Observables for two-dimensional black holes

Abstract

We consider the most general dilaton gravity theory l + l dimensions. By suitably parametrizing the metric and scalar field we find a simple expression that relates the energy of a generic solution to the magnitude of the corresponding Killing vector. In theories that admit black hole solutions, this relationship leads directly to an expression for the entropy S = ~~T o / G , where TO is the value of the scalar field (in this parametrization) at the event horizon. This result agrees with the one obtained using the more general method of Wald. Finally, we point out an intriguing connection between the black hole entropy and the imaginary part of the "phase" of the exact Dirac quantum wave functionals for the theory. PACS number(s): 04.70.D~

Quantum evolution of near-extremal Reissner–Nordström black holes

“…And from this feature it follows immediately, see (6.17) and (6.18), that x + int = x − int and then the intersection point belongs to the AdS boundary so that it gives an infinite amount of proper time in accordance with (6.16). We can also show that F ′′′ (x + int ) = F ′ (x + int ) = 0 and since for all of these three curves (6.17), (6.18), we have 20) one can conclude that the three curves meet at the end-point becoming a null line. The complete physical process is represented in Fig.5.…”

“…At this point the extremal radius curveφ = 0 is null and both outer and inner horizons meet. This means that we have arrived at the end-point of the evaporation and, since we are dealing with analytic expressions, one can check explicitly that the evaporating solution (5.14) matches smoothly along x + = x + int with a static solution for 20) which turns out to be just the extremal black hole. A conformal coordinate transformation brings the metric andφ into the form (4.8) of the extremal solution…”

“…We can also get this result studying the behavior of the metric (2.11) near extremality. The general solution of (2.10) is [20] 19) where J(φ) = φ dφV (φ). The thermodynamical variables, in terms of the twodimensional effective theory, read [21] S = 4πφ + , (2.20)…”

An action for the exact string black hole

“…Thus, for the ESBH one has to evaluate Φ at the Killing horizon. With the same convention for the Boltzmann constant as in [39] the result is…”

“…Simple thermodynamic considerations establish that entropy S is proportional to the dilaton field evaluated on the Killing horizon [39], where "the dilaton field" again refers to the factor multiplying the Ricci scalar in the action. Thus, for the ESBH one has to evaluate Φ at the Killing horizon.…”