Strength of the Null Singularity Inside Black Holes

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Strength of the null singularity inside black holes

Lior M. Burko Theoretical Astrophysics 130–33, California Institute of Technology, Pasadena, California 91125 (July 20, 1999) We study analytically the Cauchy horizon singularity inside spherically-symmetric charged black holes, coupled to a spherical self-gravitating, minimally-coupled, massless scalar field. We show that all causal geodesics terminate at the Cauchy horizon at a null singularity, which is weak according to the Tipler classification. The singularity is also deformationally-weak in the sense of Ori. Our results are valid at arbitrary points along the null singularity, in particular at late retarded times, when non-linear effects are crucial.

PACS number(s): 04.70.Bw, 04.20Dw

I. INTRODUCTION the local existence and genericity of a null and weak singularity in solutions of the vacuum Einstein equations The issue of spacetime singularities – which are known was demonstrated in [8]. This was more recently demon- to inevitably occur inside black holes under very plausible strated also in the framework of plane-symmetric space- assumptions [1] – is an intriguing puzzle of physics. The times in [9]. For the toy model of a spherical charged laws of physics, as we presently understand them (e.g., black hole, the main features of the CH singularity were classical General Relativity), are presumably invalid at first found analytically for simplified models based on singularities. Instead, some other theories (e.g., quan- null fluids [10–12], and later confirmed numerically for a tum gravity), as yet unknown, are expected to take over model with a self-gravitating scalar field [13,14]. Expres- from General Relativity and control the spacetime struc- sions for the divergence rate of the blue-shift factors for ture. The General Relativistic predictions are neverthe- that model, which are valid everywhere along the CH, less of the greatest importance, as they reveal the space- were found analytically in [15]. Those expressions are time structure under extreme conditions in the strong- exact on the CH as functions of retarded time. How- field regime. Of particular interest is the possibility that ever, they are only asymptotic expressions as functions there are two distinct ways in which General Relativity of advanced time (see below). can fail at different types of singularities; For one type The strength of the null singularity is of crucial im- of singularities the failure is through infinite destructive portance for the question of the hypothetical possibility effects on physical objects, whereas for the other type the of hyperspace travel through the CH of black holes. A failure is through breakdown of predictability. necessary condition for this possibility to be realized is Until recently, the only known generic singularity in that physical objects would traverse the CH peacefully. General relativity was the Belinsky-Khalatnikov-Lifshitz Because the CH is known to be a curvature singularity, it (BKL) singularity [2]. According to the BKL picture, is necessary that the singularity would be weak according spacetime develops a succession of Kasner epochs in to the Tipler classification of singularity strengths. For which the axes of contraction and expansion change di- the toy model of a spherical charged black hole, which rections chaotically. This succession ends at unbounded we shall study here, the properties of the CH singularity oscillations at a spacelike singularity, which is unavoid- which have been found in [11–15] are all consistent with ably destructive for any physical object – a strong sin- the picture of a Tipler weak singularity. However, the gularity. In the last several years, however, evidence has weakness of the singularity was demonstrated only for been accumulating that the BKL singularity is not the the simplified Ori model [12] and at asymptotically early only type of singularity which may evolve in General Rel- times for spinning black holes [5], where there are still ativity from generic initial data. no strong non-linear effects, such as focusing of the null The new type of singularity forms at the Cauchy hori- generators of the CH, which are crucial at later times. In zon (CH) of spinning or charged black holes. (For a re- the context of spherical charged black holes and a self- cent review see [3].) The features of this singularity are gravitating scalar field, several important features of the markedly different from those of the BKL singularity: (i) spacetime structure have been found in fully non-linear It is null (rather than spacelike). (ii) It is weak (according numerical simulations. Specifically, it was shown that for to Tipler’s classification [4]); Specifically, the tidal defor- any point along the CH singularity there existed coordi- mations which an extended physical object suffers upon nates for which the metric coefficients were finite and approaching the singularity are bounded. In the case the metric determinant was non-degenerate in an open of a spinning black hole, the evidence for the null and neighborhood to the past [13,14]. However, despite pre- weak singularity has emerged from analytical perturba- vious claims [5,14,16], this still does not guarantee that tive [5,6] and non-perturbative [7] analyses. In addition, the singularity is weak in the Tipler sense [17].

1 2 3 It is the purpose of this paper to present an analyt- + b2 [ ln( V )]− + O [ ln( V )]− (2) ical demonstration of the weakness of the singularity − − − − o for the model of a spherical charged black hole with a (nr A) n 1 1 self-gravitating, minimally-coupled, massless, real scalar Φ,V = − [ ln ( V )]− − 1+c1[ ln( V )]− field. Our results are valid at arbitrary points along the rV − − − − 2 n 3 CH singularity, in particular at late times, where strong + c [ ln( V )]− + O [ ln( V )]− . (3) 2 − − − − non-linear effects (focusing of the null generators of the o CH and growth of the blue-shift factors) are crucial. In Here, A =[r+/(2r )](r+/r + r /r+), and the expan- − − − fact, our results are valid everywhere along the CH sin- sion coefficients bi and ci are functions of retarded time gularity, down to the event where the generators of the only. Note that in the limit V 0 these are exact ex- → CH are completely focused, and the singularity becomes pressions as functions of retarded time. That is, to the 1 spacelike and Tipler strong [18]. We emphasize that al- leading order in [ ln( V )]− there is implicit depen- − − though our discussion here is analytical, we do make as- dence on retarded time through r = r(u), and along the sumptions which are based on results obtained by numer- CH singularity both r,V and Φ,V are exactly inversely ical simulations. proportional to r(u), in the following sense. Consider two outgoing null rays, and let one ray be at u = u1,say, and the other at u = u2. The ratios r,V (2)/r,V (1) and II. STRENGTH OF THE SINGUALRITY Φ (2)/Φ (1) approach r(u )/r(u )asV 0. Tak- ,V ,V 1 2 → ing now u1 to be in the asymptotically-early parts of the We write the general spherically-symmetric line- CH, where r(u1) r , we find that both r,V and Φ,V ≈ − element in the form are inversely proportional to r(u). As r(u) is monotonically decreasing as a function of retarded time along the ds2 = f(u, v) du dv + r2(u, v) dΩ2, (1) CH, we find that r and Φ grow monotonically along − ,V ,V the CH. This growth is a non-linear effect which indi- 2 2 2 2 where dΩ = dϑ +sin ϑdϕ is the line-element on the cates the strengthening of the singularity along the CH unit two-sphere. The coordinates u, v are any outgoing (although the singularity is still weak according to the and ingoing null coordinates, correspondingly. (Below, Tipler classification; see below). we shall specialize to a specific choice of gauge, and de- All the non-zero components of the Riemann- fine a particular choice of an ingoing null coordinate.) Christoffel curvature tensor Rµνρσ are given completely We consider the class of scalar field perturbations which in terms of the divergent blue-shift factors r,V , Φ,V ,and is inherent to any gravitational collapse process. These the finite quantities r, r,u, Φ,u,andF. Interestingly, are the perturbations which result from the evolution Rµνρσ does not depend on gradients of F .Thiscanbe of non-vanishing multipole moments during the collapse. understood from the following consideration. The ten- When these perturbations propagate outwards, they are sor Rµνρσ can be written as the sum of the Weyl ten- partially reflected off the spacetime curvature and cap- sor, and another tensor which is built from the Ricci tured by the black hole. This process results in a scalar and the metric tensors (but not involving their deriva- field, which at late advanced times decays along the event tives). In spherical symmetry the Weyl tensor is given horizon like an inverse power of advanced time. Specifi- completely in terms of the mass function, which is de- cally, we assume that the scalar field behaves along the fined by r r,µ =1 2m(u, v)/r + q2/r2, q being the EH n ,µ event horizon at late times according to Φ (κv )− − ∝ e charge of the black hole. In Kruskal-like coordinates the [19–21], where n is a positive integer which is related to 2 mass function m(u, V )=(r/2)(1+4r,ur,V /F )+q /(2r), the multipole moment under consideration. (We do not which depends only on r, r,V ,r,u,andF. (The divergence consider, however, other possible sources of perturbations of the mass function at the singularity, and consequently [22].) By ve we denote the usual advanced time in the also the divergence of curvature, is evident from the di- Eddington gauge, and r ,κare the outer and inner hori- vergence of r,V and the finiteness of r, r,u ,and F .) The zons and the surface gravity± of the latter, respectively, Ricci tensor Rµν =2Φ,µΦ,ν, and consequently Rµνρσ is for a Reissner-Nordström black hole having the same ex- independent of gradients of F . ternal parameters as the black hole we consider has at We find that the components of Rµνρσ which have the late times. We define the dimensionless ingoing Kruskal- strongest divergence near the CH are RVϕVϕ and RVϑVϑ. like coordinate by V exp( κve). In the Kruskal (In the Appendix we list all the non-zero independent gauge we denote the metric≡− function− g by F/2. For uV − components of the Riemann-Christoffel tensor.) It can this model, it can be shown analytically that at arbitrary be readily shown that points along the CH the following relations are satisfied, 1 2 2 2 to the leading orders in [ ln( V )]− [15]: RVϑVϑ = r (Φ,V ) =sin− ϑRVϕVϕ. (4) − − − 2 We denote these two components schematically and col- (nr A) 2n 2 1 r,V = − [ ln ( V )]− − 1+b1[ ln( V )]− lectively by . We find that, to the leading orders in V rV − − − − R n 2 1 and in [ ln ( V )]− , From the numerical results of [13,14] it is known that as − − 2 2n 2 V 0, F approaches a finite value. Consequently, in (V ) V − [ ln ( V )]− − . (5) → R ∝ − − order to have consistency with the numerical results we (The other divergent components of R , i.e., require that b1 2c1 =2n+ 2, which implies that µνρσ − RuV uV ,Rϑϕϑϕ,RuϑV ϑ,andRuϕV ϕ are proportional to 1 1 2 the leading order in V to V times a logarithmic fac- F = F0(u) 1+B[ ln ( V )]− + O [ ln ( V )]− , (13) − − − − − tor.) Because both metric functions r and F have finite n o 2 values at the CH (which is known from the numerical where B =(2n+3)b1 b2 +c1 +2c2,andF0(u)= ˜ 2 − simulations of [13,14]), it is easy to show that the depen- F0(nr A) /r. For the logarithmic derivative of F we − 1 dence of (V )onV does not change when we transform find, to the leading order in [ ln ( V )]− ,that to a parallelly-propagatedR frame. − − We next find V (τ) as a function of affine parameter 1 2 F,V /F = B [ ln ( V )]− . (14) (proper time) τ along a general null (timelike) geodesic. V − − For general causal geodesics, the geodesic equations are 1 (Higher order terms in [ ln ( V )]− are functions of − − u˙v˙ = mr2Ω˙ 2 p (6) retarded time.) We note that only b1 and c1 are − constrained. The coefficients of highr-order terms in 1 [ ln ( V )]− are immaterial near the CH for our de- v¨ +(f /f)˙v2 +2mr (fu˙v˙ + p) /(rf)=0 (7) ,v ,u termination− − of the strength of the singularity. Note that F F (u)asV 0, and that F is not analytic in u¨+(f /f)˙u2 +2mr (fu˙v˙ + p) /(rf)=0. (8) 0 ,u ,v V .→ In fact, this is an→ important property of the CH sin- Here, m = 0(1) for radial (non-radial) geodesics, and gularity: In a kruskal-like gauge the metric functions r p =0( 1) for null (timelike) geodesics. A dot denotes and F are finite at the singularity, but their gradients in differentiation− with respect to affine parameter (proper the outgoing direction diverge. The finiteness of r and time), and Ω˙ 2 = ϑ˙2 +sin2ϑϕ˙2. The geodesic equations F at the CH also implies that the metric determinant 1 can be solved to the leading order in [ln( V )]− for all is non-degenerate. This expression for F is similar to − causal geodesics. This is done by using the field equation the behavior of the guV metric function found for the [14] simplified Ori model [12]. We stress that although this 2 expression for F is exactly valid everywhere along the CH F,V /F = r,V V /r,V + r(Φ,V ) /r,V (9) singularity, it still does not allow us to find the variation to find F,V /F explicitly. Substituting Eqs. (2) and (3) of F with retarded time, as we do not know the form in Eq. (9) we find of F0(u)orr(u) along the CH. We note that near the CH the metric function F is monotonic in V . This result 1 1 1 (ln F ) = [ln ( r )] + +(2c b ) [ ln ( V )]− is in accord with the numerical results of [14]. (Notice, ,V ,V ,V V 1 1 V − − − − however, the disagreement with the numerical results of 2 2 1 2 + b b 2b c + c +2c [ ln ( V )]− [16]. It is reasonable to expect the behavior of the metric 1 2 1 1 1 2 V − − − − functions near the CH to be similar for both cases of real 1 3 + O [ ln ( V )]− . (10) and complex scalar fields. The lower panels of Figs. (3) V − − of [16] imply, however, a non-monotonic behavior of F . Integration yields That kind of behavior can be obtained from a numerical code with a specific choice of parameters if the latter is ln F =lnF˜ +ln(Vr ) (2c b ) ln [ ln ( V )] 0 ,V − 1 − 1 { − − } far from convergence near the CH.) 2 2 1 + b b 2b c + c +2c [ ln ( V )]− Let us consider first radial geodesics. (The case of 1 − 2 − 1 1 1 2 − − 2 non-radial geodesics will be treated next.) In the null + O [ ln ( V )]− . (11) − − case (m =0andp=0)itiseasytosolvethegeodesic n o equations (6)–(8). For outgoing geodesics one readily Here, ln F˜ is an integration constant, which can be a 0 finds that the solution is u = const and V˙ =const/F . function of u. Exponentiating both sides, and substitut- The metric function F canbeexpandedin[ ln( V )] 1, ing Eq. (2) for r we find − ,V despite its non-analyticity in V . To the leading− orders− in 2 1 (nr A) 2n 2+b 2c [ ln( V )]− we find that F = F˜ − [ ln ( V )]− − 1− 1 − − 0 r − − 1 1 2 F = F0(u) 1+B[ ln ( V )]− , (15) 1+b1[ ln ( V )]− + O [ ln ( V )]− − − × − − − − n o n 2 2 o 1 exp b b 2b c + c +2c [ ln ( V )]− such that × 1 − 2 − 1 1 1 2 − − const n 2 ˙ 1 + O [ ln ( V )]− . (12) V = 1 B [ ln ( V )]− . (16) − − F0 − − − o n o 3 The solution for V (τ) is then given asymptotically close does not converge as τ 0. It can be readily shown 1 → to the CH, to the leading orders in [ ln( V )]− by that when (τ) is integrated twice with respect to τ, − − (τ) convergesR in the limit τ 0. Consequently, we 1 I → V (τ )=τ 1 B[ ln ( τ)]− . (17) find that a necessary condition for any causal geodesic to − − − terminate at a Tipler strong singularity is not satisfied. n o (Recall that the affine parameter is given up to a linear Hence, all causal geodesics terminate at a Tipler weak 1 transformation.) To the leading order in [ ln ( τ)]− singularity, namely, the singularity is Tipler weak. The we can thus approximate V (τ) τ.Notethatal-− − physical content of this result is that the volume element though asymptotically V (τ)andV˙≈(τ) behave like τ and of physical objects remains bounded at the singularity. τ˙ 1, correspondingly, V¨ (τ) behaves very differently We emphasize that this result is valid everywhere along from≡ τ ¨ 0. In fact, V¨ (τ) diverges as τ 0. There- the singularity, in particular at late retarded times where fore, one≡ can approximate V (τ) τ only→ if one is in- the non-linear effects (focusing of the generators of the terested in V (τ) itself, or at the≈ most in V˙ (τ). This CH and the growth of the blue-shift factors) are crucial. approximation is invalid for V¨ (τ) or higher derivatives. For radial timelike geodesics (m =0andp= 1) one − III. CONCLUDING REMARKS uses the finiteness of F,u to find approximately that again V˙ const/F , and consequently one finds the same result≈ for V (τ). In the case of non-radial geodesics (m =1) Recently, Ori suggested to define a deformationally- one can consider a specific value of the retarded time at strong singularity in the following way. Let λ(τ)be a timelike geodesic with proper time τ along it. The which the geodesic hits the CH singularity. Than, r, r,u, and F can be approximated by their values at the sin- geodesic λ(τ) terminates at a deformationally-strong singularity. When this is done, the equations for null non- gularity at τ = 0 if at least one of the following two condi- radial (m =1andp= 0) geodesics become inhomoge- tions holds: (i) λ(τ) terminates at a Tipler strong singu- neous linear equations. The corresponding homogeneous larity, or (ii) there exists a Jacobi field J(τ) for which at equations are nothing but the equations for the radial least one parallelly-propagated tetrad component is unbounded at the limit τ 0 [24]. This definition is more geodesics, which we already solved. Particular solutions → for the inhomogeneous equations are easy to generate, physically-motivated than Tipler’s definition, because it and one finds that again V (τ) is given asymptotically as classifies a singularity as strong not only when the vol- before. The last case is the case of non-radial timelike ume element vanishes, but also when the volume element (m =1andp= 1) geodesics. In this case the geodesic diverges to infinity, or there is infinite compression in one equation becomes− (under similar assumptions) an inho- direction, and infinite stretching in a different direction, mogeneous non-linear equation. Although this equation such that the volume element remains bounded. In fact, is hard to solve directly, it can be checked that the same it can be shown that the failure of the necessary con- leading order proportionality of V (τ)andτis the solu- dition for the singularity to be Tipler strong implies not tion also for this case. We thus find that for all causal only the boundedness of the volume element, but also the 1 boundedness of the Jacobi fields themselves [24], such geodesics, to the leading order in [ ln ( τ)]− , V (τ)is proportional to τ. − − that objects are not expected to be destroyed also be- We next re-express as a function of affine parameter cause of distortions which preserve the volume element (proper time) along radialR or non-radial null (timelike) or divergence to infinity of the volume element. Conse- 1 quently, the singularity we are studying here is weak also geodesics. To the leading order in [ ln ( τ)]− we find in the sense of Ori (deformationally weak). (Ori’s defini- that in a parallelly-propagated frame− − tion does not include null geodesics. However, extended 2 2n 2 physical objects move along timelike geodesics, such that (τ) τ − [ ln ( τ)]− − . (18) R ∝ − − this deficiency does not restrict our discussion. It is con- A necessary condition for a singularity to be strong in ceivable that both Ori’s definition and theorem for the the Tipler sense is given by the following theorem [23]: necessary condition for the singularity to be Ori strong For null (timelike) geodesics, if the singularity is strong can be generalized to all causal geodesics.) in the Tipler sense, then for at least one component of We note that according to the Królak classification the Riemann-Christoffel curvature tensor in a parallelly- of singularities [25] this is a strong singularity. Specifi- propagated frame, the twice integrated component with cally, if we integrate over the divergent components of the respect to affine parameter (proper time) does not con- Riemann-Christoffel tensor only once, the integral does verge at the singularity. Specifically, the necessary con- not converge on the singularity. This means that the ex- dition for the singularity to be Tipler strong is that pansion diverges (negatively) on the singularity (Królak strong), but still the volume element (and the distortion τ τ 0 in general) remains finite (Tipler and Ori weak). One (τ)= dτ 0 dτ 00 (τ 00) (19) I |R | might be worried that even if spacetime were classically Z Z

4 2 2 2 extendible beyond the CH, this infinite negative expan- Rvϕvϕ = r Φ,v sin ϑ sion would inevitably result in unavoidable destruction of − any extended physical object subsequent to its traversing of the CH [26]. Of course, any classical extension of geometry beyond the CH is not unique. We can, however, consider an extension with a continuous (C0)metricand aunique C1 timelike geodesic, and assume that the object follows this geodesic [27]. Any extension of classical [1] S. W. Hawking and G. F. R. Ellis, The large scale struc- geometry beyond the CH (which can be modeled as a ture of space-time (Cambridge University Press, Cam- thin layer wherein the geometry is inherently quantum) bridge, 1973). requires an infinite flux of negative energy traveling along [2] V. A. Belinsky, I. M. Khalatnikov, and E. M. Lifshitz, the contracting CH. This negative energy flux may then Adv. Phys. 19, 525 (1970). act to regularize the expansion, such that the deforma- [3] L. M. Burko and A. Ori, in Internal structure of black tion rate of physical objects beyond the CH would be holes and spacetime singularities, edited by L. M. Burko bounded [27]. (The infinite expansion is likely not to de- and A. Ori (Institute of Physics, Bristol, 1997). stroy physical objects up to the CH [27], in contrast with [4] F. J. Tipler, Phys. Lett. 64A, 8 (1977). Ref. [26].) Indeed, a simplified two-dimensional quantum [5] A. Ori, Phys. Rev. Lett. 68, 2117 (1992). [6] A. Ori, Gen. Rel. Gravitation 29, 881 (1997). model shows an infinite ingoing flux of negative energy [7] P. R. Brady, S. Droz, and S. M. Morsink, Phys. Rev D along the CH [28,29]. More recent semi-classical toy mod- 58, 084034 (1998). els of a quantum field on a mass-inflation background are [8] A. Ori and E.´ E.´ Flanagan, Phys. Rev. D 53, R1754 not inconsistent with this picture [30,31]. One should (1996). not take these quantum results too seriously, however, [9] A. Ori, Phys. Rev. D 57, 4745 (1998). because in these models the semi-classical contributions [10] W. A. Hiscock, Phys. Lett. 83A, 110 (1981). are dominated by the regime where curvature is Planck- [11] E. Poisson and W. Israel, Phys. Rev. D 41, 1796 (1990). ian, such that the semi-classical approximation is not ex- [12] A. Ori, Phys. Rev. Lett, 67, 789 (1991). pected to be valid anymore. Instead, a full quantum [13] P. R. Brady and J. D. Smith, Phys. Rev. Lett. 75, 1256 theory of gravity is of need. Of course, in the absence of (1995). a valid theory of quantum gravity it is difficult to make [14] L. M. Burko, Phys. Rev. Lett. 79, 4958 (1997). [15] L. M. Burko and A. Ori, Phys. Rev. D 57, R7084 (1998). predictions on the detailed interaction of the thin layer [16] S. Hod and T. Piran, Phys. Rev. Lett. 81, 1554 (1998); of the CH with physical objects, but the evidence we Gen. Rel. Gravitation 30, 1555 (1998). currently have do not preclude the possibility of objects [17] This point was stressed by B. C. Nolan, gr-qc/9902020 traversing the CH singularity peacefully. (unpublished). Nolan tried to present an example of a I have benefited from useful discussions with Patrick strong spacelike singularity with a non-singular C0 met- Brady and Amos Ori. This work was supported by NSF ric. However, as shown in A. Ori, gr-qc/9902055 (un- grant AST-9731698. published), Nolan’s singularity is, in fact, weak. Thus, the only known examples for strong singularities with non-singular C0 metrics are pathological and unlikely to APPENDIX be realized in nature. In fact, all those singularities are strong only on subsets of measure zero. However, a non- singular C0 metric is not equivalent to the singularity The independent components of the Riemann- being Tipler weak. Christoffel curvature tensor which do not vanish iden- [18] L. M. Burko, Phys. Rev. D 59, 024011 (1999). tically are: [19] R. H. Price, Phys. Rev. D 5, 2419 (1972). [20] C. Gundlach, R. H. Price, and J. Pullin, Phys. Rev. D 1 f q2 R = 4r r + f 1 2 + fΦ Φ 49, 883 (1994). uvuv − 4 r2 ,u ,v − r2 ,u ,v [21] L. M. Burko and A. Ori, Phys. Rev. D 56, 7280 (1997). r2 [22] L. M. Burko, Phys. Rev. D 55, 2105 (1997). R =4 r r sin2 ϑ ϑϕϑϕ f ,u ,v [23] C. J. S. Clarke and A. Królak, J. Geom. Phys. 2, 127 (1985). 2 f q [24] A. Ori, in preparation. Ruϑvϑ = r,ur,v 1 − − 4 − r2 [25] A. Królak, J. Math. Phys. 28, 138 (1987). 2 2 [26] R. Herman and W. A. Hiscock, Phys. Rev. D 46, 1863 Rvϑvϑ = r Φ − ,v (1992). f q2 R = r r + 1 sin2 ϑ [27] A. Ori, in Internal structure of black holes and spacetime uϕvϕ − ,u ,v 4 − r2 singularities, edited by L. M. Burko and A. Ori (Institute R = r2Φ2 sin2 ϑ of Physics, Bristol, 1997). uϕuϕ − ,u [28] W. A. Hiscock, Phys. Rev. D 15, 3054 (1977). R = r2Φ2 [29] N. D. Birrell and P. C. W. Davis, Nature 272, 35 (1978). uϑuϑ − ,u

5 [30] R. Balbinot and E. Poisson, Phys. Rev. Lett. 70,13 (1993). [31] W. G. Anderson, P. R. Brady, W. Israel, and S. M. Morsink, Phys. Rev. Lett. 70, 1041 (1993).