Particle creation in a colliding plane wave spacetime: wave packet quantization

We use wave packet mode quantization to compute the creation of massless scalar quantum particles in a colliding plane wave spacetime. The background spacetime represents the collision of two gravitational shock waves followed by trailing gravitational radiation which focus into a Killing-Cauchy horizon. The use of wave packet modes simplifies the problem of mode propagation through the different spacetime regions which was previously studied with the use of monocromatic modes. It is found that the number of particles created in a given wave packet mode has a thermal spectrum with a temperature which is inversely proportional to the focusing time of the plane waves and which depends on the mode trajectory.


Introduction
Exact solutions representing the head on collision of two gravitational plane waves are some of the simplest exact dynamical spacetimes. They provide clear examples of highly non linear behavior in general relativity: when two plane waves collide the focusing effects of each exact plane wave lead to mutual fucusing. This is revealed by the formation of either a spacetime singularity or of a non-singular Killing-Cauchy horizon at the focusing points of the two waves [1,2,3]. They may also be useful to provide local models for processes that may be taking place in our universe as a result of gravitational waves produced in black hole collisions [4,5], the decay of cosmological inhomogeneous singularities [6], or by travelling waves in strongly gravitating cosmic strings [7,8].
Quantum effects in such dynamical spacetimes must surely be important and one expects particle production and vacuum polarization when a quantum field is coupled to such background. Yurtsever [6] was the first to study field quantization on a colliding wave background, he considered the Kahn and Penrose [1] solution which may be interpreted as the collision of two impulsive plane waves. The solution which has curvature singularities at the focusing points of the plane waves allows for the definition of two physically meanigful vacuum states: an "in" vacuum state associated to the flat space before the collision of the two plane waves and an "out" vacuum state related to the flat spacetime regions behind the shock fronts also before the collision. The Bogoliubov coefficients relating the "in" and "out" creation and annihilation operators could be found only approximately in the longwavelenght limit. In this approximation the spectrum of created particles is consistent with a thermal distribution.
In a recent paper [9] we have considered the quantization of a massless scalar field in the background of the collision of two plane waves which form a non singular Killing-Cauchy horizon. This solution describes the collision of two gravitational shock waves followed by trailing gravitational radiation [10]. The interaction region of the two waves is locally isometric to a region inside the event horizon of a Schwarzschild black hole with the Killing-Cauchy horizon corresponding to the event horizon [2,11,12]. Two unambiguous and physically meaningful quantum vacuum states may be defined: an "in" vacuum associated to the positive frequency mode solutions in the flat region before the collision of the waves and an "out" vacuum related to the positive frequency modes defined through the two null vector fields in the Killing-Cauchy horizon. Such state, which is invariant under the symmetries associated to the horizon corresponds to the unique preferred vacuum state defined by Kay and Wald [13] in spacetimes with bifurcate Killing horizons. It was found that the "in" vacuum contains a number of "out" particles which is proportional to the inverse of the frequency. In the long-wavelength limit the spectrum is consistent with a thermal spectrum at a temperature which is inversely proportional to the focusing time of the gravitational plane waves, in agreement with Yurtsever's result [6].
Our result however is exact: although the Klein-Gordon equation in the interaction region cannot be solved exactly the "in" modes become blueshifted towards the trailing points of the waves and can be propagated through this region by using the geometrical optics approximation. This is somewhat similar to the situation in a Schwarzschild black hole [14].
In this paper we want to reconsider this problem in the light of wave packet mode quantization instead of the monocromatic modes we have used in ref. [9]. There are several reasons that, we believe, justify this. One of the reasons is well known: the use of monocromatic plane wave modes leads to infinite expressions for the total number of particles created of a given frequency, whereas the number of particles created in a given wave packet mode is finite [14]. The second, and more important for us here, is that wave packet modes propagate in a simple way in a spacetime, such as ours, formed by the matching of different regions. Since the singular free coordinates of one of the regions differ from the next region, the propagation of modes which are extended through all space finds serious difficulties. In ref [9] this could be done because of the blueshift of the modes in certain regions and the use there of the geometrical optics approximation which ammounts to ray propagation. Since wave packets are localized in space and the packet maximum follows a well defined path in the spacetime, a natural approximation may be taken by which the propagation of the packets through the different regions becomes a simpler problem.
Another reason is that the wave packet formalism localize the fenomena of particle creation: the one particle states defined with the wave packet modes have two labels, one gives information on the "energy" of the particle and the other on its "trajectory" (all within a certain range of values). Finally, we know that our colliding wave spacetime can be maximally extended through the Killing-Cauchy horizon with the extended Schwarzschild spacetime [10,15]. This is possible if one of the transversal coordinates of the plane waves is made cyclic. The resulting spacetime represents the collision of two plane waves propagating in a cylindrical universe and the creation of a black hole of mass proportional to the strength (or focusing time) of the plane waves. In a forthcoming paper we want to consider particle creation in the extended spacetime. Using the results of the present paper the calculation will become somewhat similar to that of stimulated emission by black holes [16]; the use of wave packet modes has proved useful also in this case, see ref. [17].
The plan of the paper is the following. In section 2 we introduce wave packet mode quantization. In section 3 we briefly review the geometrical properties of our colliding wave spacetime with special emphasis in the coordinates which are appropriate in the different regions. In section 4 we quantize a massless scalar field on the colliding wave background and propagate the "in" wave packet modes through the different regions. The advantage of wave packet modes is clearly seen in subsection 4.4 when the modes are propagated through the interaction region. The Bogoliubov coefficients relating packet creation and annihilation operators are derived in subsection 4.6, and the creation of particles is derived in subsection 4.7 where we also compare our results with those of ref. [9].

Wave packets
Let us consider a complete orthonormal family, {f ω (x)}, of complex solutions of the Klein-Gordon equation for a massless scalar field φ (i.e. ✷φ = 0), which contain only positive frequencies with respect to a given timelike Killing vector ∂/∂t, i.e. such that L ∂/∂t f ω (x) = −iωf ω (x) with ω > 0. Then it is possible to define a positive definite inner product between these solutions. The label ω is continuous and stands for the energy (or frequency) and x stands for the spacetime coordinates. Since these solutions have a well defined value for the energy (or frequency) we can call them monocromatic modes. It is true that since the energy of the monocromatic modes is well defined their space localization is completely uncertain in concordance with the Heisenberg's uncertainty principle. Now we want to work with a complete and orthonormal set of modes that are localized in space in some sense. To achieve this we can make an adequate superposition, within a small energy range, of continuous ω-labeled monocromatic modes in order to introduce an uncertainty in the energy and gain a certain information on spatial localization. We can do this as follows [14,17,18], define where the new labelsω and n are restricted to verify thatω/ǫ ≡ j and nǫ/2π ≡ l are integers, ǫ being a small and positive parameter. We will call this superposition a wave packet. Here n is a kind of Fourier label (the phase term exp(−inω) has l periodes in the interval (ω,ω +ǫ)), ω is the lower extremum of the integration interval and gives information on the energy of the wave packet. It can be shown easily that a set of discrete (ω, n)-labeled wave packets {fω ,n (x)} is complete and orthonormal if the set of continuous ω-labeled monocromatic modes {f ω (x)} is complete and orthonormal. It is worth noticing that a general set of wave packets, {fω ,n (x)}, given by (1), satisfy the following property, which follows from the equality, Note that the sums in (2) and (3) are over the integer labels (j, l) which could also be used to label the modes, i.e. f j,l (x), instead of the labels (ω, n) which we will use throughout.
We can now see in what sense the wave packets (1) are localized. Let us consider the generic wave packet (1) in terms of the modulus and phase of f ω (x), i.e. f ω (x) = |f ω (x)| e iθω(x) , as: If the interval of integration in (1) is small enough so that |f ω (x)| can be taken as approximatly constant (this will be true in general provided that |f ω (x)| has no singularities in the interval of integration), the |f ω (x)| can be factorized out and we have an integral over the phase only. Now if the integrand's phase, i.e. Θ ω (x) ≡ −nω + θ ω (x), oscillates rapidly over the range of integration (at least when nω is big enough, and this is true for l >> 1) then the integral roughly vanishes except at the stationary phase points, that is when and we can use the stationary phase method [19] to evaluate (4). Note that in the equation (5) we have set ω =ω, after derivation, which is accurate provided the range of energy superposition is small enough. Equation (5) stablishes a relation between the labels (ω, n) of the wave packet and the spacetime coordinates x and so it represents a three dimensional hypersurface. It determines the geometric locus of the spacetime points which give the main contribution to the integral (1), this will be a spacetime region labeled byω and n and we will identify it as the wave packet trajectory, because outside this region the integral (1) roughly vanishes. It is worth noticing that if the monocromatic modes are labeled with some other continuous parameters, besides the energy, we can construct double or triple wave packets by superposition of monocromatic modes in a small range of these parameters. In this case we obtain surfaces or curves as the trajectories of the double or triple wave packet, respectively. Finally we can make the following remark about the uncertainty of the energy and position of a wave packet. When we construct a wave packet we take a superposition of Klein-Gordon solutions with a well defined energy ω (the monocromatic modes) over a small interval (ω, ω + ǫ) , and so the uncertainty in the energy of the wave packet is ∆ω ≃ ǫ.
To know the uncertainty in the position of the wave packet let us assume the following simple form for the monocromatic modes f ω (x) = e −iωt F ω (x j ) (where x j , j = 1, 2, 3 are space coordinates) which verify L ∂/∂t f ω (x) = −iωf ω (x). Then with the use of the stationary phase method we easily see that the wave packets (4) are peaked arround values of time given by t = −2πlǫ −1 , with width 2πǫ −1 (l is an integer) and so the time uncertainty is ∆t ≃ 2πǫ −1 , which reflects Heisenberg's uncertainty principle ∆ω ∆t ≃ 2π.

Colliding plane waves geometry
We will work in a spacetime that describes the head on collision of two linearly polarized gravitational plane waves propagating in the z-direction. This spacetime has four regions (see Fig. 1): a flat region (or region IV) at the past, before the arrival of the waves, two plane wave regions (regions II and III) and an interaction region (region I) where the waves collide and interact nonlinearly. The geometry of these regions is given by the following four metrics, in coordinates which are adapted to the Killing vectors ∂ x and ∂ y of the spacetime by (see ref. [2] and references therein), where u and v are two dimensionless null coordinates (v + u is a time coordinate and v − u a space coordinate) and L 1 , L 2 are two arbitrary positive lenght parameters, which represent the inverse of the strength (focusing time) of the waves. The boundaries of these four regions are: {u = 0, v ≤ 0} between regions IV and II, {v = 0, u ≤ 0} between regions IV and III, {v = 0, 0 ≤ u < π/2} between regions II and I and {u = 0, 0 ≤ v < π/2} between regions III and I. At the surfaces u = π/2 and v = π/2 on regions II and III, respectively, the determinants of the respective metrics vanish, this marks the focusing points of the waves and a coordinate singularity. This singularity can be avoided with the use of appropriate coordinates (harmonic coordinates) in which the causal structure of these spacetime regions is well posed. In these coordinates the surfaces u = π/2 and v = π/2 become spacetime lines, see [9,10] for details.
Region I (in Fig. 1, this is the triangle bounded by the lines {v = 0, 0 ≤ u < π/2}, {u = 0, 0 ≤ v < π/2} and u + v = π/2 ) is locally isometric to a region of the interior of the Schwarzschild metric. This is easily seen with the coordinate transformation, The metric (6) becomes which is the interior of the Schwarzshild metric. The surface u + v = π/2 corresponds to the black hole event horizon. The boundary v = 0 corresponds to r = M(1 + cos θ) and u = 0 corresponds to r = M(1 − cos θ). These are the boundaries of the plane waves, these boundaries join at r = M (spacetime point of the collision) and also at the surface u + v = π/2 at θ = 0 and θ = π. This region of the Schwarzschild interior does not include the singularity r = 0 and thus the interaction region has no curvature singularities. The above local isometry is not global however, the coordinates θ and φ are cyclic in the black hole case but in the plane wave case, −∞ < y < ∞ and −∞ < v − u < ∞. As in the the Schwarzschild case it is convenient to introduce a set of Kruskal-Szekeres like coordinates to describe the interaction region, because the (u, v, x, y) coordinates become singular at the horizon. These coordinates will play an important role in the quantization of the field. First we introduce dimensionless time and space coordinates (ξ, η) with the range 0 ≤ ξ < π/2, −π/2 ≤ η < π/2 (we shall later see that in these coordinates the Klein-Gordon equation can be separated). Then we introduce a new time coordinate ξ * related to the dimensionless time coordinate ξ by and a new set of null coordinatesŨ Note that the transversal coordinate x appears in the coordinate transformation because it behaves badly at the horizon. Finally, we define, and the metric in the interaction region (6) reads, The curves ξ = const. and x = const. are, respectively, hyperbolae and straight lines through the origin of coordinates (U ′ = V ′ = 0), see Fig. 2. The Schwarzshild horizon (which is a Killing-Cauchy horizon for the spacetime) corresponds to the limit of the hyperbolae when ξ → π/2 i.e. the "roofs" V ′ = 0 or U ′ = 0. Notice that the problem with the transversal coordinate x at the horizon is that all the lines x = const. go through the origin of the (U ′ , V ′ ) coordinates, so that all the range of x collapses into the point V ′ = U ′ = 0, whereas the lines U ′ = 0 and V ′ = 0 represent x = −∞ and x = ∞ respectively. One should recall that we have not represented the coordinate x in our picture of the collision ( Fig. 1) in which only the (u, v) coordinates are shown, x is a transversal coordinate perpendicular to the propagation and adapted to the Killing vector ∂ x .
To understand the global geometry of the spacetime one needs a tridimensional picture where the boundary surfaces between the different regions have to be written in terms of appropriate nonsingular coordinates adapted to each region. See [9,10] for details.
It is worth noticing that most of the plane wave collisions produce true curvature singularities [2], but the spacetime described above is an example of a collision where the curvature singularity has been substituted by a Killing-Cauchy horizon (i.e. the surface U ′ V ′ = 0 in the Kruskal-Szekeres like coordinates). 4 Wave packets in the collision of two gravitational plane waves

Monocromatic mode quantization (summary)
In ref. [9] the quantization of a massless scalar field φ was considered in the spacetime described above representing the head on collision of two gravitational plane waves. Let us now briefly summarize this quantization sheme. We introduce a massless scalar field φ on the colliding spacetime background, which satisfies the Klein-Gordon equation, where ✷φ = (−g) −1/2 (−g) 1/2 g µν φ ,ν ,µ , and g is the determinant of the metric. It has the generic plane wave solution where the labels k x and k y are two separation constants which are interpreted physically as the momenta in the directions given by the Killing vectors ∂ x and ∂ y , which generate the plane symmetry of the whole spacetime. Introducing (17) in (16) one obtains, where the coefficients F , G and e −N , come from the adoption of a generic metric for all colliding plane wave spacetime, adapted to ∂ x and ∂ y , i.e.
One defines an initial vacuum state constructed with a complete orthonormal set of solutions of the Klein-Gordon equation (16) which are defined in the flat IV region, before the arrival of the plane waves, to be of positive frequency with respect to the timelike Killing vector ∂ u+v ; these are the "in" monocromatic modes. Then one propagates these modes throughout the spacetime up to the horizon of region I (solving the appropriate boundary conditions imposed by the different classes of solutions for the Klein-Gordon equation, (18), in the different regions of the spacetime). It is possible to define another "natural" vacuum state on the horizon of region I from a complete and orthonormal set of solutions of the Klein-Gordon equation (16), the "out" monocromatic modes, which are positive frequency solutions with respect to the vectors ∂/∂U ′ and ∂/∂V ′ , which are two null Killing fields over the horizon.
Comparing the propagated "in" monocromatic modes and the "out" monocromatic modes on the horizon via a Bogoliubov transformation, lead us to show that there is spontaneous creation of particles in this spacetime with a spectrum of "out" particles given by the formula [9], Here N out ω − is the usual number operator for "out" particles, ω − , l ′ and m ′ are labels of the "out" modes (ω − is the energy label),k − , k and m are dimensionless labels of the "in" modes and (2M) 3 /π|C l ′ | 2 is a geometric factor whose coefficient, |C l ′ |, depends on l ′ , m ′ ,k − and k. This spectrum is inversely proportional to the inverse of the energy of the "out" particles which are produced and it is consistent, in the long wavelenght limit, i.e. 8πMω − << 1, with a thermal espectrum with temperature T = (8πM) −1 . The temperature is inversely proportional to the focusing time of the plane waves, given by the parameter M = √ L 1 L 2 , and (4M) −1 is also the surface gravity of the horizon. Note that the spectrum contains a logarithmic divergence, this divergence appears because the spectrum which has units of (length) 3 , given by the factor (2M) 3 , describes the total number of particles created in the whole spacetime volume. As it is well known [23] this is characteristic of the use of continuous labeled modes and can be avoided using wave packets.
The propagation of the "in" monocromatic modes throughout the spacetime is a difficult task. This is due, essentially, to the fact that the monocromatic modes are defined in the whole spacetime and there is not a single singularity free coordinate chart for all the four spacetime regions. Basically the problem is that the matching has to be done through all the spacetime points of the boundaries. Due to their spatial localization wave packet modes will be more easily matched. The matching will be done approximately on a single boundary point for each wave packet.
In what follows we introduce the wave packet formalism in the colliding spacetime background. As the first step we construct a complete set of "in" wave packets, from a superposition of positive frequency monocromatic "in" modes, in the four spacetime regions. Then we will propagate these "in" wave packets throughout the spacetime up to the horizon, with appropriate matching conditions. These matching conditions will ensure that the trajectory of the "in" wave packet will be smoothly connected through the boundaries between the different regions. Next we will contruct a complete set of "out" wave packets on the surface of the horizon. Finally, we will relate the "in" and "out" wave packets via a Bogoliubov transformation and compute creation of particles in this formalism.

Flat region (region IV)
In the flat region the complete set of "in"-modes is, where the labels k x , k y and k − are independent separation constants for the Klein-Gordon equation (16) and where u ′ , v ′ are two dimensional null coordinates related to the dimensionless null coordinates u, v by: It was shown in ref [9] that these modes are well normalized on the hypersurface The labels k x and k − are continuous but k y is discrete if we take a cyclic spacetime in the y-direction (this is not necessary but it is convenient if we want to maximally extend the collidig wave spacetime later on). We identify k y with m/M where m is an integer. Our aim now is to discretize the continuous labels k x and k − by constructing a double wave packet as, Note that since we are only integrating over positive frequency monocromatic modes the vacuum associated with the wave packets is the same as that defined by the monocromatic modes (21).
The integrand's phase is where the relation (22) has been used. The trajectory of the double wave packet, in the sense given by the stationary phase method, i.e. ∂Θ (IV) /∂k x = ∂Θ (IV) /∂k − = 0, is given by, which define a null geodesic in the flat region. That is, this double wave packet moves on a null trajectory.

Plane wave region (region II)
The complete set of solutions of the Klein-Gordon equation (16) in region II is easily found (see ref. [9]) and is given by The labels k x , k y and k − have the same meaning that in the flat region IV because this expression for the solutions of the Klein-Gordon equation in region II match smootly (i.e. in a continuous and diferentiable way) with the respective solutions (21) on the boundary between regions II and IV, i.e. {u = 0, v ≤ 0}.
In analogy with the previous case, we now construct a double wave packet The integrand's phase is given by, and the trajectory of the double wave packet, i.e. ∂Θ (II) /∂k x = ∂Θ (II) /∂k − = 0, is which represents a null geodesic travelling throughout the plane wave region II. Note that this trajectory matches smoothly (i.e. in a continuous and diferentiable way) with the trajectory of the wave packet in region IV, i.e. (25), (26), on the boundary {u = 0, v ≤ 0}.
Since f (0) = g(0) = 0 and df /du(0) = dg/du(0) = 0 the following matching conditions are satisfied: This is not surprising because the monocromatic modes which we have used to define the double wave packets in the two regions, match smoothly on the boundary {u = 0, v ≤ 0}.
Before going further, we can extract some information on the meaning of the new discrete labels n and q. First, note that at the boundary between regions II and IV, {u = 0, v ≤ 0}, we have from (25), (26) and (31), (32), which means that a wave packet labeled by n and q has a peak on the null geodesic which crosses the flat region into the plane wave region at the spacetime coordinates x 0 = n and v ′ 0 = −q. Note that for wave packets in region III we can repeat the same discussion.

Interaction region (region I)
In this region mode propagation is more difficult; instead of relaying on the calculations given in ref. [9], we will start the discussion from the beginning. First, let us consider that the Klein-Gordon equation (16) in this region can be separed by taking and reduced to equations for ψ αkx (ξ) and ϕ αky (η). The coordinates ξ, η are related to the usual null coordinates u, v, by (10), and the two new equations read, where α is a dimensionless separation constant and k x , k y are the same labels as in regions IV or II. We use the notationâ = 2Ma, thereforek x ,k y are dimensionless parameters. These differential equations have singular points at ξ = π/2, i.e. on the horizon of region I, and at η = ±π/2 respectively. To avoid them we can perform the following change of variables: dξ * = M(1 + sin ξ) 2 cos −1 ξ dξ, dη * = M cos −1 η dη. With appropriate integration constants, ξ * reduces to (11) and η * to: We can still introduce a new function γ αkx (ξ) instead of ψ αkx (ξ) by, γ = (1/2)(1 + sin ξ)ψ, and we arrive at the following equations for γ αkx (ξ) and ϕ αky (η), which have no singular points in the regions of interest.
Recall that we are looking for a solution of the scalar field φ in the interaction region, restricted to satisfy certain boundary conditions imposed by the wave packets travelling from regions II (or III) into region I on the boundary {v = 0, 0 ≤ u < π/2} (or {u = 0, 0 ≤ v < π/2}) Thus the general solution for φ in terms of the new functions γ αkx (ξ), ϕ αky (η) is given by, where the coefficients C α depend on α and the separation constants used to label the monocromatic modes in region II, i.e. k x , k y and k − . Now we will try to obtain all possible information on the coefficients C α in (39). We know that expression (39) gives the general solution (once k x and k y are fixed) for a massless scalar field in the interaction region, and the coefficients C α have to be such that the appropriate boundary conditions are satisfied. If we want to match the monocromatic modes defined in region I with those of region II (27) (or III) we have to perform a sum over all possible values of α in (39), this was one of the main difficulties in ref. [9]. In the following discussion we will show that when we match the spatially localized wave packets travelling from region II (or III) into region I (29) with wave packets defined using the general solution (39), there is only one coefficient C α which carries the main contribution. That is, we will find a single value of α for which the infinite linear combination (39) can be approximated by a single term. Furthermore, we will be able to find the value of the coefficient C α for that particular value of α. We will proceed as follows. First, we will find an approximate solution of equations (37) and (38) for γ αkx (ξ) and ϕ αky (η) respectively, near the boundary of region I with region II which will lead to an approximate solution of (39) near this boundary. With this solution we will construct a wave packet in analogy with (23) or (29) and then we will find the particular value of α and the phase of the coefficient C α which allows the matching of this wave packet with the wave packet (29) travelling from region II. Next, we will find an approximate expression for the general solution (39) near the horizon of region I, i.e. the surface ξ = π/2, with the particular values of α and the phase of C α calculated before. Such solution becomes exact on the surface of the horizon and it will be used to construct a wave packet there.
Let us start with the approximate solution of (39) near the boundary between regions II and I. First note that the solution φ of the Klein-Gordon equation in the plane wave regions (regions II or III) takes an exact WKB form, i.e. it can be written as φ = C exp iS, where C and S are two real functions of the spacetime coordinates. This is directly related to the fact that the geometrical optics approximation is exact in the single plane wave regions, i.e. the rays of the Klein-Gordon solutions (the lines perpendicular to the constant phase surfaces) follow null geodesics. Thus we can expect that an approximate solution of the massless scalar field in the interaction region close to the boundaries with the plane wave regions can be obtained with the WKB method, this is physically related to the fact that near the boundaries the colliding plane waves superpose linearly. In fact from equations (37) and (38) we can see that they admit WKB solutions given by, see for example [20], where Q(ξ * ) and R(η * ) are, These WKB solutions can be used provided Q(ξ * ) and R(η * ) do not vanish and they become accurate solutions when Q(ξ * ) and R(η * ) change slowly, i.e. when |dQ(ξ * )/dξ * | << Q 2 (ξ * ) and |dR(η * )/dη * | << R 2 (η * ). Note that equation (37) admits a WKB approximation of this kind throughout region I. Furthermore this approximation becomes asymptotically exact on the surface of the horizon of region I, i.e. ξ = π/2, and it is a really good approximation near the boundaries between regions I and II (or I and III) provided the separation constant α is large enough: in fact, we will show that to ensure the correct matching between wave packets travelling from region II into I (or from region III into I) with a single term in the infinite sum (39), it is necessary that the terms containing α in (42) and (43), which also contain factors cos 2 ξ or cos 2 η, be dominant. Recall that when we aproach the horizon, i.e. the surface ξ = π/2, on the boundary between regions I and II (or I and III), then both cos 2 ξ and cos 2 η go rapidly to zero. This means that α must be big enough to compense the decreasing behavior of cos 2 ξ and cos 2 η. Equation (38) on the other hand admits a WKB approximation only near the boundary between regions I and II (or I and III) under the assumption of large α (this is because of the minus term in (43)). Fortunately, we know its exact solution when it is written in the form (35), this solution is given in terms of the associated Legendre polynomials and that will be the solution we will take in the regions where the WKB approximation is not valid. Let us now evaluate the form of γ αkx (ξ) and ϕ αky (η) near the boundary between regions I and II, i.e. {ξ = −η, 0 < ξ < π/2}. Although Q(ξ * ) and R(η * ), as given by (42) and (43), are the complete terms that appear in the WKB formulae, if we assume that α is large only the first two terms in powers of α −1 are relevant. With this expansion, performing the integrals in (40) and (41) and choosing the minus sign in the exponent of (40) and the plus sign in the exponent of (41), we have: where the functions f and g are given by (28). Next we construct a double wave packet as and write the coefficient C α as and phase by The integrand's phase in (45) is then, NowΘ (I) provides us with a natural way of matching the wave packets in region II and the wave packets in region I. Let us assume that the matching is good enough if the trajectories of the wave packets in regions II and I are joined in a continuous and diferentiable way on the point (u = u 0 , v = 0) of the boundary between these regions. This means thatΘ (I) must satisfy the three constraints,Θ Using (47) and (30), these constraints give, respectively, whereḟ = df /du,ġ = dg/du are, using (28), Note that the constraint (51) gives the value of the phase θ α for the coefficient C α in (39), and the constraints (52) and (53) define the value of α. These two constraints are compatible, this is not surprising because we have already noticed that the WKB approximation is the natural approximation near those boundaries. The value of α is, where we assume an expansion in even powers of cos u 0 and we recall thatḟ (u 0 ) andġ(u 0 ) are of the order (cos u 0 ) −2 . Then α goes like (cos u 0 ) −4 , this justifies our assumption that α cos 2 ξ >> 1 or α cos 2 η >> 1 because both terms go like (cos u 0 ) −2 in the matching point (u = u 0 , v = 0). Note that (cos u 0 ) −2 increases rapidly when u 0 differs from zero, but even when u 0 = 0 (55) with the use of (22) gives √ α ≃ k + L 1 and the wave packet (45) still matches with (29). It is important to note that this matching fixes the phase of the wave packet but does not ensure its normalization. In fact, the normalization condition fixes the modulus of the coefficient C α , but it is convenient to postpone the calculation of this term until we have the form of the wave packet close to the horizon of region I, where the wave packet normalization condition is much simpler. Our next step is to identify the integrand's phase defining the wave packet (45) near the horizon ξ = π/2. To do this we can still use the WKB approximation (40) of equation (37) near the horizon, this means that we can take asymptoticaly, where we have choosen the minus sign in the phase in order to be consistent with (44). From (39) the scalar field φ is then given by where ξ * − x =Ũ , ξ * + x =Ṽ and U ′ = −2M exp(−Ũ /2M), V ′ = −2M exp(−Ṽ /2M), following equations (12), (13). Notice that when k x ≥ 0, the scalar field reaches the "roof" V ′ = 0 of the horizon (strictly speaking, the rays of φ, i.e. the lines normal to the constant phase surfaces of φ, reach the "roof" V ′ = 0), and when k x ≤ 0, they reach the "roof" U ′ = 0 of the horizon. This asymptotic solution has been obtained because near to the horizon ξ = π/2 the dominant term in (42) is k 2 x . However, it is not possible to obtain such an asymptotic solution for equation (38), because of the minus term in (43). Fortunately, as we have said, we can go back to the untransformed equation (35) which is the equation for the associated Legendre polynomials P αky (sin η). Note that in the cyclic case Mk y = m is an integer and we can take α = l(l + 1) for l integer, then e ikyy P αky (sin η) is the spherical harmonic Y m l (y/M, π/2 − η). The matching of the packets requires that α is given by (55), therefore we can take l = l α ≃ √ α, where l α is the closest integer to √ α for large α. Now using the asymptotic form (57) of φ near the horizon we can construct a double packet as in (45). The integrand's phase defining the wave packet is given by, where the upper sign in (58) stands for k x ≥ 0 and the lower sign stands for k x ≤ 0, and θ α is the phase for the coefficient C α given in (51). As we have said the coefficients C α are restricted to satisfy a normalization condition, to make sure that the scalar field φ is well normalized on the horizon. The appropriate inner product of any two solutions φ 1 , φ 2 of the Klein-Gordon equation is given on the horizon by, see [9]: Since, as we have just seen, where l(k x , k y , k − ) = l α and θ α is given by (51), the normalization condition with respect to (59) leads to, Finally with the asymptotic value (58) for the phase near the horizon (ξ = π/2) we can use ∂Θ (I) /∂k x = ∂Θ (I) /∂k − = 0 to obtain the trajectory of the wave packet near the horizon: Notice that n + (k x /k − )L 1 f (u 0 ) ≡ x (I) (0) is the value of the coordinate x when the wave packet trajectory coming from region II crosses the boundary v = 0 into region I, as one can see from (31), then (62) reads, This is the equation for a null geodesic in a region close to the horizon as it can be seen by writing it in the form x (I) ≈ x (I) (0) ∓ ln(cos ξ), where x (I) (0) is a constant for a fixed wave packet ((11) has been used). It is worth noticing that this is the equation for null geodesics close to the horizon as one can see in the Appendix of [9]. Equation (63) can be written, using (12) and (13), as That is, the trajectories of the "in" wave packets, in the region near the horizon, are straight

"Out" wave packets
Since at the horizon the fields ∂/∂U ′ and ∂/∂V ′ become two null Killing vector fields we can define a new complete set of "out" modes on region I. We define solutions with positive frequency, ω + , with respect to the vector ∂/∂U ′ on the V ′ = 0 "roof" of the horizon and positive frequency, ω − , with respect to the vector ∂/∂V ′ on the U ′ = 0 "roof" of the horizon [9,13,21]. These modes, close to the horizon, are given by, where the labels ω − , m ′ and l ′ are the three independents separation constants of the Klein-Gordon equation (16) using the asymptotic metric on the horizon of region I, and the label ω + is given by, If we restrict, now, to the cyclic case the labels l ′ , m ′ are both discrete but ω − is continuous. We can transform ω − into a discrete label by constructing a single wave packet as follows, The integrand's phase is given by, and the wave packet trajectory (i.e.. ∂Θ/∂ω − = 0), using relation (67), is: Let us discuss these trajectories. On the "roof" U ′ = 0 the trajectory (70) is V ′ = −n ′ = const ≤ 0, the coordinate V ′ is always negative, see (13), thus n ′ is positive and, for a fixed wave packet, constant. This relation means that on the "roof" U ′ = 0 the wave packet trajectories are straight lines with V ′ = const. = −n ′ . Similarly on the "roof" V ′ = 0, the trajectories, (70), are (ω + /ω − )U ′ = n ′ , this means that they are straight lines with coordinate U ′ = const. = (ω − /ω + )n ′ ; now the label n ′ is negative since the coordinate U ′ is negative, see (13). Therefore wave packets with n ′ positive are localized on the U ′ = 0 "roof" of the horizon and wave packets with n ′ negative are localized on the V ′ = 0 "roof" of the horizon. These "roofs" of the horizon are depicted in Fig. 3.

Bogoliubov coefficients
Using the well defined inner product on the surface of the horizon (59), we can compute the Bogoliubov transformation coefficients relating the propagated "in" wave packet modes from the flat region (region IV) up to the horizon of region I and the "out" wave packet modes defined on the horizon. The two Bogoliubov coefficients, defined in the usual way (i.e.. αk x,k−,n,q,m;ω−,n ′ ,l ′ ,m ′ = u iñ kx,k − ,n,q,m , u out ω − ,n ′ ,l ′ ,m ′ and βk x,k−,n,q,m;ω−,n ′ ,l ′ ,m ′ = − u iñ kx,k − ,n,q,m , u out * ω − ,n ′ ,l ′ ,m ′ ) are, αk x,k−,n,q,m;ω−,n where l ′ ≃ √ α and θ α are given by (55) and (51) respectively. The upper signs in (71), except for the label ω ± , stand for the α coefficient and the lower signs for the β coefficient.
The label ω ± is the same for both coefficents and its double sign means k x ≥ 0 for the upper sign and k x ≤ 0 for the lower sign.
In the approximation of small integration intervals ǫ, ǫ ′ and δ, and following the stationary phase technique one can take the non oscillating terms in the integrals as constants. For that reason we can perform a convenient separation in the Bogoliubov coefficients into modulus and phase as, where we have made use of Stirling's formula for the gamma function [22], i.e.
it can be written as for values y > 0, and can be written as for y > 1. We have also defined R α/β as, where Ω = 4M|k x | ln (4M|k x |) − 4M|k x | + π/4, it comes from Stirling's formula (74), where we have assumed 4M|k x | > 1 for simplicity. This assumption does not afect our results because the purpose is to relate a single "in" wave packet with a fix labelk x to a single "out" wave packet with a fix labelω − . Of course we can follow the same steps for the case 4M|k x | > 0 using the more accurate version of the Stirling's formula (73). We have kept the terms |C l ′ | and (ω − ) −1/2 in the integrands for convergence.
The terms, R α and R β act like localizing terms in the sense that the three integrals, of which they consist, have a well localized peak. The integrand's phase of R α and R β (Θ α and Θ β , respectively) are and the peaks of R α and R β are in the spacetime points which satisfy ∂Θ α/β /∂k x = ∂Θ α/β /∂k − = ∂Θ α/β /∂ω − = 0. Solving these three equations for Θ α we find, respectively, where the upper sign in (77) stands for k x ≥ 0 and the lower sign for k x ≤ 0. Similarly, from the equations for Θ β , we obtain:n where the two first equations are functionally the same as (77) and (78) but with a bar over the "in" labels, i.e.k x ,n,q, since these three equations come from the β coefficient which relates an "out" wave packet with an "in" anti wave packet (i.e. a wave packet constructed by the superposition of monocromatic modes of negative frequency). On the other hand (77), (78) and (79) come from an α coefficient and so relate an "out" wave packet with an "in" wave packet. In fact, notice that equations (77), (78) and (79) give a relation between the labels of the "in" wave packets (i.e.k x ,k y ,k − , n and q) and the labels of the "out" wave packets (i.e. ω − , n ′ , l ′ and m ′ ) and equations (80), (81) and (82) give a relation between the labels of the "in" anti wave packets (i.e.k x ,k y ,k − ,n andq) and the labels of the "out" wave packets (i.e. ω − , n ′ , l ′ and m ′ ). When the previous equations are satisfied the terms R α and R β have a peak, otherwise they roughly vanish.
Let us now extract some more information from equations (77)-(82). For instance, equation (77) (or (80)) can be written as, where we have used (51), (64), (65) and the fact that is the value of the coordinate x when the "in" wave packet coming from region II crosses the boundary {v = 0, 0 ≤ u < π/2} into region I (see (31)). Note that U ′ 0 or V ′ 0 are the coordinates on the horizon reached for the wave packets. From (83) equations (77) and (80) can be seen as redshift formulae, because they relate the energy of an "out" wave packet (i.e. ω − ) with the energies of an "in" wave packet and an "in" anti wave packet, respectively, (the energy label for the "in" wave packets in the region near the horizon is |k x |). Note also from (83), that the redshift coefficient is given by the value of the coordinate U or V on the horizon reached by the "in" wave or anti wave packets and by the surface gravity of the horizon, i.e. (4M) −1 . Equations (79) and (82) are position formulae because they give information on the trajectories of the "in" wave and anti wave packets which are related to a given "out" wave packet. In fact, the label n ′ appearing in these formulae can be written, in analogy to (83), as Recall that the label n ′ of the "out" wave packet is related to its position on the horizon, see (70), when n ′ ≥ 0 the "out" wave packet is localized on the U ′ = 0 "roof" at coordinate V ′ ≡ V ′ max = −n ′ = const, and when n ′ ≤ 0 it is localized on the V ′ = 0 "roof" at coordinate (79) we see that when n ′ ≥ 0 thenk x ≤ 0, and viceversa, and from (82), that when n ′ ≥ 0 thenk x ≥ 0, and viceversa. Equation (78) (or (81)), with the use of (51), is the same as (61) and it does not give us any additional information.

Particle creation
Following the formalism of quantum field theory on curved spacetime, spontaneous particle creation is directly related to the β Bogoliubov coefficient. In fact, the number of "out" particles in a given wave packet mode with labels (ω − , n ′ , l ′ , m ′ ) (i.e. the number of quanta in the wave packet mode (ω − , n ′ , l ′ , m ′ )) in the "in" vacuum is given by the sum over the "in" labels of the squared modulus of the β coefficient [23], (72), that is, Here N out ω − ,n ′ ,l ′ ,m ′ is the number operator of "out" particles defined in the standard way as N out ω − ,n ′ ,l ′ ,m ′ = a †out ω − ,n ′ ,l ′ ,m ′ a out ω − ,n ′ ,l ′ ,m ′ , where a †out ω − ,n ′ ,l ′ ,m ′ and a out ω − ,n ′ ,l ′ ,m ′ are the "out" wave packet creation and annihilation operators respectively. With these operators we can write the field operator φ, as a combination of "out" wave and anti wave packets, i.e.
To evaluate the sum in (85) it is worth noticing that the R β coefficient in (72), given by (75), satisfy the following two equalities: These equalities follow from the general property (2) of wave packets and from the form of C l ′ given by (60). Note also, in order to make sense of the square of C l ′ according to (60) we consider the product of a Dirac's delta with a Kronecker's delta. These two equalities and the fact that the R β term has a peak when the relations (80)-(82), between the "in" labels (i.e. k x , k y , k − , n, q) and the "out" labels (i.e. ω − , n ′ , l ′ , m ′ ) are satisfied, allow us to approximately write where n =n(ω − , n ′ , l ′ , m ′ ) is given by (80), q = q(ω − , n ′ , l ′ , m ′ ) by (81), and δ k − , k − (ω − ,n ′ ,l ′ ,m ′ ) is given by δ l ′ ,lα which appears from the squared modulus of (60). Then the number of "out" particles created in the "out" packet mode (ω − , n ′ , l ′ , m ′ ), equation (85), is simply, which can be interpreted as a thermal spectrum for each fixed value of the label n ′ , with a temperature, This spectrum depends on the dimensionless label n ′ /4M, i.e. on the trajectory of the wave packet, but it is thermal for all wave packet modes with the same trajectory. This is quite different from the black hole case [14] where the temperature is independent of the packet trajectory [17] and depends only on the surface gravity, κ = (4M) −1 . In ref. [9] we discuss how the black hole case can be seen in some sense as the time reversal of the colliding wave case. The physical interpretation of this n ′ -dependent temperature follows from the fact that we are computing the particles produced on the wave packet mode (ω − , n ′ , l ′ , m ′ ), which is localized on the horizon by n ′ , given by (70), so that these particles may be "localized" in the same position on the horizon. Note that when n ′ ≃ 0, i.e. near the bifurcation point U ′ = V ′ = 0 of the horizon, the temperature is higher. This spectrum is in agreement with the spectrum of particles created on the monocromatic modes (20), i.e. particles with a well defined momentum but not localized in space, because the one particle Fock space can be decomposed on a basis given in terms of the monocromatic labels, |ω , or, alternatively, on a basis given in terms of the wave packet labels, |ω, n ′ . The space is the same but, of course, the particle interpretation is different. For the monocromatic modes, discused in ref. [9], the spectrum of particles created (20) is inversely proportional to the energy of the "out" modes (i.e. ω − ), with a proportionality factor (8πM) −1 . The relation between the number operators N out ω − and N out ω − ,n ′ is given in terms of the Bogoliubov coefficients relating the monocromatic modes and the wave packet modes. Such transformation has no β Bogoliubov coefficient because the positive frequency wave packets are constructed with positive frequency monocromatic modes only. The α Bogoliubov coefficient is where, here, θ(x) is the usual Heaviside step function, thus the relation between the monocromatic and wave packet annihilation operators is given by Then the monocromatic number operator can be written as and we can approximate as the phase term exp[−i(n ′ −ñ ′ )ω − ] is fast oscillating (except when n ′ =ñ ′ ), because n ′ −ñ ′ = (l −l)2π/ǫ ′ with l andl integers and ǫ ′ a small positive parameter. Then where we have assumed 8πMω − << 1 in order to approximate k ≡ (8πMω − )n ′ /4M as a continuous dimensionless variable. The inverse proportionality in ω − and the logarithmic divergence, in (20), have been recovered. The extra factor (2M) 2 in (20) is due to the fact that two monocromatic labels in ref. [9] were, in fact, discrete and this factor guarantees the correct normalizations. This figure shows a projection of the colliding plane wave spacetime in the (u, v) plane. One can see four regions: region IV is the flat region before the waves collide, region II and III are the plane wave regions and region I is the interaction region. The interaction starts at u = v = 0, the lines u = 0, v < 0 and v = 0, u < 0 are the boundaries of region IV with regions II and III respectively, and v = 0, 0 ≤ u < π/2 and u = 0, 0 ≤ v < π/2 are the boundaries of region I with regions II and III respectively. At u + v = π/2 a Killing-Cauchy horizon is formed which one can see as a coordinate singularity of the metric in region I.

Fig.2
In this figure we represent the coordinates (ξ, x) in terms of the Kruskal-Szekeres like coordinates (U ′ , V ′ ) in the interaction region. The lines ξ = constant are hyperbolae and the x = constant are straight lines crossing the origin U ′ = V ′ = 0. The Cauchy horizon is {U ′ = 0, V ′ < 0} ∪ {V ′ = 0, U ′ < 0} which corresponds to the limit of the hyperbolae as ξ → 0. The "roof" U ′ = 0 corresponds to x → −∞ and the "roof" V ′ = 0 to x → ∞.

Fig.3
This is a 3-dimensional plot of the interaction region (region I) in which all its boundaries are shown, using non singular Kruskal-Szekeres like coordinates. The surface S 3 and the lines M 1 , M 2 , M ′ 1 and M ′ 2 , are the boundaries of region I and the two plane wave regions II and III, the points P and P ′ correspond to U ′ = 0, V ′ = 0, η = −π/2 and to U ′ = 0, V ′ = 0, η = π/2, respectively, and they are folding singularities [9,10,24]. The Cauchy horizon is the "roof" {U ′ = 0, V ′ < 0} ∪ {V ′ = 0, U ′ < 0} and region I is enclosed between the surface S 3 and the "roof".