A Simple Line Element to the Dilaton Gravity

The metric to the two-dimensional dilaton gravity can be writen in an alternative form, similar to the two-dimensional Schwarzschild metric, and allow us the identification of some quantitiies with those equivalent in the Schwarzschild solution. This new form, howewer, presents a non-physical singularity at the horizon in the same way that in the realistic four dimensional case. We show a procedure to eliminate this horizon singularity and, as an application, the resulting metric is used to obtain the associated Hawking temperature. We discuss also some differents between this metric and the Schwarzcchild one. . PACS: 04.60.+n; 11.17.+y; 97.60.Lf (a)[email protected] Work supported in part by Fundação Universitária José Bonifácio, FUJB. 1 1INTRODUCTION In the heuristic picture often proposed to visualize the origin of the Hawking-Bekenstein effect[1], the radiation of the black-hole arises by the combination of the pair creation and tunneling processes. At the same time, the simplest way to describe these objects to wit, the Schwarzschild solution has an (unphysical) singularity at the horizon. So, since we need to describe the tunneling as an across-horizon phenomena, it is necessary to choose coordinates that, unlike Schwarzschild ones, are not singular at the horizon. These coordinates are the well known Kruskal-Szekeres coordinates. We have the same situation in the two-dimensional dilaton gravity in the Schwarzrschild form where there is a coordinate singularity that has to be avoided to calculate some quantities at the horizon. As in the realistic case above mentioned, we can put the metric in such a way that this singularity is removed. This is the purpose of this note: to write out a metric to the dilaton gravity without this coordinate singularity. As an application, we calculate the associate Hawking temperature in a very simple way. The article is organized as follows. We show the coordinate transformation for the 2d Schwarzschild case. Then, with a similar transformation, we obtain the desired form to the dilaton metric. We finish with the application of this line element to the calculation of the associated temperature of the black-hole. 2THE NEW COORDINATES TO THE TWO-DIMENSIONAL GRAVITY We start this section with the Schwarzschild solution where the angular parts have been discarded: ds = gμνdx dx = − ( 1− 2M r ) dts + ( 1− 2M r )−1 dr (1)