black holes

ar X iv :h ep -th /9 40 11 09 v 1 2 1 Ja n 94 The Superscattering Matrix For Two Dimensional Black Holes S. W. Hawking Department of Applied Mathematics and Theoretical Physics University of Cambridge Silver Street Cambridge CB3 9EW UK California Institute of Technology Pasadena CA 91125 USA November 1993 Abstract A consistent Euclidean semi classical calculation is given for the superscattering op- erator $ in the RST model for states with a constant flux of energy. The $ operator is CPT invariant. There is no loss of quantum coherence when the energy flux is less than a critical rate and complete loss when the energy flux is critical. 1 1. Introduction In classical general relativity a lot of information is lost in the formation of a black hole. A black hole will settle down rapidly to a stationary state. The classical no hair theorems say that this state is characterized just by the values of conserved charges, like mass, angular momentum, and electric charge, that are coupled to gauge fields. In other words, a black hole of a given mass, angular momentum, and electric charge, can be formed by the collapse of a very large number of different objects. This loss of information was not a worry in the purely classical theory, because one could say the information was still inside the black hole, even if one couldn’t get at it. But the situation changed when it was discovered that according to quantum theory, black holes should radiate and slowly evaporate. This made the question about the information in a black hole much more pressing. If the black hole evaporated and disappeared completely, what would happen to the information? There were three possible ways the information might be preserved: 1 The information might come out again at the end of the evaporation. The problem was that information requires energy to carry it and there wouldn’t be much energy left in the final stages. 2 The information might come out continuously during the evaporation. The difficulty here was that the information would be carried by the in-falling matter, far beyond the apparent horizon. So, if it were to appear outside the horizon, it would violate causality. If one could send information faster than light like this, one could also send it back in time. 3 The black hole might not evaporate completely, but might leave some long lived rem- nant that could be said still to contain the information. But this would violate CPT, if black holes could form, but never disappear completely. In this paper I want to take seriously the idea that information is lost. As I showed some time ago[1], this would imply a loss of quantum coherence. One could form a black hole from matter in a pure quantum state, but it would decay into radiation in a mixed quantum state described by a density matrix. In ordinary quantum field theory the evo- lution is described by an S matrix. This can be thought of as a two index tensor on the Hilbert space that maps initial states to final states: ψf A = SABψi B However when one goes to quantum gravity, the possibility of forming real or virtual black holes means that the evolution is given by what I call a super scattering operator, $. This 2 can be thought of as a four index tensor on Hilbert space that maps initial density matrices to final ones: ρf A B = $ A BC Dρi C D In general, the $ operator will not be the product of an S matrix with its complex conjugate: SABC D 6= SAC S¯BD This proposal of a non unitary evolution was greeted with outrage by most particle physi- cists. I was accused of violating quantum mechanics. That is not the case. One gets loss of quantum coherence and a mixed state whenever there is part of a system you don’t measure. All I have done, is point out you can’t measure the part of the quantum state that is inside the black hole. The argument about whether quantum coherence is lost in black hole evaporation has dragged on inconclusively over the years because general relativity is not renormalizable. This has meant we have not been able to calculate what happens in the final stages of black hole evaporation. Maybe the answer lies in supergravity or superstring theory, but we don’t know how to use them to do the calculation. However, in the last two years, there has been a revival of interest in two dimensional black holes. These have the great advantage of being renormalizable, so it should be possible to use them to decide the issue. Those who felt a mission to defend quantum purity, hoped that these two dimensional models would show that information and quantum coherence were preserved. However, they have been disappointed with the results so far: all calculations to date have shown loss of information and quantum coherence. This has left the unitary S matrix camp mumbling something about the breakdown of the large N approximation, but with no real argument. Most of the calculations that have been done on two dimensional black holes, have been carried out in Lorentzian spacetime. They have assumed no horizons or singularities in the past. One can show[2] that this implies that there must be horizons or naked singularities in the future. Thus the calculations are manifestly not CPT invariant. This is reflected in the fact that the outgoing energy flux they predict is always below a certain critical level. However, the ingoing energy flux can have any value. Thus the super scattering operator given by these calculations will not be CPT invariant. To lose quantum coherence is bad enough, but to lose CPT as well, sounds like carelessness. I shall therefore outline a different approach to calculating the super scattering oper- ator which is guaranteed to give CPT invariant results. It is the Euclidean path integral method, coupled with the no boundary condition[3]. In the cosmological case, the no 3 boundary condition literally meant that spacetime had no boundary. In other words, the quantum state is defined by a path integral over all compact positive definite metrics. But in the particle scattering case, the appropiate quantum state is defined by a path integral over all positive definite metrics that have one of more asymptotically Euclidean regions but no other boundary. One can show that the asymptotic Green functions in this quan- tum state are CPT invariant[4]. It then follows that the super scattering operator is CPT invariant, in the sense of detailed balance[5]: the probability to go from the initial pure state A to a final state B, is the same as the probability to go from the CPT conjugate of B to the CPT conjugate of A. 2. Two dimensional black holes The starting point for two dimensional black holes is the CGHS model[6]. This has a metric, g, and dilaton field, φ, coupled to N minimal scalar fields, fi. L = 1 2pi √−g [ e−2φ(R + 4(∇φ)2 + 4λ2)− 1 2 N∑ i=1 (∇fi)2 ] The quantum effective action of the minimal scalars in the curved metric, g, can be eval- uated exactly and added to the action of the classical CGHS model. One can then define new field variables, χ and Ω, in which the theory is a conformally invariant quantum field theory and the action looks rather like that of the Liouville model: S = 1 pi ∫ d2x [ −∂+χ∂−χ+ ∂+Ω∂−Ω+ λ2 exp [ 2√ κ (χ− Ω) ] + 1 2 N∑ i=1 ∂+fi∂−fi ] In making the theory, a conformal field theory, one has to modify the action. One can either change the kinetic term, as Russo, Susskind, and Thorlacius do[7], or the non derivative term, as Bilal and Callan[8], and de Alwis[9] do. Both modifications end up with the same Liouville like theory, but the relation between the Liouville fields, χ and Ω, and the physical fields, g and φ, is different. The RST version has the great advantage that it admits the linear dilaton as a solution. The other, ABC, version does not seem to have any natural ground state. I shall therefore work with the RST version. In the RST model the Liouville fields are related to the physical fields by Ω = √ κ 2 φ+ e−2φ√ κ χ = √ κρ− √ κ 2 φ+ e−2φ√ κ 4 where the metric is in the conformal gauge ds2 = −e2ρdx+dx− The Liouville model looks to be almost linear, but this is deceptive. Both in the ABC and RST versions, the field, Ω, is bounded below as a function of the physical field φ. It takes its minimum value, Ωc, when φ = φc. There are two possible attitudes one can take to this. One, propounded by de Alwis[9], is that one should take the quantum theory to be defined by a path integral over the full range of χ and Ω from minus infinity to plus infinity. The other attitude, put forward by RST[7], is that the path integral should be restricted to the range of Ω that corresponds to real physical fields. I shall adopt this latter approach. The physical fields are what are important. The Liouville fields, χ and Ω, can be regarded just as mathematical constructions that help to solve the field equations, but are of no physical significance of themselves. The field equations for the Liouville model are simple. ∂+∂−(χ−Ω) = 0 ∂+∂−(χ+Ω) = −2λ 2 √ κ exp ( 2√ κ (χ− Ω) ) The first equation corresponds to the freedom to choose coordinates. One can therefore take χ− Ω = 0 at least locally. It is then easy to write down the general solution in this coordinate gauge: Ω = − λ 2 √ κ x+x− + F (x+) +G(x−) To make this solution Euclidean, introduce a complex coordinate z in the plane with z = x+, z¯ = −x− This very simple theory becomes highly non trivial if one imposes the requirement that the physical fields be non singular. One way to get a non singular solution is if omega is always above its minimum value, Ωc. The only solutions like this that are also static and with only an asymptotically flat region can be written down in the χ = Ω gauge: Ω = λ2√ κ zz¯ +M + √ κ 2 5 Ω is a function only of r in polar coordinates, with a minimum at the origin. These solutions represent Euclidean black holes with the horizon at the origin. The polar angle, θ corresponds to imaginary time. Because this is periodic, the black hole is in thermal equilibrium. There is a steady flow of energy in at infinity and a similar flow out. Suppose now that one has a field configuration in which Ω actually reaches the critical value on some curve, C. Then Ω will drop below the critical value in a neighbourhood of C and the physical fields will be singular unless the gradient of Ω vanishes on C. Thus the necessary condition for the physical fields to be non singular is ∇Ω|C = 0 I shall refer to this as the RST boundary condition. With this boundary condition, no signal can propagate through the curve C. One can therefore cut off the spacetime beyond C and consider only the region between C and asymptotically flat infinity. One can regard the curve C as being like the axis of symmetry in the dimensional reduction of a spherically symmetric spacetime to two dimensions. On the basis of this analogy, the appropiate boundary condition on the minimal scalars would be that their normal gradient should vanish on C. If C is time like, the normal direction will be space like. The field equation ∂+∂−Ω = − λ 2 √ κ exp ( 2√ κ (χ− Ω) ) will then imply that the second derivative of Ω in the normal direction will be positive. This means that Ω ≥ Ωc in a neighbourhood of C and the physical fields can be non singular. On the other hand, if C is space like, the normal direction will be time like and the second derivative of Ω in the normal direction will be negative. Thus Ω will go below Ωc and the physical fields will be singular. Thus the RST boundary condition can not be imposed in Lorentzian spacetime if the boundary is space like. By contrast, in the Euclidean regime the normal direction to C is always space like. This means one can always satisfy the boundary condition. Thus the Liouville model with the RST condition and the no boundary condition, is a well defined quantum theory. One should therefore be able to calculate the superscattering operator and see whether it involves loss of quantum coherence. I shall give evidence that it does, in at least some situations. 6 3. The quantum RST model If one just had the Liouville model without any restriction on the range of Ω, the theory would be linear and the semi classical approximation would be exact. This means that the quantum theory would be determined completely by solutions of the quantum effective action, which will be the same as the original Liouville action. However, if the range of Ω is restricted, and the RST boundary condition imposed, the theory becomes effectively non linear. This means that the solutions of the Liouville theory, are no longer the whole story. But one can still hope that they will give a first approximation. As well as the field equations obtained by varying φ and ρ, there are constraint equa- tions obtained by varying the components of the metric that are zero in the conformal gauge. In terms of the physical fields, the constraint equations can be written in a form like the Einstein equations: − [ e−2φ + κ 4 ] (4∂±ρ∂±φ− 2∂2±φ) = 1 2 N∑ i=1 ∂±fi∂±fi − κ(∂±ρ∂±ρ− ∂2± + t±) The term on the left that is the analogue of the Einstein tensor, is the trace free part of the second covariant derivative of φ. The first term on the right is the classical energy momentum tensor of the scalar fields. The remaining terms on the right can be regarded as the quantum induced energy momentum tensor. It is non local because it contains the functions, t±(x±), that have to be determined by boundary conditions. Given a solution of the field equations, one can always find functions t±(x±) that will satisfy the constraint equations. I shall therefore regard the left hand side of the constraint equations as the definition of the total energy momentum tensor, classical plus quantum, of the scalar fields. In other words, given a solution of the φ and ρ field equations, one can read off from the constraint equations what the energy momentum flow is. In the χ = Ω gauge there are a two parameter family of solutions which have a Killing vector in the direction of the polar angle, θ. The two parameters are the coefficient of the log r term and the constant term in Ω. Imposing the RST boundary condition that ∇Ω = 0 where Ω = Ωc puts one relation between the two parameters. One therefore has a one parameter family of static Euclidean solutions. Ω = λ2√ κ zz¯ − P log(λ2zz¯)− P [log (P√κ)+ 1]− √κ 4 [ log (κ 4 ) + 1 ] The linear dilaton is the member of this family with P = √ κ 4 . In it, Ω has a minimum of Ωc on a circle around the origin. The total energy momentum tensor of the scalar fields 7 calculated in the manner I described, is zero. Thus this solution represents the vacuum with no energy flux at infinity. However for 0 ≤ P < √ κ 4 , there is a family of solutions with a constant positive flux of energy coming in at infinity and a similar flux going out. As the energy flux increases from zero, the circle at which Ω = Ωc shrinks in radius. When the flux reaches the critical level, the circle shrinks to zero and the solution becomes a Euclidean black hole with no boundary and with Ω > Ωc everywhere. There is a one parameter family of black hole solutions, but all have the same energy flux at infinity. There are no regular static Euclidean solutions with an energy flux greater than this value. That is what one might expect because if one sent in energy at such a rate, one would create a black hole that would grow faster than it could radiate itself away. So one couldn’t get a static solution. 4. The superscattering operator One would expect these Euclidean solutions to give the dominant contributions to the superscattering operator for incoming states with a steady flux of energy. Consider first the solutions, like the linear dilaton, that have a boundary at Ωc In these, the region inside the boundary circle is not regarded as part of the spacetime. This means there’s no reason to identify the polar angle, θ, with any particular period. It is therefore natural to take θ to run the full range from minus infinity to infinity. It is straightforward to calculate Green functions for the scalar fields in these space- times. The minimal scalars propagate in the flat background metric independently of the dilaton and the conformal factor. They are however affected by the global structure such as whether θ is periodically identified and by the presence of the boundary. On the basis that the boundary is like an axis of symmetry, it is natural to impose Neumann boundary con- ditions on the minimal scalars: the normal gradient of each field vanishes on the boundary. It is then straightforward to obtain the Green functions by the method of images. The procedure to calculate the superscattering operator from the asymptotic Green functions was described in [5]. One first defines creation and annihilation operators for incoming and outgoing particles in terms of integrals of the field operators at infinity, e.g. ai(ω) = − i√ 2pi ∫ Σ pi(ω, x) ↔ ∇µf(x)dΣµ where ai(ω) is the annihilation operator for a particle in the incoming mode pi(ω, x) of frequency ω, Σ is a time like line near infinity and the integral is carried out over real Lorentzian time. The superscattering operator is then the expectation value of a string of creation and annihilation operators for the initial and final states and their complex 8 conjugates: $ABC D = 〈OCOAOBOD〉 where OD is a string of creation operators for the initial state |ψiD〉 = OD|0〉 and OC is a string of annihilation operators for the initial state 〈ψiC | = 〈0|OC . Similarly OA and OB are the creation and annihilation operators for the final state. One interprets the expectation values 〈f(x1), f(x2), ...f(xn)〉 as the analytic contin- uation of the Euclidean Green function G(x1, x2, ...xn) keeping each of the differences xi+1 − xi as a vector with a small future directed imaginary part. In other words, to evaluate the superscattering matrix element $ABC D one does a multiple integral of the an- alytically continued Green function on the time like line Σ near infinity with the contours of integration arranged in increasing order of imaginary time reading from left to right in the expectation value. In ordinary non gravitational quantum field theory, the superscattering operator fac- torizes $ABC D = 〈OCOA〉〈OBOD〉 The second factor on the right is defined to be the S matrix and the first factor is its complex conjugate. There is no loss of quantum coherence and the S matrix is unitary. But in quantum gravity, the superscattering matrix may not factorize leading to loss of quantum coherence, as I shall show. The static solutions with a boundary are conformal to half of flat space. This means that the Green functions for the scalar fields are the same as those for flat space with a reflecting wall. They will propagate a positive frequency wave function purely in the negative imaginary time direction, and a negative frequency wave function purely in the positive imaginary time direction. This means that there will be propagation only between the top two contours corresponding to OD and OB and between the bottom two contours corresponding to OA and OC . Thus the superscattering operator will factorize and there will be no loss of quantum coherence. One can adjust the solution so that the ingoing flux of the initial state is equal to the energy flux of the solution, calculated in the way I described. This gives a consistent semi classical calculation of the superscattering operator for initial states with constant energy fluxes less than the critical flux. The situation is very different however, if one considers Green functions on a black hole background. In this case, the point at the origin of Euclidean