Radial fall of a test particle onto an evaporating black hole

A test particle falling into a classical black hole crosses the event horizon and ends up in the singularity within finite eigentime. In the ‘more realistic’ case of a ‘classical’ evaporating black hole, an observer falling onto a black hole observes a sudden evaporation of the hole. This illustrates the fact that the discussion of the classical process commonly found in the literature may become obsolete when the black hole has a finite lifetime. The situation is basically the same for more complex cases, e.g. where a particle collides with two merging black holes. It should be pointed out that the model used in this paper is mainly of academic interest, since the description of the physics near a black hole horizon still presents a difficult problem which is not yet fully understood, but our model provides a valuable possibility for students to enter the interesting field of black hole physics and to perform numerical calculations of their own which are not very involved from the computational point of view. PACS numbers: 04.25.-g; 04.70.-s; 04.70.Dy The Schwarzschild metric generated by an uncharged non-rotating classical black hole is given by ds = ( 1− rs r ) cdt − ( 1− rs r ) −1 dr − rdΩ, (1) where rs = 2GM0/c 2 is the Schwarzschild radius, G the gravitational constant, M0 the mass of the black hole, c the speed of light, and dΩ denotes the line element squared on a two-dimensional unit sphere. Due to the Hawking radiation, the black hole evaporates, if we exclude any accretion of energy. For macroscopic black holes, the luminosity is proportional to the inverse mass squared. The corresponding ansatz for the time-dependent black hole mass M(t) Ṁ(t) = −kM(t), (2) leads directly to M(t) = k 3 (t0 − t), (3) where t0 = 3M 3 0 /k is the lifetime of the black hole. Accordingly, we have rs(t) = k ′(t0 − t). (4) It is clear that limited knowledge is available about the actual evaporation process of black holes at present, which is based mainly on the classic works of Unruh and Hawking [1, 2], and there is an ongoing interest in the subject [4, 5, 6]. Therefore, our model is purely academic, but we consider it worthwhile to investigate it due to its simplicity and because it exhibits the interesting feature that a particle falling onto a black hole does not vanish from our universe, since it experiences an immediate black hole evaporation. A realistic treatment of the process presented here would involve detailed 1 knowledge about the physics in the very close vicinity of the black hole horizon. E.g., Babichev et al. [6] discussed the case of black holes accreting dark energy in the form of a phantom energy, where the energy density ρ > 0 and the negative pressure p < 0 fulfill the condition ρc +p < 0. For a black hole surrounded by a constant phantom energy bath, the black hole mass decreases like (t + τ), where τ > 0 is a characteristic evolution time. Therefore, we could have chosen decay laws for our discussion given below that differ from eq. (4), but the qualitative results would nevertheless remain the same. It is also clear that equation (3) will fail to describe the evaporation process of the black hole in its final stage. We also point out that an introduction to the description of evaporating black holes can be found in a known paper by William Hiscock [7]. There, different decreasing functions M(t) in a slightly different framework using the so-called Vaidya metric [8, 9] were considered. The main drawback of using simple approaches like those in [7] is the fact that the impact of the emitted radiation is not taken into account, because the black hole does not evaporate due to the emission of radiation but by artificially adding negative energy to the black hole. We give here a handwaving argument in order to illustrate the strange physical conditions that might be expected in the vicinity of the black hole horizon. For a distant observer, a massive black hole is a very cold object with Hawking temperature TBH = h̄c 4kBGM , (5) where kB is the Boltzmann constant. It is well-known that the condition for thermal equilibrium in a static system is √ g00(~r)T (~r) =const., where T (~r) is the temperature measured by a local static observer. This condition can be most easily derived when two systems in thermal equilibrium are considered, which are coupled only via their thermal radiation, such that the black body radiation of the two systems has to travel through the gravitational field. For the region near the horizon with r = rs + δ, where δ > 0 is small, g00 is given approximately by g00 = (