Kolmogorov-Sinai entropy and black holes
aa r X i v : . [ g r- q c ] A ug Kolmogorov-Sinai entropy and black holes ‡ Kostyantyn Ropotenko
State Department of communications, Ministry of transport and communications ofUkraine, 22, Khreschatyk, 01001, Kyiv, Ukraine
Abstract.
It is shown that stringy matter near the event horizon of a Schwarzschildblack hole exhibits chaotic behavior (the spreading effect) which can be characterizedby the Kolmogorov-Sinai entropy. It is found that the Kolmogorov-Sinai entropy ofa spreading string equals to the half of the inverse gravitational radius of the blackhole. But the KS entropy is the same for all objects collapsing into the black hole.The nature of this universality is that the KS entropy possesses the main propertyof temperature: it is the same for all bodies in thermal equilibrium with the blackhole. The Kolmogorov-Sinai entropy measures the rate at which information aboutthe string is lost as it spreads over the horizon. It is argued that it is the maximumrate allowed by quantum theory. A possible relation between the Kolmogorov-Sinaiand Bekenstein-Hawking entropies is discussed.PACS numbers: 04.70.Dy ‡ This is an extended version of my paper arXiv:0711.3131. olmogorov-Sinai entropy and black holes
1. Introduction
It is well known that some general relativistic systems described by the Einsteinequations can exhibit chaotic behavior [1, 2]. But it is still not clear whether suchgeneral relativistic systems as black holes can do it. In series of insightful papers,Susskind [3, 4, 6] and Mezhlumian, Peet and Thorlacius [5] showed that a string fallingtoward a black hole spreads over the stretched horizon. In doing so, it exhibits chaoticbehavior and the rate of spreading is determined by the inverse gravitational radius ofthe black hole.In this paper I propose to use the conceptions of chaos theory to describe thebehavior of stringy matter near the event horizon of a Schwarzschild black hole. Inthe following sections we will introduce the main conceptions of chaotic dynamics,demonstrate chaotic behavior of stringy matter near the event horizon of a black hole,and find its Kolmogorov-Sinai entropy. In the end we discuss a possible relation betweenthe Kolmogorov-Sinai (KS) and Bekenstein-Hawking (BH) entropies. This paper is anextended version of [7].
2. The main conceptions of chaotic dynamics
We begin with definitions. According to chaos theory [8, 9] the chaotic behavior ofdynamical systems that is, systems whose state evolves with time, has its origin inthe so-called local instability, when a small change in initial conditions leads to anexponential divergence of phase space trajectories. As a result of this sensitivity toinitial conditions, the behavior of dynamical systems appears unpredictable. Supposethat two nearby trajectories in the phase space of a system start off with a separation d (0) at time t = 0. Then, if there exists a local instability, it follows that there exists adirection in phase space along which the trajectories diverge exponentially: d ( t ) = d (0) e ht . (1)The parameter h is called the Lyapunov exponent for the trajectories. If h is positive,then we say the behavior of the system is chaotic. The sum of all the positive Lyapunovexponents is called the KS entropy h KS . It describes the rate of change of informationabout the phase space trajectories as a system evolves. Following Zaslavsky [8], wecan define the KS entropy in not so formal way and connect it with thermodynamicalentropy. Suppose that the phase space of a dynamical system is finite (in at least onephase space direction). Then, since there exists the local instability (1), it follows thatan initial volume in phase space ∆Γ(0) becomes very complicated like a fractal. Thuswe should perform coarse graining, and if we coarse grain the region, it will appear thatit is growing ∆Γ( t ) = ∆Γ(0) e ht , (2) olmogorov-Sinai entropy and black holes h is the averaged over phase space Lyapunov exponent. As a result, the coarse-grained Boltzmann entropy increases S ( t ) = ht. (3)The quantity ∆Γ(0) can be taken as simply equal to a volume of the coarse-graining.Then the expression h KS = lim t →∞ lim ∆Γ(0) → t ln (cid:18) ∆Γ( t )∆Γ(0) (cid:19) (4)just defines the KS entropy. It is clear that the KS entropy h KS is not really an entropybut an entropy per unit time, or entropy rate, dS/dt . Note also that by order ofmagnitude h KS ∼ h . As is easily seen, if ∆Γ(0) → S ( t ) = ht at t → ∞ ,the Boltzmann entropy doesn’t reach a maximum on a compact phase space. But thesituation changes if we fix a finite volume of coarse-graining. In this case the exponentialseparation of trajectories (1) and the consequent increase in volume (2) can last only onfinite time-scale. Just that very case will be discussed below. In general, the calculationof the KS entropy offers great mathematical difficulties. Our task is facilitated by thefact that all our relevant quantities have exponential dependence on time.The BH entropy of a black hole, on the other hand, S BH = A G = πR g l P = 4 πGM , (5)is obtained from the thermodynamical relation dE = T dS , where the energy of theblack hole is its mass M , the temperature is given by T H = 1 / πGM and the area ofthe event horizon A is related to the gravitational radius R g , R g = 2 GM , in the usualway A = 4 πR g . The BH entropy is defined in the reference frame of an external distantobserver at fixed static position above the horizon (an external observer). It is wellestablished that, from her/his point of view, the classical physics of a quasistationaryblack hole can be described in terms of a ’stretched horizon’ which is a membraneplaced near the event horizon and endowed with certain mechanical, electrical andthermal properties [10]. The exact distance of this membrane above the event horizonis somewhat arbitrary. In the context of string theory - the subject of our research -the stretched horizon is most naturally thought of as lying at the string scale above theevent horizon. In what follows we will deal with this conception (sometimes referred tosimply as horizon).It may seem that the event horizon itself exhibits the desired instability: any lightray which is just not on the horizon separates from it exponentially. This is very clearin Eddington-Finkelstein coordinates. But it is not the local instability in the phasespace (1). Our purpose is to find a kinematic effect caused by the black hole geometrywith respect to which a system evolves like (1), (2). For this purpose we repeat, forcompleteness, some well-known facts from [11] concerning the behavior of matter nearthe horizon without proofs, thus making our exposition self-contained. olmogorov-Sinai entropy and black holes
3. Chaotic behavior of a relativistic string collapsing into a black hole
My proposal rests on stringy matter having unusual kinematic properties near the eventhorizon of a black hole. According to string theory [12], the most promising candidatefor a fundamental theory of matter, all particles are excitations of a one-dimensionalobject - a string. String theory is characterized by two fundamental parameters: thestring scale l s , and the string coupling constant g ; if l P is the Planck length then l P = gl s .An important fact is that behavior of a string in weakly coupled string theory is veryprecisely described in terms of the random-walk model [12]-[15]. We can imagine astring as simply built by joining together bits of string, each of which is of length l s .Suppose that the total length of the string is L and each string bit can point in any of n possible directions. Then the number of bits is L/l s and the number of states of thestring is N s ∼ n L/l s ∼ exp ( L ln n/l s ) , (6)For notational simplicity the factor ln n will be omitted henceforward. There is no lossof generality in doing so because we can always redefine g , l S and L . We can also definethe mean squared radius of the string h R s i = Ll s . Then N s ∼ exp ( h R s i /l s ) , (7)and the entropy of the string is S s ∼ Ll s ∼ h R s i l s . (8)The mass of the string is given by M s ∼ T L , where T ∼ /l s is the string tension. Soin terms of the mass S s ∼ M s l s . (9)Note that in fundamental contrast to the black hole entropy (5), the entropy of a stringgoes like the mass.Another important fact is that strings behave very differently from the ordinaryparticles. The crucial difference is that the size and shape of a string are sensitiveto the time resolution. It is a smearing time over which the internal motions of thestring are averaged. Susskind showed [3] that zero-point fluctuations of a string makethe size of the string depend on a time resolution; the shorter the time over whichthe oscillations of a string are averaged the larger is its spatial extent. In low energyphysics, resolutions times are always large and this phenomenon is not important. Let,for example, an observation of a string lasts a time τ res . Then Susskind argued that thecontributions of modes with frequency ≫ /τ res in the normal mode expansion for thesize of the string should be averaged out. As a result of such a coarse-graining or time-smoothing, Susskind found that the mean squared radii of the string in the transverseand longitudinal directions in Planck units are h R ⊥ i = l s ln(1 /τ res ) and h R k i = l s /τ res ,respectively. Similar calculations can be performed for the total length of the string,and Susskind found that L = l s /τ res . Now consider a string falling toward a black hole. olmogorov-Sinai entropy and black holes τ and the Schwarzschildtime of an external observer t are related through τ ∼ exp( − t/ R g ) due to the redshiftfactor. This means that the transverse size of the string will increase linearly: h R ⊥ i = l s t R g , (10)while its longitudinal size and total length - exponentially: h R k i = l s exp( t/ R g ), L = l s exp( t/ R g ). But the longitudinal growth is rapidly canceled by the Lorentzlongitudinal contraction. Thus the string approaching the horizon spreads only in thetransverse directions (in this connection the subscript ’ ⊥ ’ at the the mean squared radiusof the string will be replaced by the ’s’ henceforward).In (10) we can immediately recognize the linear dependence of the squareddisplacement of a Brownian particle from the origin on time. The theory of Brownianmotion is closely related to that of random walks. One normally associates diffusionwith the Brownian motion of a particle. The Schr¨odinger equation describing thestring has the same mathematical structure as the diffusion equation of a Brownianparticle. In fact the spreading appears to behave as if the string was diffusing awayfrom its original transverse location. As is well known, the Brownian motion is achaotic process (moreover, it appears that one can even infers a positive KS entropyfrom the Brownian motion (see § N bit = N bit (0) exp( t/ R g ) . (11)The authors interpreted this as a branching diffusion process, where every string bitdiffuses independently of others over the whole horizon and bifurcates into two bits andso on. In this connection it is relevant to remark the following. First the diffusion isa distinctive random process. But in our case there are no real random forces. Thebehavior of a string near the horizon is very well described by the Hamilton dynamics.If there are exact equations of motion no true randomness is possible. Second the stringis a fundamental object. It is not a dissipative system. In the spreading process nopoints of a string should be lost or gained. The irreversible character of the spreadingeffect arises, as has been shown above, exceptionally from the coarse-grained or time-smoothed procedure, in which the fine details of the string motion (over the Planck timescale) have been wiped out. As an ordinary classical body, a string undergos Lorentztransformations. As is well known [17], under the Lorentz transformations the phasespace volume of a classical body and, consequently, its entropy does not change. Butdue to the coarse-grained procedure the phase space volume of a string can increase. Wecan give the following interpretation of (11). Initially phase points of a string occupy olmogorov-Sinai entropy and black holes N cell ( t ) = N cell (0) exp( t/ R g ) . (12)In turn this number is proportional to the distance between the trajectories of phasepoints that all initially occupy one cell (1), as required.As we have seen, the growth of string (10) is linear. But as noted by Susskindhimself [6, 11], this result was obtained in the framework of free string theory. It doesn’ttake into account such a nonperturbative phenomenon as string interactions; there areindications [6, 11] that a true growth must be exponential: h R s i = l s exp( t/ R g ) . (13)How does the phase space of string evolve during the spreading process? The spreadingprocess begins to occur when the string reaches the horizon at distance of order of thestring scale l s from the horizon in a thin layer ∼ l s . But in string theory at such scalesthe mirror symmetry should takes place [18, 19]. In general it relates the complex andK¨ahler structures of some manifolds. In the simplest case for closed strings it exchangesthe winding number around some circle with the corresponding momentum quantumnumber (T-duality) or, roughly speaking, coordinates with momenta. At the scales ≫ l s we can always single out the configuration space and the phase space is its cotangentbundle. At the scales ∼ l s this is not the case: at such scales there is a replacementof the configuration space of a string by its phase space [19]. A similar phenomenon inquantum mechanics - a particle in magnetic field [20]: on the distances of order of themagnetic length l mag ∼ p ~ c/eH a replacement of the configuration plane transversalto the direction of the magnetic field by the phase plane takes place so that the numberof states is A/l mag , where A is the area of the transversal plane. A string is not a point-like particle and has its own phase space which can be very complex. But in any caseduring the spreading process the configuration space of string increases exponentially(in transverse directions). Taking into account the replacement of the configurationspace of a string by its phase space we can conclude that the phase space of string willincrease exponentially in the corresponding directions:∆Γ s ( t ) = ∆Γ s (0) exp( t/ R g ) . (14)It is also true in the framework of the random-walk model. Since the size of stringincreases exponentially, the relative velocities of string bits should obey the Hubble law.It is obvious that in this case the velocity subspace does not decrease and as a result thewhole phase space increases exponentially. We can also estimate the number of statesof a spreading string with the help of the random-walk model. For example, in the caseof the linear growth, substituting (10) in (7), we obtain N s ∼ exp ( h R s i /l s ) ∼ exp( t/ R g ) . (15) olmogorov-Sinai entropy and black holes S s ∼ t R g , (16)and the entropy rate dS s dt = 12 R g . (17)But for the exponential growth of the size the same formulas give N s ∼ exp(exp( t/ R g ))(the same result is obtained and for the total length). In this case the entropy S s ∼ exp( t/ R g ) and the entropy rate dS s dt = S s R g . (18)This is a comparatively large rate. The point is that we have used the simple random-walk model which is valid only in the framework of the free string theory. Apparentlythe interactions will impose constraints on the total number of string states and thehuge entropy rate (18) should reduce. But the precise calculation of the entropy rate inthe nonperturbative regime is beyond the current technology of string theory. Despitethis difficulty we can estimate an upper limit on the entropy rate. As is well known,the maximum rate at which information I may be transmitted at a temperature T inquantum theory is given by Pendry’s formula [21, 22]: dIdt ∼ T. (19)So the maximum rate is determined by temperature. Taking into account thefundamental equivalence relation between information and entropy we can rewrite (19)in terms of entropy [21, 22]: dSdt ∼ T. (20)This expression can be immediately applied to the spreading process near the horizonof a black hole with the Hawking temperature T H = 1 / πR g . Thus we obtain dS s dt ≤ πR g . (21)This is the maximum rate at which entropy of string can increase during the spreadingprocess. Up to a factor 1 / π it coincides with the entropy rate for the linear growth(17). It is obvious that the entropy rate for the exponential spreading is greater or equalto (17) but less or equal to (21). So for the exponential spreading we obtain dS s dt = 12 R g . (22)Strictly speaking, the rate at which entropy of string (or any body made of strings)increases during the spreading process saturates the bound (21). olmogorov-Sinai entropy and black holes h KS = 12 R g . (23)Obviously the KS entropy is the same for all objects collapsing into a black hole. So, itis universal. The nature of this universality is that the KS entropy possesses the mainproperty of temperature: it is the same for all bodies in thermal equilibrium with ablack hole at the Hawking temperature T H = 1 / πR g .It is relevant to remark that the behavior of string mentioned above just correspondsto the thermal properties of a black hole and the second law of thermodynamics. Sincethe temperature of the black hole radiation depends on the radial position, T ( r ) = T H /χ ,where χ is the the redshift factor, χ = (1 − R g /r ) / , it follows that from the viewpointof the external observer the string falls into an increasingly hot region. A thermalinterchange will take place. So the string should ’melt’ and spread throughout thehorizon. Obviously during this process the phase volume and entropy of the bodyshould increase. Thus (15), (16) is a natural response to the hot horizon.How does the KS entropy reach its maximum? As mentioned above, there is nopoint to define the configuration space of string with accuracy better than l s . Thesurface of the horizon is a compact manifold. Since there exists the finite size of coarse-graining l s (note that in strong coupling regime l s ∼ l P ), a string covers the horizon ina finite time [6] t spread ∼ R g ln R g l P . (24)At this time a spreading string completely covers the entire horizon of a black hole. Thenumber of states of the string becomes N ∼ exp( R g /l P ) and the entropy of the stringreaches that of the black hole, S ∼ R g /l P . Note that in doing so the total length of thestring becomes L ∼ R g /l P and the corresponding mass M s ∼ R g /l P . It is a huge mass.But the redshift factor reduces it to the black hole mass. Since the spreading processbegins to occur when the string reaches the stretched horizon at the proper distance l P from the event horizon, the redshift factor is χ ≈ l P / R g , and we obtain for thestring mass M s ∼ χ ( R g /l P ) ∼ R g / (2 l P ) ∼ R g / G . Thus, from the string theory pointof view [11], a black hole is nothing but a single string. According to Susskind [3, 4, 6]and Mezhlumian, Peet and Thorlacius [5], at the time t spread the string spreads over theentire horizon and can no longer expand due to the nonperturbative effects. The resultis crucial for the relaxation of the string to statistical equilibrium: to reach a statisticalequilibrium in a finite time we should have the finite time of spreading. According tochaos theory, this is an average time over which the state of a string can be predicted;after the time t spread all information contained in the string will be lost and we willable only to make statistical predictions. This time is comparable to the characteristictime of a black hole R g but is smaller than the black hole lifetime ∼ R g . Hence the KS olmogorov-Sinai entropy and black holes x ∆ τ ∼ l P . Then, taking into account the redshift factor, we can obtain theexponential growth of the transverse size of matter similar to (10), as required.
4. Is there a relation between the KS and BH entropies?
As mentioned above, a falling string spreads over the stretched horizon until its entropybecomes equal to the black hole entropy S s = S BH = πR g l P . (25)The spreading ends and only a new falling string or any other perturbation can startit again. The next string falling toward the horizon interacts with a previous onelying on the horizon in such a way that the formation of a single (new) string isthermodynamically favored, etc. So the stretched horizon is a single string made out ofall strings whenever fallen into it. From the string theory point of view [11], a black holeis nothing but a single string lying on the sphere of the radius R g . But since a black holeabsorbs a string, its gravitational radius must increase. According to the teleologicalnature of the event horizon (see Chapter VI in Thorne et al [10]), before a fall of thenext string, the gravitational radius and the horizon area increase like exp ( t/ R g ). Thismeans that the spreading takes place. In this case the entropy rate is d ln S BH dt = 12 R g . (26)It coincides with (18) and does not obey the bound (21). To avoid this, we could assumethat a true rate is given by (22). It seems, however, that it can still satisfy Lloyd’s limit.According to Lloyd [23], the maximum rate at which information can be moved in andout of a system with size R and entropy S is dI/dt ∼ S/R (attained by taking all theinformation S ln 2 in the system and moving it outward at the speed of light). In myopinion, this aspect deserves further investigation. In any case, the KS entropy of ablack hole measures the rate at which information about the black hole (or a stringforming the stretched horizon) is lost during a perturbation. olmogorov-Sinai entropy and black holes
5. Conclusions
In this paper we have shown that stringy matter near the event horizon of a black holewith the gravitational radius R g exhibits instability (the spreading effect) and chaoticbehavior which can be characterized by the Kolmogorov-Sinai entropy h KS . We havefound that for a spreading string h KS = 1 / R g . But it is the same for all objectscollapsing into the black hole. It is universal. The nature of this universality is that theKS entropy possesses the main property of temperature: it is the same for all bodies inthermal equilibrium with the black hole at the Hawking temperature T H = 1 / πR g . TheKS entropy of a spreading string measures the rate at which information about the string(or any body made of strings) collapsing into a black hole is lost as the string (the body)spreads over the horizon. It is argued that it is the maximum rate allowed by quantumtheory. We have also discussed a possible relation between the Kolmogorov-Sinai andBekenstein-Hawking entropies and suggested that h KS = d ln S BH /dt . References [1] Barrow J D 1982
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