Choptuik scaling and Quantum e ects in 2 D

We study numerically the collapse of massless scalar elds in two-dimensional dilaton gravity, both classically and semiclassically. At the classical level, we nd that the black hole mass scales at threshold like M bh / jp ? p j , where ' 0:53. At the semiclassical level, we nd that in general M bh approaches a non-zero constant as p ! p. Thus, quantum eeects produce a mass gap not present classically at the onset of black hole formation. 1 The discovery of universality and scaling at the onset of black hole formation 1] may have important implications in understanding the structure of solution-space in gravitational theory. Choptuik studied numerically the collapse of a spherically symmetric self-gravitating real scalar eld in four-dimensional (4D) Einstein gravity and considered one-parameter families of initial data, Sp], where p is a parameter specifying the strength of the gravita-tional self-interaction of the scalar eld. He found that there exists a critical value, p = p , such that for p < p , the \subcritical case", no black hole is formed and the solution is regular, while for p > p , the \supercritical case", a black hole is formed. Furthermore, by careful analysis of the solutions near criticality, p = p , he found that as p approaches p from above, the mass of the created black hole approaches zero, and when a black hole just forms, its mass scales as M bh / jp ? p j , where the critical exponent ' 0:37. The critical solutions also exhibit discrete self-similarity 1]. Similar behavior was found in other models of non-linear gravity 2]. In the semiclassical scenario, i.e., for a quantum eld propagating on a classical dynamical background metric, the created black hole of mass M radiates in 4D with the Hawking