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.Comment: 6 pages, 3 figures, LATE

Andreas Aste

Babichev V.

Dirk Trautmann

Hajicek W.

Stephens G.

English

arXiv

A test particle falling onto 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, for example, 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 that 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 that are not very involved from the computational point of view

Aste, A.

Trautmann, D.

edoc

arXiv:gr-qc/0509007v1  1 Sep 2005Radial fall of a test particle onto an evaporating black holeAndreas Aste and Dirk TrautmannDepartment for Physics and Astronomy, University of Basel,4056 Basel, SwitzerlandE-mail: andreas.aste@unibas.chJune 8, 2005AbstractA test particle falling into a classical black hole crosses the event horizon and ends up in thesingularity 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 illustratesthe fact that the discussion of the classical process commonly found in the literature may becomeobsolete when the black hole has a finite lifetime. The situation is basically the same for morecomplex cases, e.g. where a particle collides with two merging black holes. It should be pointedout that the model used in this paper is mainly of academic interest, since the description of thephysics 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 holephysics and to perform numerical calculations of their own which are not very involved from thecomputational point of view.PACS numbers: 04.25.-g; 04.70.-s; 04.70.DyThe Schwarzschild metric generated by an uncharged non-rotating classical black hole is given byds2 =(1− rsr)c2dt2 −(1− rsr)−1dr2 − r2dΩ2, (1)where rs = 2GM0/c2 is the Schwarzschild radius, G the gravitational constant, M0 the mass of theblack hole, c the speed of light, and dΩ2 denotes the line element squared on a two-dimensional unitsphere. Due to the Hawking radiation, the black hole evaporates, if we exclude any accretion ofenergy. For macroscopic black holes, the luminosity is proportional to the inverse mass squared. Thecorresponding ansatz for the time-dependent black hole mass M(t)M˙(t) = −kM(t)−2, (2)leads directly toM(t)3 =k3(t0 − t), (3)where t0 = 3M30 /k is the lifetime of the black hole. Accordingly, we havers(t) = k′(t0 − t)1/3. (4)It is clear that limited knowledge is available about the actual evaporation process of black holesat present, which is based mainly on the classic works of Unruh and Hawking [1, 2], and there is anongoing interest in the subject [4, 5, 6]. Therefore, our model is purely academic, but we consider itworthwhile to investigate it due to its simplicity and because it exhibits the interesting feature that aparticle falling onto a black hole does not vanish from our universe, since it experiences an immediateblack hole evaporation. A realistic treatment of the process presented here would involve detailed1knowledge 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 theenergy density ρ > 0 and the negative pressure p < 0 fulfill the condition ρc2+p < 0. For a black holesurrounded by a constant phantom energy bath, the black hole mass decreases like (t + τ)−1, whereτ > 0 is a characteristic evolution time. Therefore, we could have chosen decay laws for our discussiongiven 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 itsfinal stage.We also point out that an introduction to the description of evaporating black holes can be foundin a known paper by William Hiscock [7]. There, different decreasing functions M(t) in a slightlydifferent framework using the so-called Vaidya metric [8, 9] were considered. The main drawback ofusing simple approaches like those in [7] is the fact that the impact of the emitted radiation is nottaken into account, because the black hole does not evaporate due to the emission of radiation but byartificially adding negative energy to the black hole.We give here a handwaving argument in order to illustrate the strange physical conditions thatmight be expected in the vicinity of the black hole horizon. For a distant observer, a massive blackhole is a very cold object with Hawking temperatureTBH =h¯c34kBGM, (5)where kB is the Boltzmann constant. It is well-known that the condition for thermal equilibrium ina static system is√g00(~r)T (~r) =const., where T (~r) is the temperature measured by a local staticobserver. This condition can be most easily derived when two systems in thermal equilibrium areconsidered, which are coupled only via their thermal radiation, such that the black body radiation ofthe two systems has to travel through the gravitational field. For the region near the horizon withr = rs + δ, where δ > 0 is small, g00 is given approximately byg00 =(1− rsr)=(1− rsrs + δ)≃ δrs. (6)Furthermore, the spatial distance d to the horizon isd =rs+δ∫rs(1− rsr)−1/2dr ≃δ∫0√rsδdδ = 2√rsδ. (7)Therefore, a hypothetical static observer very close to the horizon with d = lP =√h¯G/c3 measures atemperature of the order of the Planck temperature TP , sincekBTBH√g00(δ = l2P /4rs)=h¯c34GM2rslP=h¯clP= kBTP . (8)For an external observer, the test particle gets arbitrarily close to the horizon within finite time, butin contrast to our considerations made above, it is not in a static state, but accelerated, and doestherefore not experience the extreme Unruh effect as a static observer. Nevertheless, it will be difficultto make any reasonable predictions about the fate of a particle approaching the horizon, where thephysics becomes Planckian.In the following, we assume that the metric generated by the evaporating black hole is given byexpression (1), but with rs replaced by the time-dependent expression given by eq. (4). In view ofthe very slow evaporation process and the low energy density of the emitted Hawking radiation, thisis certainly a satisfactory approximation for our purpose. Near the black hole horizon, the validity ofthe approximation is dubious at best.2From now on, we will use geometric units (G=c=1) throughout. If we parametrize the (geodesic)worldline of a massive test particle by its eigentime τ , then the equation of motion of the particlefalling radially onto the black hole can be derived from the LagrangianL = B(r, t)t˙2 −B(r, t)−1r˙2, B(r, t) =(1− k′(t0 − t)1/3r), (9)where the dot denotes differentiation with respect to the eigentime τ . The Euler-Lagrange equations∂τδLδ(t˙, r˙)=δLδ(r, t)(10)then read∂τ (2B(r, t)t˙) = ∂tB t˙2 − ∂t(B−1) r˙2, (11)∂τ (2B(r, t)−1 r˙) = −∂rB t˙2 + ∂r(B−1) r˙2. (12)Eq. (11) leads tor¨ =∂rB2Br˙2 +∂tBBr˙t˙− B2∂rBt˙2, (13)t¨ = −∂rBBr˙t˙− ∂tB2Bt˙2 − ∂t(B−1)2Br˙2. (14)Additionally, we can take into account that the four-velocity of the test particle is constantBt˙2 −B−1r˙2 = 1, (15)since we used the eigentime τ as curve parameter.We solved eq. (12) in conjunction with eq. (15) numerically. We considered first the analyticallysolvable case of a particle falling radially on a stable black hole (with mass M) starting at rest at adistance Rdrdτ∣∣∣∣∣r=R= 0, r(τ = 0) = R. (16)Introducing the parameter η defined byη = arccos(2rR− 1), (17)one obtainsr =R2(1 + cos η), τ =√R38M(η + sin η),t = 2M√R2M− 1[η +R4M(η + sin η)]+ 2M ln∣∣∣∣∣√R2M − 1 + tan η2√R2M − 1− tan η2∣∣∣∣∣. (18)Such a particle reaches the horizon after an eigentime τHτH =√R34rs[arccos(2rsR− 1)+ 2√rsR− r2sR2]. (19)This value served as a practical independent check for the accuracy of our numerical calculation.A simple explicit finite difference Euler forward scheme is fully sufficient for the numerical inte-gration of eq. (13), where one can eliminate t˙ by using eq. (15). But it is necessary to adapt theeigentime step ∆τ during the integration process in order to obtain an accurate solution. As long as30 2 4 6 8 10 1200.511.52External time t or eigentime τrFigure 1: Test particle falling into a stable black hole with Schwarzschild radius rs = 1. This graphcan be found in many standard textbooks.the particle has a relatively large distance to the horizon, the time step can be held constant, but inthe vicinity of the horizon, a update of the time step according to∆τn+1 =r˙n−1r˙n∆τn (20)for each integration step is a good choice. The effect is a reduction of the time step when the radialcoordinate r starts to change faster with respect to the eigentime τ , and the resolution of the trajectorywith respect to the external observer time t is maintained.Fig. 1 shows the trajectory of the particle falling into a stable black hole with rs = 1 and R = 2with respect to the particle’s eigentime τ and the external observer time t. In this case, the particlereaches the horizon for τH =√2(π/2 + 1) ≃ 3.6357 and the singularity for τS = pi2√R3/rs ≃ 4.4429.Fig. 2 shows the typical situation for an unstable black hole. In order to obtain reasonable plots,we used ’short’ black hole lifetimes, but the qualitative picture remains the same independently fromthe actual lifetime of the black hole which may be very large for massive holes. From the externalobserver’s point of view, the particle impinges on the black hole horizon, where it remains captureduntil the black hole evaporates. In the final evaporation phase, the particle even looses contact to thehorizon due to the fast decrease of the black hole radius.From the point of view of the test particle, the black hole seems to evaporate as soon as the particleapproaches the horizon. The amusing point in our toy model is the fact that from the ’test particle’spoint of view’, the evaporation time of the black hole is larger for black holes with a short lifetime.This is due to the fact that the evaporation process continuously reduces the distance between thehorizon to the particle, such that the particle experiences a smaller time contraction near the horizon,when the horizon shrinks faster. For massive black holes, the black hole evaporation time approachesthe time that the particle would need to reach the horizon of a stable black hole with rs = rs(τ = 0),given by eq. (19). Fig. 3 shows τ(t0) − τH, i.e. basically the eigentime of the particle with the sameinitial conditions as in Fig. 2 at which the evaporation process of the black hole with initial lifetimet0 comes to the end.We conclude with the remark that within our simplified framework one should expect basically thesame result for more complex processes, e.g. for a test particle approaching two merging black holeswhich evaporate subsequently. The test particle cannot cross the classical horizon, and an asymptoticobserver will never observe that the particle impinges the horizon within finite time [10].40 10 2000.511.52trs, r0 2 400.511.52τrs, rFigure 2: Typical trajectories for a test particle falling onto a black hole with initial Schwarzschildradius rs(0) = 1 and a lifetime t0=20.10 15 20 25 3000.050.10.150.20.250.3Black hole lifetime t0τ(t0)−τ HFigure 3: Eigentime τ(t0) reduced by the eigentime τH needed by a test particle to reach the horizonof a stable black hole, versus the black hole lifetime t0.5References[1] W.G. Unruh, Phys. Rev. D14 (1976) 870.[2] S.W. Hawking, Commun. Math. Phys. 43 (1975) 199, Nature 248 (1974) 30.[3] W.G. Unruh, Phys. Rev. D51 (1995) 2827.[4] T. Jacobson, Phys. Rev. D44 (1991) 1731.[5] C. R. Stephens, G.’t Hooft, B.F. Whiting, Class. Quant. Grav. 11 (1994) 621.[6] E. Babichev, V. Dokuchaev, Yu. Eroshenko, Phys. Rev. Lett. 93 (2004) 021102.[7] W. A. Hiscock, Phys. Rev. D23 (1980) 2813.[8] P. C. Vaidya, Proc. Indian. Acad. Sci. A33 (1951) 264.[9] P. Hajicek, W. Israel, Phys. Lett. 80A (1980) 9.[10] N. Straumann, General Relativity with Applications to Astrophysics, Springer (Berlin) 2004.6

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