A quantum model of Schwarzschild black hole evaporation

Abstract

We construct a one-loop effective metric describing the evaporation phase of a Schwarzschild black hole in a spherically symmetric nulldust model. This is achieved by quantising the Vaidya solution and by chosing a time dependent quantum state. This state describes a black hole which is initially in thermal equilibrium and then the equilibrium is switched off, so that the black hole starts to evaporate, shrinking to a zero radius in a finite proper time. The naked singularity appears, and the Hawking flux diverges at the end-point. However, a static metric can be imposed in the future of the end-point. Although this end-state metric cannot be determined within our construction, we show that it cannot be a flat metric.

One-loop corrections for a Schwarzschild black hole via 2D dilaton gravity

“…Since the matter in this model behaves as a free 2D scalar field, we take that T 00 at infinity (in the zero-th order in κ) has the temperature dependence of a 1D free bose-gas (π/6)T 2 H , where T H is the classical Hawking temperature 4πT H = 1 a . This is consistent with the fact that for spherical null-dust T uu = T vv = 1/48π(4M) 2 in the Hartle-Hawking vacuum, where u and v are the asymptotically flat Schwarzschild coordinates [11]. This gives C 2 = 1 a 2 .…”

One-loop effective action for spherical scalar field collapse

“…This asymptotics was taken because it appears in the 2d dilaton gravity models which describe the spherical general relativity [2,3], as well as in the CGHS model of 2d black holes [4]. Therefore the effective action derived in [1] can describe the back-reaction effects for a realistic 4d black hole.…”

One-loop effective action for a generic 2d dilaton gravity theory

“…When α = 4 and U(Φ) =const.= 4λ 2 the action (1.1) is the CGHS model, while α = 2 and U(Φ) = e 2Φ gives the spherically symmetric general relativity coupled to null-dust (in units G = 1, where G is the Newton constant) [16], to which we refere as the SSND model. If the dilaton is redefined as φ = e −2Φ , and after an appropriate rescaling of the metric (g µν = e αΦ/2 g µν ), the action (1.1) simplifies…”

“…The form (1.2) is more convenient for the calculation of the effective action, and the only restriction on the dilaton potential will be that for large φ the potential V will behave as φ −k , where k ≥ 0. This ensures that in the weak-coupling limit φ → ∞ (which for the spherically symmetric general relativity means large radius r, since in that case φ = r 2 [16]) we can ignore the contributions to the effective action proportional to d n V /dφ n or to V n+1 for n ≥ 1. In this limit we will calculate the complete one-loop effective action, which will include the loop contributions from the metric, the dilaton and the ghosts.…”

“…where g, R and ∇ stand for the determinant, the scalar curvature and the covariant derivative associated to the physical 2d metric gµν , α is a numerical constant and i = 1, ..., N. Φ is the dilaton scalar field, while f i are the matter scalar fields. When α = 4 and U(Φ) =const.= 4λ 2 the action (1.1) is the CGHS model, while α = 2 and U(Φ) = e 2Φ gives the spherically symmetric general relativity coupled to null-dust (in units G = 1, where G is the Newton constant) [16], to which we refere as the SSND model. If the dilaton is redefined as φ = e −2Φ , and after an appropriate rescaling of the metric (g µν = e αΦ/2 g µν ), the action (1.1) simplifies…”