and the following results, ap- plicable to different spacetimes, are obtained at one go. (i) The concept of temperature associated with the horizon is derived in a unified manner and is shown to arise from purely kinematic considerations. (ii) QFT near any horizon is mapped to a conformal field theory without introducing concepts from string theory. (iii) For spherically sym- metric spacetimes (in D = 1 + 3) with a horizon at r = l, the partition function has the generic form Z ∝ exp[S − βE], where S = (1/4)4πl 2 and |E| = (l/2). This analysis reproduces the conventional result for the blackhole spacetimes and provides a simple and consistent interpretation of entropy and energy for deSitter spacetime. (iv) For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. (v) In the case of a Schwarzschild black hole there exist quantum states (like Unruh vacuum) which are not invariant under time reversal and can describe blackhole evaporation. There also exist quantum states (like Hartle-Hawking vacuum) in which temperature is well-defined but there is no flow of radiation to infinity. In the case of deSitter universe or Rindler patch in flat spacetime, one usually uses quantum states analogous to Hartle-Hawking vacuum and obtains a temperature without the cor- responding notion of evaporation. It is, however, possible to construct the analogues of Unruh vacuum state in the other cases as well. Associating an entropy or a radiating vacuum state with a general horizon raises conceptual issues which are briefly discussed.