Annalen der Physik. 7. Folge, Band 41, Heft 4/5, 1984, S. 353-356 J. A. Barth, Leipzig Do Black Holes Physically Exist? By P. F. GONZLLEZ-D~AZ Instituto de Optica “Daza de Valdhs”, Madrid, Spain Abstract. It is shown that, if the quantum resolution limit in the region of a collapsing body near the horizon is of the order of the Schwarzschild radius, a black hole in vacuum evaporates away instantaneously; hence white and black holes ought to be physically indistinguishable and should not physically exist. Existieren Schwarze Locher physikalisch? Inhaltsiibersicht. Es wird gezeigt, daB ein Schwarzes Loch augenblicklichim Vakuum verdampft, wenn die quantenmechanische Unbestimmtheit in der Gegend einea kollabierenden K6rpem nahe dem Horizont von der GroBenordnung des Schwarzschildradius ist. Schwarze und w e i h Locher werden so physikalisch unterscheidbar und sollten physikalisch nicht existieren. The discovery by HAWKING [l]of the quantum mechanically induced thermal emission by black holes has some implications of the greatest interest. According to HAWKING himself [2], the state of thermal equilibrium of a sufficiently large amount of energy in a small container with perfectly reflecting walls is achieved by a single spherical hole in thermal equilibrium with its surrounding radiation. Since the essential physical theories involved are time-symmetric, the equilibrium state will be time-symmetric also. Thus, Hawking claims, white holes ought to be physically indistinguishable from black holes. who has argued Difficulties with this view have been raised mainly by PENROSE [3, 41 that the spacetime geometry of black and white holes are quite different, even in the case that the entire portions inside the horizons are deleted. An answer to this difficulty is that it does not make any physical sense to talk about a uniquely defined objectively determined spacetime geometry when one considers quantum mechanically induced particle creation processes ; because of the back-reaction on geometry the spacetime has itself become subject to quantum mechanical uncertainties. However, although it becomes clear that one cannot consistently talk about a classical geometry for black holes of the Planck size or so (for which the detection resolution A x should be of the order of the hole size), it is not so clear that the spacetime of a black hole with a solar mass or more has the required degree of indeterminacy to allow its geometry to be identifiable to that of a white hole of the same mass. It appears that, in order to render the Hawking’s view acceptable, one would need to assume a detection resolution of about the hole dimension (i.e. Ax- G M )for the spacetime of a black hole with any large mass M . On the other hand, white holes ought to be completely unstable [5- 71 to the particle creation taking place near the white hole singularity, evaporating away instantaneously ; thus, such a particle creation can by no means be re-interpreted a8 the Hawking radiation from black holes, and time would have to be essentially asymmetric. Ann. Physik Leipzig 41 (1984)415 354 I n this letter we shall show that, if one allows the indeterminacy of the black hole geometry to be large enough t o render acceptable the Hawking's white-black hypothesis, then the radiative lifetime of black holes is largely shortened in such a way that the Hawking radiation could be naturally reinterpreted as the particle creation occurring near the white hole singularity. The above difficulty should be then ruled out in a very natural way. Let us then assume that the detection resolution in the spacetime region of a collapsing body with mass M is A x N G M when the collapse process has progressed up t o just the black hole horizon, and d x ~ G l(Planck / ~ length) before the onset of the collapse process and for empty space. We estimate now the effects on the solution (see Ref. [l]), caused by the change in the resolution limit when one reaches the horizon. I n (1)the subscript h refers to values measured in the region inside the collapsing body just outside the horizon; C , D are constants which depend on the details of the collapse, x is the surface gravity, and P; is the value for the radial function on the past event horizon in the analytically continued Schwarzschild solution where no matter is present, so that o is for empty space. HAWKINGtook [1] the values labeled h the same as those in empty space. Then, pi;L)w= p z ) , i.e. just the Hawking solution. However, this is not actually the case for black holes with mass other than the Planck mass if A x - GM just outside the horizon. ABBOTTand WISE have recently shown [8] that the path of a quantum mechanical particle can be considered as a MANDELBROTfractal [9] of dimension 2. If the path has to be self-similar, the time interval At between two consecutive measurements must be scaled so that A t oc AX)^, where A x is the detection resolution. I n this case, the average distance which the particle travels in a time At is A1 oc A x . Then, if we assume A x N G1r2for empty space, and Ax,& GM for the region inside the collapsing matter just outside the black hole horizon, we obtain Ath N (wo - w ) N ~ N(w, w), oOh N N-lo, and x, N N-lx, where N G M 2 is the normalized entropy (entropy divided by Boltzmann constant) of the black hole. The transformation relation taken for surface gravity is motivated by the definition x = 8n aMIaA, in which A is the area of the event horizon ; hence we get x Ak2. Thus, if the geometries of black and white holes go t o be the same, solution (1) through the collapsing body near the horizon and out on the past null infinity becomes: - - - = N1/2p,(,2) exp (-imx-'G-l In N ) where (if we assume Hawking solution t o be approximately valid for sufficiently large black holes when the detection limit is taken t o be about the Planck length) p c ) is the Hawking solution, with wo - v of the order of the Planck time and positive. Our solution ( 2 ) may be interpreted as follows. The key point in HAWKING theory [l]is to consider that just outside the horizon there is a crowding of an infinite number of waves, and hence short wavelengths dominate. Because its effective frequency was arbitrarily large, the waves would propagate by geometric optics through the centre of the body and out on the past null infinity. However, if one introduces quantum fluctuations in geometry comparable to the Schwarzschild radius, geometric optics becomes inapplicable. One cannot then choose arbitrarily high frequencies through the collapsing body near the horizon. I n such a regime, wave optics should be the appropriate theory to describe propagation. Then owing to the wave-optics nature of the overall propaga- P. F. G O N Z ~ E Z - DDo ~ ABlack z, Holes Physically Exist ? 355 tion, there will be a given number of elementary sources of secondary disturbance whose spacetime location is uncertain; the quantum indeterminacy in the geometry of the sources will lead to a quantum indeterminacy in the phase of the resulting waves. For the jth wave, we have p c ) exp (iYi)in which p(,2)should be the Hawking solution and !Pi is the quantum random phase for the jth wave. The total wave would be then p(,2)2 exp (iYi)= p',2)aexp (iY). 3 Average over all quantum internal states of the cycle-averaged beam intensity in free space leads to: I N (8n2)-l o - l ~ ~ (r I- ~ exp (iYj)12) (3) 3 N (8n2)-' o - ' ~ , r - ~ N , where the second line is the cycle-averaged intensity obtained from (2). The intensity i in (3) is just N times the cycle-averaged intensity due to a single wave, i.e. since, owing to the intrinsic spacetime indeterminacy of the secondary sources, the total solution must look like a completely chaotic mixture of secondary waves, the number of sources of secondary disturbance ought to be of the order of the normalized black hole entropy N . Let us analyze now the physical consequences implied by solution (2). The basis functions, f, before the onset of the collapse process (early times), in a regime where the geometry and matter distribution are assumed static, should be defined in the usual way [l,101. In such a regime, semiclassical approximation is still applicable for black holes much larger than the Planck size. Hence the inner products betwaen functions f and p' can be consistently defined at early times in the realm of that approximation, and we can compute the coefficients 01,,,, and ,3/., for the conventional Bogoliubov transformations. Following DE WITT[lo], we obtain from (2) 1 x 0 -iO'/xc- -2 T (1 + io'/xG) (4) 1 XW-iw'/xG-- 2 F(l + io'/xG). If the quantum state a t early times is taken to be the vacuum state relative to the basis functions f and we restore Ylm(cos0), the expectation value of the stress tensor in the steady state region obtained from (4) will be: (TP")- -(4n2)-1NsinOC (Y, ( ~ o s @ ) ) ~ J A (e2no/xG ~ ( o ) - l ) - l w d o , (5) Im where the coefficient Al(w)behaves as a filter for every mode. Thus, the obtained spectrum looks N times Planckian a t temperature T = xG/2n. This result seems to suggest that, if the time has to be essentially symmetric, a quantum mechanical black hole with mass M behaves like if it would be formed by N GM2 chaotically assembled identical black bodies all in thermal equilibrium at temperature (8nGM)-l and having an individual entropy of the order unity. The total luminosity of a black hole with mass M turns out to be [lo, 111: - 366 Ann. Physik Leipzig 41 (1984) 4/5 Hence we obtain a black hole lifetimes zMN G M . In the vacuum, the black hole ought to be then completely unstable to the process of thermal emission, evaporating away instantaneously. Note that zM should actually be the time the emitted radiation lasts in traveling the region of the collapsing matter just outside the horizon. In this way, it appears quite natural to re-interpret the black hole thermal radiation as the particle creation process taking place near the white hole singularity. In conclusion, our arguments suggest that if, independent of the hole size, one cannot at all talk about a classically objective black hole geometry, black holes ought to be physically indistinguishable from white holes, though, contrary to the Hawking’s opinion [2], in a vacuum situation neither black nor white holes should physically exist. Time-symmetry would be so preserved in quantum gravit.y. References [l] HAWKINQ,S. W.: Commun. Math. Phys. 43 (1975) 199. [2] HAWKINQ, S. W,: Phys. Rev. D 13 (1976) 191; D 14 (1976) 2460. R.: An Einstein Centenary Survey. In: General Relativity (eds. S. W. HAWKING [3] PENROSE, and W. ISRAEL) Cambridge 1979. [4] PENROSE, R.: A Second Oxford Symposium. In: Quantum Gravity 2 (eds. C. J. I s m , R. PENROSE and D. W. SCIAMA) Oxford 1981. [5] ZEL‘DOVICH, YA. B. : IAU Symposium. In: Gravitational Radiation and Gravitational Collapse (ed. C. M. DE WITT) Boston 1974. [6] EARDLEY, D. M.: Phys. Rev. Lett. 83 (1974) 442. [7] RANASWAMY, S.; WALD,R. M.: Phys. Rev. D 21 (1980) 2736. [8] ABBOTT,L. F.; WISE, M. B.: Am. J. Phys. 49 (1981) 37. [9] MANDELBROT,B. B.: The Fractal Geometry of Nature. Freeman, New York, 1983. [lo] DE WITT,B. S.: Phys. Rep. 19 (1975) 295. [ l l ] PAGE,D. N.: Phys. Rev. D 13 (1976) 198. Bei der Redaktion eingegsngen am 11.September 1984. Anschr. d. Verf. : P. F. GONZLLEZ-D~AZ Instituto de Optica “Daza de ValdBs”, C.S.I.C., Serrano 121 Madrid-6, Spain ~~ ~ Verlag Johann Ambrosius Barth, DDR-7010 Leipzig, SalomonstraBe 18b; Ruf 70131. Verlagsdirektor: K. WIECKE Chefredakteure: Prof. Dr. Dr. h.c. mult. H.-J. TREDER,DDR-1502 Potsdam-Babelsberg, Rosa-Luxemburg-Str. 17a , Berlin-Adlershof, Rudower Chauesee 5 Prof. Dr. G. R I ~ T E RDDR-1199 Veroffentlicht unter der Lizenznummer 1396 des Preaseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen’Republik Satz, Druck und Einband: VEB Druckhaus Kothen, DDR-4370 Kothen AN (EDV) 61216 6mal jahrlich, DDR 8,60 M

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